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Semantic Rate-Distortion Theory: Deductive Compression and Closure Fidelity
Jianfeng Xu
Intelligence
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Summary
The paper introduces a semantic rate-distortion theory where the fidelity criterion is based on the preservation of deductive closure in a knowledge base. It defines an 'irredundant core' of a knowledge base, showing that redundant states can be reconstructed by the receiver at zero cost, leading to a semantic leverage phenomenon and a tighter rate-distortion bound than classical Shannon theory.
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Relation Signals (3)
Semantic Rate-Distortion Theory → utilizes → Closure Fidelity
confidence 95% · This paper develops a rate–distortion theory under this criterion [closure fidelity].
Irredundant Core → reduces → Source Rate
confidence 90% · under closure fidelity the required source rate is reduced, with an asymptotic i.i.d. leverage factor
Datalog → verifies → Semantic Rate-Distortion Theory
confidence 90% · All results are verified on Datalog instances with up to 24 000 base facts.
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Abstract
Abstract:Shannon's rate-distortion theory treats source symbols as unstructured labels. When the source is a knowledge base equipped with a logical proof system, a natural fidelity criterion is closure fidelity: a reconstruction is acceptable if it preserves the deductive closure of the original. This paper develops a rate-distortion theory under this criterion. Central to the theory is the irredundant core-a canonical generating set extracted by a fixed-order deletion procedure, from which the full deductive closure can be rederived. We prove that the zero-distortion semantic rate equals a quantity that is strictly below the classical entropy rate whenever the knowledge base contains redundant states. More generally, the full semantic rate-distortion function depends only on the core; redundant states are invisible to both rate and distortion. We derive a semantic source-channel separation theorem showing a semantic leverage phenomenon: under closure fidelity, the required source rate is reduced by an asymptotic leverage factor greater than one, allowing the same knowledge base to be communicated with proportionally fewer channel uses-not by violating Shannon capacity, but because redundant states become free. We also prove a strengthened Fano inequality that exploits core structure. For heterogeneous multi-agent communication, an overlap decomposition gives necessary and sufficient conditions for closure-reliable transmission and identifies a semantic bottleneck in broadcast settings that persists even over noiseless channels. All results are verified on Datalog instances with up to 24,000 base facts.
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- Source: https://arxiv.org/abs/2604.11204v1
- Canonical: https://arxiv.org/abs/2604.11204v1
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Semantic Rate–Distortion Theory: Deductive Compression and Closure Fidelity Jianfeng Xu1 1Koguan School of Law, China Institute for Smart Justice, School of Computer Science, Shanghai Jiao Tong University, Shanghai 200030, China. Email: xujf@sjtu.edu.cn Abstract Shannon’s rate–distortion theory treats source alphabets as unstructured sets. When the source is a knowledge base equipped with a proof system, a natural fidelity criterion is closure fidelity: a reconstruction is acceptable if it preserves the deductive closure of the original. This paper develops a rate–distortion theory under this criterion. The irredundant core Atom(SO)Atom(S_O)—a canonical irredundant generating set, extracted by a fixed-order deletion procedure, from which the full deductive closure can be re-derived—plays a central role. We prove that the zero-distortion semantic rate equals PAH(πA)P_A\,H( _A), where PAP_A is the source probability of the core and πA _A the conditional core distribution; this is strictly below the classical H(PO)H(P_O) whenever the knowledge base contains redundant states. More generally, the full semantic rate–distortion function decomposes into a contribution from the core alone: redundant states are invisible to both rate and distortion. We derive a semantic source–channel separation theorem exhibiting a semantic leverage phenomenon: under closure fidelity the required source rate is reduced, with an asymptotic i.i.d. leverage factor Λ∞=log|SO|/(PAH(πA))>1 _∞= |S_O|/(P_A\,H( _A))>1, enabling the same knowledge base to be communicated with proportionally fewer channel uses—not by violating the Shannon capacity C(W)C(W), which remains an upper bound, but because closure-based fidelity renders redundant states free. We also prove a strengthened Fano inequality exploiting core structure. For heterogeneous multi-agent communication, an overlap decomposition yields necessary and sufficient conditions for closure-reliable transmission and identifies a semantic bottleneck in broadcast settings that persists even over noiseless carriers. All results are verified on Datalog instances with up to 24 000 base facts. I Introduction Shannon’s mathematical theory of communication [29] deliberately sets aside the meaning of messages, treating source and channel alphabets as unstructured label sets. This abstraction has yielded the fundamental limits of data compression, channel coding, and network information theory [8, 9], and remains the foundation of modern digital communications. Yet already in the companion essay by Weaver [32], a three-level hierarchy was envisaged: accurate symbol transmission (Level A), conveying intended meaning (Level B), and achieving the desired effect (Level C). Shannon’s theory addresses Level A with unmatched precision; the present paper develops a rigorous rate–distortion theory for Level B that is mathematically compatible with—and strictly generalizes—Level A. Crucially, the “meaning” captured here is not a generic similarity measure or a black-box task metric, but the deductive content induced by a shared proof system: the fidelity criterion is preservation of the deductive closure, and the compression gain arises from the receiver’s ability to re-derive redundant knowledge—a mechanism that is structurally distinct from both classical symbol-level coding and the task/effect-oriented metrics studied at Level C. Motivation The need for such a theory has intensified with the rise of knowledge-intensive communication systems. In multi-agent coordination, retrieval-augmented generation pipelines, and federated knowledge-base synchronization, communicating parties exchange not raw symbols but structured semantic states—ground facts, rules, or queries drawn from a shared or partially overlapping logical vocabulary. A symbol-level error that leaves the deductive content unchanged is harmless, while a symbol-level “success” that corrupts a single irredundant axiom may destroy an entire branch of derivable knowledge. Classical information theory cannot make this distinction because it treats every source symbol as equally “expensive” to protect. The key observation motivating this work is that a fixed proof system shared by communicating agents induces a natural partition of the source alphabet into an irredundant core Atom(SO)Atom(S_O)—a canonical irredundant generating set, extracted via a deterministic deletion procedure under a fixed canonical order, from which the full deductive closure can be re-derived—and a collection of stored shortcuts J=SO∖Atom(SO)J=S_O (S_O) that are deductively redundant. Note that Atom(SO)Atom(S_O) is irredundant (no element can be removed without losing closure equivalence) but need not have minimum cardinality among all generating subsets; the order-dependent extraction procedure of Definition 2.1 determines a unique canonical representative. Under a closure-based fidelity criterion (which deems a reconstruction acceptable if it preserves the deductive closure), only the |Atom(SO)||Atom(S_O)| core elements need to be transmitted reliably; the remaining |J||J| states can be recovered by the receiver’s inference engine at zero additional channel cost. This suggests that the effective source entropy for closure-reliable communication should be strictly less than the Shannon entropy H(PO)H(P_O)—a qualitative prediction that we make precise and tight in this paper. Related Work and Gap Identification Semantic information theory. The formal study of semantic information dates to Carnap and Bar-Hillel [6], who measured information by the set of possible worlds excluded, and to Floridi [11], who required information to be truthful. Kolchinsky and Wolpert [17] proposed a viability-based formulation. Recently, Niu and Zhang [25, 38] established a mathematical theory based on synonymous mappings, deriving semantic entropy, capacity (Cs≥C_s≥ C), and rate–distortion functions by partitioning source alphabets into equivalence classes. Their framework elegantly extends Shannon theory through source-side collapsing and leaves room for integrating logical inference structure—the direction pursued here. We note that their result Cs≥C_s≥ C quantifies a semantic throughput gain under a synonymous-mapping fidelity criterion; it does not assert that the Shannon channel capacity C(W)C(W) is violated, since the data processing inequality Csem≤C(W)C_sem≤ C(W) (Theorem 3.1(i)) remains in force. Our framework produces an analogous but distinct gain via receiver-side deductive reconstruction rather than source-side equivalence-class collapsing. Semantic coding algorithms. Ma et al. [21] proved a semantic channel coding theorem for many-to-one sources via a generalized Fano inequality. Han et al. [14] proposed an extended Blahut–Arimoto algorithm for computing the semantic rate–distortion function under synonymous mappings. Liang et al. [19] introduced semantic arithmetic coding achieving higher compression efficiency over synonymous sets. These works provide coding-theoretic results within the synonymous-mapping paradigm; our work complements them by grounding the compression mechanism in receiver-side deductive inference rather than source-side equivalence-class collapsing. Multi-agent and goal-oriented communication. Seo et al. [28] addressed inverse contextual reasoning via Bayesian inference. Alshammari and Bennis [3] proposed a logic-driven framework for resilient multi-agent semantic communication. Li et al. [18] and Wu et al. [33] developed goal-oriented metrics unifying significance measures. Deep-learning-based semantic communication systems [35, 26, 12, 20] motivate the formal guarantees developed here; our overlap decomposition and semantic bottleneck results provide coding-theoretic complements to these approaches. Logical and database foundations. The logical substrate draws on descriptive complexity (FO(LFP)FO(LFP) capturing PTIME on ordered finite structures [15]), Datalog fixpoint theory [7, 1], and knowledge compilation [10]. Marx [22] introduced submodular width for conjunctive query evaluation; Abo Khamis and Chen [2] recently achieved near-optimal query-evaluation complexity. Mu [23] studied formula roles in inconsistency, paralleling our core/redundant distinction. These results provide the technical substrate on which our communication-theoretic analysis builds. Gap. The works reviewed above illuminate complementary facets of semantic communication, yet a unified framework that derives tight rate–distortion bounds from the proof-system structure of the communicated knowledge base—and that handles the vocabulary heterogeneity arising in multi-agent settings—has not been established. The present paper addresses this gap. Approach and Key Insight Our approach originates from an ontological perspective on information [37, 16, 36, 27]: we model a source as a finite knowledge base SOS_O equipped with a deductive closure operator Cn(⋅)Cn(·) induced by a fixed proof system PS, and define closure distortion dCnd_Cn—a Jaccard-distance-based measure under which replacing a source state by any deductively equivalent substitute incurs zero penalty. Under this distortion, redundant states (those in Cn(SO∖s)Cn(S_O \s\)) are “free”: they can be mapped to any output in the closure without cost. Only the irredundant core elements carry genuine rate, because they contribute unique deductive content that cannot be recovered from the remaining knowledge base. The central quantitative consequence is a tight zero-distortion semantic rate: Rsem(0;dCn,PO)=PAH(πA),R_sem(0;\,d_Cn,\,P_O)\;=\;P_A\,H( _A), (1.1) where PA=PO(Atom(SO))P_A=P_O(Atom(S_O)) and πA(a)=PO(a)/PA _A(a)=P_O(a)/P_A is the conditional core distribution (Theorem 4.1). This is strictly below the classical R(0;dH)=H(PO)R(0;d_H)=H(P_O) whenever |J|>0|J|>0, and the ratio PAH(πA)/H(PO)P_A\,H( _A)/H(P_O) quantifies the deductive compression gain. Under a uniform source, the ratio reduces to klogk/(|SO|log|SO|)k k/(|S_O| |S_O|) where k=|Atom(SO)|k=|Atom(S_O)|. Main Contributions C1. Axiomatic information model (Section I, Appendix A): We introduce a framework comprising ℒsemL_sem-definable state sets linked by computable enabling maps, a deductive closure operator Cn with a fixed proof system PS, and an irredundant semantic core Atom(SO)Atom(S_O) with derivation-depth stratification Dd(⋅∣B)Dd(· B), both shown to be computable invariants. C2. Semantic channel (Section I): We define the semantic channel as a composition of Markov kernels whose supports respect the enabling structure of composable information models, and introduce four distortion functions of increasing semantic depth: Hamming, closure, depth, and a parameterized composite. C3. Tight semantic rate–distortion function (Section IV): We prove that the zero-distortion rate under closure fidelity is exactly PAH(πA)P_A\,H( _A) (Theorem 4.1), and that the full Rsem(D)R_sem(D) decomposes into a contribution from the core sub-source alone (Theorem 4.2). C4. Semantic source–channel separation and semantic leverage (Section IV-C): We derive a separation theorem showing that under closure fidelity the required source rate is reduced by a semantic leverage factor Λ1=log|SO|/log|Atom(SO)|>1 _1= |S_O|/ |Atom(S_O)|>1 in the single-shot regime, and Λ∞=log|SO|/(PAH(πA))≥Λ1 _∞= |S_O|/(P_A\,H( _A))≥ _1 in the asymptotic i.i.d. regime (Theorem 4.3), enabling the same knowledge base to be communicated with proportionally fewer channel uses—not by exceeding the Shannon capacity C(W)C(W), but because closure fidelity renders redundant states free. C5. Strengthened semantic Fano inequality (Section IV-D): We prove a Fano bound involving log|A| |A| rather than log|SO| |S_O|, yielding a tighter constraint by log(|SO|/|A|) (|S_O|/|A|) bits (Theorem 4.4). C6. Rate–delay–distortion tradeoff and semantic sampling theorem (Section IV-E): When the receiver’s derivation budget is bounded by δ steps of T_ PS, we define a δ-irredundant core filtration SO=Atom0⊇Atom1⊇⋯⊇Atom=Atom(SO)S_O=Atom_0 _1 ·s _ D_d=Atom(S_O) and prove that the zero-distortion rate under delay budget δ is Rsem(0,δ)=PδH(πδ)R_sem(0,δ)=P_δ\,H( _δ), yielding a rate–delay–distortion surface that smoothly interpolates between the classical H(PO)H(P_O) at δ=0δ=0 and the semantic rate PAH(πA)P_A\,H( _A) at δ=δ= D_d (Theorem 4.5). A semantic sampling theorem identifies the critical delay δ∗δ^* below which closure-reliable communication is impossible (Corollary 4.1). An expected-budget relaxation via time-sharing yields a convex rate–computation tradeoff whose Lagrangian dual prices receiver computation in bits of communication rate (Remark 4.10). C7. Heterogeneous multi-agent communication (Section V): We introduce a pairwise overlap decomposition, derive necessary and sufficient conditions for closure-reliable communication, show that the deductive compression ratio is invariant under vocabulary heterogeneity, and identify a semantic bottleneck phenomenon in broadcast settings. All results are verified on explicit Datalog instances with up to 24 000 base facts. Paper Organization Section I presents the system model: the deductive closure operator, irredundant cores, derivation depth, and closure fidelity. Section I builds the probabilistic layer: enabling kernels, the semantic channel, distortion measures, channel invariants, and preliminary coding theorems. Section IV derives the tight rate–distortion bounds, the source–channel separation theorem, and the strengthened Fano inequality. Section V instantiates the framework for heterogeneous multi-agent communication and presents the numerical validation. Section VI concludes. Appendix A provides the full axiomatic foundations: the logical language, information model axioms, and synonymous state sets. Notation. SOS_O: semantic state set (knowledge base); Cn(⋅)Cn(·): deductive closure; Atom(SO)Atom(S_O): irredundant core; Dd(s∣B)Dd(s B): derivation depth; κ:X↝Yκ:X Y: Markov kernel; H(⋅)H(·), I(⋅;⋅)I(·;·): Shannon entropy and mutual information (base 2, bits); dH,dCn,dDd,dsemd_H,d_Cn,d_Dd,d_sem: distortion functions; PA:=PO(Atom(SO))P_A:=P_O(Atom(S_O)); πA _A: conditional core distribution. A notation summary for the multi-agent application appears in Table I. I System Model This section introduces the deterministic and logical substrate on which the probabilistic structure of the semantic channel (Section I) and the rate–distortion analysis (Section IV) are built. The presentation focuses on the three concepts that enter directly into the main theorems: the deductive closure operator Cn, the irredundant core Atom(SO)Atom(S_O), and the derivation-depth stratification Dd(⋅∣B)Dd(· B). The full axiomatic development—the many-sorted logical language ℒL, the semantic sublanguage ℒsemL_sem, the information model with time-indexed state sets and enabling maps, and the synonymous state set formalism—is deferred to Appendix A. Throughout, the operational inference substrate is a Datalog or Horn-clause proof system over a finite active domain; the FO(LFP)FO(LFP) language of Appendix A-A serves as a background framework for defining expressible state sets and does not enter the rate–distortion analysis directly. I-A Proof System and Deductive Closure We fix a finite set OS_O of potential semantic states (the ambient semantic universe) equipped with an injective encoding encO:O→0,1∗enc_O:S_O→\0,1\^* and a fixed canonical order. Each element of OS_O is identified with a ground atom of an inference fragment ℒkb⊆ℒsemL_kb _sem (typically Datalog or a Horn fragment; see Appendix A-A for the full logical language). Additional closure properties of OS_O (closure under definable recodings, effective representability) are stated in Assumption A.3 of Appendix A. Assumption 2.1 (Fixed effective proof system). We fix an effective proof system PS over the syntax of ℒkbL_kb such that proof checking is decidable. For any finite Γ⊆ℒkb _kb and any φ∈ℒkb _kb, write Γ⊢kbφ _kb for derivability in PS, and define the deductive closure operator Cn(Γ):=φ∈ℒkb:Γ⊢kbφ.Cn( )\;:=\; \ _kb:\ _kb \. The operator Cn satisfies three standing properties used throughout without further comment: (Cn1) Reflexivity: Γ⊆Cn(Γ) ( ). (Cn2) Monotonicity: Γ⊆Γ′⇒Cn(Γ)⊆Cn(Γ′) ( ) ( ). (Cn3) Idempotence: Cn(Cn(Γ))=Cn(Γ)Cn (Cn( ) )=Cn( ). When elements of a state set SO⊆OS_O _O serve as premises for Cn, each state is identified with the corresponding ground atom of ℒkbL_kb. Assumption 2.2 (Finite and effectively listable knowledge bases). The knowledge bases SO⊆OS_O _O considered in this paper are finite and effectively listable under the fixed canonical order. Throughout, OS_O denotes the active-domain semantic universe: for each problem instance (logical structure, domain, rule set), OS_O is the finite set of all ground atoms over the active domain—typically the Herbrand base restricted to the constants appearing in the instance [1]. The finiteness of OS_O is thus a per-instance property, not a restriction on the logical language itself. Assumption 2.3 (Effective redundancy test). For the knowledge bases considered, the predicate s∈Cn(Γ)s ( ) is decidable whenever Γ⊆SO S_O is finite and s∈SOs∈ S_O. This holds in Datalog/Horn settings and bounded-domain theories [1, 10]. I-B Irredundant Core and Derivation Depth Definition 2.1 (Irredundant core). Let SOS_O be a finite knowledge base. Define Atom(SO)Atom(S_O) by the following deterministic procedure: initialize A←SOA← S_O; scan elements of SOS_O in canonical order; for each s, if s∈Cn(A∖s)s (A \s\), set A←A∖sA← A \s\; output A. Proposition 2.1 (Core correctness). Under Assumptions 2.2–2.3, the set A:=Atom(SO)A:=Atom(S_O) satisfies: (i) Cn(A)=Cn(SO)Cn(A)=Cn(S_O); (i) for every a∈Aa∈ A, a∉Cn(A∖a)a (A \a\) (irredundancy); (i) Atom(SO)Atom(S_O) is uniquely determined by SOS_O and the canonical order; (iv) SO⊆Cn(Atom(SO))S_O (Atom(S_O)). Each removal preserves the closure: if s∈Cn(A∖s)s (A \s\), then A⊆Cn(A∖s)A (A \s\), so Cn(A)⊆Cn(Cn(A∖s))=Cn(A∖s)Cn(A) (Cn(A \s\))=Cn(A \s\) by (Cn2) and (Cn3). By induction over the scan, Cn(A)=Cn(SO)Cn(A)=Cn(S_O), giving (i). Canonicality (i) is immediate from the determinism of the procedure and the fixed canonical order. Part (iv) follows from (i): reflexivity (Cn1) gives SO⊆Cn(SO)=Cn(A)S_O (S_O)=Cn(A). Irredundancy (i): let AscanA_scan denote the current set at the moment a is scanned, and let AfinalA_final denote the output. Since a was retained, a∉Cn(Ascan∖a)a (A_scan \a\). Subsequent removals only shrink the set, so Afinal⊆AscanA_final A_scan and hence Afinal∖a⊆Ascan∖aA_final \a\ A_scan \a\. By monotonicity (Cn2), Cn(Afinal∖a)⊆Cn(Ascan∖a)Cn(A_final \a\) (A_scan \a\), so a∉Cn(Afinal∖a)a (A_final \a\). ∎ Definition 2.2 (Core and shortcuts). For a knowledge base SOS_O, write A:=Atom(SO)A:=Atom(S_O) (core premises) and J:=SO∖AJ:=S_O A (stored shortcuts). Remark 2.1 (Proof-system dependence of the core). The irredundant core Atom(SO)Atom(S_O) depends on two choices: the proof system PS (which determines the closure operator Cn) and the canonical order on SOS_O (which resolves ties in the deletion procedure of Definition 2.1). Changing either may alter Atom(SO)Atom(S_O) while preserving the closure equivalence Cn(Atom(SO))=Cn(SO)Cn(Atom(S_O))=Cn(S_O). All information-theoretic quantities in this paper—the semantic rate PAH(πA)P_AH( _A), the leverage factor Λ , the compression ratio—are therefore proof-system-relative structural invariants. This is a feature, not a limitation: it captures the fact that the “value” of a semantic state for communication depends on the inference capabilities shared by sender and receiver. Derivation depth via the immediate consequence operator. Definition 2.3 (Immediate consequence operator). Write Γ⊢kb1s _kb^1s for single-step derivability. Define T(Γ):=Γ∪s∈O:Γ⊢kb1sT_ PS( ):= ∪\s _O: _kb^1s\, with iteration T0(Γ):=ΓT^0_ PS( ):= and Tn+1(Γ):=T(Tn(Γ))T^n+1_ PS( ):=T_ PS(T^n_ PS( )). Axiom 2.1 (Properties of T_ PS). The operator satisfies: (IC1) monotonicity; (IC2) computability (finite output for finite input); (IC3) closure characterization: Cn(Γ)=⋃n≥0Tn(Γ)Cn( )= _n≥ 0T^n_ PS( ); (IC4) finite stabilization. These hold in Datalog and Horn-clause settings over finite domains [1]. Definition 2.4 (Derivation depth). For finite B⊆OB _O and s∈Os _O, Dd(s∣B):=minn≥0:s∈Tn(B),Dd(s B)\;:=\; \n≥ 0:s∈ T^n_ PS(B) \, with Dd(s∣B):=∞Dd(s B):=∞ if s∉Cn(B)s (B). Lemma 2.1 (Properties of derivation depth). Under Axiom 2.1, for finite B⊆OB _O: (i) Dd(s∣B)Dd(s B) is a unique, finite, computable non-negative integer for every s∈Cn(B)s (B); (i) Dd(s∣B)=0Dd(s B)=0 iff s∈Bs∈ B; (i) if B⊆B′B B and s∈Cn(B)s (B), then Dd(s∣B′)≤Dd(s∣B)Dd(s B ) (s B). Part (i): by (IC4), the chain stabilizes at Cn(B)Cn(B) in finitely many steps, so n:s∈Tn(B)≠∅\n:s∈ T^n(B)\≠ and its minimum is finite; computability follows from (IC2). Part (i): T0(B)=BT^0(B)=B. Part (i): monotonicity (IC1) gives Tn(B)⊆Tn(B′)T^n(B) T^n(B ) by induction, so minn:s∈Tn(B′)≤minn:s∈Tn(B) \n:s∈ T^n(B )\≤ \n:s∈ T^n(B)\. ∎ Definition 2.5 (Intrinsic and operational depths). For q∈Cn(A)=Cn(SO)q (A)=Cn(S_O), define nint(q):=Dd(q∣A)n_int(q):=Dd(q A) and nop(q):=Dd(q∣SO)n_op(q):=Dd(q S_O). Definition 2.6 (Semantic atomicity). The semantic atomicity of an information model ℐI with semantic space SOS_O is (ℐ):=|Atom(SO)| A(I):=|Atom(S_O)|. Definition 2.7 (Maximum intrinsic derivation depth). The maximum intrinsic derivation depth of ℐI is (ℐ):=maxq∈SODd(q∣Atom(SO)), D_d(I)\;:=\; _q\,∈\,S_O\;Dd (q (S_O) ), with the convention max∅:=0 :=0. Theorem 2.1 (Computable semantic invariants). Under Assumptions 2.2–2.3 and Axiom 2.1: (i) A and D_d are uniquely determined, finite, and computable. (i) nop(q)≤nint(q)≤n_op(q)≤ n_int(q)≤ D_d for every q∈SOq∈ S_O. (i) =0 D_d=0 if and only if Atom(SO)=SOAtom(S_O)=S_O. Part (i): Atom(SO)Atom(S_O) is uniquely determined by Proposition 2.1(i) and computable by the deterministic procedure of Definition 2.1; each Dd(q∣A)Dd(q A) is finite and computable by Lemma 2.1(i); the maximum over the finite set SOS_O is computable by enumeration. Part (i): A⊆SOA S_O gives Dd(q∣SO)≤Dd(q∣A)Dd(q S_O) (q A) by Lemma 2.1(i); the bound by D_d is immediate from the definition. Part (i): if A=SOA=S_O, every q∈SO=T0(A)q∈ S_O=T^0(A) has depth 0; conversely, depth 0 for all q implies SO⊆AS_O A by Lemma 2.1(i), and A⊆SOA S_O by construction. ∎ I-C Noisy Information and Closure Fidelity When the sender’s knowledge base SOS_O and the receiver’s reconstructed space S^O S_O differ, the discrepancy is captured by a noise pair. Definition 2.8 (Noisy semantic base). A noisy semantic base of SOS_O is any set S~O:=(SO∖SO−)∪SO+ S_O:=(S_O S_O^-)∪ S_O^+, where SO−⊆SOS_O^- S_O (lost states) and SO+⊆O∖SOS_O^+ _O S_O (spurious states). The pair (SO−,SO+)(S_O^-,S_O^+) is the noise pair; it is trivial when both sets are empty. Definition 2.9 (Closure fidelity). For finite S,S^⊆OS, S _O, Cn(S,S^):=|Cn(S)∩Cn(S^)||Cn(S)∪Cn(S^)|, F_Cn(S, S)\;:=\; |Cn(S) ( S)||Cn(S) ( S)|, with 0/0:=10/0:=1. We have Cn=1 F_Cn=1 iff Cn(S)=Cn(S^)Cn(S)=Cn( S). Definition 2.10 (Core preservation ratio). For A=Atom(SO)A=Atom(S_O) and any S^⊆O S _O, ρAtom(SO,S^):=|A∩S^|/|A| _Atom(S_O, S):=|A∩ S|/|A| (with 0/0:=10/0:=1). Proposition 2.2 (Noise pair, core preservation, and closure fidelity). Let S~O=(SO∖SO−)∪SO+ S_O=(S_O S_O^-)∪ S_O^+ and A=Atom(SO)A=Atom(S_O). (i) ρAtom(SO,S~O)=1 _Atom(S_O, S_O)=1 iff A∩SO−=∅A∩ S_O^-= . (i) If A∩SO−=∅A∩ S_O^-= , then Cn(SO)⊆Cn(S~O)Cn(S_O) ( S_O). (i) If A∩SO−=∅A∩ S_O^-= and SO+⊆Cn(SO)S_O^+ (S_O), then Cn(SO)=Cn(S~O)Cn(S_O)=Cn( S_O) and Cn(SO,S~O)=1 F_Cn(S_O, S_O)=1. (iv) Trivial noise implies ρAtom=1 _Atom=1 and Cn=1 F_Cn=1. (i): Since A⊆SOA S_O and SO+⊆O∖SOS_O^+ _O S_O, we have A∩SO+=∅A∩ S_O^+= , hence A∩S~O=A∖SO−A∩ S_O=A S_O^- and ρAtom=|A∖SO−|/|A|=1 _Atom=|A S_O^-|/|A|=1 iff A∩SO−=∅A∩ S_O^-= . (i): If A∩SO−=∅A∩ S_O^-= , then A⊆SO∖SO−⊆S~OA S_O S_O^- S_O. By (Cn2), Cn(A)⊆Cn(S~O)Cn(A) ( S_O), and Cn(A)=Cn(SO)Cn(A)=Cn(S_O) by Proposition 2.1(i). (i): By (i), Cn(SO)⊆Cn(S~O)Cn(S_O) ( S_O). For the reverse: SO∖SO−⊆SO⊆Cn(SO)S_O S_O^- S_O (S_O) by (Cn1), and SO+⊆Cn(SO)S_O^+ (S_O) by hypothesis, so S~O⊆Cn(SO) S_O (S_O). By (Cn2) and (Cn3), Cn(S~O)⊆Cn(Cn(SO))=Cn(SO)Cn( S_O) (Cn(S_O))=Cn(S_O). (iv): Immediate from S~O=SO S_O=S_O. ∎ Remark 2.2 (Zero-distortion property of redundant states). Proposition 2.2(i) has a per-state counterpart crucial for the rate–distortion analysis: if j∈J=SO∖Atom(SO)j∈ J=S_O (S_O), then j∈Cn(SO∖j)j (S_O \j\), so replacing j by any s^∈Cn(SO) s (S_O) preserves the deductive closure. Errors on redundant states incur zero closure distortion—a property absent from any classical distortion measure and the source of the deductive compression gain formalized in Section IV. I Semantic Channel This section erects the probabilistic layer on the structural framework of Section I. The central object is the semantic channel: a composition of Markov kernels—encoding, carrier transmission, and decoding—each constrained by the enabling structure of an underlying information model (Definition A.3). Notation. All state spaces are finite. A probability distribution on a nonempty finite set S is a function P:S→[0,1]P:S→[0,1] with ∑sP(s)=1 _sP(s)=1; write Δ(S) (S) for the probability simplex and supp(P):=s:P(s)>0supp(P):=\s:P(s)>0\. A Markov kernel κ:X↝Yκ:X Y is a function κ:X×Y→[0,1]κ:X× Y→[0,1] with κ(⋅∣x)∈Δ(Y)κ(· x)∈ (Y) for each x; kernels compose by (κ2∘κ1)(z∣x):=∑yκ1(y∣x)κ2(z∣y)( _2 _1)(z x):= _y _1(y x)\, _2(z y). A kernel is deterministic if |supp(κ(⋅∣x))|=1|supp(κ(· x))|=1 for all x. Shannon entropy, conditional entropy, and mutual information are denoted H(⋅)H(·), H(⋅∣⋅)H(· ·), and I(⋅;⋅)I(·\,;·) (base 2, bits) [8]. Random variables are in sans-serif (o,^o S_o, S_o); expectations are [⋅]E[·]. I-A Enabling Kernels and the Semantic Channel Definition 3.1 (Semantic source). A semantic source is a pair (SO,PO)(S_O,P_O) with PO∈Δ(SO)P_O∈ (S_O). It is full-support if supp(PO)=SOsupp(P_O)=S_O and uniform if PO≡1/|SO|P_O≡ 1/|S_O|. Definition 3.2 (Enabling kernel). Let ℐI be an information model with enabling map ℰ:SO⇒SCE:S_O S_C (Axiom A.3). An enabling kernel for ℐI is a Markov kernel κ:SO↝SCκ:S_O S_C satisfying supp(κ(⋅∣so))⊆ℰ(so),∀so∈SO.supp (κ(· s_o) )\; \;E(s_o), ∀\,s_o∈ S_O. (3.1) Write (ℐ)K(I) for the set of all enabling kernels for ℐI. By Axiom A.3(E3), the deterministic kernel κe(sc∣so):=[sc=e(so)] _e(s_c s_o):=1[s_c=e(s_o)] belongs to (ℐ)K(I), so (ℐ)≠∅K(I)≠ . Proposition 3.1 (Enabling kernels compose). If (ℐ1,ℐ2)(I_1,I_2) is composable (Definition A.4) and κi∈(ℐi) _i (I_i) for i=1,2i=1,2, then κ2∘κ1∈(ℐ2∘ℐ1) _2 _1 (I_2 _1). If (κ2∘κ1)(s′∣so)>0( _2 _1)(s s_o)>0, there exists scs_c with κ1(sc∣so)>0 _1(s_c s_o)>0 and κ2(s′∣sc)>0 _2(s s_c)>0. The enabling constraints give sc∈ℰ1(so)s_c _1(s_o) and s′∈ℰ2(sc)⊆ℰ2∘1(so)s _2(s_c) _2 1(s_o). ∎ We now define the semantic channel as a three-stage composition. Definition 3.3 (Semantic channel). A semantic channel is a tuple ℭ=(ℐ,ℐch,ℐdec,κenc,W,D) C=(I,\,I_ch,\,I_dec,\, _enc,\,W,\,D), where: (i) ℐI is the sender’s information model (Definition A.3) with encoding kernel κenc∈(ℐ) _enc (I), mapping SO↝SCS_O S_C; (i) ℐchI_ch is a carrier channel model with SO(ℐch)=SCS_O(I_ch)=S_C, carrier state set S^C S_C, and carrier channel kernel W∈(ℐch)W (I_ch), mapping SC↝S^CS_C S_C; (i) ℐdecI_dec is a decoding model with SO(ℐdec)=S^CS_O(I_dec)= S_C, reconstructed space S^O⊆O S_O _O, and decoding kernel D∈(ℐdec)D (I_dec), mapping S^C↝S^O S_C S_O. The end-to-end kernel is κsem:=D∘W∘κenc:SO↝S^O. _sem:=D W _enc:S_O S_O. (3.2) By Proposition 3.1 (applied twice), κsem∈(ℐsem) _sem (I_sem) where ℐsem:=ℐdec∘ℐch∘ℐI_sem:=I_dec _ch is the composite information model (Definition A.4, Remark A.1). Since S^O⊆O S_O _O, the proof system (,T,Cn)( PS,T_ PS,Cn) acts on S^O S_O, making Cn(S^O)Cn( S_O), Atom(S^O)Atom( S_O), and Dd(⋅∣Atom(S^O))Dd(· ( S_O)) well-defined. Definition 3.4 (End-to-end noise pair). Setting S~O:=S^O S_O:= S_O, the reconstructed space is a noisy semantic base of SOS_O (Definition 2.8) with SO−:=SO∖S^OS_O^-:=S_O S_O and SO+:=S^O∖SOS_O^+:= S_O S_O. By Proposition 2.2, the core preservation and closure fidelity properties hold with this noise pair. Definition 3.5 (Ideal semantic channel). A semantic channel is ideal if all three constituent models are ideal (Definition A.7) and the kernels are the deterministic bijections induced by the synonymy witnesses. In this case κsem _sem is a deterministic bijection and SO≡ℒsemS~OS_O _L_sem S_O (Proposition A.2). When additionally S~O=SO S_O=S_O and τe2e=idSO _e2e=id_S_O, the noise pair is trivial. I-B Semantic Distortion Definition 3.6 (Distortion function). A distortion function is any d:SO×S^O→[0,∞)d:S_O× S_O→[0,∞) with d(s,s)=0d(s,s)=0 for s∈SO∩S^Os∈ S_O∩ S_O. It is normalized if d≤1d≤ 1. Definition 3.7 (Hamming distortion). dH(so,s^o):=[so≠s^o]d_H(s_o, s_o):=1[s_o≠ s_o]. Definition 3.8 (Closure distortion). For a reference base Γ⊆O _O (typically Γ=SO =S_O), write Γ−s:=Γ∖s _-s:= \s\, Cs:=Cn(Γ−s∪s)C_s:=Cn( _-s∪\s\), and Cs^:=Cn(Γ−s∪s^)C_ s:=Cn( _-s∪\ s\). The closure distortion is the Jaccard distance dCn(so,s^o∣Γ):=1−|Cs∩Cs^||Cs∪Cs^|,d_Cn(s_o, s_o ):=1- |C_s∩ C_ s||C_s∪ C_ s|, (3.3) with 0/0:=00/0:=0. We abbreviate dCn(so,s^o):=dCn(so,s^o∣SO)d_Cn(s_o, s_o):=d_Cn(s_o, s_o S_O). Remark 3.1 (Zero distortion on redundant states). Two properties follow from the Cn-axiomatics: (a) if so∈Cn(Γ−so)s_o ( _-s_o) (redundant), then any s^o∈Cn(Γ−so) s_o ( _-s_o) yields dCn=0d_Cn=0; (b) if so∈Atom(SO)s_o (S_O) and Γ=SO =S_O, then so∉Cn(Γ−so)s_o ( _-s_o), so the replacement genuinely matters. Definition 3.9 (Depth distortion). Let A=Atom(SO)A=Atom(S_O) and dmax:=(ℐ)d_ := D_d(I) (Definition 2.7). Define dDd(so,s^o):=min(|Dd(so∣A)−Dd(s^o∣A)|max(dmax,1), 1)if s^o∈Cn(A),1otherwise.d_Dd(s_o, s_o):= cases \! ( |Dd(s_o A)-Dd( s_o A)| (d_ ,1),\,1 )&if s_o (A),\\ 1&otherwise. cases Definition 3.10 (Composite semantic distortion). For weights α,β,γ≥0α,β,γ≥ 0 with α+β+γ=1α+β+γ=1, dsem(so,s^o):=αdH(so,s^o)+βdCn(so,s^o)+γdDd(so,s^o).d_sem(s_o, s_o):=α\,d_H(s_o, s_o)+β\,d_Cn(s_o, s_o)+γ\,d_Dd(s_o, s_o). (3.4) Setting (α,β,γ)=(1,0,0)(α,β,γ)=(1,0,0) recovers Hamming distortion; (0,1,0)(0,1,0) yields a purely deductive-content measure. Definition 3.11 (Expected distortion). For a semantic source (SO,PO)(S_O,P_O) and distortion d, d¯(ℭ,PO):=∑so,s^oPO(so)κsem(s^o∣so)d(so,s^o). d( C,P_O):= _s_o, s_oP_O(s_o)\, _sem( s_o s_o)\,d(s_o, s_o). (3.5) Definition 3.12 (Per-input expected distortion). For any so∈SOs_o∈ S_O and distortion d, d¯(so∣ℭ):=∑s^oκsem(s^o∣so)d(so,s^o). d(s_o C)\;:=\; _ s_o _sem( s_o s_o)\,d(s_o, s_o). When d=dCnd=d_Cn, we write d¯Cn(so∣ℭ) d_Cn(s_o C). Proposition 3.2 (Noise-pair bounds on closure distortion). Let A=Atom(SO)A=Atom(S_O). If A∩SO−=∅A∩ S_O^-= and SO+⊆Cn(SO)S_O^+ (S_O), then for every so∈SO∖As_o∈ S_O A and s^o∈S~O s_o∈ S_O, dCn(so,s^o∣SO)=0d_Cn(s_o, s_o S_O)=0. Consequently, d¯Cn(ℭ,PO)≤PO(A)⋅maxa∈Ad¯Cn(a∣ℭ) d_Cn( C,P_O)≤ P_O(A)· _a∈ A d_Cn(a C). For so∈SO∖As_o∈ S_O A, Cn(SO∖so)=Cn(SO)Cn(S_O \s_o\)=Cn(S_O). By Proposition 2.2(i), Cn(S~O)=Cn(SO)Cn( S_O)=Cn(S_O). Since SO∖SO−⊆SO⊆Cn(SO)S_O S_O^- S_O (S_O) (by (Cn1)) and SO+⊆Cn(SO)S_O^+ (S_O) by hypothesis, S~O=(SO∖SO−)∪SO+⊆Cn(SO) S_O=(S_O S_O^-)∪ S_O^+ (S_O), so in particular s^o∈Cn(SO) s_o (S_O). Since so∈Js_o∈ J, we have Cn(SO∖so)=Cn(SO)Cn(S_O \s_o\)=Cn(S_O). Because s^o∈Cn(SO)=Cn(SO∖so) s_o (S_O)=Cn(S_O \s_o\), the set (SO∖so)∪s^o⊆Cn(SO∖so)(S_O \s_o\)∪\ s_o\ (S_O \s_o\), so by monotonicity (Cn2) and idempotence (Cn3), Cn((SO∖so)∪s^o)⊆Cn(SO∖so)=Cn(SO)Cn((S_O \s_o\)∪\ s_o\) (S_O \s_o\)=Cn(S_O). The reverse inclusion follows from SO∖so⊆(SO∖so)∪s^oS_O \s_o\ (S_O \s_o\)∪\ s_o\ and monotonicity. Hence Cso=Cs^o=Cn(SO)C_s_o=C_ s_o=Cn(S_O) and dCn=0d_Cn=0. Non-core states contribute zero to the expected distortion; the bound follows. ∎ I-C Semantic Channel Invariants Definition 3.13 (Semantic mutual information and capacity). For a semantic source (SO,PO)(S_O,P_O) and semantic channel ℭ C, the semantic mutual information is Isem(PO,ℭ):=I(o;^o)I_sem(P_O, C):=I( S_o; S_o) under the joint PO(so)κsem(s^o∣so)P_O(s_o)\, _sem( s_o s_o). The Shannon capacity of the carrier channel is C(W):=maxPCI(c;^c)C(W):= _P_CI( S_c; S_c). The semantic channel capacity is Csem(W):=maxPO,κenc∈(ℐ),D∈(ℐdec)Isem(PO,ℭ).C_sem(W):= _P_O,\, _enc (I),\,D (I_dec)I_sem(P_O, C). (3.6) Remark 3.2 (Source of “semantic” in IsemI_sem). The semantic mutual information IsemI_sem is defined as the standard Shannon mutual information and satisfies the same algebraic properties. Its “semantic” qualifier refers not to the functional form but to the context in which it is evaluated: the encoding and decoding kernels are constrained by enabling maps (Definition 3.2), and the performance criteria are closure-based distortion measures (Section I-B) rather than symbol-level metrics. The semantic novelty thus enters through the feasible set and the fidelity criterion, not through a redefinition of mutual information itself. Theorem 3.1 (Data processing bound). (i) Csem(W)≤C(W)C_sem(W)≤ C(W). (i) Csem(W)≤log|SO|C_sem(W)≤ |S_O|. (i) If the enabling maps are full (ℰ(so)=SCE(s_o)=S_C, ℰdec(s^c)=S^OE_dec( s_c)= S_O for all inputs) and the alphabet sizes satisfy |SO|≥|SC||S_O|≥|S_C| and |S^O|≥|S^C|| S_O|≥| S_C| (so that a deterministic surjection f:SO→SCf:S_O→ S_C and a deterministic injection g:S^C→S^Og: S_C→ S_O exist), then Csem(W)=C(W)C_sem(W)=C(W). (i): The Markov chain o→c→^c→^o S_o→ S_c→ S_c→ S_o and the data processing inequality [8] give I(o;^o)≤I(c;^c)≤C(W)I( S_o; S_o)≤ I( S_c; S_c)≤ C(W). (i): Isem≤H(o)≤log|SO|I_sem≤ H( S_o)≤ |S_O|. (i): Let PC∗P_C^* achieve C(W)C(W). Choose a deterministic surjection f:SO→SCf:S_O→ S_C, set POP_O so that PC∗=f#POP_C^*=f_\#P_O, and take a deterministic injection g:S^C→S^Og: S_C→ S_O. Then I(o;^o)=I(o;g(^c))=I(o;^c)=I(c;^c)=C(W)I( S_o; S_o)=I( S_o;g( S_c))=I( S_o; S_c)=I( S_c; S_c)=C(W), where the second equality uses the invertibility of g, and the third uses the Markov chain o→c→^c S_o→ S_c→ S_c (since the encoding is deterministic, H(^c∣o)=H(^c∣c)H( S_c S_o)=H( S_c S_c), giving I(o;^c)=I(c;^c)I( S_o; S_c)=I( S_c; S_c)). ∎ Remark 3.3 (Role of the size condition in Theorem 3.1(i)). The condition |SO|≥|SC||S_O|≥|S_C| ensures the existence of a surjection f:SO→SCf:S_O→ S_C used to push forward the capacity-achieving input distribution; it is a sufficient condition for the single-letter equality Csem=C(W)C_sem=C(W). All block-coding results in this paper (Theorems 3.4–3.5 and Section V) use C(W)C(W) directly via SCnS_C^n and do not require this condition. Proposition 3.3 (Enabling-constrained capacity bound). If the encoding enabling map satisfies |ℰ(so)|≤kenc|E(s_o)|≤ k_enc for all so∈SOs_o∈ S_O, then Csem(W)≤log|⋃so∈SOℰ(so)|≤log(|SO|⋅kenc).C_sem(W)\;≤\; \! | _s_o∈ S_OE(s_o) |\;≤\; (|S_O|· k_enc ). (3.7) In particular, when the enabling map is singleton-valued (kenc=1k_enc=1), as in an ideal channel (Definition 3.5), Csem(W)≤log|SO|C_sem(W)≤ |S_O|, reproducing the source entropy bound of Theorem 3.1(i). By the enabling support constraint (3.1), supp(PC)⊆⋃soℰ(so)supp(P_C) _s_oE(s_o), so H(c)≤log|⋃soℰ(so)|H( S_c)≤ | _s_oE(s_o)|. Data processing gives I(o;^o)≤I(c;^c)≤H(c)I( S_o; S_o)≤ I( S_c; S_c)≤ H( S_c). The second inequality uses |⋃soℰ(so)|≤|SO|⋅kenc| _s_oE(s_o)|≤|S_O|· k_enc. ∎ Definition 3.14 (Structural quality indices). The semantic fidelity index and depth expansion index of ℭ C are (ℭ) F( C) :=1−maxso∈SO∑s^oκsem(s^o∣so)dCn(so,s^o∣SO), :=1- _s_o∈ S_O _ s_o _sem( s_o s_o)\,d_Cn(s_o, s_o S_O), (3.8) (ℭ) E( C) :=maxso∈SO∑s^oκsem(s^o∣so)dDd(so,s^o). := _s_o∈ S_O _ s_o _sem( s_o s_o)\,d_Dd(s_o, s_o). (3.9) Both lie in [0,1][0,1]. For any POP_O, d¯Cn≤1− d_Cn≤ 1- F and d¯Dd≤ d_Dd≤ E. Corollary 3.1 (Fidelity concentration on core). Under the conditions of Proposition 3.2, (ℭ)=1−maxa∈Atom(SO)d¯Cn(a∣ℭ) F( C)=1- _a (S_O) d_Cn(a C): the worst-case closure distortion is attained at a core element. Definition 3.15 (Noise-pair indices). Let A=Atom(SO)A=Atom(S_O) and S~O∩:=SO∩S~O=SO∖SO− S_O^∩:=S_O∩ S_O=S_O S_O^- denote the preserved region. The probabilistic core preservation index is ΦAtom(ℭ):=mina∈Aκsem(a∣a)if A∩SO−=∅,0otherwise. _Atom( C):= cases _a∈ A _sem(a a)&if A∩ S_O^-= ,\\[4.0pt] 0&otherwise. cases The spurious probability index is Ψ+(ℭ):=maxso∈SO∑s^o∈SO+κsem(s^o∣so) _+( C):= _s_o∈ S_O _ s_o∈ S_O^+ _sem( s_o s_o). Proposition 3.4 (Properties of noise-pair indices). (i) Both indices lie in [0,1][0,1]. (i) For an ideal channel with S~O=SO S_O=S_O and τe2e=id _e2e=id: ΦAtom=1 _Atom=1, Ψ+=0 _+=0. (i) For any PO∈Δ(SO)P_O∈ (S_O), d¯H(ℭ,PO)≤ 1−ΦAtom(ℭ)⋅PO(A). d_H( C,P_O)\;≤\;1- _Atom( C)· P_O(A). (3.10) Part (i) is immediate from the definitions. Part (i): under the identity kernel, κsem(a∣a)=1 _sem(a a)=1 and SO+=∅S_O^+= . Part (i): d¯H=1−∑so∈S~O∩PO(so)κsem(so∣so) d_H=1- _s_o∈ S_O^∩P_O(s_o)\, _sem(s_o s_o). When A∩SO−≠∅A∩ S_O^-≠ , ΦAtom=0 _Atom=0 and the bound is trivial. When A∩SO−=∅A∩ S_O^-= , one has A⊆S~O∩A S_O^∩, so ∑so∈S~O∩PO(so)κsem(so∣so)≥ _s_o∈ S_O^∩P_O(s_o)\, _sem(s_o s_o)\;≥\; ∑a∈APO(a)κsem(a∣a) _a∈ AP_O(a)\, _sem(a a) ≥ \;≥\; PO(A)⋅ΦAtom, P_O(A)· _Atom, giving d¯H≤1−PO(A)ΦAtom d_H≤ 1-P_O(A)\, _Atom. ∎ Definition 3.16 (Receiver-side comparison indices). The atomicity shift is Δ:=|Atom(S~O)|−|Atom(SO)| A:=|Atom( S_O)|-|Atom(S_O)|; the depth shift is Δ:=maxq∈S~ODd(q∣Atom(S~O))−(ℐ) D_d:= _q∈ S_ODd(q ( S_O))- D_d(I). Both are computable (Remark A.2). Proposition 3.5 (Structural comparison properties). (i) Trivial noise: Δ=Δ=0 A= D_d=0. (i) Core-preserving noise (A∩SO−=∅A∩ S_O^-= , SO+⊆Cn(SO)S_O^+ (S_O)): the set A is an irredundant generating subset of S~O S_O for Cn(S~O)=Cn(SO)Cn( S_O)=Cn(S_O). The canonical irredundant core Atom(S~O)Atom( S_O) satisfies Cn(Atom(S~O))=Cn(A)Cn(Atom( S_O))=Cn(A), but in general |Atom(S~O)||Atom( S_O)| may be larger or smaller than |A||A|, depending on the canonical order and the surplus elements SO+S_O^+. When SO+=∅S_O^+= , Atom(S~O)=AAtom( S_O)=A and hence Δ=0 A=0. (i): S~O=SO S_O=S_O. (i): By Proposition 2.2(i), Cn(S~O)=Cn(SO)Cn( S_O)=Cn(S_O) and A⊆S~OA S_O. Since A is irredundant (Proposition 2.1(i)) and Cn(A)=Cn(SO)=Cn(S~O)Cn(A)=Cn(S_O)=Cn( S_O), the set A is an irredundant generating subset of S~O S_O. However, the canonical irredundant core Atom(S~O)Atom( S_O)—computed by the deletion procedure of Definition 2.1 applied to S~O S_O under the fixed canonical order—may differ from A because surplus elements in SO+S_O^+ can render core elements redundant before they are scanned, while themselves remaining irredundant. When SO+=∅S_O^+= , S~O⊆SO S_O S_O and every element of S~O∖A S_O A lies in J⊆Cn(A)J (A). We show Atom(S~O)=AAtom( S_O)=A. For any a∈Aa∈ A: at scan time the current set Bscan⊆S~O⊆SOB_scan S_O S_O, so Bscan∖a⊆SO∖aB_scan \a\ S_O \a\. By monotonicity (Cn2), Cn(Bscan∖a)⊆Cn(SO∖a)Cn(B_scan \a\) (S_O \a\). Since a∈A=Atom(SO)a∈ A=Atom(S_O), a∉Cn(SO∖a)a (S_O \a\), hence a∉Cn(Bscan∖a)a (B_scan \a\) and a survives. For any s∈S~O∖As∈ S_O A: since all elements of A survive (those before s by the preceding argument; those after s not yet scanned), A⊆Bscan∖sA B_scan \s\, so s∈Cn(A)⊆Cn(Bscan∖s)s (A) (B_scan \s\) and s is removed. Hence Atom(S~O)=AAtom( S_O)=A and Δ=0 A=0. ∎ Theorem 3.2 (Semantic Fano bound). Let (SO,PO)(S_O,P_O) be full-support, ϵ:=d¯H(ℭ,PO)ε:= d_H( C,P_O), and hbh_b the binary entropy. Then Isem(PO,ℭ)≥H(o)−hb(ϵ)−ϵlog(|SO|−1).I_sem(P_O, C)≥ H( S_o)-h_b(ε)-ε (|S_O|-1). (3.11) The Fano penalty log(|SO|−1) (|S_O|-1) is determined by the source alphabet size |SO||S_O| and is independent of the reconstruction alphabet S~O S_O; this is the standard form of Fano’s inequality [8, Theorem 2.10.1]. By Proposition 3.4(i), ϵ≤1−ΦAtom⋅PO(A)ε≤ 1- _Atom· P_O(A), so high core preservation forces high mutual information. For an ideal channel with S~O=SO S_O=S_O and τe2e=id _e2e=id, ϵ=0ε=0 and Isem=H(o)I_sem=H( S_o). Apply Fano’s inequality [8] to the source variable o∈SO S_o∈ S_O and the reconstruction ^o∈S~O S_o∈ S_O: H(o∣^o)≤hb(ϵ)+ϵlog(|SO|−1)H( S_o S_o)≤ h_b(ε)+ε (|S_O|-1). Since Isem=H(o)−H(o∣^o)I_sem=H( S_o)-H( S_o S_o), (3.11) follows. ∎ Corollary 3.2 (Irredundant source, trivial noise). If Atom(SO)=SOAtom(S_O)=S_O, S~O=SO S_O=S_O, and the enabling maps are full with |SO|≥|SC||S_O|≥|S_C| and |S^O|≥|S^C|| S_O|≥| S_C|, then Csem(W)=C(W)C_sem(W)=C(W), =0 D_d=0, dsem=αdH+βdCnd_sem=α\,d_H+β\,d_Cn for every reachable pair, and Ψ+=0 _+=0. Theorem 3.3 (Invariant summary). Under the standing assumptions, all invariants in families I–VI are well-defined, finite, and computable: I. Source-side: , A, D_d. I. Set-level: ρAtom,Cn _Atom, F_Cn. I. Noise-pair: ΦAtom,Ψ+ _Atom, _+. IV. Quality: , F, E. V. Comparison: Δ,Δ A, D_d. VI. Info-theoretic: Isem,Csem,C(W)I_sem,C_sem,C(W). Key relationships: (a) Isem≤Csem≤C(W)I_sem≤ C_sem≤ C(W); (b) d¯Cn≤1− d_Cn≤ 1- F, d¯Dd≤ d_Dd≤ E; (c) d¯H≤1−ΦAtomPO(A) d_H≤ 1- _Atom\,P_O(A); (d) the Fano bound (3.11); (e) fidelity concentrates on core under core-preserving noise; (f) ideal collapse: all distortion invariants vanish and all fidelity invariants are maximal. I-D Semantic Channel Coding For the carrier channel kernel W:SC↝S^CW:S_C S_C, the n-fold memoryless extension is W⊗n(s^cn∣scn):=∏i=1nW(s^c(i)∣sc(i))W n( s_c^n s_c^n):= _i=1^nW( s_c^(i) s_c^(i)). Definition 3.17 (Semantic block code). An (n,M)(n,M) semantic block code consists of a message set ℳ⊆SOM S_O with |ℳ|=M|M|=M, an encoding function fn:ℳ→SCnf_n:M→ S_C^n, and a decoding function gn:S^Cn→S^Og_n: S_C^n→ S_O. The rate is R:=(logM)/nR:=( M)/n. Definition 3.18 (Reliability criteria). For each m∈ℳm , let ^o(m):=gn(^cn) S_o^(m):=g_n( S_c^n) be the reconstruction when fn(m)f_n(m) is sent through W⊗nW n. (i) Hamming: Pe(n):=maxmPr[^o(m)≠m]P_e^(n):= _m [ S_o^(m)≠ m]. (i) Closure: Pe,Cn(n):=maxmPr[dCn(m,^o(m)∣SO)>0]P_e,Cn^(n):= _m [d_Cn(m, S_o^(m) S_O)>0]. Assumption 3.1 (Deductive independence of core elements). For distinct a1,a2∈Atom(SO)a_1,a_2 (S_O), RCn(a1)∩RCn(a2)=∅R_Cn(a_1)∩ R_Cn(a_2)= , where RCn(m):=s^o∈S^O:dCn(m,s^o∣SO)=0R_Cn(m):=\ s_o∈ S_O:d_Cn(m, s_o S_O)=0\. This holds when core elements contribute disjoint deductive increments to the closure. Remark 3.4 (General case: confusability structure and relaxation of core-disjointness). Assumption 3.1 is the simplest sufficient condition guaranteeing that each core element is uniquely identifiable from its zero-distortion reconstruction set. When the assumption fails—i.e., RCn(a1)∩RCn(a2)≠∅R_Cn(a_1)∩ R_Cn(a_2)≠ for some distinct a1,a2∈Aa_1,a_2∈ A—the overlapping acceptable sets induce a confusability graph GCn=(A,ECn)G_Cn=(A,E_Cn) with a1,a2∈ECn\a_1,a_2\∈ E_Cn iff RCn(a1)∩RCn(a2)≠∅R_Cn(a_1)∩ R_Cn(a_2)≠ . In the general case, zero-error source coding theory [9] suggests that Rsem(0;dCn)R_sem(0;d_Cn) is determined by the graph entropy of this confusability graph under πA _A, scaled by PAP_A. Specifically, the zero-distortion rate is sandwiched between a graph-entropy lower bound and the achievable upper bound PAH(πA)P_A\,H( _A) of Theorem 4.1; a precise characterization under general confusability is left to future work. Under Assumption 3.1, GCnG_Cn has no edges, so the graph entropy reduces to the Shannon entropy H(πA)H( _A) and the two bounds coincide, recovering (4.1). All results in this paper are stated under Assumption 3.1, which holds whenever core elements contribute disjoint deductive increments to the closure—a condition verified computationally in the instances of Section V-E, where the core facts were observed to produce pairwise-disjoint closure increments. Theorem 3.4 (Converse bounds). Let A=Atom(SO)A=Atom(S_O). (i) Any (n,M)(n,M) code with Pe(n)≤ϵP_e^(n)≤ε satisfies logM≤(nC(W)+1)/(1−ϵ) M≤(nC(W)+1)/(1-ε). (i) (Under Assumption 3.1) Any (n,|SO|)(n,|S_O|) code with Pe,Cn(n)≤ϵP_e,Cn^(n)≤ε satisfies log|A|≤(nC(W)+1)/(1−ϵ). |A|≤(nC(W)+1)/(1-ε). (3.12) (i): Standard Fano argument [8]: for uniform M, logM≤I(;^)+H(∣^)≤nC(W)+1+ϵlogM M≤ I( M; M)+H( M M)≤ nC(W)+1+ε M. (i): Under Assumption 3.1, the sets RCn(a):=s^o:dCn(a,s^o∣SO)=0R_Cn(a):=\ s_o:d_Cn(a, s_o S_O)=0\ for a∈Aa∈ A are pairwise disjoint. The preimages Ba:=yn:gn(yn)∈RCn(a)B_a:=\y^n:g_n(y^n)∈ R_Cn(a)\ are pairwise disjoint in S^Cn S_C^n with W⊗n(Ba∣fn(a))≥1−ϵW n(B_a f_n(a))≥ 1-ε. The Fano argument applied to the |A||A|-message sub-problem yields the bound: assign a uniform prior on A, so log|A|=H()≤I(;^)+H(∣^)≤nC(W)+1+ϵlog|A| |A|=H( A)≤ I( A; A)+H( A A)≤ nC(W)+1+ε |A|, where A is the core estimate and Pr[^≠]≤ϵ [ A≠ A]≤ε follows from the disjointness of the decoding regions. Rearranging gives (3.12). ∎ Theorem 3.5 (Achievability). Let C(W)>0C(W)>0 and A=Atom(SO)A=Atom(S_O). (i) Hamming: The full knowledge base SOS_O can be communicated Hamming-reliably using n>log|SO|/C(W)n> |S_O|/C(W) channel uses. (i) Closure: If S^O⊇SO S_O S_O and the encoding enabling is full, then Pe,Cn(n)→0P_e,Cn^(n)→ 0 provided log|A|/n<C(W) |A|/n<C(W). (i): By Shannon’s channel coding theorem [29, 8], for any rate R<C(W)R<C(W) there exists a sequence of (n,⌈2nR⌉)(n, 2^nR ) codes with maximal error probability Pe(n)→0P_e^(n)→ 0. Setting M=|SO|M=|S_O| and R=log|SO|/nR= |S_O|/n, the hypothesis n>log|SO|/C(W)n> |S_O|/C(W) ensures R<C(W)R<C(W), giving Hamming-reliable communication of SOS_O. (i): The code is constructed in two layers. Layer 1 (core code). Since log|A|/n<C(W) |A|/n<C(W), the channel coding theorem yields a sequence of (n,|A|)(n,|A|) codes (fnA,gnA)(f_n^A,g_n^A) with fnA:A→SCnf_n^A:A→ S_C^n and gnA:S^Cn→Ag_n^A: S_C^n→ A satisfying Pe(n)(A):=maxa∈APr[gnA(^cn)≠a∣fnA(a) sent]→0P_e^(n)(A):= _a∈ A [g_n^A( S_c^n)≠ a f_n^A(a) sent]→ 0. Since A⊆SO⊆S^OA S_O S_O by hypothesis, every output of gnAg_n^A is a valid element of S^O S_O. Layer 2 (redundant extension). Fix a0∈Aa_0∈ A. Define the full encoder fn:SO→SCnf_n:S_O→ S_C^n by fn(m):=fnA(m)f_n(m):=f_n^A(m) for m∈Am∈ A and fn(m):=fnA(a0)f_n(m):=f_n^A(a_0) for m∈Jm∈ J. The decoder is gn:=gnA:S^Cn→Ag_n:=g_n^A: S_C^n→ A. Closure-reliability analysis. For m=a∈Am=a∈ A: since dCn(a,a∣SO)=0d_Cn(a,a S_O)=0, Pr[dCn(a,gn(^cn)∣SO)>0]≤Pr[gn(^cn)≠a]≤Pe(n)(A). [d_Cn (a,g_n( S_c^n) S_O )>0 ]\;≤\; [g_n( S_c^n)≠ a ]\;≤\;P_e^(n)(A). For m=j∈Jm=j∈ J: the decoder outputs some a^∈A a∈ A, determined by the channel realization. Regardless of which a a is output, a^∈A⊆SO⊆Cn(SO) a∈ A S_O (S_O). Since j∈Jj∈ J, we have j∈Cn(SO∖j)j (S_O \j\), so Cn(SO∖j)=Cn(SO)Cn(S_O \j\)=Cn(S_O) (by (Cn1), (Cn2), and (Cn3)). Because a^∈Cn(SO)=Cn(SO∖j) a (S_O)=Cn(S_O \j\), the set (SO∖j)∪a^⊆Cn(SO∖j)(S_O \j\)∪\ a\ (S_O \j\), giving Cn((SO∖j)∪a^)=Cn(SO∖j)=Cn(SO)Cn ((S_O \j\)∪\ a\ )\;=\;Cn(S_O \j\)\;=\;Cn(S_O) by (Cn2) and (Cn3). The reverse inclusion follows from SO∖j⊆(SO∖j)∪a^S_O \j\ (S_O \j\)∪\ a\ and monotonicity. Hence dCn(j,a^∣SO)=0d_Cn(j, a S_O)=0 deterministically for every channel output. Combining: Pe,Cn(n)=maxm∈SOPr[dCn(m,gn(^cn)∣SO)>0]≤Pe(n)(A)→0P_e,Cn^(n)= _m∈ S_O [d_Cn(m,g_n( S_c^n) S_O)>0]≤ P_e^(n)(A)→ 0. ∎ Corollary 3.3 (Minimum blocklength). Under Assumption 3.1, the minimum blocklength for closure-reliable communication satisfies n∗≈log|Atom(SO)|/C(W)n^*≈ |Atom(S_O)|/C(W), while Hamming reliability requires nH∗≈log|SO|/C(W)n_H^*≈ |S_O|/C(W). The deductive compression ratio is log|Atom(SO)|/log|SO|<1 |Atom(S_O)|/ |S_O|<1 whenever |J|>0|J|>0. Definition 3.19 (Semantic rate–distortion function). For a semantic source (SO,PO)(S_O,P_O) and bounded distortion d, Rsem(D):=minPS^∣S:[d(So,S^o)]≤DI(o;^o).R_sem(D):= _ subarraycP_ S S:\,E[d(S_o, S_o)]≤ D subarrayI( S_o; S_o). (3.13) When d=dCn(⋅,⋅∣SO)d=d_Cn(·,· S_O), the reference knowledge base SOS_O is a fixed model parameter: each source symbol So(t)S_o^(t) takes values in SOS_O, and every evaluation dCn(So(t),S^o(t)∣SO)d_Cn(S_o^(t), S_o^(t) S_O) uses the same SOS_O, so dCnd_Cn is a standard single-letter distortion parameterized by SOS_O. For i.i.d. source blocks of length m, the block distortion is d(m)(sm,s^m):=m−1∑t=1md(st,s^t)d^(m)(s^m, s^m):=m^-1 _t=1^md(s_t, s_t), and the rate–distortion coding theorem [8, Theorem 10.2.1] guarantees achievability at any rate exceeding Rsem(D)R_sem(D). Proposition 3.6 (Properties of RsemR_sem). (i) The minimum exists. (i) RsemR_sem is non-increasing and convex. (i) Under Hamming distortion, Rsem(0)=H(PO)R_sem(0)=H(P_O). (iv) Under closure distortion with S^O⊇Cn(SO)∩O S_O (S_O) _O, Rsem(0;dCn)≤log|Atom(SO)|R_sem(0;d_Cn)≤ |Atom(S_O)|. (v) RsemR_sem is computable. Parts (i)–(i) are standard [8]. For (iv), fix a0∈Aa_0∈ A and define ϕ(so):=soφ(s_o):=s_o if so∈As_o∈ A, ϕ(so):=a0φ(s_o):=a_0 if so∈Js_o∈ J. The deterministic conditional PS^∣S(⋅∣so)=δϕ(so)P_ S S(· s_o)= _φ(s_o) achieves [dCn]=0E[d_Cn]=0 (redundant states yield zero distortion by Remark 3.1(a)) and I(o;^o)=H(^o)≤log|A|I( S_o; S_o)=H( S_o)≤ |A|. ∎ Remark 3.5 (Computation and source–channel separation). The semantic rate–distortion function Rsem(D;dCn)R_sem(D;d_Cn) can be computed to arbitrary precision via a variant of the Blahut–Arimoto algorithm [8]: the standard alternating minimization applies because the feasible set is a compact subset of a finite-dimensional probability simplex and the objective I(o;^o)I( S_o; S_o) is convex in the conditional distribution for fixed POP_O. Han et al. [14] have recently developed an extended Blahut–Arimoto algorithm for computing semantic rate–distortion functions under synonymous mappings. Their alternating-minimization framework is conceptually related to ours and can potentially be adapted to the closure-distortion setting; a formal verification that the feasibility structure of dCnd_Cn satisfies the regularity conditions of their convergence proof is an interesting algorithmic direction. The classical source–channel separation theorem [29, 8] carries over verbatim to the semantic framework: semantic source coding at rate Rsem(D)R_sem(D) followed by channel coding at rate C(W)C(W) is optimal. The semantic novelty is that Rsem(D;dCn)<R(D;dH)R_sem(D;d_Cn)<R(D;d_H) for D near zero whenever |J|>0|J|>0, yielding a strict improvement in the achievable distortion–capacity operating point. IV Fundamental Limits of Semantic Communication This section derives the central information-theoretic results. The key ingredient is the zero-distortion property of redundant states (Remark 3.1(a)): under closure distortion only the core Atom(SO)Atom(S_O) contributes to rate and distortion, enabling a decomposition with no classical counterpart. Section roadmap. Sections IV-A–IV-B derive the tight zero-distortion semantic rate and its extension to all distortion levels, treating the receiver’s inference engine as unconstrained. Section IV-C establishes the semantic source–channel separation theorem and quantifies the semantic leverage factor. Section IV-D proves the strengthened semantic Fano inequality. Section IV-E introduces a new degree of freedom—the receiver’s derivation budget δ, measured in T_ PS-iterations—and replaces the closure distortion dCnd_Cn by a budget-constrained variant dCnδd_Cn^δ (Definition 4.2), which requires the receiver to reconstruct every stored state within δ iterations rather than merely preserving the abstract closure. This leads to a δ-irredundant core filtration AδA_δ and a rate–delay–distortion surface that smoothly interpolates between the classical rate H(PO)H(P_O) at δ=0δ=0 and the semantic rate PAH(πA)P_A\,H( _A) at δ=δ= D_d. Throughout, SOS_O is a finite knowledge base with irredundant core A:=Atom(SO)A:=Atom(S_O), k:=|A|k:=|A|, and stored shortcuts J:=SO∖AJ:=S_O A with |J|=|SO|−k|J|=|S_O|-k. We write PA:=PO(A)P_A:=P_O(A), PJ:=1−PAP_J:=1-P_A, and πA(a):=PO(a)/PA _A(a):=P_O(a)/P_A for the conditional core distribution. The carrier channel is W:SC↝S^CW:S_C S_C with C(W)>0C(W)>0. Assumption 3.1 is in force for converse bounds. In block-coding results, n denotes the blocklength (number of channel uses); in rate expressions for the uniform source, we write |SO||S_O| explicitly to avoid ambiguity. IV-A Tight Semantic Rate–Distortion Function Theorem 4.1 (Tight zero-distortion semantic rate). Let (SO,PO)(S_O,P_O) be a semantic source with irredundant core A=Atom(SO)A=Atom(S_O). Under closure distortion dCn(⋅,⋅∣SO)d_Cn(·,· S_O) with S^O⊇Cn(SO)∩O S_O (S_O) _O and Assumption 3.1: Rsem(0;dCn,PO)=PAH(πA).R_sem(0;\,d_Cn,\,P_O)\;=\;P_A\,H( _A). (4.1) Under the uniform source: Rsem(0)=k|SO|logkR_sem(0)= k|S_O| k. The semantic compression gain relative to Hamming fidelity is Rsem(0;dCn)R(0;dH)=PAH(πA)H(PO)< 1whenever k<|SO|. R_sem(0;\,d_Cn)R(0;\,d_H)\;=\; P_A\,H( _A)H(P_O)\;<\;1 k<|S_O|. (4.2) We establish the lower and upper bounds separately. Lower bound. Let PS^∣SP_ S S be any conditional distribution achieving [dCn(So,S^o∣SO)]=0E[d_Cn(S_o, S_o S_O)]=0. Define the auxiliary random variable T:SO→A∪∗T:\,S_O→ A∪\*\ by T(so):=soT(s_o):=s_o if so∈As_o∈ A and T(so):=∗T(s_o):=* if so∈Js_o∈ J. Since T is a deterministic function of SoS_o, the data processing inequality gives I(So;S^o)≥I(T;S^o).I(S_o; S_o)\;≥\;I(T; S_o). (4.3) We now lower-bound the right-hand side. Under Assumption 3.1, the zero-distortion constraint forces supp(PS^∣S(⋅∣a))⊆RCn(a)supp(P_ S S(· a)) R_Cn(a) for each a∈Aa∈ A, where RCn(a):=s^o∈S^O:dCn(a,s^o∣SO)=0R_Cn(a):=\ s_o∈ S_O:d_Cn(a, s_o S_O)=0\ (Assumption 3.1), with the sets RCn(a)a∈A\R_Cn(a)\_a∈ A pairwise disjoint. For redundant states j∈Jj∈ J, PS^∣S(⋅∣j)P_ S S(· j) is unconstrained on Cn(SO)∩S^OCn(S_O)∩ S_O (Remark 3.1(a)). Let Qj:=PS^∣S(⋅∣j)Q_j:=P_ S S(· j) for each j∈Jj∈ J, and define the average redundant output distribution Q:=∑j∈J[PO(j)/PJ]QjQ:= _j∈ J[P_O(j)/P_J]\,Q_j. Write π¯:=∑a∈AπA(a)PS^∣S(⋅∣a) π:= _a∈ A _A(a)\,P_ S S(· a) for the average core output distribution. Since the supports RCn(a)R_Cn(a) are pairwise disjoint, the entropy of the mixture π¯ π decomposes by the standard disjoint-support identity [8, Theorem 2.7.3]: H(π¯)=H(πA)+∑a∈AπA(a)H(PS^∣S(⋅∣a)).H( π)\;=\;H( _A)+ _a∈ A _A(a)\,H (P_ S S(· a) ). (4.4) The conditional entropy of S^o S_o given T is H(S^o∣T)=PA∑aπA(a)H(PS^∣S(⋅∣a))+PJH(Q),H( S_o T)=P_A _a _A(a)\,H(P_ S S(· a))+P_J\,H(Q), and the output marginal is PS^=PAπ¯+PJQP_ S=P_A\, π+P_J\,Q. By the concavity of Shannon entropy [8, Theorem 2.7.3], H(PAπ¯+PJQ)≥PAH(π¯)+PJH(Q).H(P_A\, π+P_J\,Q)\;≥\;P_A\,H( π)+P_J\,H(Q). (4.5) Therefore, using (4.4): I(T;S^o) I(T; S_o) =H(PAπ¯+PJQ)−H(S^o∣T) =H(P_A\, π+P_J\,Q)-H( S_o T) ≥PAH(π¯)+PJH(Q)−PA∑aπA(a)H(PS^∣S(⋅∣a))−PJH(Q) ≥ P_A\,H( π)+P_J\,H(Q)-P_A\! _a _A(a)\,H(P_ S S(· a))-P_J\,H(Q) =PA[H(π¯)−∑aπA(a)H(PS^∣S(⋅∣a))] =P_A [H( π)-\! _a _A(a)\,H(P_ S S(· a)) ] =PAH(πA). =P_A\,H( _A). (4.6) Since (4.6) holds for every feasible PS^∣SP_ S S (the distribution Q is determined by the conditional, not chosen by us), combining with (4.3) gives Rsem(0)≥PAH(πA)R_sem(0)≥ P_A\,H( _A). Upper bound (achievability). We exhibit a feasible conditional achieving I(So;S^o)=PAH(πA)I(S_o; S_o)=P_A\,H( _A) at zero distortion. Define PS^∣S(s^o∣so):=[s^o=so]if so∈A,πA(s^o)if so∈J,P_ S S( s_o s_o):= cases1[ s_o=s_o]&if s_o∈ A,\\ _A( s_o)&if s_o∈ J, cases where πA _A is supported on A. Distortion check. For so=a∈As_o=a∈ A: s^o=a s_o=a, giving dCn(a,a∣SO)=0d_Cn(a,a S_O)=0. For so=j∈Js_o=j∈ J: s^o s_o is drawn from πA _A on A, so s^o∈A⊆Cn(SO) s_o∈ A (S_O); since j is redundant, dCn(j,s^o∣SO)=0d_Cn(j, s_o S_O)=0 (Remark 3.1(a)). Rate computation. Under this conditional, Q=πAQ= _A and the output marginal is PS^=PAπA+PJπA=πAP_ S=P_A _A+P_J _A= _A. All redundant inputs produce the same conditional output distribution πA _A, so S^o⟂So∣T S_o \!\!\! S_o T, giving I(So;S^o∣T)=0I(S_o; S_o T)=0. Hence I(So;S^o)=I(T;S^o)=H(πA)−PJH(πA)=PAH(πA).∎I(S_o; S_o)=I(T; S_o)=H( _A)-P_J\,H( _A)=P_A\,H( _A). Remark 4.1 (Relaxed acceptable sets). The lower bound PAH(πA)P_A\,H( _A) in the proof of Theorem 4.1 holds for arbitrary acceptable-set sizes |RCn(a)|≥1|R_Cn(a)|≥ 1, provided the pairwise-disjoint support condition of Assumption 3.1 holds; the key step is the entropy splitting identity (4.4). The achievability construction of the upper bound applies unchanged. Remark 4.2 (Role of the reconstruction alphabet assumption). Theorem 4.1 assumes S^O⊇Cn(SO)∩O S_O (S_O) _O, ensuring that the decoder can output any state in the sender’s closure. This is an idealized “full reconstruction alphabet” setting that maximizes the deductive compression gain; it corresponds to a receiver whose vocabulary contains at least all closure elements expressible in the ambient universe. When the reconstruction alphabet is restricted to S^O⊊Cn(SO)∩O S_O (S_O) _O—as in the heterogeneous setting of Section V, where S^O=SO(j) S_O=S_O^(j) may exclude some closure elements—the zero-distortion rate can only increase: Rsem(0;S^O)≥Rsem(0;Cn(SO)∩O)=PAH(πA)R_sem(0; S_O)≥ R_sem(0;Cn(S_O) _O)=P_AH( _A). The heterogeneous achievability results of Theorem 5.1 show that the rate PAH(πA)P_AH( _A) remains achievable as long as the core coverage condition A(i)⊆S^OA^(i) S_O holds; when it fails, Corollary 5.3 provides the impossibility characterization. Remark 4.3 (Necessity of the proof-system structure). The result (4.1) depends on three concepts absent from classical rate–distortion theory: the irredundant core Atom(SO)Atom(S_O), the deductive closure Cn, and the core/redundant partition of source symbols. Replacing dCnd_Cn by any distortion that assigns positive cost to all symbol errors eliminates the zero-distortion property of redundant states and recovers R(0)=H(PO)R(0)=H(P_O). IV-B Rate–Distortion Core Decomposition Theorem 4.1 extends to all distortion levels: the full semantic rate–distortion function decomposes into a contribution from the core alone. Theorem 4.2 (Core decomposition of RsemR_sem). Under the hypotheses of Theorem 4.1, Rsem(D;dCn,PO)=PA⋅R(A)(DPA;dCn,πA),D≥0,R_sem(D;\,d_Cn,\,P_O)\;=\;P_A· R^(A)\! ( DP_A;\,d_Cn,\, _A ), D≥ 0, (4.7) where R(A)(D′;dCn,πA)R^(A)(D ;\,d_Cn,\, _A) is the rate–distortion function of the core sub-source (A,πA)(A, _A) with distortion dCn(⋅,⋅∣SO)d_Cn(·,· S_O) restricted to core inputs. Consequences: (i) At D=0D=0, Rsem(0)=PAH(πA)R_sem(0)=P_A\,H( _A), recovering Theorem 4.1. (i) RsemR_sem is convex, non-increasing, and computable. (i) Redundant states are invisible: the function (4.7) depends on SOS_O only through A, PAP_A, πA _A, and the closure-distortion structure on A. Distortion decomposition. By Remark 3.1(a), dCn(j,s^o∣SO)=0d_Cn(j, s_o S_O)=0 for all j∈Jj∈ J and s^o∈Cn(SO) s_o (S_O). The expected distortion therefore decomposes as [dCn]=∑a∈APO(a)[dCn(a,S^o∣SO)∣So=a]=PAπA[d¯A],E[d_Cn]= _a∈ AP_O(a)\,E[d_Cn(a, S_o S_O) S_o=a]=P_A\,E_ _A[ d_A], where d¯A(a):=[dCn(a,S^o∣SO)∣So=a] d_A(a):=E[d_Cn(a, S_o S_O) S_o=a]. The constraint [dCn]≤DE[d_Cn]≤ D is thus equivalent to πA[d¯A]≤D/PAE_ _A[ d_A]≤ D/P_A. Rate decomposition. Using the auxiliary variable T from the proof of Theorem 4.1: I(So;S^o)=I(T;S^o)+I(So;S^o∣T)≥I(T;S^o).I(S_o; S_o)=I(T; S_o)+I(S_o; S_o T)≥ I(T; S_o). The residual I(So;S^o∣T)I(S_o; S_o T) is minimized to zero by choosing identical conditionals for all j∈Jj∈ J (as in the achievability of Theorem 4.1). Hence the minimum of I(So;S^o)I(S_o; S_o) subject to πA[d¯A]≤D/PAE_ _A[ d_A]≤ D/P_A equals the minimum of I(T;S^o)I(T; S_o) subject to the same constraint. The distribution of T concentrates probability PAP_A on A (with conditional distribution πA _A) and probability PJP_J on the singleton ∗\*\. The problem minPS^∣TI(T;S^o):πA[d¯A]≤D/PA _P_ S T\I(T; S_o):E_ _A[ d_A]≤ D/P_A\ is a rate–distortion problem whose source is the mixture of πA _A (with weight PAP_A) and a point mass (with weight PJP_J). The point-mass component contributes zero mutual information and zero distortion, so the minimum equals PA⋅R(A)(D/PA;dCn,πA)P_A· R^(A)(D/P_A;\,d_Cn,\, _A) by the standard rate–distortion scaling for mixtures with a “free” component [8, Problem 10.8]. ∎ IV-C Semantic Source–Channel Separation and Semantic Leverage Theorem 4.3 (Semantic source–channel separation). Let W:SC↝S^CW:S_C S_C with C(W)>0C(W)>0, and let (SO,PO)(S_O,P_O) be a semantic source with core A=Atom(SO)A=Atom(S_O). Part A (Single-shot / message-set regime). Consider an (n,|SO|)(n,|S_O|) semantic block code (Definition 3.17) with message set ℳ=SOM=S_O. (A1) Converse. Closure-reliable communication (Pe,Cn(n)→0P_e,Cn^(n)→ 0) requires blocklength n≥(1−o(1))log|A|/C(W)n≥(1-o(1)) |A|/C(W) (Theorem 3.4(i)). (A2) Achievability. The two-layer code (Theorem 3.5(i)) achieves Pe,Cn(n)→0P_e,Cn^(n)→ 0 whenever log|A|/n<C(W) |A|/n<C(W). (A3) Single-shot leverage. The minimum blocklength for closure reliability is nCn∗≈log|A|/C(W)n^*_Cn≈ |A|/C(W) (Corollary 3.3). Define the single-shot semantic leverage factor Λ1(SO):=log|SO|log|Atom(SO)|> 1whenever k<|SO|. _1(S_O)\;:=\; |S_O| |Atom(S_O)|\;>\;1 k<|S_O|. (4.8) Since identifying one of |SO||S_O| knowledge-base states requires log|SO| |S_O| bits, closure reliability conveys this identification in nCn∗≈log|A|/C(W)n^*_Cn≈ |A|/C(W) channel uses, yielding an effective task-identification rate Reff:=log|SO|nCn∗≈C(W)⋅Λ1(SO).R_eff\;:=\; |S_O|n^*_Cn\;≈\;C(W)· _1(S_O). (4.9) The quantity ReffR_eff is a task-throughput index that counts knowledge-base index bits conveyed per channel use under closure fidelity; it does not represent a Shannon mutual-information rate, and its exceeding C(W)C(W) is consistent with the data processing bound Csem≤C(W)C_sem≤ C(W) (Theorem 3.1(i)), because the closure-based fidelity criterion treats |J||J| redundant states as free. Part B (Asymptotic i.i.d. source-coding regime). Consider a length-m i.i.d. source block from (SO,PO)(S_O,P_O) mapped to n channel uses of W⊗nW n, with m→∞m→∞. The fidelity criterion is zero expected closure distortion (D=0D=0 in (3.13)). We work under the same hypotheses as Theorem 4.1: S^O⊇Cn(SO)∩O S_O (S_O) _O and Assumption 3.1. The relevant source rate is therefore Rsem(0;dCn)=PAH(πA)R_sem(0;d_Cn)=P_A\,H( _A). (B1) Necessary condition. Achieving zero expected closure distortion requires PAH(πA)≤C(W).P_A\,H( _A)\;≤\;C(W). (4.10) (B2) Sufficient condition. If the strict inequality holds in (4.10), then zero expected closure distortion is achievable: by the classical rate–distortion coding theorem [8, Theorem 10.2.1], for every R>Rsem(0)=PAH(πA)R>R_sem(0)=P_A\,H( _A) there exist length-m block source codes achieving [dCn]=0E[d_Cn]=0 at rate R; cascading with a capacity-achieving channel code completes the separation argument. (B3) Asymptotic leverage. Encoding m source symbols at distortion D=0D=0 requires at least mPAH(πA)m\,P_A\,H( _A) bits, and each channel use carries at most C(W)C(W) bits, so lim infm→∞n/m≥PAH(πA)/C(W) _m→∞n/m≥ P_A\,H( _A)/C(W), yielding the asymptotic semantic leverage factor Λ∞(SO,PO):=log|SO|PAH(πA)≥Λ1(SO), _∞(S_O,P_O)\;:=\; |S_O|P_A\,H( _A)\;≥\; _1(S_O), (4.11) with equality when πA _A is uniform on A and PA=1P_A=1. Like Λ1 _1, the factor Λ∞ _∞ uses task-identification normalization: the numerator log|SO| |S_O| counts the index bits needed to specify one knowledge-base state, not the source entropy H(PO)H(P_O). The entropy-based compression gain is the distinct ratio PAH(πA)/H(PO)<1P_A\,H( _A)/H(P_O)<1 of (4.2). Under the uniform source: Λ1=log|SO|/logk _1= |S_O|/ k and Λ∞=|SO|log|SO|/(klogk) _∞=|S_O| |S_O|/(k k). Part A. (A1) is Theorem 3.4(i). (A2) is Theorem 3.5(i). (A3) follows by dividing log|SO| |S_O| by nCn∗≈log|A|/C(W)n^*_Cn≈ |A|/C(W). Part B. (B1): the closure distortion dCn≥0d_Cn≥ 0 is a bounded single-letter distortion on finite alphabets, so the classical source–channel separation theorem [29, 8] applies: achieving expected distortion D=0D=0 over the i.i.d. source requires source rate Rsem(0;dCn)=PAH(πA)≤C(W)R_sem(0;d_Cn)=P_A\,H( _A)≤ C(W), giving (4.10). (B2): by the rate–distortion coding theorem for finite alphabets [8, Theorem 10.2.1], for every ϵ>0ε>0 there exists a block source code of rate PAH(πA)+ϵP_A\,H( _A)+ε achieving [dCn]=0E[d_Cn]=0; a capacity-achieving channel code transmits the compressed index reliably whenever PAH(πA)+ϵ<C(W)P_A\,H( _A)+ε<C(W). (B3): by (B1), every achievable pair (m,n)(m,n) satisfies mPAH(πA)≤nC(W)m\,P_A\,H( _A)≤ nC(W), hence lim infm→∞n/m≥PAH(πA)/C(W) _m→∞n/m≥ P_A\,H( _A)/C(W). The leverage ordering Λ∞≥Λ1≥1 _∞≥ _1≥ 1 follows from PAH(πA)≤log|A|≤log|SO|P_A\,H( _A)≤ |A|≤ |S_O|. ∎ Remark 4.4 (Comparison with the synonymous-mapping semantic leverage). The leverage factors Λ1 _1 and Λ∞ _∞ are the closure-fidelity counterparts of the semantic throughput gain Cs≥C_s≥ C of Niu and Zhang [25]. The two mechanisms are complementary: [25] achieves its gain through source-side equivalence-class collapsing; our framework achieves it through receiver-side deductive reconstruction. Neither mechanism violates the Shannon channel capacity C(W)C(W); both exploit task-specific fidelity criteria to reduce the effective source rate below H(PO)H(P_O). A unified theory combining both mechanisms—synonymous collapsing of the irredundant core followed by deductive expansion at the receiver—would compound the two gains. IV-D Strengthened Semantic Fano Inequality The classical Fano inequality bounds H(X∣X^)H(X X) in terms of Pr[X≠X^] [X≠ X] and the full alphabet size |X||X|. The core/redundant decomposition enables a tighter bound whose penalty term involves log|A| |A| rather than log|SO| |S_O|. Theorem 4.4 (Semantic Fano inequality). Let (SO,PO)(S_O,P_O) be a full-support semantic source, ℭ C a semantic channel, and ϵA:=Pr[S^o≠So,So∈A] _A:= [ S_o≠ S_o,\;S_o∈ A] the core error probability. Then I(So;S^o)≥PAH(πA)−hb(ϵA)−ϵAlog(|A|−1),I(S_o; S_o)\;≥\;P_A\,H( _A)-h_b( _A)- _A (|A|-1), (4.12) where hbh_b is the binary entropy. When |J|=0|J|=0, PA=1P_A=1, πA=PO _A=P_O, and (4.12) reduces to the classical Fano bound with alphabet |A|=|SO||A|=|S_O|. When |J|>0|J|>0, the penalty involves log|A| |A| rather than log|SO| |S_O|; additionally, the reference level is the semantic rate PAH(πA)P_A\,H( _A) rather than the full source entropy H(PO)H(P_O), which is a tightening by H(PO)−PAH(πA)>0H(P_O)-P_A\,H( _A)>0 bits. Define B:=[So∈A]B:=1[S_o∈ A]. Since B is a deterministic function of SoS_o, H(B∣So)=0H(B S_o)=0 and hence I(So;B)=H(B)=hb(PA)I(S_o;B)=H(B)=h_b(P_A). The chain rule for mutual information gives I(So;S^o)+I(So;B∣S^o)=I(So;B)+I(So;S^o∣B).I(S_o; S_o)+I(S_o;B S_o)\;=\;I(S_o;B)+I(S_o; S_o B). Since B=f(So)B=f(S_o), we have I(So;B∣S^o)=H(B∣S^o)−H(B∣So,S^o)=H(B∣S^o)≥0I(S_o;B S_o)=H(B S_o)-H(B S_o, S_o)=H(B S_o)≥ 0. Hence I(So;S^o)=hb(PA)−H(B∣S^o)+I(So;S^o∣B)≥I(So;S^o∣B),I(S_o; S_o)\;=\;h_b(P_A)-H(B S_o)+I(S_o; S_o B)\;≥\;I(S_o; S_o B), (4.13) where the inequality uses H(B∣S^o)≤H(B)=hb(PA)H(B S_o)≤ H(B)=h_b(P_A). Expanding the right-hand side: I(So;S^o∣B) I(S_o; S_o B) =PAI(So;S^o∣B=1)+PJI(So;S^o∣B=0) =P_A\,I(S_o; S_o B=1)+P_J\,I(S_o; S_o B=0) ≥PAI(So;S^o∣B=1), ≥ P_A\,I(S_o; S_o B=1), (4.14) since mutual information is non-negative. Core term. Conditionally on B=1B=1, the source SoS_o takes values in A with distribution πA _A, so H(So∣B=1)=H(πA)H(S_o B=1)=H( _A). Let peA:=Pr[S^o≠So∣So∈A]=ϵA/PAp_e^A:= [ S_o≠ S_o S_o∈ A]= _A/P_A. By the standard Fano inequality applied to the |A||A|-valued variable SoS_o given S^o S_o conditioned on B=1B=1: H(So∣S^o,B=1)≤hb(peA)+peAlog(|A|−1),H(S_o S_o,\,B=1)\;≤\;h_b(p_e^A)+p_e^A (|A|-1), where |A|−1|A|-1 is the standard Fano penalty for the |A||A|-valued source So|B=1S_o|_B=1; this form is valid regardless of the range of the reconstruction S^o S_o, since the penalty is determined by the source alphabet size [8, Theorem 2.10.1]. Hence I(So;S^o∣B=1) I(S_o; S_o B=1) =H(πA)−H(So∣S^o,B=1) =H( _A)-H(S_o S_o,\,B=1) ≥H(πA)−hb(peA)−peAlog(|A|−1). ≥ H( _A)-h_b(p_e^A)-p_e^A (|A|-1). Substituting into (4.14) and (4.13): I(So;S^o)≥PA[H(πA)−hb(ϵA/PA)−(ϵA/PA)log(|A|−1)].I(S_o; S_o)≥ P_A [H( _A)-h_b( _A/P_A)-( _A/P_A) (|A|-1) ]. This equals PAH(πA)−PAhb(ϵA/PA)−ϵAlog(|A|−1)P_A\,H( _A)-P_A\,h_b( _A/P_A)- _A (|A|-1). It remains to bound the first penalty term. Define the random variable Z that takes value ϵA/PA _A/P_A with probability PAP_A and 0 with probability PJP_J. Then [Z]=ϵAE[Z]= _A and [hb(Z)]=PAhb(ϵA/PA)E[h_b(Z)]=P_A\,h_b( _A/P_A). By Jensen’s inequality (concavity of hbh_b): PAhb(ϵA/PA)=[hb(Z)]≤hb([Z])=hb(ϵA).∎P_A\,h_b( _A/P_A)=E[h_b(Z)]≤ h_b(E[Z])=h_b( _A). Remark 4.5 (Operational significance). Bound (4.12) provides a converse for semantic source coding: any encoder–decoder pair with low core error ϵA _A must transmit mutual information at least PAH(πA)−hb(ϵA)−ϵAlog(|A|−1)P_A\,H( _A)-h_b( _A)- _A (|A|-1). Compared with the classical Fano bound I≥H(PO)−hb(ϵ)−ϵlog(|SO|−1)I≥ H(P_O)-h_b(ε)-ε (|S_O|-1), the semantic improvement is twofold: (a) the reference level drops from H(PO)H(P_O) to the semantic rate PAH(πA)P_A\,H( _A), absorbing all redundant-state entropy; (b) the Fano penalty involves log|A| |A| instead of log|SO| |S_O|. Moreover, under the optimal semantic code (which maps all redundant inputs to core elements), the total Hamming error is ϵ=ϵA+PJε= _A+P_J, substantially larger than ϵA _A; the classical bound applied with this inflated ϵε is much looser than the semantic bound (4.12) applied with ϵA _A alone. Combined with Theorem 4.3, this characterizes the operational regime where semantic compression strictly outperforms symbol-level compression. IV-E Rate–Delay–Distortion Tradeoff and Semantic Sampling Theorem The rate–distortion results of Sections IV-A–IV-B treat the receiver’s inference engine as unconstrained: it may iterate T_ PS arbitrarily many times to reconstruct the full closure. In practice, communication is subject to delay constraints: the receiver has a bounded computation budget δ≥0δ≥ 0 (measured in T_ PS-iterations) within which it must reconstruct the deductive closure. This subsection shows that the interaction between derivation depth and delay budget gives rise to a rate–delay–distortion surface that smoothly interpolates between the classical zero-distortion rate H(PO)H(P_O) (no inference, δ=0δ=0) and the semantic rate PAH(πA)P_A\,H( _A) (full inference, δ=δ= D_d), and yields a semantic analogue of the Nyquist sampling theorem. The key structural ingredient is the derivation-depth stratification of Section I-B: the partition of SOS_O into strata Ld:=s∈SO:Dd(s∣A)=dL_d:=\s∈ S_O:Dd(s A)=d\ for d=0,1,…,d=0,1,…, D_d. When the receiver’s derivation budget is δ, states at depth greater than δ from the transmitted base cannot be reconstructed within the budget; conversely, states within depth δ are “free” and need not be transmitted. This depth-for-rate exchange is formalized below. Preview of key objects and main result. The subsection introduces three new objects: (a) the δ-constrained closure distortion dCnδd_Cn^δ (Definition 4.2), a 0,1\0,1\-valued fidelity measure that requires the receiver to reconstruct every stored state within δ derivation steps; (b) the δ-irredundant core Aδ:=Atomδ(SO)A_δ:=Atom_δ(S_O) (Definition 4.3), the subset of SOS_O whose elements cannot be rederived within δ steps from the remaining knowledge base; (c) the rate–delay function Rsem(0,δ)=PδH(πδ)R_sem(0,δ)=P_δ\,H( _δ) (Theorem 4.5), the minimum communication rate when the receiver has a derivation budget of δ steps. The central finding is that AδA_δ shrinks monotonically as δ increases (Proposition 4.1): at δ=0δ=0 no state is free and the rate equals the classical H(PO)H(P_O); at δ=δ= D_d all redundant states become free and the rate drops to the semantic rate PAH(πA)P_A\,H( _A). Assumption 4.1 (Derivation-path completeness; used only for the computable bound in Theorem 4.5(iv) and Lemma 4.2). The knowledge base SOS_O contains all intermediate derivation elements: Tn(A)⊆SOT^n_ PS(A) S_O for every n∈0,1,…,n∈\0,1,…, D_d\. Equivalently, B(≤m)=Tm(A)B^(≤ m)=T^m_ PS(A) for every m≤m≤ D_d. This holds whenever SOS_O is closed under the inference rules up to the maximum derivation depth—in particular, when SOS_O includes all materialized IDB facts up to depth D_d. When this assumption fails, the auxiliary bounds of Proposition 4.1(iv) require replacement of B(≤m)B^(≤ m) by Tm(A)T^m_ PS(A); the tight rate–delay result (4.17) and the rate–delay–distortion surface (4.23) remain valid, as their proofs depend only on Lemma 4.1 and the δ-redundancy definition. Definition 4.1 (Depth-stratified stored base). For m∈0,1,…,m∈\0,1,…, D_d\, define the m-deep stored base B(≤m):=s∈SO:Dd(s∣A)≤m=Tm(A)∩SO.B^(≤ m)\;:=\; \s∈ S_O:Dd(s A)≤ m \\;=\;T^m_ PS(A)∩ S_O. Note: B(≤0)=AB^(≤ 0)=A and B(≤)=SOB^(≤ D_d)=S_O (by Proposition 2.1(iv) and Lemma 2.1(i)). Definition 4.2 (δ-constrained closure distortion). For a reference base Γ=SO =S_O, Γ−s:=SO∖s _-s:=S_O \s\, and integer δ≥0δ≥ 0, the δ-constrained closure distortion is dCnδ(so,s^o∣SO):=0if SO⊆Tδ(Γ−so∪s^o),1otherwise.d_Cn^δ(s_o, s_o S_O)\;:=\; cases0&if S_O T^δ_ PS ( _-s_o∪\ s_o\ ),\\[4.0pt] 1&otherwise. cases (4.15) Remark 4.6 (Properties of dCnδd_Cn^δ). In operational terms: dCnδ(so,s^o∣SO)=0d_Cn^δ(s_o, s_o S_O)=0 means that after replacing the single stored state sos_o by s^o s_o, the receiver can re-derive every element of SOS_O within δ immediate-consequence iterations. The distortion dCnδd_Cn^δ takes values in 0,1\0,1\. When dCnδ(so,s^o∣SO)=0d_Cn^δ(s_o, s_o S_O)=0 and s^o∈Cn(SO) s_o (S_O)—a condition satisfied by every constructive coding scheme in Sections I–IV, whose decoders output elements of SOS_O (or subsets such as A, AδA_δ, or BδB_δ, all contained in Cn(SO)Cn(S_O)), as well as by the heterogeneous two-layer code of Theorem 5.1 (with SO:=SO(i)S_O:=S_O^(i) as reference base), whose decoder outputs elements of A(i)⊆Cn(SO(i))A^(i) (S_O^(i))—the chain SO⊆Tδ(Γ−so∪s^o)⊆Cn(Γ−so∪s^o)S_O T^δ( _-s_o∪\ s_o\) ( _-s_o∪\ s_o\) gives Cn(SO)⊆Cn(Γ−so∪s^o)Cn(S_O) ( _-s_o∪\ s_o\) by (Cn2), while Γ−so∪s^o⊆Cn(SO) _-s_o∪\ s_o\ (S_O) gives the reverse inclusion by (Cn2) and (Cn3); hence the closures coincide and dCn(so,s^o∣SO)=0d_Cn(s_o, s_o S_O)=0. Without the condition s^o∈Cn(SO) s_o (S_O), neither direction of implication between dCnδ=0d_Cn^δ=0 and dCn=0d_Cn=0 holds in general. The following auxiliary property is used solely to verify the boundary case (i) of Theorem 4.5 and does not assert a universal equivalence between dCnδd_Cn^δ and dCnd_Cn: for Cn-redundant states so∈Js_o∈ J and s^o∈SO s_o∈ S_O, if δ≥δ≥ D_d, then dCnδ(so,s^o∣SO)=0d_Cn^δ(s_o, s_o S_O)=0, since A⊆Γ−soA _-s_o and monotonicity give Tδ(Γ−so∪s^o)⊇T(A)⊇SOT^δ( _-s_o∪\ s_o\) T D_d(A) S_O. For general (non-redundant) states and δ<δ< D_d, the two distortion measures dCnδd_Cn^δ and dCnd_Cn need not coincide. The core mechanism of the rate–delay theory is the δ-redundancy–based free substitution of Lemma 4.1, which holds for arbitrary δ≥0δ≥ 0 without reference to dCnd_Cn. Operationally, the condition SO⊆Tδ(Γ−so∪s^o)S_O T^δ( _-s_o∪\ s_o\) requires every explicitly stored state in SOS_O to be re-derivable within δ T_ PS-iterations from the modified base—a stronger requirement than merely preserving the closure Cn(SO)Cn(S_O), reflecting the constraint that the receiver must reconstruct each element of the sender’s knowledge base within δ T_ PS-iterations. Definition 4.3 (δ-redundancy and δ-irredundant core). A state s∈SOs∈ S_O is δ-redundant if s∈Tδ(SO∖s)s∈ T^δ_ PS(S_O \s\). The δ-irredundant core is Atomδ(SO):=s∈SO:s∉Tδ(SO∖s).Atom_δ(S_O)\;:=\; \s∈ S_O:s∉ T^δ_ PS(S_O \s\) \. Lemma 4.1 (δ-redundant free substitution). If s∈SOs∈ S_O is δ-redundant (Definition 4.3), i.e., s∈Tδ(SO∖s)s∈ T^δ_ PS(S_O \s\), then for every s^∈O s _O: dCnδ(s,s^∣SO)= 0.d_Cn^δ(s, s S_O)\;=\;0. Write Γ−s:=SO∖s _-s:=S_O \s\. We must show SO⊆Tδ(Γ−s∪s^)S_O T^δ( _-s∪\ s\). Since Γ−s⊆Γ−s∪s _-s _-s∪\ s\, monotonicity (IC1) gives Tδ(Γ−s)⊆Tδ(Γ−s∪s^)T^δ( _-s) T^δ( _-s∪\ s\). The δ-redundancy hypothesis s∈Tδ(Γ−s)s∈ T^δ( _-s) then yields s∈Tδ(Γ−s∪s^)s∈ T^δ( _-s∪\ s\). Furthermore, Γ−s⊆T0(Γ−s∪s^)⊆Tδ(Γ−s∪s^) _-s T^0( _-s∪\ s\) T^δ( _-s∪\ s\). Hence SO=Γ−s∪s⊆Tδ(Γ−s∪s^)S_O= _-s∪\s\ T^δ( _-s∪\ s\). ∎ Remark 4.7 (Output insensitivity for δ-redundant states). Lemma 4.1 gives dCnδ(s,s^∣SO)=0d_Cn^δ(s, s S_O)=0 for every s^∈O s _O when s is δ-redundant. This universality is by design: dCnδd_Cn^δ tests whether the receiver can reconstruct all of SOS_O within δ T_ PS-iterations from Γ−s∪s _-s∪\ s\, not whether s s itself is meaningful. Since δ-redundancy gives SO⊆Tδ(Γ−s)S_O T^δ( _-s), monotonicity yields SO⊆Tδ(Γ−s∪s^)S_O T^δ( _-s∪\ s\) regardless of s s. Operationally, every constructive coding scheme in Sections I–IV outputs elements of SOS_O (or subsets such as A, AδA_δ, BδB_δ, all contained in Cn(SO)Cn(S_O)); the universality over OS_O is needed only in the lower bound of Theorem 4.5, where it ensures that δ-redundant states contribute zero mutual information under any feasible conditional. Assumption 4.2 (Deductive independence of δ-irredundant core). For each δ∈0,1,…,δ∈\0,1,…, D_d\ and reconstruction alphabet S^O⊇Atomδ(SO) S_O _δ(S_O), the δ-constrained zero-distortion sets Rδ(a):=s^o∈S^O:dCnδ(a,s^o∣SO)=0,a∈Atomδ(SO),R_δ(a)\;:=\; \ s_o∈ S_O:d_Cn^δ(a, s_o S_O)=0 \, a _δ(S_O), are pairwise disjoint: Rδ(a1)∩Rδ(a2)=∅R_δ(a_1)∩ R_δ(a_2)= for distinct a1,a2∈Atomδ(SO)a_1,a_2 _δ(S_O). Under the condition S^O⊆Cn(SO) S_O (S_O), Assumption 3.1 implies the present assumption for every δ (Remark 4.8). When S^O⊈Cn(SO) S_O (S_O), the present assumption must be verified independently. Remark 4.8 (Proof that core-disjointness implies δ-core-disjointness). Suppose S^O⊆Cn(SO) S_O (S_O) and let s^o∈Rδ(a) s_o∈ R_δ(a), so SO⊆Tδ(Γ−a∪s^o)⊆Cn(Γ−a∪s^o)S_O T^δ( _-a∪\ s_o\) ( _-a∪\ s_o\). By monotonicity (Cn2), Cn(SO)⊆Cn(Γ−a∪s^o)Cn(S_O) ( _-a∪\ s_o\). For the reverse: since s^o∈S^O⊆Cn(SO) s_o∈ S_O (S_O) and Γ−a⊆SO⊆Cn(SO) _-a S_O (S_O), we have Γ−a∪s^o⊆Cn(SO) _-a∪\ s_o\ (S_O), giving Cn(Γ−a∪s^o)⊆Cn(SO)Cn( _-a∪\ s_o\) (S_O) by (Cn2) and (Cn3). Hence Cn(Γ−a∪s^o)=Cn(SO)=Cn(Γ−a∪a)Cn( _-a∪\ s_o\)=Cn(S_O)=Cn( _-a∪\a\), so dCn(a,s^o∣SO)=0d_Cn(a, s_o S_O)=0 and s^o∈RCn(a) s_o∈ R_Cn(a). Thus Rδ(a)⊆RCn(a)R_δ(a) R_Cn(a), and the pairwise disjointness of Assumption 3.1 transfers to the δ-constrained sets. At δ=δ= D_d and under the same condition S^O⊆Cn(SO) S_O (S_O), Aδ=A_δ=A (Proposition 4.1(i)) and the present assumption coincides with Assumption 3.1. Lemma 4.2 (Depth-budget derivability). Under Assumption 4.1, let m,δ≥0m,δ≥ 0 with m+δ≤m+δ≤ D_d, and let B⊇B(≤m)B B^(≤ m). Then: (i) For every s∈SOs∈ S_O with Dd(s∣A)≤m+δDd(s A)≤ m+δ: s∈Tδ(B)s∈ T^δ_ PS(B). (i) In particular, setting m:=−δm:= D_d-δ: every s∈SOs∈ S_O satisfies s∈Tδ(B(≤−δ))s∈ T^δ_ PS(B^(≤ D_d-δ)). (i): Under Assumption 4.1, B(≤m)=Tm(A)B^(≤ m)=T^m(A), so Tδ(B)⊇Tδ(B(≤m))=Tδ(Tm(A))=Tm+δ(A)T^δ(B) T^δ(B^(≤ m))=T^δ(T^m(A))=T^m+δ(A). If Dd(s∣A)≤m+δDd(s A)≤ m+δ, then s∈Tm+δ(A)⊆Tδ(B)s∈ T^m+δ(A) T^δ(B). (i): Every s∈SOs∈ S_O has Dd(s∣A)≤=m+δDd(s A)≤ D_d=m+δ; apply (i). ∎ Proposition 4.1 (Irredundancy filtration). Under the standing assumptions, parts (i)–(i) and (v) hold unconditionally; part (iv) additionally requires Assumption 4.1. (i) Atom0(SO)=SOAtom_0(S_O)=S_O (no derivation budget: every state is irredundant). (i) For every δ≥δ≥ D_d, Atomδ(SO)=Atom(SO)=AAtom_δ(S_O)=Atom(S_O)=A. (i) The δ-irredundant cores form a non-increasing filtration: SO=Atom0(SO)⊇Atom1(SO)⊇⋯⊇Atom(SO)=A.S_O=Atom_0(S_O) _1(S_O) ·s _ D_d(S_O)=A. (4.16) (iv) (Under Assumption 4.1.) A⊆Atomδ(SO)⊆B(≤−δ)A _δ(S_O) B^(≤ D_d-δ) for every 0≤δ≤0≤δ≤ D_d. The first inclusion holds without Assumption 4.1; the second requires it. (v) Each Atomδ(SO)Atom_δ(S_O) is computable. (i): T0(SO∖s)=SO∖s∌sT^0(S_O \s\)=S_O \s\ s. (i): If δ≥δ≥ D_d, then for any s∈Js∈ J: since SO∖s⊇AS_O \s\ A, monotonicity gives Tδ(SO∖s)⊇Tδ(A)⊇T(A)⊇SO∋sT^δ(S_O \s\) T^δ(A) T D_d(A) S_O s, so s is δ-redundant. For a∈Aa∈ A: since Tδ(SO∖a)⊆Cn(SO∖a)T^δ(S_O \a\) (S_O \a\) (by (IC3)) and a∉Cn(SO∖a)a (S_O \a\) (irredundancy of a), we have a∉Tδ(SO∖a)a∉ T^δ(S_O \a\), so a is δ-irredundant. Hence Atomδ(SO)=AAtom_δ(S_O)=A. (i): If s∉Tδ+1(SO∖s)s∉ T^δ+1(S_O \s\), then since Tδ(⋅)⊆Tδ+1(⋅)T^δ(·) T^δ+1(·), s∉Tδ(SO∖s)s∉ T^δ(S_O \s\). (iv): The inclusion A⊆Atomδ(SO)A _δ(S_O) follows from (i) applied to elements of A: each a∈Aa∈ A satisfies a∉Cn(SO∖a)⊇Tδ(SO∖a)a (S_O \a\) T^δ(S_O \a\). For the second inclusion, under Assumption 4.1: let s∈SO∖B(≤−δ)s∈ S_O B^(≤ D_d-δ), so Dd(s∣A)>−δDd(s A)> D_d-δ. By Lemma 4.2(i) with B=SO∖s⊇B(≤−δ)=T−δ(A)B=S_O \s\ B^(≤ D_d-δ)=T D_d-δ(A) (since s∉T−δ(A)s∉ T D_d-δ(A)), and noting m+δ=≥Dd(s∣A)m+δ= D_d (s A), we get s∈Tδ(SO∖s)s∈ T^δ(S_O \s\), i.e., s is δ-redundant. (v): Iterate T_ PS from SO∖sS_O \s\ for δ steps and check membership; repeat for each s∈SOs∈ S_O. ∎ We now state the main result of this subsection: the zero-distortion rate under a delay constraint. Theorem 4.5 (Tight zero-distortion rate–delay function). Let (SO,PO)(S_O,P_O) be a semantic source with core A=Atom(SO)A=Atom(S_O) and maximum derivation depth D_d. For δ∈0,1,…,δ∈\0,1,…, D_d\, write Aδ:=Atomδ(SO)A_δ:=Atom_δ(S_O), Pδ:=PO(Aδ)P_δ:=P_O(A_δ), and πδ(s):=PO(s)/Pδ _δ(s):=P_O(s)/P_δ for s∈Aδs∈ A_δ. Under the 0,1\0,1\-valued δ-constrained closure distortion dCnδd_Cn^δ (Definition 4.2), with reconstruction alphabet S^O S_O satisfying Aδ⊆S^O⊆OA_δ S_O _O, and under Assumption 4.2 (Assumption 4.1 enters only in part (iv); the tight rate (4.17) and parts (i)–(i) are independent of it): Rsem(0,δ;dCnδ,PO)=PδH(πδ).R_sem(0,δ;\,d_Cn^δ,\,P_O)\;=\;P_δ\,H( _δ). (4.17) Boundary values and monotonicity: (i) No inference (δ=0δ=0): A0=SOA_0=S_O, P0=1P_0=1, π0=PO _0=P_O, so Rsem(0,0)=H(PO)R_sem(0,0)=H(P_O), recovering the classical zero-distortion rate. (i) Full inference (δ=δ= D_d): A=A_ D_d=A, P=PAP_ D_d=P_A, π=πA _ D_d= _A, so Rsem(0,)=PAH(πA)R_sem(0, D_d)=P_A\,H( _A), recovering Theorem 4.1. (i) Monotonicity: Rsem(0,δ)R_sem(0,δ) is non-increasing in δ: H(PO)=Rsem(0,0)≥Rsem(0,1)≥⋯≥Rsem(0,)=PAH(πA).H(P_O)=R_sem(0,0)≥ R_sem(0,1)≥·s≥ R_sem(0, D_d)=P_A\,H( _A). (4.18) (iv) Depth-stratified upper bound: If additionally S^O⊇Bδ:=B(≤−δ) S_O B_δ:=B^(≤ D_d-δ), then Rsem(0,δ)≤PBδH(πBδ)R_sem(0,δ)≤ P_B_δ\,H( _B_δ), where PBδ:=PO(Bδ)P_B_δ:=P_O(B_δ) and πBδ _B_δ is the conditional distribution on BδB_δ. This bound is computable directly from the derivation-depth stratification without extracting Atomδ(SO)Atom_δ(S_O). (Since Aδ⊆BδA_δ B_δ by Proposition 4.1(iv), the condition S^O⊇Bδ S_O B_δ strengthens the standing hypothesis S^O⊇Aδ S_O A_δ.) The proof parallels Theorem 4.1 with the substitutions A→AδA→ A_δ, πA→πδ _A→ _δ, PA→PδP_A→ P_δ, and Cn-redundancy replaced by δ-redundancy. Write PJδ:=1−PδP_J_δ:=1-P_δ for the probability of the δ-redundant set SO∖AδS_O A_δ. Lower bound. Let PS^∣SP_ S S be any conditional distribution achieving [dCnδ(So,S^o∣SO)]=0E[d_Cn^δ(S_o, S_o S_O)]=0. Define the auxiliary random variable Tδ:SO→Aδ∪∗T_δ:S_O→ A_δ∪\*\ by Tδ(s):=sT_δ(s):=s if s∈Aδs∈ A_δ and Tδ(s):=∗T_δ(s):=* if s∈SO∖Aδs∈ S_O A_δ. Since TδT_δ is a deterministic function of SoS_o, data processing gives I(So;S^o)≥I(Tδ;S^o).I(S_o; S_o)\;≥\;I(T_δ; S_o). (4.19) Under Assumption 4.2, the δ-constrained zero-distortion sets Rδ(a):=s^o∈S^O:dCnδ(a,s^o∣SO)=0R_δ(a):=\ s_o∈ S_O:d_Cn^δ(a, s_o S_O)=0\ for distinct a∈Aδa∈ A_δ are pairwise disjoint. Since dCnδ∈0,1d_Cn^δ∈\0,1\, the constraint [dCnδ(So,S^o∣SO)]=0E[d_Cn^δ(S_o, S_o S_O)]=0 is equivalent to dCnδ(so,s^o∣SO)=0d_Cn^δ(s_o, s_o S_O)=0 holding PO(so)PS^∣S(s^o∣so)P_O(s_o)\,P_ S S( s_o s_o)-almost surely; in particular, supp(PS^∣S(⋅∣a))⊆Rδ(a)supp(P_ S S(· a)) R_δ(a) for each a∈Aδa∈ A_δ. For δ-redundant states j∈SO∖Aδj∈ S_O A_δ, Lemma 4.1 gives dCnδ(j,s^o∣SO)=0d_Cn^δ(j, s_o S_O)=0 for every s^o∈O s_o _O, so PS^∣S(⋅∣j)P_ S S(· j) is unconstrained. Let Qj:=PS^∣S(⋅∣j)Q_j:=P_ S S(· j) for each j∈SO∖Aδj∈ S_O A_δ, and define the average δ-redundant output distribution Q:=∑j∈SO∖Aδ[PO(j)/PJδ]QjQ:= _j∈ S_O A_δ[P_O(j)/P_J_δ]\,Q_j. Write π¯δ:=∑a∈Aδπδ(a)PS^∣S(⋅∣a) π_δ:= _a∈ A_δ _δ(a)\,P_ S S(· a) for the average core output distribution. Since the supports Rδ(a)R_δ(a) are pairwise disjoint, the standard disjoint-support entropy splitting identity [8, Theorem 2.7.3] gives H(π¯δ)=H(πδ)+∑a∈Aδπδ(a)H(PS^∣S(⋅∣a)).H( π_δ)\;=\;H( _δ)+ _a∈ A_δ _δ(a)\,H (P_ S S(· a) ). (4.20) The conditional entropy of S^o S_o given TδT_δ is H(S^o∣Tδ)=Pδ∑aπδ(a)H(PS^∣S(⋅∣a))+PJδH(Q),H( S_o T_δ)=P_δ _a _δ(a)\,H(P_ S S(· a))+P_J_δ\,H(Q), and the output marginal is PS^=Pδπ¯δ+PJδQP_ S=P_δ\, π_δ+P_J_δ\,Q. By concavity of entropy: H(Pδπ¯δ+PJδQ)≥PδH(π¯δ)+PJδH(Q).H(P_δ\, π_δ+P_J_δ\,Q)\;≥\;P_δ\,H( π_δ)+P_J_δ\,H(Q). Therefore, using (4.20): I(Tδ;S^o) I(T_δ; S_o) =H(PS^)−H(S^o∣Tδ) =H(P_ S)-H( S_o T_δ) ≥PδH(π¯δ)+PJδH(Q)−Pδ∑aπδ(a)H(PS^∣S(⋅∣a))−PJδH(Q) ≥ P_δ\,H( π_δ)+P_J_δ\,H(Q)-P_δ\! _a _δ(a)\,H(P_ S S(· a))-P_J_δ\,H(Q) =Pδ[H(π¯δ)−∑aπδ(a)H(PS^∣S(⋅∣a))] =P_δ [H( π_δ)-\! _a _δ(a)\,H(P_ S S(· a)) ] =PδH(πδ). =P_δ\,H( _δ). (4.21) Combining with (4.19) gives Rsem(0,δ)≥PδH(πδ)R_sem(0,δ)≥ P_δ\,H( _δ). Upper bound (achievability). Define PS^∣S(s^o∣so):=[s^o=so]if so∈Aδ,πδ(s^o)if so∈SO∖Aδ,P_ S S( s_o s_o):= cases1[ s_o=s_o]&if s_o∈ A_δ,\\ _δ( s_o)&if s_o∈ S_O A_δ, cases where πδ _δ is supported on AδA_δ. Distortion check. For so=a∈Aδs_o=a∈ A_δ: s^o=a s_o=a and (Γ−a∪a)=SO( _-a∪\a\)=S_O, so Tδ(SO)⊇T0(SO)=SOT^δ(S_O) T^0(S_O)=S_O, giving dCnδ(a,a∣SO)=0d_Cn^δ(a,a S_O)=0. For so=j∈SO∖Aδs_o=j∈ S_O A_δ: j is δ-redundant, so Lemma 4.1 gives dCnδ(j,s^o∣SO)=0d_Cn^δ(j, s_o S_O)=0 for every s^o∈Aδ s_o∈ A_δ. Rate computation. Under this conditional, Q=πδQ= _δ and the output marginal is PS^=Pδπδ+PJδπδ=πδP_ S=P_δ _δ+P_J_δ _δ= _δ. All δ-redundant inputs produce the same conditional output distribution πδ _δ, so S^o⟂So∣Tδ S_o \!\!\! S_o T_δ, giving I(So;S^o∣Tδ)=0I(S_o; S_o T_δ)=0. Hence I(So;S^o)=I(Tδ;S^o)=H(πδ)−PJδH(πδ)=PδH(πδ).I(S_o; S_o)=I(T_δ; S_o)=H( _δ)-P_J_δ\,H( _δ)=P_δ\,H( _δ). Combining with the lower bound establishes (4.17). Part (i). At δ=0δ=0: T0(SO∖s)=SO∖s∌sT^0(S_O \s\)=S_O \s\ s, so every s is 0-irredundant (Proposition 4.1(i)): A0=SOA_0=S_O, P0=1P_0=1, π0=PO _0=P_O, and Rsem(0,0)=H(PO)R_sem(0,0)=H(P_O). Part (i). At δ=δ= D_d: Atom(SO)=AAtom_ D_d(S_O)=A (Proposition 4.1(i)), so A=A_ D_d=A, P=PAP_ D_d=P_A, π=πA _ D_d= _A, and Rsem(0,)=PAH(πA)R_sem(0, D_d)=P_A\,H( _A). Part (i). If dCnδ(so,s^o∣SO)=0d_Cn^δ(s_o, s_o S_O)=0, then SO⊆Tδ(Γ−so∪s^o)⊆Tδ+1(Γ−so∪s^o)S_O T^δ( _-s_o∪\ s_o\) T^δ+1( _-s_o∪\ s_o\), so dCnδ+1(so,s^o∣SO)=0d_Cn^δ+1(s_o, s_o S_O)=0. Every conditional distribution feasible at budget δ is therefore feasible at budget δ+1δ+1. The minimum of I(So;S^o)I(S_o; S_o) over a larger feasible set cannot increase: Rsem(0,δ+1)≤Rsem(0,δ)R_sem(0,δ+1)≤ R_sem(0,δ). Part (iv). This part uses Assumption 4.1. Since A⊆Bδ⊆SOA B_δ S_O (every a∈Aa∈ A satisfies Dd(a∣A)=0≤−δDd(a A)=0≤ D_d-δ), the two-layer achievability argument above applies with Bδ:=B(≤−δ)B_δ:=B^(≤ D_d-δ) in place of AδA_δ: elements of BδB_δ are encoded losslessly, and each s∈SO∖Bδs∈ S_O B_δ is mapped to πBδ _B_δ. By Proposition 4.1(iv), s∉Bδs∉ B_δ implies s∉Atomδ(SO)s _δ(S_O), so s is δ-redundant and Lemma 4.1 gives dCnδ(s,s^∣SO)=0d_Cn^δ(s, s S_O)=0 for any s^∈SO s∈ S_O. The resulting rate is PBδH(πBδ)P_B_δ\,H( _B_δ), giving Rsem(0,δ)≤PBδH(πBδ)R_sem(0,δ)≤ P_B_δ\,H( _B_δ). ∎ The rate–delay function (4.17) reveals a fundamental depth-for-rate exchange: each additional unit of derivation budget δ renders a new stratum of states δ-redundant, reducing the effective source entropy. Definition 4.4 (Marginal rate of delay). The marginal rate of delay at budget δ is ΔR(δ):=Rsem(0,δ−1)−Rsem(0,δ),δ=1,…,. R(δ)\;:=\;R_sem(0,δ-1)-R_sem(0,δ), δ=1,…, D_d. This quantifies the rate saving per additional derivation step. Remark 4.9 (Extension of Pδ,πδP_δ, _δ beyond D_d). The δ-irredundant core Atomδ(SO)Atom_δ(S_O) is well-defined for all integers δ≥0δ≥ 0 (Definition 4.3). By Proposition 4.1(i), Atomδ(SO)=AAtom_δ(S_O)=A for every δ≥δ≥ D_d, so Pδ=PAP_δ=P_A and πδ=πA _δ= _A for all δ≥δ≥ D_d. The function δ↦PδH(πδ)δ P_δ\,H( _δ) is therefore constant on ,+1,…\ D_d, D_d+1,…\, and the minimum in (4.22) is well-defined over all non-negative integers. Corollary 4.1 (Semantic sampling theorem). Let W:SC↝S^CW:S_C S_C with C(W)>0C(W)>0. The minimum derivation delay for δ-constrained closure-reliable communication of SOS_O is δ∗(SO,PO,W):=minδ≥0:PδH(πδ)≤C(W).δ^*(S_O,P_O,W)\;:=\; \δ≥ 0:P_δ\,H( _δ)≤ C(W) \. (4.22) In particular: (i) If H(PO)≤C(W)H(P_O)≤ C(W) (the channel can carry the full classical rate), then δ∗=0δ^*=0: no inference is needed. (i) If PAH(πA)>C(W)P_A\,H( _A)>C(W) (even full inference cannot reduce the rate below capacity), then the feasible set in (4.22) is empty: no finite derivation budget suffices, and δ-constrained closure-reliable communication is impossible for every δ∈0,…,δ∈\0,…, D_d\. (i) Otherwise, 1≤δ∗≤1≤δ^*≤ D_d, and—under the Assumption 4.2 (at δ=δ∗δ=δ^*) and the condition Atomδ∗(SO)⊆S^OAtom_δ^*(S_O) S_O—the minimum blocklength at delay δ∗δ^* is n∗(δ∗)≈log|Atomδ∗(SO)|C(W).n^*(δ^*)\;≈\; |Atom_δ^*(S_O)|C(W). The critical delay δ∗δ^* plays a role analogous to the Nyquist sampling period: below this delay, the “semantic bandwidth” Rsem(0,δ)R_sem(0,δ) exceeds the channel capacity, and faithful closure reconstruction becomes impossible. The parallel is interpretive rather than a formal equivalence; its value lies in highlighting the depth-for-rate exchange as a resource tradeoff. Since Rsem(0,δ)R_sem(0,δ) is non-increasing in δ (Theorem 4.5(i)), the minimum exists. Part (i): δ=0δ=0 is feasible. Part (i): no δ satisfies the condition. Part (i): the two-layer code of Theorem 3.5(i), adapted with Atomδ∗Atom_δ^* in place of A, achieves the stated blocklength. ∎ Remark 4.10 (Communication–computation exchange). The hard-budget function δ↦Rsem(0,δ)δ R_sem(0,δ) can be relaxed to an expected-budget model: for each transmitted base B with A⊆B⊆SOA B S_O, the two-layer code of Theorem 4.1 applied with B in place of A achieves zero closure distortion at rate R(B):=PO(B)H(πB)R(B):=P_O(B)\,H( _B), and the expected receiver inference cost is C¯(B):=∑s∈SOPO(s)Dd(s∣B) C(B):= _s∈ S_OP_O(s)\,Dd(s B). Time-sharing among bases convexifies the achievable rate–computation region; the lower boundary of this convex hull is a non-increasing convex function of the expected computation budget Δ , with boundary values H(PO)H(P_O) at Δ=0 =0 and PAH(πA)P_A\,H( _A) at Δ=C¯(A) = C(A). Standard LP duality yields a Lagrangian formulation in which the multiplier λ≥0λ≥ 0 prices one unit of receiver computation (a single T_ PS-iteration) in bits of communication rate—an exchange rate absent from classical information theory, where the decoder’s computation is treated as a free resource. A full characterization, including converse bounds and the connection to source coding with structured decoder side information, is deferred to future work. Theorem 4.6 (Full rate–delay–distortion surface). Under the hypotheses of Theorem 4.5, for every δ∈0,…,δ∈\0,…, D_d\ and D≥0D≥ 0: Rsem(D,δ)=Pδ⋅R(Aδ)(DPδ;dCnδ,πδ),R_sem(D,δ)\;=\;P_δ· R^(A_δ)\! ( DP_δ;\,d_Cn^δ,\, _δ ), (4.23) where R(Aδ)R^(A_δ) is the rate–distortion function of the δ-irredundant sub-source (Aδ,πδ)(A_δ, _δ). This is a two-parameter family of rate–distortion functions indexed by δ that generalizes Theorem 4.2: at δ=δ= D_d it recovers (4.7); at δ=0δ=0 it recovers the classical rate–distortion function R(D;dH,PO)R(D;d_H,P_O) (since A0=SOA_0=S_O, P0=1P_0=1, π0=PO _0=P_O, and dCn0=dHd_Cn^0=d_H). The argument parallels Theorem 4.2 with the substitutions A→AδA→ A_δ, PA→PδP_A→ P_δ, πA→πδ _A→ _δ, and Cn-redundancy replaced by δ-redundancy (Lemma 4.1). Distortion decomposition. For every δ-redundant state j∈SO∖Aδj∈ S_O A_δ and every s^o∈O s_o _O, dCnδ(j,s^o∣SO)=0d_Cn^δ(j, s_o S_O)=0, so [dCnδ]=∑a∈AδPO(a)[dCnδ(a,S^o∣SO)∣So=a]=Pδπδ[d¯Aδ],E [d_Cn^δ ]\;=\; _a∈ A_δP_O(a)\,E [d_Cn^δ(a, S_o S_O) S_o=a ]\;=\;P_δ\,E_ _δ[ d_A_δ], and the constraint [dCnδ]≤DE[d_Cn^δ]≤ D reduces to πδ[d¯Aδ]≤D/PδE_ _δ[ d_A_δ]≤ D/P_δ. Rate decomposition. Define Tδ(s):=sT_δ(s):=s if s∈Aδs∈ A_δ, Tδ(s):=∗T_δ(s):=* otherwise. The point mass at ∗* carries probability PJδP_J_δ but contributes zero distortion (Lemma 4.1) and, when all δ-redundant inputs share the same conditional output distribution, zero mutual information; hence I(So;S^o∣Tδ)I(S_o; S_o T_δ) vanishes and minPS^∣S:πδ[d¯Aδ]≤D/PδI(So;S^o)=Pδ⋅R(Aδ)(DPδ;dCnδ,πδ), _ subarraycP_ S S:\\ E_ _δ[ d_A_δ]≤ D/P_δ subarrayI(S_o; S_o)\;=\;P_δ· R^(A_δ)\! ( DP_δ;\,d_Cn^δ,\, _δ ), by the same scaling argument as Theorem 4.2. The boundary values follow from A0=SOA_0=S_O, dCn0=dHd_Cn^0=d_H (at δ=0δ=0) and A=A_ D_d=A (at δ=δ= D_d). ∎ V Application: Heterogeneous Multi-Agent Semantic Communication The theoretical framework developed in Sections I and I is fully general: the semantic state space SOS_O, the reconstructed space S^O S_O, and the enabling structures that constrain encoding and decoding are left as abstract parameters. This section instantiates the framework in a concrete and practically motivated setting—heterogeneous multi-agent semantic communication—and derives new results that illustrate the framework’s applicability in a setting where classical channel coding theory does not capture the deductive structure of the communicated content. The distinguishing feature of the heterogeneous setting is that the sender and receiver maintain different knowledge bases: the sender’s semantic space is SOS_O while the receiver’s reconstructed space S^O S_O may differ from SOS_O both in vocabulary (the set of expressible states) and in inferential structure (the irredundant core and derivation-depth stratification). In the terminology of Section I-C, the end-to-end noise pair (SO−,SO+)(S_O^-,S_O^+) is generically non-trivial: SO−≠∅S_O^-≠ captures sender concepts absent from the receiver’s vocabulary (vocabulary loss), and SO+≠∅S_O^+≠ captures receiver concepts absent from the sender’s intent (vocabulary surplus). Classical Shannon theory, which treats sender and receiver alphabets as abstract label sets, cannot distinguish vocabulary loss from vocabulary surplus, nor can it exploit shared deductive structure to reduce communication cost. The semantic channel invariants of Section I-C are precisely the tools needed to make these distinctions precise and quantitative. The section is organized as follows. Section V-A describes the multi-agent communication scenario and identifies the key design questions. Section V-B formalizes the scenario within the information model framework and states the standing assumptions specific to this application. Section V-C instantiates the semantic channel machinery and derives closed-form relationships between knowledge-base overlap structure and semantic channel invariants. Section V-D presents the main analytical results: conditions for closure-reliable heterogeneous communication, a heterogeneous deductive compression theorem, and a broadcast extension to one-sender–multi-receiver scenarios. Section V-E verifies all results on an explicit Datalog knowledge-base instance with full numerical computation of every invariant. V-A Problem Description: Heterogeneous Agent Communication Consider a network of K+1K+1 autonomous agents—indexed by i∈0,1,…,Ki∈\0,1,…,K\—that must coordinate by exchanging semantic states over noisy physical links. Each agent i maintains a finite knowledge base SO(i)⊆OS_O^(i) _O, where OS_O is the common ambient semantic universe introduced in Section I-B. All agents share the same proof system (,T,Cn)( PS,\,T_ PS,\,Cn) and the same semantic sublanguage ℒsemL_sem; they differ, however, in the sets of semantic states they store and operate on. Agent i can derive consequences within Cn(SO(i))Cn(S_O^(i)) and possesses the irredundant core A(i):=Atom(SO(i))A^(i):=Atom(S_O^(i)) together with the associated derivation-depth stratification (Definitions 2.1 and 2.4). This knowledge-base heterogeneity is the defining feature of the scenario and the source of all phenomena that distinguish it from classical Shannon-theoretic communication [29, 8]. Recent work on semantic communication [20, 30, 4] has highlighted the need for frameworks that go beyond symbol-level fidelity, but a rigorous logical-information-theoretic treatment of heterogeneous knowledge bases has been lacking. When agent i (the sender) transmits a semantic state so∈SO(i)s_o∈ S_O^(i) to agent j (the receiver), the receiver reconstructs a state s^o∈SO(j) s_o∈ S_O^(j)—not necessarily in SO(i)S_O^(i)—because agent j can only produce outputs expressible in its own vocabulary. Unless SO(i)=SO(j)S_O^(i)=S_O^(j), the end-to-end noise pair (SO−,SO+)(S_O^-,S_O^+) (Definition 3.4) is generically nontrivial: states in SO(i)∖SO(j)S_O^(i) S_O^(j) have no direct counterpart in the receiver’s vocabulary (vocabulary loss), while states in SO(j)∖SO(i)S_O^(j) S_O^(i) can appear in the receiver’s output without having been intended by the sender (vocabulary surplus). Classical channel coding theory, which treats the source and reconstruction alphabets as unstructured label sets, is blind to this distinction: it can detect that a symbol error has occurred, but cannot determine whether the error represents a genuine loss of semantic content or a harmless reformulation within the receiver’s richer (or merely different) vocabulary. Communication Sub-Scenarios Three sub-scenarios of increasing structural complexity arise naturally in the multi-agent setting; they are listed below in decreasing order of analytical depth in this paper. Definition 5.1 (Pairwise unicast scenario). Fix a sender–receiver pair (i,j)(i,j) with i≠ji≠ j. Agent i wishes to communicate its full knowledge base SO(i)S_O^(i) to agent j over a noisy carrier channel Wij:SC↝S^CW_ij:S_C S_C (cf. Definition 3.3(i)). Agent j reconstructs a state in its own vocabulary S^O:=SO(j) S_O:=S_O^(j), using a decoding kernel D∈(ℐdec(j))D (I_dec^(j)). The end-to-end semantic channel is ℭij=(ℐ(i),ℐchij,ℐdec(j),κenc,Wij,D), C^ij\;=\; (\,I^(i),\;I_ch^ij,\;I_dec^(j),\; _enc,\;W_ij,\;D\, ), with semantic source space SO(i)S_O^(i), reconstructed space SO(j)S_O^(j), and noise pair SO−=SO(i)∖SO(j),SO+=SO(j)∖SO(i).S_O^-=S_O^(i) S_O^(j), S_O^+=S_O^(j) S_O^(i). (5.1) The constituent information models ℐ(i)I^(i), ℐchijI_ch^ij, and ℐdec(j)I_dec^(j) are formalized in Section V-B. The pairwise unicast scenario is the primary focus of Sections V-B–V-D. All new theorems are stated and proved for this case first; generalizations to the broadcast setting are given as corollaries. Definition 5.2 (Broadcast scenario). A designated sender (agent 0) communicates its knowledge base SO(0)S_O^(0) simultaneously to K receivers (agents 1,…,K1,…,K) over a common carrier channel W. Each receiver j maintains a distinct vocabulary S^O(j):=SO(j) S_O^(j):=S_O^(j) and observes a (possibly receiver-specific) noise pair (SO−,j,SO+,j)=(SO(0)∖SO(j),SO(j)∖SO(0)).(S_O^-,j,\;S_O^+,j)\;=\; (S_O^(0) S_O^(j),\;S_O^(j) S_O^(0) ). All K receivers observe the same channel output S^C S_C; the receiver-specific noise pairs arise solely from vocabulary mismatch, not from different physical channel realizations. The broadcast semantic channel is a family ℭ0jj=1K\ C^0j\_j=1^K of pairwise channels sharing the same sender, the same encoding kernel, and the same carrier channel, but differing in decoding model and noise pair. The broadcast scenario reveals a phenomenon absent from classical broadcast channel theory [8]: even over a noiseless carrier (W=idSCW=id_S_C), the achievable fidelity at each receiver is constrained by its vocabulary overlap with the sender—a purely semantic bottleneck (see Proposition 5.9 in Section V-D). Remark 5.1 (Relay scenario (future work)). A third sub-scenario arises when an intermediate agent k acts as a relay: agent i transmits to agent k, which performs inference within Cn(SO(k))Cn(S_O^(k)) and then re-encodes and forwards the result to agent j. This is naturally modeled as a composition of two pairwise channels, ℭrelayij=ℭkj∘ℭik C^ij_relay= C^kj C^ik, using the information-model composition machinery of Definition A.4 and Remark A.1. The relay setting raises the question of whether intermediate inference can change the effective capacity of the end-to-end link—a possibility that has no direct classical analogue, since in classical relay channel theory [8] the relay cannot exploit logical structure to reduce the message set. A complete treatment requires multi-letter extensions and is deferred to future work; the pairwise and broadcast results of this section provide the necessary building blocks. Key Questions The heterogeneous multi-agent setting gives rise to four design and analysis questions that the classical framework cannot address. Q1 (Closure reliability from vocabulary overlap). Under what conditions on the overlap between SO(i)S_O^(i) and SO(j)S_O^(j) can agent j reconstruct the deductive closure of agent i’s knowledge base, i.e., achieve Cn(SO(i),SO(j))=1 F_Cn(S_O^(i),S_O^(j))=1? Proposition 2.2 provides abstract conditions; the task is to translate them into explicit predicates on the knowledge-base pair. Q2 (Heterogeneous deductive compression). How many channel uses are needed to communicate SO(i)S_O^(i) to agent j under closure reliability, and does the deductive compression ratio log|Atom(SO)|/log|SO| |Atom(S_O)|/ |S_O| of the homogeneous setting (Corollary 3.3) persist under vocabulary heterogeneity? Q3 (Invariant diagnosis). How do the six families of semantic channel invariants (Theorem 3.3) depend on the knowledge-base overlap, and which invariants can be evaluated a priori from knowledge-base metadata alone? Q4 (Broadcast bottleneck). In the broadcast scenario, which receiver determines the minimum blocklength? Does a purely semantic axis of receiver weakness, distinct from physical channel degradation, arise? Preview of Main Results The answers to Q1–Q4 are developed in full in Sections V-C–V-D. In brief: for Q1, closure fidelity Cn(SO(i),SO(j))=1 F_Cn(S_O^(i),S_O^(j))=1 holds if and only if every sender core element is derivable from the receiver’s knowledge base and every receiver surplus state is derivable from the sender (Proposition 5.3); the operational two-layer code requires the stronger literal containment A(i)⊆SO(j)A^(i) S_O^(j) (Theorem 5.1). For Q2, the deductive compression ratio is invariant under vocabulary heterogeneity: the minimum blocklength remains n∗≈log|Atom(SO(i))|/C(Wij)n^*≈ |Atom(S_O^(i))|/C(W_ij), identical to the homogeneous case (Theorem 5.2); when core coverage fails, no code of any blocklength achieves Cn=1 F_Cn=1 (Corollary 5.3). For Q3, each invariant family is expressed as a function of the overlap decomposition (Propositions 5.2–5.6); set-level invariants depend on the knowledge-base pair alone, while probabilistic indices are additionally constrained by the channel kernel. For Q4, the broadcast blocklength depends only on the sender’s core and is independent of the number of receivers, provided every receiver covers the sender’s core (Theorem 5.3); a receiver violating this condition becomes a semantic bottleneck whose fidelity limitation persists even over a noiseless carrier (Proposition 5.9). Remark 5.2 (Classical recovery as a special case). When all agents share the same knowledge base (SO(i)=SO(j)S_O^(i)=S_O^(j) for all i,ji,j), the noise pair is trivial, the two-condition criterion is vacuously satisfied, and all results of this section reduce to the homogeneous theory of Section I-D (and, in the irredundant case, to classical Shannon theory via Corollary 3.2). The heterogeneous analysis thus strictly generalizes rather than replaces the earlier results. V-B Formal Model and Standing Assumptions This subsection formalizes the multi-agent communication scenario of Section V-A within the information model framework of Sections I–I, and introduces the overlap decomposition—the combinatorial structure through which all semantic channel invariants are expressed as functions of the sender–receiver knowledge-base pair. Definition 5.3 (Agent knowledge base). Fix a set of K+1K+1 agents indexed by i∈0,1,…,Ki∈\0,1,…,K\. Each agent i is associated with a finite knowledge base SO(i)⊆OS_O^(i) _O that is ℒsemL_sem-definable in the ambient structure ℜ R and satisfies Assumptions 2.2–2.3. The irredundant core and stored shortcuts of agent i are A(i):=Atom(SO(i)),J(i):=SO(i)∖A(i).A^(i):=Atom (S_O^(i) ), J^(i):=S_O^(i) A^(i). By Proposition 2.1, Cn(A(i))=Cn(SO(i))Cn (A^(i) )=Cn (S_O^(i) ), the core A(i)A^(i) is irredundant, and SO(i)⊆Cn(A(i))S_O^(i) (A^(i) ). Assumption 5.1 (Common proof system and ambient universe). All agents share: (CP1) the same proof system (,T,Cn)( PS,\,T_ PS,\,Cn) and inference fragment ℒkbL_kb (Assumption 2.1, Axiom 2.1); (CP2) the same ambient semantic universe OS_O with its injective encoding encOenc_O and canonical order (Section I-B, Assumption A.3); (CP3) the same semantic sublanguage ℒsemL_sem (Assumption A.2). The agents differ only in the knowledge bases SO(0),SO(1),…,SO(K)⊆OS_O^(0),S_O^(1),…,S_O^(K) _O that they store and operate on. Definition 5.4 (Pairwise overlap decomposition). For a fixed sender–receiver pair (i,j)(i,j) with i≠ji≠ j, define the following subsets of OS_O. Three-way partition of SO(i)∪SO(j)S_O^(i)∪ S_O^(j): S∩ij S_∩^ij :=SO(i)∩SO(j) \;:=\;S_O^(i)∩ S_O^(j) (common states), (common states), (5.2) S−ij S_-^ij :=SO(i)∖SO(j) \;:=\;S_O^(i) S_O^(j) (lost states), (lost states), (5.3) S+ij S_+^ij :=SO(j)∖SO(i) \;:=\;S_O^(j) S_O^(i) (surplus states). (surplus states). (5.4) Core partition: A∩ij A_∩^ij :=A(i)∩SO(j) \;:=\;A^(i)∩ S_O^(j) (preserved core), (preserved core), (5.5) A−ij A_-^ij :=A(i)∖SO(j) \;:=\;A^(i) S_O^(j) (lost core). (lost core). (5.6) Surplus stratification: S+,dij S_+,d^ij :=S+ij∩Cn(SO(i)) \;:=\;S_+^ij (S_O^(i) ) (derivable surplus), (derivable surplus), (5.7) S+,nij S_+,n^ij :=S+ij∖Cn(SO(i)) \;:=\;S_+^ij (S_O^(i) ) (non-derivable surplus). (non-derivable surplus). (5.8) Proposition 5.1 (Overlap partition properties). The overlap decomposition of Definition 5.4 satisfies: (i) Three-way disjoint union: SO(i)∪SO(j)=S−ij∪˙S∩ij∪˙S+ijS_O^(i)∪ S_O^(j)=S_-^ij\; ∪\;S_∩^ij\; ∪\;S_+^ij. (i) Sender decomposition: SO(i)=S∩ij∪˙S−ijS_O^(i)=S_∩^ij\; ∪\;S_-^ij. (i) Receiver decomposition: SO(j)=S∩ij∪˙S+ijS_O^(j)=S_∩^ij\; ∪\;S_+^ij. (iv) Core partition: A(i)=A∩ij∪˙A−ijA^(i)=A_∩^ij\; ∪\;A_-^ij. (v) Surplus partition: S+ij=S+,dij∪˙S+,nijS_+^ij=S_+,d^ij\; ∪\;S_+,n^ij. (vi) Noise-pair consistency: S−ij=SO−S_-^ij=S_O^- and S+ij=SO+S_+^ij=S_O^+, where (SO−,SO+)(S_O^-,S_O^+) is the end-to-end noise pair of Definition 5.1. (vii) Core loss refines state loss: A−ij⊆S−ijA_-^ij S_-^ij. (viii) Computability: All seven sets and their cardinalities are computable from the finite knowledge bases SO(i),SO(j)S_O^(i),S_O^(j) under Axiom 2.1. Parts (i)–(i) are standard set partition identities: S∩ijS_∩^ij, S−ijS_-^ij, S+ijS_+^ij are pairwise disjoint by construction, and their union equals SO(i)∪SO(j)S_O^(i)∪ S_O^(j); restricting to SO(i)S_O^(i) yields (i), and to SO(j)S_O^(j) yields (i). Part (iv): since A(i)⊆SO(i)A^(i) S_O^(i), A(i)=(A(i)∩SO(j))∪˙(A(i)∖SO(j))=A∩ij∪˙A−ijA^(i)=(A^(i)∩ S_O^(j))\; ∪\;(A^(i) S_O^(j))=A_∩^ij\; ∪\;A_-^ij. Part (v): immediate from the definition of S+,dijS_+,d^ij and S+,nijS_+,n^ij as complementary subsets of S+ijS_+^ij. Part (vi): comparing (5.3)–(5.4) with (5.1) gives the identification directly. Part (vii): A−ij=A(i)∖SO(j)⊆SO(i)∖SO(j)=S−ijA_-^ij=A^(i) S_O^(j) S_O^(i) S_O^(j)=S_-^ij, since A(i)⊆SO(i)A^(i) S_O^(i). Part (viii): since SO(i)S_O^(i) and SO(j)S_O^(j) are finite and effectively listable (Assumption 2.2), membership is decidable by exhaustive comparison. The sets S∩ijS_∩^ij, S−ijS_-^ij, S+ijS_+^ij, A∩ijA_∩^ij, A−ijA_-^ij are then computable by enumeration. The surplus stratification requires testing s∈Cn(SO(i))s (S_O^(i)) for each s∈S+ijs∈ S_+^ij, which is decidable by iterating T_ PS from SO(i)S_O^(i) until stabilization (Axiom 2.1(IC2)–(IC4)). ∎ Remark 5.3 (Core preservation is tested against SO(j)S_O^(j), not against A(j)A^(j)). The preserved core A∩ij=A(i)∩SO(j)A_∩^ij=A^(i)∩ S_O^(j) tests whether each sender core element is present in the receiver’s vocabulary—that is, an element of SO(j)S_O^(j)—not whether it belongs to the receiver’s irredundant core A(j)A^(j). A sender core element a∈A(i)a∈ A^(i) may appear in SO(j)S_O^(j) as a redundant stored shortcut (i.e., a∈J(j)a∈ J^(j)); it is still counted as preserved, because the receiver can produce it as a decoding output regardless of its redundancy status in SO(j)S_O^(j). Remark 5.4 (Key scalar summaries of the overlap). Although the overlap decomposition produces seven subsets, the set-level conditions governing closure fidelity (Proposition 5.3) reduce to two binary tests: |A−ij|=0|A_-^ij|=0 (no core loss) and |S+,nij|=0|S_+,n^ij|=0 (no non-derivable surplus). The coding-theoretic results of Section V-D (blocklength, compression ratio) depend additionally on |A(i)||A^(i)| and C(Wij)C(W_ij), while the probabilistic indices (ΦAtom _Atom, Ψ+ _+, F, E) depend further on the channel kernel κsemij _sem^ij. The remaining overlap cardinalities are related by simple accounting: |A∩ij|=|A(i)|−|A−ij||A_∩^ij|=|A^(i)|-|A_-^ij|, |S+ij|=|S+,dij|+|S+,nij||S_+^ij|=|S_+,d^ij|+|S_+,n^ij|; the quantities |S−ij||S_-^ij|, |S∩ij||S_∩^ij|, and |SO(j)||S_O^(j)| are mutually determined once any one of them is known, via |S∩ij|=|SO(i)|−|S−ij||S_∩^ij|=|S_O^(i)|-|S_-^ij| and |SO(j)|=|S∩ij|+|S+ij||S_O^(j)|=|S_∩^ij|+|S_+^ij|. Remark 5.5 (Broadcast overlap decomposition). In the broadcast scenario (Definition 5.2), the sender is agent 0 and the receivers are agents 1,…,K1,…,K. For each receiver j, the overlap decomposition (Definition 5.4) is applied to the pair (0,j)(0,j), yielding receiver-specific quantities A∩0jA_∩^0j, A−0jA_-^0j, S+,d0jS_+,d^0j, S+,n0jS_+,n^0j, etc. The broadcast core coverage condition—that A−0j=∅A_-^0j= for every j∈1,…,Kj∈\1,…,K\—plays a central role in Theorem 5.3 and Proposition 5.9. Definition 5.5 (Heterogeneous semantic channel (formal)). For a sender–receiver pair (i,j)(i,j) with i≠ji≠ j, the heterogeneous semantic channel is the semantic channel (Definition 3.3) ℭij=(ℐ(i),ℐchij,ℐdec(j),κenc,Wij,D), C^ij\;=\; (\,I^(i),\;I_ch^ij,\;I_dec^(j),\; _enc,\;W_ij,\;D\, ), where the constituent models are defined as follows. (i) Sender information model. ℐ(i)=⟨O(i),TO(i),SO(i),C,TC,SC,Rℰ(i)⟩I^(i)= O^(i),T_O^(i),S_O^(i),C,T_C,S_C,R_E^(i) is an information model (Definition A.3) with semantic state set SO(i)S_O^(i), carrier state set SCS_C, and enabling map ℰ(i):SO(i)⇒SCE^(i):S_O^(i) S_C. (i) Carrier channel model. ℐchijI_ch^ij is a carrier channel information model (cf. Definition 3.3(i)) with input SCS_C, output S^C S_C, and carrier channel kernel Wij:SC↝S^CW_ij:S_C S_C. (i) Receiver decoding model. ℐdec(j)=⟨C^,TC^,S^C,O^(j),TO^(j),SO(j),Rℰdec,(j)⟩I_dec^(j)= C,T_ C, S_C, O^(j),T_ O^(j),S_O^(j),R_E^dec,(j) is a decoding information model (cf. Definition 3.3(i)) with reconstructed space S^O:=SO(j)⊆O S_O:=S_O^(j) _O and enabling map ℰdec(j):S^C⇒SO(j)E_dec^(j): S_C S_O^(j). (iv) Kernels. κenc∈(ℐ(i)) _enc (I^(i)) is the encoding kernel and D∈(ℐdec(j))D (I_dec^(j)) is the decoding kernel. The end-to-end kernel is κsemij:=D∘Wij∘κenc:SO(i)↝SO(j), _sem^ij\;:=\;D W_ij _enc\;:\;S_O^(i) S_O^(j), (5.9) and the end-to-end noise pair is (SO−,SO+)=(S−ij,S+ij)(S_O^-,S_O^+)=(S_-^ij,S_+^ij) (Proposition 5.1(vi)). Remark 5.6 (Inherited proof-system structure at the receiver). Since SO(j)⊆OS_O^(j) _O and the proof system PS acts on all of OS_O (Assumption 5.1), the receiver inherits the deductive closure Cn(SO(j))Cn(S_O^(j)), the irredundant core A(j)=Atom(SO(j))A^(j)=Atom(S_O^(j)), and the derivation-depth stratification Dd(⋅∣A(j))Dd(· A^(j)). The semantic invariants (ℐ(j))=|A(j)| A(I^(j))=|A^(j)| and (ℐ(j))=maxq∈SO(j)Dd(q∣A(j)) D_d(I^(j))= _q∈ S_O^(j)Dd(q A^(j)) are therefore well-defined and computable (Theorem 2.1). In general, A(j)≠A(i)A^(j)≠ A^(i), Cn(SO(j))≠Cn(SO(i))Cn(S_O^(j)) (S_O^(i)), and (ℐ(j))≠(ℐ(i)) D_d(I^(j))≠ D_d(I^(i)); the overlap decomposition (Definition 5.4) quantifies each of these discrepancies. Assumption 5.2 (Full enabling (heterogeneous setting)). For the heterogeneous semantic channel ℭij C^ij of Definition 5.5: (FE1) Full encoding enabling: ℰ(i)(so)=SCE^(i)(s_o)=S_C for every so∈SO(i)s_o∈ S_O^(i). (FE2) Full decoding enabling: ℰdec(j)(s^c)=SO(j)E_dec^(j)( s_c)=S_O^(j) for every s^c∈S^C s_c∈ S_C. Remark 5.7 (Role of the full enabling assumption). Under Assumption 5.2, all vocabulary-mismatch effects are captured entirely by the noise pair (S−ij,S+ij)(S_-^ij,S_+^ij). This is the heterogeneous counterpart of the full enabling condition in Theorem 3.1(i). When the enabling is constrained, additional capacity reductions follow from the data-processing bound (Theorem 3.1(i)). Remark 5.8 (Carrier alphabet size and semantic capacity). Assumption (SA4) requires |SC|≥maxi|SO(i)||S_C|≥ _i|S_O^(i)| and |S^C|≥|SC|| S_C|≥|S_C|, ensuring that block codes over SCnS_C^n can represent any message set of size |SO(i)||S_O^(i)| for large enough n. This does not, however, imply the single-letter capacity equality Csemij=C(Wij)C_sem^ij=C(W_ij), which requires the reverse size condition |SO|≥|SC||S_O|≥|S_C| and |S^O|≥|S^C|| S_O|≥| S_C| (Theorem 3.1(i)). Under (SA4), only the data processing bound Csemij≤C(Wij)C_sem^ij≤ C(W_ij) (Theorem 3.1(i)) and the source entropy bound Csemij≤log|SO(i)|C_sem^ij≤ |S_O^(i)| (Theorem 3.1(i)) are guaranteed. All achievability and converse results in this section use C(Wij)C(W_ij) directly via block coding and are unaffected by the single-letter capacity gap. Assumption 5.3 (Standing assumptions for Section V). Throughout Sections V-B–V-D, the following conditions are in force unless explicitly stated otherwise: (SA1) all standing assumptions of Sections I–I, including Assumptions A.1, A.2, 2.1, 2.2, 2.3, Axiom 2.1, and Assumption A.3; (SA2) the common proof system assumption (Assumption 5.1); (SA3) the full enabling assumption (Assumption 5.2); (SA4) the carrier channel satisfies C(Wij)>0C(W_ij)>0 and the carrier alphabet sizes satisfy |SC|≥maxi|SO(i)||S_C|≥ _i|S_O^(i)| and |S^C|≥|SC|| S_C|≥|S_C|; (SA5) the deductive independence of core elements (Assumption 3.1) holds for the sender’s knowledge base SO(i)S_O^(i) when converse bounds are invoked. Table I collects the notation introduced in this subsection for convenient reference throughout Section V. TABLE I: Notation summary for Section V. All quantities are defined relative to a fixed sender–receiver pair (i,j)(i,j). Symbol Meaning SO(i)S_O^(i) Knowledge base (semantic state set) of agent i A(i),J(i)A^(i),\;J^(i) Irredundant core and stored shortcuts of agent i S∩ijS_∩^ij Common states: SO(i)∩SO(j)S_O^(i)∩ S_O^(j) S−ijS_-^ij Lost states: SO(i)∖SO(j)S_O^(i) S_O^(j) (=SO−)(\,=S_O^-) S+ijS_+^ij Surplus states: SO(j)∖SO(i)S_O^(j) S_O^(i) (=SO+)(\,=S_O^+) A∩ijA_∩^ij Preserved sender core: A(i)∩SO(j)A^(i)∩ S_O^(j) A−ijA_-^ij Lost sender core: A(i)∖SO(j)A^(i) S_O^(j) S+,dijS_+,d^ij Derivable surplus: S+ij∩Cn(SO(i))S_+^ij (S_O^(i)) S+,nijS_+,n^ij Non-derivable surplus: S+ij∖Cn(SO(i))S_+^ij (S_O^(i)) ℭij C^ij Heterogeneous semantic channel from i to j WijW_ij Carrier channel kernel from i to j κsemij _sem^ij End-to-end semantic kernel for pair (i,j)(i,j) V-C Instantiation of Semantic Channel Invariants This subsection applies the invariant machinery of Section I to the heterogeneous pair (i,j)(i,j), expressing each invariant of Theorem 3.3 as a function of the overlap decomposition of Section V-B. Throughout, we fix a sender–receiver pair (i,j)(i,j) with the heterogeneous semantic channel ℭij C^ij of Definition 5.5, under the standing assumptions of Assumption 5.3. Proposition 5.2 (Set-level invariants from overlap). For the heterogeneous semantic channel ℭij C^ij with noise pair (SO−,SO+)=(S−ij,S+ij)(S_O^-,S_O^+)=(S_-^ij,S_+^ij) (Proposition 5.1(vi)): (i) Preserved region: S~O∩=SO(i)∩SO(j)=S∩ij S_O^∩=S_O^(i)∩ S_O^(j)=S_∩^ij. (i) Core preservation ratio: ρAtom(SO(i),SO(j))=|A∩ij||A(i)|= 1−|A−ij||A(i)|. _Atom (S_O^(i),S_O^(j) )\;=\; |A_∩^ij||A^(i)|\;=\;1- |A_-^ij||A^(i)|. (5.10) In particular, ρAtom=1 _Atom=1 if and only if A−ij=∅A_-^ij= . (i) Spurious derivability: SO+⊆Cn(SO(i))S_O^+ (S_O^(i)) if and only if S+,nij=∅S_+,n^ij= . (i): SO−=S−ijS_O^-=S_-^ij (Proposition 5.1(vi)), so S~O∩=SO(i)∖SO−=SO(i)∖(SO(i)∖SO(j))=SO(i)∩SO(j)=S∩ij S_O^∩=S_O^(i) S_O^-=S_O^(i) (S_O^(i) S_O^(j))=S_O^(i)∩ S_O^(j)=S_∩^ij. (i): By Definition 2.10, ρAtom=|A(i)∩SO(j)|/|A(i)|=|A∩ij|/|A(i)| _Atom=|A^(i)∩ S_O^(j)|/|A^(i)|=|A_∩^ij|/|A^(i)|. By Proposition 5.1(iv), |A∩ij|=|A(i)|−|A−ij||A_∩^ij|=|A^(i)|-|A_-^ij|. (i): SO+=S+ijS_O^+=S_+^ij and S+ij⊆Cn(SO(i))S_+^ij (S_O^(i)) iff S+ij∖Cn(SO(i))=∅S_+^ij (S_O^(i))= iff S+,nij=∅S_+,n^ij= . ∎ Proposition 5.3 (Closure fidelity: necessary and sufficient conditions). For the sender–receiver pair (i,j)(i,j), Cn(SO(i),SO(j))=1⟺Cn(SO(i))=Cn(SO(j)), F_Cn (S_O^(i),\,S_O^(j) )=1 (S_O^(i) )=Cn (S_O^(j) ), and this holds if and only if both of the following conditions are satisfied: (F1) Sender core derivable from receiver: A(i)⊆Cn(SO(j))A^(i) (S_O^(j) ). (F2) No non-derivable surplus: S+,nij=∅S_+,n^ij= (equivalently, S+ij⊆Cn(SO(i))S_+^ij (S_O^(i))). The first equivalence is the definition of Cn F_Cn (Definition 2.9): Cn=1 F_Cn=1 iff the Jaccard index of the two closures equals 11, i.e., the closures coincide. It remains to show Cn(SO(i))=Cn(SO(j))Cn(S_O^(i))=Cn(S_O^(j)) iff (F1)+(F2). Sufficiency. From (F1): A(i)⊆Cn(SO(j))A^(i) (S_O^(j)). By monotonicity (Cn2), Cn(A(i))⊆Cn(Cn(SO(j)))=Cn(SO(j))Cn(A^(i)) (Cn(S_O^(j)))=Cn(S_O^(j)) (idempotence (Cn3)). Since Cn(A(i))=Cn(SO(i))Cn(A^(i))=Cn(S_O^(i)) (Proposition 2.1(i)), Cn(SO(i))⊆Cn(SO(j))Cn(S_O^(i)) (S_O^(j)). From (F2): S+ij⊆Cn(SO(i))S_+^ij (S_O^(i)). Since S∩ij⊆SO(i)⊆Cn(SO(i))S_∩^ij S_O^(i) (S_O^(i)) (reflexivity (Cn1)), SO(j)=S∩ij∪S+ij⊆Cn(SO(i))S_O^(j)=S_∩^ij∪ S_+^ij (S_O^(i)). By monotonicity and idempotence, Cn(SO(j))⊆Cn(SO(i))Cn(S_O^(j)) (S_O^(i)). Combining the two inclusions yields Cn(SO(i))=Cn(SO(j))Cn(S_O^(i))=Cn(S_O^(j)). Necessity. Suppose Cn(SO(i))=Cn(SO(j))Cn(S_O^(i))=Cn(S_O^(j)). For (F1): A(i)⊆Cn(A(i))=Cn(SO(i))=Cn(SO(j))A^(i) (A^(i))=Cn(S_O^(i))=Cn(S_O^(j)). For (F2): S+ij⊆SO(j)⊆Cn(SO(j))=Cn(SO(i))S_+^ij S_O^(j) (S_O^(j))=Cn(S_O^(i)), so S+,nij=S+ij∖Cn(SO(i))=∅S_+,n^ij=S_+^ij (S_O^(i))= . ∎ Remark 5.9 (Strong vs. weak core coverage). Condition (F1) requires that each sender core element be derivable from the receiver’s knowledge base; it does not require the element to be literally present in SO(j)S_O^(j). A strictly stronger condition is (F1′) A−ij=∅A_-^ij= , i.e., A(i)⊆SO(j)A^(i) S_O^(j). Condition (F1′) implies (F1) (since SO(j)⊆Cn(SO(j))S_O^(j) (S_O^(j))) but not conversely: a core element a∈A−ija∈ A_-^ij may satisfy a∈Cn(SO(j))a (S_O^(j)) even though a∉SO(j)a∉ S_O^(j). For set-level closure fidelity, the weak condition (F1) is both necessary and sufficient (Proposition 5.3). For operational closure reliability via the two-layer code of Theorem 3.5(i), the decoder outputs core elements directly and therefore requires the strong condition (F1′) so that A(i)⊆SO(j)=S^OA^(i) S_O^(j)= S_O. When only (F1) holds with A−ij≠∅A_-^ij≠ , a more sophisticated decoding strategy is needed; this is addressed in Section V-D. Throughout the remainder of this subsection, results are stated under whichever version is required, with the distinction noted explicitly. Corollary 5.1 (Sufficient condition via overlap scalars). If A−ij=∅A_-^ij= and S+,nij=∅S_+,n^ij= , then Cn(SO(i),SO(j))=1 F_Cn(S_O^(i),S_O^(j))=1. This is the instantiation of Proposition 2.2(i) in the overlap language and the condition used in the achievability results of Section V-D. A−ij=∅A_-^ij= gives A(i)⊆SO(j)⊆Cn(SO(j))A^(i) S_O^(j) (S_O^(j)), so (F1) holds. S+,nij=∅S_+,n^ij= is (F2). Apply Proposition 5.3. ∎ Proposition 5.4 (Noise-pair probabilistic indices from overlap). Let ℭij C^ij be a heterogeneous semantic channel with kernel κsemij:SO(i)↝SO(j) _sem^ij:S_O^(i) S_O^(j). (i) Core preservation index: ΦAtom(ℭij)=mina∈A(i)κsemij(a∣a)if A−ij=∅,0if A−ij≠∅. _Atom( C^ij)\;=\; cases _a∈ A^(i) _sem^ij(a a)&if A_-^ij= ,\\[6.0pt] 0&if A_-^ij≠ . cases (5.11) (i) Spurious probability index: Ψ+(ℭij)=maxso∈SO(i)∑s^o∈S+ijκsemij(s^o∣so). _+( C^ij)\;=\; _s_o∈ S_O^(i)\; _ s_o∈ S_+^ij _sem^ij( s_o s_o). (5.12) In particular, Ψ+=0 _+=0 whenever S+ij=∅S_+^ij= (i.e., SO(j)⊆SO(i)S_O^(j) S_O^(i)). (i) Noiseless deterministic case: If Wij=idSCW_ij=id_S_C and both κenc _enc and D are deterministic with induced end-to-end function f:=D∘id∘κenc:SO(i)→SO(j)f:=D _enc:S_O^(i)→ S_O^(j), then ΦAtom(ℭij) _Atom( C^ij) =1if A−ij=∅ and f(a)=a∀a∈A(i),0otherwise, = cases1&if A_-^ij= \\ & and f(a)=a\;∀ a∈ A^(i),\\ 0&otherwise, cases (5.13) Ψ+(ℭij) _+( C^ij) =0if f(SO(i))⊆S∩ij,1otherwise. = cases0&if f(S_O^(i)) S_∩^ij,\\ 1&otherwise. cases (5.14) (i): By Proposition 5.1(vi), A(i)∩SO−=A(i)∩S−ij=A−ijA^(i)∩ S_O^-=A^(i)∩ S_-^ij=A_-^ij. When A−ij≠∅A_-^ij≠ , Definition 3.15 sets ΦAtom:=0 _Atom:=0. When A−ij=∅A_-^ij= , A(i)⊆SO(j)=S~OA^(i) S_O^(j)= S_O, so π(a)=κsemij(a∣a)π(a)= _sem^ij(a a) is well-defined for every a∈A(i)a∈ A^(i) and ΦAtom=minaπ(a) _Atom= _aπ(a). (i): Direct from Definition 3.15 with SO+=S+ijS_O^+=S_+^ij. When S+ij=∅S_+^ij= , the sum is empty and Ψ+=0 _+=0. (i): When κsemij _sem^ij is deterministic, κsemij(s^o∣so)∈0,1 _sem^ij( s_o s_o)∈\0,1\ for all so,s^os_o, s_o. For ΦAtom _Atom: π(a)=κsemij(a∣a)=1π(a)= _sem^ij(a a)=1 iff f(a)=af(a)=a, and this must hold for all a∈A(i)a∈ A^(i) (which requires a∈SO(j)a∈ S_O^(j), i.e., A−ij=∅A_-^ij= ). For Ψ+ _+: p+(so)=[f(so)∈S+ij]p_+(s_o)=1[f(s_o)∈ S_+^ij], which is 0 for all sos_o iff f(SO(i))⊆S∩ijf(S_O^(i)) S_∩^ij, and 11 for some sos_o otherwise; the maximum is therefore 0 or 11. ∎ Remark 5.10 (Kernel dependence of probabilistic indices). Unlike the set-level invariants ρAtom _Atom and Cn F_Cn, which depend only on the knowledge-base pair (SO(i),SO(j))(S_O^(i),S_O^(j)), the probabilistic indices ΦAtom _Atom and Ψ+ _+ depend on the channel kernel κsemij _sem^ij and hence on the specific encoder, physical channel, and decoder. The overlap decomposition constrains the range of these indices—ΦAtom=0 _Atom=0 whenever A−ij≠∅A_-^ij≠ , and Ψ+=0 _+=0 whenever S+ij=∅S_+^ij= —but their precise values within the feasible range are determined by the kernel. Proposition 5.5 (Structural quality indices from overlap). Let ℭij C^ij be a heterogeneous semantic channel with kernel κsemij:SO(i)↝SO(j) _sem^ij:S_O^(i) S_O^(j) and let A=A(i)A=A^(i). (i) Fidelity concentration under core-preserving overlap: If A−ij=∅A_-^ij= and S+,nij=∅S_+,n^ij= , then by Corollary 3.1, (ℭij)= 1−maxa∈A(i)d¯Cn(a∣ℭij). F( C^ij)\;=\;1- _a∈ A^(i) d_Cn(a C^ij). (5.15) That is, the worst-case closure distortion is attained at a sender core element; redundant states contribute zero. (i) Depth expansion under vocabulary match: If S−ij=S+ij=∅S_-^ij=S_+^ij= (i.e., SO(i)=SO(j)S_O^(i)=S_O^(j)), then every reachable s^o∈SO(j)=SO(i) s_o∈ S_O^(j)=S_O^(i) lies in Cn(A)Cn(A) and the depth distortion reduces to its first branch (Definition 3.9). In particular, (ℭij)=0 E( C^ij)=0 if and only if Dd(s^o∣A)=Dd(so∣A)Dd( s_o A)=Dd(s_o A) holds κsemij _sem^ij-almost surely for every so∈SO(i)s_o∈ S_O^(i). (i): By Corollary 5.1, Cn(SO(i))=Cn(SO(j))Cn(S_O^(i))=Cn(S_O^(j)) and A(i)∩SO−=A−ij=∅A^(i)∩ S_O^-=A_-^ij= . Proposition 3.2 gives dCn(so,s^o∣SO(i))=0d_Cn(s_o, s_o S_O^(i))=0 for every so∈SO(i)∖As_o∈ S_O^(i) A and s^o∈SO(j) s_o∈ S_O^(j). Hence maxso∈SO(i)d¯Cn(so∣ℭij)=maxa∈Ad¯Cn(a∣ℭij) _s_o∈ S_O^(i) d_Cn(s_o C^ij)= _a∈ A d_Cn(a C^ij), and the conclusion follows from Definition 3.14. (i): When SO(i)=SO(j)S_O^(i)=S_O^(j), the noise pair is trivial and every s^o∈SO(j)=SO(i) s_o∈ S_O^(j)=S_O^(i) satisfies s^o∈Cn(A) s_o (A) (Proposition 2.1(iv)). The claim follows from Proposition 3.2 and Definition 3.14. ∎ Proposition 5.6 (Receiver-side structural comparison from overlap). Let ℭij C^ij be a heterogeneous semantic channel with S~O=SO(j) S_O=S_O^(j). (i) Atomicity shift: Δ(ℭij)=|A(j)|−|A(i)| A( C^ij)=|A^(j)|-|A^(i)|. (i) Depth shift: Δ(ℭij)=(ℐ(j))−(ℐ(i)) D_d( C^ij)= D_d(I^(j))- D_d(I^(i)). (i) Under core-preserving conditions (A−ij=∅A_-^ij= , S+,nij=∅S_+,n^ij= ): (a) Cn(SO(j))=Cn(SO(i))Cn(S_O^(j))=Cn(S_O^(i)) (Corollary 5.1). (b) The set A(i)A^(i) is an irredundant generating subset of SO(j)S_O^(j) for Cn(SO(j))=Cn(SO(i))Cn(S_O^(j))=Cn(S_O^(i)) (Proposition 3.5(i)). In general, |A(j)||A^(j)| may differ from |A(i)||A^(i)|: surplus states in S+,dijS_+,d^ij can make sender core elements redundant in the receiver’s vocabulary, potentially changing the canonical core size in either direction. (c) Equality Δ=0 A=0 holds when S+,dij=∅S_+,d^ij= (equivalently, S+ij=∅S_+^ij= , i.e., SO(j)⊆SO(i)S_O^(j) S_O^(i)), because then SO(j)=A(i)∪(J(i)∖S−ij)S_O^(j)=A^(i)∪(J^(i) S_-^ij) (noting that A−ij=∅A_-^ij= already implies S−ij⊆J(i)S_-^ij J^(i)) and A(i)A^(i) remains irredundant in SO(j)S_O^(j). (iv) Trivial noise pair (SO(i)=SO(j)S_O^(i)=S_O^(j)): Δ=0 A=0 and Δ=0 D_d=0 (Proposition 3.5(i)). Parts (i) and (i) are direct from Definition 3.16 with S~O=SO(j) S_O=S_O^(j). (i)(a): Corollary 5.1. (i)(b): By Proposition 3.5(i), applied with A=A(i)A=A^(i), SO−=S−ijS_O^-=S_-^ij, SO+=S+ijS_O^+=S_+^ij. When S+,dij≠∅S_+,d^ij≠ , some d∈S+,dij⊆Cn(A(i))d∈ S_+,d^ij (A^(i)) may make a previously irredundant a∈A(i)a∈ A^(i) redundant in SO(j)S_O^(j) (if a∈Cn((SO(j)∖a))a ((S_O^(j) \a\) ) due to the presence of d). (i)(c): When S+,dij=∅S_+,d^ij= , together with S+,nij=∅S_+,n^ij= (from the hypothesis of (i)), we have S+ij=∅S_+^ij= and SO(j)⊆SO(i)S_O^(j) S_O^(i). Since A−ij=∅A_-^ij= , A(i)⊆SO(j)A^(i) S_O^(j) and S−ij⊆J(i)S_-^ij J^(i) (because A(i)∩S−ij=A−ij=∅A^(i)∩ S_-^ij=A_-^ij= ). Hence SO(j)=A(i)∪(J(i)∖S−ij)S_O^(j)=A^(i)∪(J^(i) S_-^ij). We show Atom(SO(j))=A(i)Atom(S_O^(j))=A^(i) by a simultaneous induction on the canonical order of SO(j)S_O^(j). Let s1<s2<⋯<sms_1<s_2<·s<s_m be the elements of SO(j)S_O^(j) in canonical order, and let BkB_k denote the current set after the irredundantization procedure (Definition 2.1) has scanned s1,…,sks_1,…,s_k. Inductive claim. After scanning s1,…,sks_1,…,s_k: (a) every a∈A(i)a∈ A^(i) with a≤ska≤ s_k has survived (remains in BkB_k); and (b) every j′∈J(i)∖S−ij ∈ J^(i) S_-^ij with j′≤skj ≤ s_k has been removed. The base case k=0k=0 is vacuous. For the inductive step, suppose the claim holds through sks_k and consider sk+1s_k+1. Case 1: sk+1=a∈A(i)s_k+1=a∈ A^(i). By the inductive hypothesis, every element of J(i)∖S−ijJ^(i) S_-^ij preceding a has been removed, and every element of A(i)A^(i) preceding a has survived. Hence the current set satisfies Bk∖a=(A(i)∩s1,…,sk)∪sk+2,…,sm.B_k \a\\;=\; (A^(i)∩\s_1,…,s_k\ )\;∪\;\s_k+2,…,s_m\. In the irredundantization of SO(i)S_O^(i), when a was scanned, the current set B(i)B^(i) had precisely the same structure—surviving core elements before a plus all elements of SO(i)S_O^(i) after a—because elements of J(i)J^(i) before a were likewise removed at their own scan steps (they are in J(i)J^(i) by definition). Since sk+2,…,sm⊆elements of SO(i) after a\s_k+2,…,s_m\ \elements of S_O^(i) after a\ (the former is a subset, possibly missing elements of S−ijS_-^ij after a), Bk∖a⊆B(i)∖aB_k \a\ B^(i) \a\. By monotonicity (Cn2), Cn(Bk∖a)⊆Cn(B(i)∖a)Cn(B_k \a\) (B^(i) \a\). Since a survived in the irredundantization of SO(i)S_O^(i), a∉Cn(B(i)∖a)a (B^(i) \a\), hence a∉Cn(Bk∖a)a (B_k \a\), and a survives. Case 2: sk+1=j′∈J(i)∖S−ijs_k+1=j ∈ J^(i) S_-^ij. By the inductive hypothesis, A(i)∩s1,…,sk⊆BkA^(i)∩\s_1,…,s_k\ B_k, and all elements of A(i)A^(i) after sks_k are in BkB_k (not yet scanned). Hence A(i)⊆Bk∖j′A^(i) B_k \j \ (noting j′∉A(i)j ∉ A^(i)). By monotonicity, Cn(A(i))⊆Cn(Bk∖j′)Cn(A^(i)) (B_k \j \). Since j′∈J(i)⊆SO(i)⊆Cn(A(i))j ∈ J^(i) S_O^(i) (A^(i)) (Proposition 2.1(iv)), j′∈Cn(Bk∖j′)j (B_k \j \), and j′j is removed. By induction, the output of the irredundantization of SO(j)S_O^(j) retains exactly A(i)A^(i). For the reverse inclusion Atom(SO(j))⊆A(i)Atom(S_O^(j)) A^(i): suppose for contradiction that b∈Atom(SO(j))∖A(i)b (S_O^(j)) A^(i). Then b∈J(i)∖S−ijb∈ J^(i) S_-^ij, but the induction shows that every such element is removed—a contradiction. Hence Atom(SO(j))=A(i)Atom(S_O^(j))=A^(i) and Δ=0 A=0. (iv): Immediate from Proposition 3.5(i). ∎ Corollary 5.2 (Heterogeneous semantic Fano bound). Let PO∈Δ(SO(i))P_O∈ (S_O^(i)) be full-support and let ϵ:=d¯H(ℭij,PO)ε:= d_H( C^ij,P_O). Then Isemij(PO,ℭij)≥H(o)−hb(ϵ)−ϵlog(|SO(i)|−1),I_sem^ij(P_O, C^ij)\;≥\;H( S_o)-h_b(ε)-ε (|S_O^(i)|-1 ), (5.16) where hbh_b is the binary entropy. Moreover, by Proposition 3.4(i), ϵ≤ 1−ΦAtom(ℭij)⋅PO(A(i)).ε\;≤\;1- _Atom( C^ij)· P_O(A^(i)). (5.17) Consequently, when A−ij=∅A_-^ij= and ΦAtom(ℭij) _Atom( C^ij) is close to 11 (e.g., because the carrier channel is reliable), the right-hand side of (5.16) is close to H(o)H( S_o), i.e., nearly all source entropy is transmitted. When A−ij≠∅A_-^ij≠ , ΦAtom=0 _Atom=0 by (5.11) and the bound yields only the trivial lower bound Isemij≥0I_sem^ij≥ 0. Equation (5.16) is Theorem 3.2 applied with source alphabet SO(i)S_O^(i) and S~O=SO(j) S_O=S_O^(j). Equation (5.17) is Proposition 3.4(i). When A−ij≠∅A_-^ij≠ , ΦAtom=0 _Atom=0 gives ϵ≤1ε≤ 1, so hb(ϵ)+ϵlog(|SO(i)|−1)≤log|SO(i)|h_b(ε)+ε (|S_O^(i)|-1)≤ |S_O^(i)| and the lower bound cannot exceed zero in a nontrivial way. ∎ Proposition 5.7 (Semantic capacity from overlap). Let Wij:SC↝S^CW_ij:S_C S_C be the carrier channel kernel for the pair (i,j)(i,j). Under the full enabling assumption (Assumption 5.2) and (SA4) of Assumption 5.3: (i) Data processing chain: Isemij≤Csemij(Wij)≤C(Wij)I_sem^ij≤ C_sem^ij(W_ij)≤ C(W_ij). (i) Source entropy bound: Csemij(Wij)≤log|SO(i)|C_sem^ij(W_ij)≤ |S_O^(i)|. (i) Capacity equality (reverse size condition): If additionally |SO(i)|≥|SC||S_O^(i)|≥|S_C| and |SO(j)|≥|S^C||S_O^(j)|≥| S_C|, then Csemij(Wij)=C(Wij)C_sem^ij(W_ij)=C(W_ij). (iv) Mutual information bound: Isemij≤min(log|SO(i)|,log|SO(j)|)I_sem^ij≤ ( |S_O^(i)|,\; |S_O^(j)| ). Part (i) is Theorem 3.1(i). Part (i) is Theorem 3.1(i). Part (i) follows from Theorem 3.1(i) under the stated size conditions, which ensure the existence of a deterministic surjection f:SO(i)→SCf:S_O^(i)→ S_C and a deterministic injection g:S^C→SO(j)g: S_C→ S_O^(j). Part (iv) follows from I(o;^o)≤min(H(o),H(^o))I( S_o; S_o)≤ (H( S_o),\,H( S_o)). ∎ Remark 5.11 (Invariant–overlap correspondence). The six invariant families of Theorem 3.3 partition into three tiers of dependence on the overlap decomposition. Tier 1 (knowledge-base pair only): the set-level invariants ρAtom _Atom, Cn F_Cn (Proposition 5.2) and the structural comparison indices Δ A, Δ D_d (Proposition 5.6) are fully determined by (SO(i),SO(j))(S_O^(i),S_O^(j)) and the proof system PS. Tier 2 (knowledge-base pair ++ channel kernel): the noise-pair indices ΦAtom _Atom, Ψ+ _+ (Proposition 5.4) and the quality indices F, E (Proposition 5.5) are constrained by the overlap scalars (e.g., ΦAtom=0 _Atom=0 whenever A−ij≠∅A_-^ij≠ ; Ψ+=0 _+=0 whenever S+ij=∅S_+^ij= ) but additionally depend on κsemij _sem^ij. Tier 3 (kernel ++ carrier channel): the information-theoretic invariants IsemijI_sem^ij, CsemijC_sem^ij, C(Wij)C(W_ij); under full enabling and (SA4), Csemij≤min(C(Wij),log|SO(i)|)C_sem^ij≤ (C(W_ij),\, |S_O^(i)| ) (Proposition 5.7(i)–(i)), with equality under the reverse size condition of (i). Remark 5.12 (Diagnostic use). The three-tier correspondence enables a two-stage diagnostic workflow: first, compute the overlap decomposition offline and test the binary conditions A−ij=∅A_-^ij= and S+,nij=∅S_+,n^ij= to determine whether Cn=1 F_Cn=1 is achievable (Corollary 5.1); second, given a specific channel kernel, compute ΦAtom _Atom, Ψ+ _+, F, E and apply the Fano bound (Corollary 5.2) to obtain a lower bound on IsemijI_sem^ij. V-D Main Results: Heterogeneous Compression and Broadcast This subsection derives the main analytical results for the heterogeneous setting. Part 1 addresses the pairwise unicast scenario (Definition 5.1): closure-reliable achievability, a heterogeneous deductive compression theorem, an impossibility result when core coverage fails, and a vocabulary design criterion. Part 2 extends the theory to the broadcast scenario (Definition 5.2). Throughout, the standing assumptions of Assumption 5.3 are in force. Two notions of closure fidelity. The set-level closure fidelity Cn(SO(i),SO(j))=1 F_Cn(S_O^(i),S_O^(j))=1 (Definition 2.9) requires Cn(SO(i))=Cn(SO(j))Cn(S_O^(i))=Cn(S_O^(j)) and is independent of any code; the closure error probability Pe,Cn(n)→0P_e,Cn^(n)→ 0 (Definition 3.18(i)) is a property of a specific (n,M)(n,M) code. The former requires both (H1) and (H2); the latter requires only (H1) (Remark 5.13). Part 1: Pairwise Heterogeneous Communication Theorem 5.1 (Closure reliability for a heterogeneous pair). Let (i,j)(i,j) be a sender–receiver pair with heterogeneous semantic channel ℭij C^ij (Definition 5.5), carrier channel kernel Wij:SC↝S^CW_ij:S_C S_C with C(Wij)>0C(W_ij)>0, and overlap decomposition (Definition 5.4). Assume: (H1) A−ij=∅A_-^ij= (the sender’s irredundant core is contained in the receiver’s vocabulary: A(i)⊆SO(j)A^(i) S_O^(j)); (H2) S+,nij=∅S_+,n^ij= (all surplus states in the receiver’s vocabulary are derivable from the sender’s knowledge base: S+ij⊆Cn(SO(i))S_+^ij (S_O^(i))). Then: (i) Set-level closure fidelity: Cn(SO(i),SO(j))=1 F_Cn(S_O^(i),S_O^(j))=1 (Corollary 5.1). (i) Achievability: There exists a sequence of (n,|SO(i)|)(n,|S_O^(i)|) semantic block codes (Definition 3.17) with message set ℳ=SO(i)M=S_O^(i), encoding into SCnS_C^n, decoding into S^O=SO(j) S_O=S_O^(j), and Pe,Cn(n)→0P_e,Cn^(n)→ 0 as n→∞n→∞, provided log|A(i)|n<C(Wij). |A^(i)|n\;<\;C(W_ij). (5.18) (i) Converse (under Assumption 3.1 for SO(i)S_O^(i) with output space SO(j)S_O^(j)): Any (n,|SO(i)|)(n,|S_O^(i)|) code with Pe,Cn(n)≤ϵP_e,Cn^(n)≤ε satisfies log|A(i)|≤nC(Wij)+11−ϵ. |A^(i)|\;≤\; nC(W_ij)+11-ε. (5.19) (i): Immediate from Corollary 5.1. (i): The code is constructed in two layers, adapting Theorem 3.5(i) to the heterogeneous output alphabet S^O=SO(j) S_O=S_O^(j). Layer 1 (core code). Since log|A(i)|/n<C(Wij) |A^(i)|/n<C(W_ij), the classical channel coding theorem [29, 8] yields an (n,|A(i)|)(n,|A^(i)|) block code (fnA,gnA)(f_n^A,g_n^A) for WijW_ij with message set A(i)A^(i) and Pe(n)(A(i))→0P_e^(n)(A^(i))→ 0. By (H1), A(i)⊆SO(j)=S^OA^(i) S_O^(j)= S_O, so the decoder can output elements of A(i)A^(i). Layer 2 (redundant extension). Fix an arbitrary a0∈A(i)a_0∈ A^(i). For each redundant state j∈J(i)=SO(i)∖A(i)j∈ J^(i)=S_O^(i) A^(i), set fn(j):=fnA(a0)f_n(j):=f_n^A(a_0). The decoder first applies gnAg_n^A to recover a^∈A(i) a∈ A^(i) (or an incorrect element in the error event), and outputs a a. Closure analysis. For m∈A(i)m∈ A^(i): if the core code decodes correctly (a^=m a=m), then dCn(m,m∣SO(i))=0d_Cn(m,m S_O^(i))=0. Error probability: Pe(n)(A(i))→0P_e^(n)(A^(i))→ 0. For m=j∈J(i)m=j∈ J^(i): the decoder outputs some a^∈A(i)⊆SO(i)⊆Cn(SO(i)) a∈ A^(i) S_O^(i) (S_O^(i)). Since j is redundant in SO(i)S_O^(i), Cn(SO(i)∖j)=Cn(SO(i))Cn(S_O^(i) \j\)=Cn(S_O^(i)). Because a^∈Cn(SO(i))=Cn(SO(i)∖j) a (S_O^(i))=Cn(S_O^(i) \j\), monotonicity and idempotence of Cn give Cn((SO(i)∖j)∪a^)=Cn(SO(i)∖j)=Cn(SO(i))Cn ((S_O^(i) \j\)∪\ a\ )=Cn(S_O^(i) \j\)=Cn(S_O^(i)), so dCn(j,a^∣SO(i))=0d_Cn(j, a S_O^(i))=0. The closure error probability for redundant messages is zero for all n. Combining: Pe,Cn(n)≤Pe(n)(A(i))→0P_e,Cn^(n)≤ P_e^(n)(A^(i))→ 0. (i): The argument is identical to the proof of Theorem 3.4(i), with SOS_O replaced by SO(i)S_O^(i), S^O S_O replaced by SO(j)S_O^(j), and A=Atom(SO(i))=A(i)A=Atom(S_O^(i))=A^(i). Under Assumption 3.1 (applied to A(i)A^(i) with acceptable sets in SO(j)S_O^(j)), the pairwise-disjoint decoding regions and the Fano argument yield (5.19). ∎ Remark 5.13 (Role of conditions (H1) and (H2)). Condition (H1) is used only in the achievability proof to ensure that the decoder can output core elements (A(i)⊆SO(j)A^(i) S_O^(j)). Condition (H2) is used only to establish set-level closure fidelity Cn=1 F_Cn=1 in part (i); it does not enter the achievability or converse proofs, which depend only on the closure distortion dCn(⋅,⋅∣SO(i))d_Cn(·,· S_O^(i)) measured relative to the sender’s knowledge base. Hence the coding-theoretic conclusions (i) and (i) hold under (H1) alone. Condition (H2) provides the additional guarantee that the receiver’s overall knowledge base generates the same deductive closure as the sender’s. Remark 5.14 (Weak core coverage and alternative decoding). Theorem 5.1 uses the strong core coverage condition (H1): A(i)⊆SO(j)A^(i) S_O^(j). As noted in Remark 5.9, set-level closure fidelity Cn=1 F_Cn=1 requires only the weaker condition (F1): A(i)⊆Cn(SO(j))A^(i) (S_O^(j)). When (F1) holds but (H1) fails (i.e., some core element a∈A−ija∈ A_-^ij is derivable from SO(j)S_O^(j) but not literally present), the two-layer code cannot directly output a. A modified decoder could instead output a proxy element a^∈SO(j) a∈ S_O^(j) satisfying dCn(a,a^∣SO(i))=0d_Cn(a, a S_O^(i))=0—i.e., a state in the receiver’s vocabulary whose substitution for a preserves the sender’s deductive closure. Such a proxy exists whenever a∈Cn(SO(j))a (S_O^(j)), but identifying it requires knowledge of the sender’s closure structure at the decoder, making the code design more involved. A complete treatment of proxy-based decoding is deferred to future work; the results of this section focus on the operationally simpler setting where (H1) holds. The proxy-based decoding strategy shares conceptual affinity with the inverse contextual reasoning of Seo et al. [28], who address the problem of inferring a sender’s communication context from noisy observations using Bayesian methods. Theorem 5.2 (Heterogeneous deductive compression). Under the hypotheses of Theorem 5.1, the minimum blocklength for closure-reliable communication of the full knowledge base SO(i)S_O^(i) to agent j satisfies, for sufficiently small ϵ>0ε>0: (i) Closure blocklength: (1−ϵ)log|A(i)|−1C(Wij)≤ (1-ε) |A^(i)|-1C(W_ij)\;≤\; n∗(SO(i),Wij,Pe,Cn,ϵ) n^* (S_O^(i),W_ij,P_e,Cn,ε ) ≤ \;≤\; ⌈log|A(i)|C(Wij)−δ(ϵ)⌉, |A^(i)|C(W_ij)-δ(ε) , (5.20) where δ(ϵ)→0δ(ε)→ 0 as ϵ→0ε→ 0. (i) Hamming baseline: Under the additional hypothesis SO(i)⊆SO(j)S_O^(i) S_O^(j) (i.e., S−ij=∅S_-^ij= , which strengthens (H1) to full vocabulary containment), n∗(SO(i),Wij,Pe,ϵ)≥(1−ϵ)log|SO(i)|−1C(Wij).n^* (S_O^(i),W_ij,P_e,ε )\;≥\; (1-ε) |S_O^(i)|-1C(W_ij). (5.21) (i) Deductive compression ratio: When both bounds apply, n∗(Pe,Cn)n∗(Pe)≈log|A(i)|log|SO(i)|, n^*(P_e,Cn)n^*(P_e)\;≈\; |A^(i)| |S_O^(i)|\,, (5.22) identical to the homogeneous ratio of Corollary 3.3. Part (i) combines Theorem 5.1(i) (upper bound) and (i) (lower bound). Part (i) is Theorem 3.4(i) applied with M=|SO(i)|M=|S_O^(i)|; the condition SO(i)⊆SO(j)S_O^(i) S_O^(j) ensures that the Hamming criterion is meaningful (each sent state has a valid identity reconstruction in the receiver’s vocabulary). Part (i) follows by dividing the bounds. ∎ Remark 5.15 (Heterogeneity does not degrade the compression ratio). The deductive compression ratio (5.22) depends only on the sender’s knowledge-base structure (|A(i)||A^(i)| vs. |SO(i)||S_O^(i)|) and not on the receiver’s vocabulary SO(j)S_O^(j), provided the core coverage condition (H1) holds. This invariance is a consequence of the two-layer code structure: the core sub-code operates identically regardless of the receiver’s surplus states, and the redundant extension incurs zero closure distortion by the algebraic properties of Cn. The Hamming baseline (5.21) does, however, depend on the receiver’s vocabulary: it requires S−ij=∅S_-^ij= (full vocabulary containment), a strictly stronger condition than (H1). When S−ij≠∅S_-^ij≠ , perfect Hamming reconstruction is impossible (some sent states have no counterpart in the receiver’s vocabulary), while closure reliability may still be achievable under (H1). This gap illustrates the advantage of semantic fidelity criteria over symbol-level criteria in heterogeneous settings. Corollary 5.3 (Impossibility under core loss). Let (i,j)(i,j) be a sender–receiver pair. (i) Set-level impossibility: If condition (F1) of Proposition 5.3 fails, i.e., A(i)⊈Cn(SO(j))A^(i) (S_O^(j) ), then Cn(SO(i),SO(j))<1 F_Cn(S_O^(i),S_O^(j))<1. This is a property of the knowledge-base pair, independent of the channel, the blocklength, and the coding strategy. (i) Quantitative bound: Under the hypothesis of (i), the closure fidelity satisfies Cn(SO(i),SO(j))=|Cn(SO(i))∩Cn(SO(j))||Cn(SO(i))∪Cn(SO(j))|< 1. F_Cn (S_O^(i),S_O^(j) )\;=\; |Cn(S_O^(i)) (S_O^(j))||Cn(S_O^(i)) (S_O^(j))|\;<\;1. (5.23) (i) Core preservation ratio: If A−ij≠∅A_-^ij≠ , then ρAtom(SO(i),SO(j))=1−|A−ij|/|A(i)|<1 _Atom(S_O^(i),S_O^(j))=1-|A_-^ij|/|A^(i)|<1 (Proposition 5.2(i)). Similarly, if condition (F2) fails (S+,nij≠∅S_+,n^ij≠ ), then Cn<1 F_Cn<1 regardless of any coding strategy. Part (i) is the contrapositive of Proposition 5.3. Part (i) is the definition of Cn F_Cn (Definition 2.9); the strict inequality follows from (i). Part (i) is Proposition 5.2(i). The final claim follows from Proposition 5.3 (necessity of (F2)). ∎ Proposition 5.8 (Minimum receiver vocabulary for closure-reliable communication). Given a sender knowledge base SO(i)S_O^(i), the minimum-cardinality subset V⊆SO(i)V S_O^(i) serving as a receiver vocabulary (S^O=V S_O=V) that simultaneously achieves: (a) the two-layer code of Theorem 5.1(i) achieves Pe,Cn(n)→0P_e,Cn^(n)→ 0, and (b) Cn(SO(i),V)=1 F_Cn(S_O^(i),V)=1, is V∗=Atom(SO(i))=A(i)V^*=Atom(S_O^(i))=A^(i), with |V∗|=(ℐ(i))|V^*|= A(I^(i)). Sufficiency. Set V=A(i)V=A^(i). Since A(i)⊆SO(i)A^(i) S_O^(i), the overlap with sender i and “receiver” V gives S+ij=V∖SO(i)=∅S_+^ij=V S_O^(i)= and A−ij=A(i)∖V=∅A_-^ij=A^(i) V= (condition (H1)). Condition (H2) holds trivially since S+ij=∅S_+^ij= . Closure fidelity: Cn(A(i))=Cn(SO(i))Cn(A^(i))=Cn(S_O^(i)) (Proposition 2.1(i)), giving Cn(SO(i),A(i))=1 F_Cn(S_O^(i),A^(i))=1. The two-layer code of Theorem 5.1(i) applies with S^O=V=A(i) S_O=V=A^(i). Minimality. Let V⊆SO(i)V S_O^(i) satisfy both conditions (a) and (b). Condition (a) requires the two-layer code of Theorem 5.1(i) to succeed. That code’s decoder outputs elements of A(i)A^(i), so the output alphabet S^O=V S_O=V must contain every core element: A(i)⊆VA^(i) V. Hence |V|≥|A(i)||V|≥|A^(i)|. Since V=A(i)V=A^(i) achieves this bound, it is minimal. ∎ Remark 5.16 (Vocabulary design rule). Proposition 5.8 yields a principled vocabulary-selection rule for receiver design: the receiver need store only the sender’s irredundant core. All remaining semantic states (the sender’s stored shortcuts J(i)J^(i)) can be reconstructed by the receiver’s inference engine via Cn(A(i))Cn(A^(i)). The channel-use cost of this strategy is n∗≈log|A(i)|/C(Wij)n^*≈ |A^(i)|/C(W_ij), the minimum achievable under closure reliability. When the receiver already maintains a richer vocabulary SO(j)⊋A(i)S_O^(j) A^(i), the additional states are harmless provided S+,nij=∅S_+,n^ij= (condition (H2)); they do not increase the blocklength. When some surplus states are non-derivable (S+,nij≠∅S_+,n^ij≠ ), set-level closure fidelity drops below 11 (Corollary 5.3), but the coding-theoretic closure reliability may still hold if the two-layer code is used (since it ignores the surplus entirely; see Remark 5.13). Part 2: Broadcast Extension We now extend the pairwise results to the broadcast scenario of Definition 5.2. Agent 0 (the sender) communicates its knowledge base SO(0)S_O^(0) to K receivers over a common carrier channel W:SC↝S^CW:S_C S_C with C(W)>0C(W)>0. Theorem 5.3 (Broadcast deductive compression). Suppose that for every receiver j∈1,…,Kj∈\1,…,K\, the overlap conditions hold: (BH1) A−0j=∅A_-^0j= (the sender’s core is contained in every receiver’s vocabulary: A(0)⊆SO(j)A^(0) S_O^(j)); (BH2) S+,n0j=∅S_+,n^0j= (every receiver’s surplus is derivable from the sender: S+0j⊆Cn(SO(0))S_+^0j (S_O^(0))). Then: (i) Simultaneous closure fidelity: Cn(SO(0),SO(j))=1 F_Cn(S_O^(0),S_O^(j))=1 for every j∈1,…,Kj∈\1,…,K\. (i) Broadcast achievability: There exists a single sequence of (n,|SO(0)|)(n,|S_O^(0)|) semantic block codes (with a common encoding function fnf_n) such that Pe,Cn(n,j)→0P_e,Cn^(n,j)→ 0 simultaneously for all receivers j∈1,…,Kj∈\1,…,K\, provided log|A(0)|n<C(W). |A^(0)|n\;<\;C(W). (5.24) (i) Blocklength independence from K: The minimum blocklength for broadcast closure reliability is nbc∗≈log|A(0)|C(W),n^*_bc\;≈\; |A^(0)|C(W), (5.25) independent of the number of receivers K. (iv) Broadcast converse: Under Assumption 3.1 for SO(0)S_O^(0) with output space SO(j)S_O^(j) for each j, any code achieving maxjPe,Cn(n,j)≤ϵ _jP_e,Cn^(n,j)≤ε satisfies log|A(0)|≤(nC(W)+1)/(1−ϵ) |A^(0)|≤(nC(W)+1)/(1-ε). (i): For each j, conditions (BH1)–(BH2) instantiate (H1)–(H2) of Theorem 5.1, giving Cn(SO(0),SO(j))=1 F_Cn(S_O^(0),S_O^(j))=1 by Corollary 5.1. (i): Construct a single two-layer code as in the proof of Theorem 5.1(i), with the common core code (fnA,gnA)(f_n^A,g_n^A) for message set A(0)A^(0). All K receivers observe the same channel output S^Cn S_C^n and each independently applies the same core decoder gnAg_n^A. Since A(0)⊆SO(j)A^(0) S_O^(j) for every j (by (BH1)), the decoded core element a^∈A(0) a∈ A^(0) is a valid output for every receiver. The Layer 2 redundant extension and closure analysis are identical to the pairwise case (using the sender’s closure structure only), so Pe,Cn(n,j)≤Pe(n)(A(0))→0P_e,Cn^(n,j)≤ P_e^(n)(A^(0))→ 0 simultaneously for all j. (i): The blocklength is determined by the core code, which has rate log|A(0)|/n |A^(0)|/n, independent of K. (iv): Fix any receiver j. Theorem 5.1(i) applied to the pair (0,j)(0,j) gives log|A(0)|≤(nC(W)+1)/(1−ϵ) |A^(0)|≤(nC(W)+1)/(1-ε) whenever Pe,Cn(n,j)≤ϵP_e,Cn^(n,j)≤ε. Since this must hold for every j, the bound holds under maxjPe,Cn(n,j)≤ϵ _jP_e,Cn^(n,j)≤ε. ∎ Proposition 5.9 (Broadcast semantic bottleneck). In the broadcast scenario of Definition 5.2, suppose there exists a receiver j∗∈1,…,Kj^*∈\1,…,K\ such that condition (F1) of Proposition 5.3 fails for the pair (0,j∗)(0,j^*): A(0)⊈Cn(SO(j∗)).A^(0) (S_O^(j^*) ). Then: (i) Cn(SO(0),SO(j∗))<1 F_Cn(S_O^(0),S_O^(j^*))<1, regardless of the carrier channel W, the blocklength n, and the encoding/decoding strategy. (i) Even if the carrier channel is noiseless (W=idSCW=id_S_C), the closure fidelity at receiver j∗j^* is bounded by Cn(SO(0),SO(j∗))=|Cn(SO(0))∩Cn(SO(j∗))||Cn(SO(0))∪Cn(SO(j∗))|< 1. F_Cn (S_O^(0),S_O^(j^*) )\;=\; |Cn(S_O^(0)) (S_O^(j^*))\!||Cn(S_O^(0)) (S_O^(j^*))\!|\;<\;1. (5.26) (i) Receiver j∗j^* is a semantic bottleneck: its performance limitation arises from vocabulary mismatch, not from the physical channel. The other receivers j≠j∗j≠ j^* satisfying (BH1)–(BH2) achieve Cn=1 F_Cn=1 and closure reliability simultaneously, unaffected by j∗j^*. Parts (i) and (i) follow from Corollary 5.3(i)–(i) applied to the pair (0,j∗)(0,j^*). Part (i): the common encoding and core code are shared by all receivers; the failure at j∗j^* is due solely to the mismatch A(0)⊈Cn(SO(j∗))A^(0) (S_O^(j^*)), which is independent of the channel. Receivers satisfying (BH1)–(BH2) are handled by Theorem 5.3. ∎ Remark 5.17 (Semantic bottleneck vs. classical channel degradation). In classical broadcast channel theory [8, 9], the weakest receiver is the one with the noisiest channel, and the rate region depends only on channel transition probabilities. Proposition 5.9 reveals a second, purely structural axis of weakness invisible to classical theory: even over a noiseless carrier, a receiver with A(0)⊈Cn(SO(j∗))A^(0) (S_O^(j^*)) cannot achieve Cn=1 F_Cn=1. This semantic bottleneck is irreducible by coding: resolving it requires a design-time vocabulary augmentation (e.g., pre-loading A(0)A^(0) into SO(j∗)S_O^(j^*) per Proposition 5.8), fundamentally different from the encoder/decoder optimization that suffices in classical settings. Remark 5.18 (Classical recovery). When SO(j)=SO(0)S_O^(j)=S_O^(0) for all j, (BH1)–(BH2) hold vacuously and Theorem 5.3 reduces to K independent applications of Theorem 3.5(i). If additionally A(0)=SO(0)A^(0)=S_O^(0) (irredundant source), the broadcast blocklength becomes n∗≈log|SO(0)|/C(W)n^*≈ |S_O^(0)|/C(W), recovering the classical channel coding theorem [8]. Remark 5.19 (Summary of answers to Q1–Q4). The results of this subsection answer the four key questions posed in Section V-A: Q1 (closure reliability from overlap): Proposition 5.3 and Corollary 5.1 provide the necessary and sufficient conditions; the operational achievability under the strong condition (H1) is Theorem 5.1. Q2 (heterogeneous compression): Theorem 5.2 establishes that the deductive compression ratio is invariant under vocabulary heterogeneity. Corollary 5.3 characterizes the impossibility regime. Q3 (invariant diagnosis): Section V-C (Propositions 5.2–5.6 and Remark 5.11) expresses every invariant family in terms of the overlap decomposition. Q4 (broadcast bottleneck): Theorem 5.3 shows blocklength independence from K under core coverage; Proposition 5.9 identifies the semantic bottleneck phenomenon. V-E Numerical Validation This subsection verifies the theoretical results on two scales: a small Datalog instance permitting closed-form computation of every invariant, and medium-scale supply-chain knowledge graphs with up to approximately 24 000 base facts. Small-Scale Datalog Instance Example 5.1 (Path-reachability knowledge bases). Fix a domain =a,b,c,dD=\a,b,c,d\ with relation symbols Edge and ℎPath and two Datalog rules: ℎ(x,y)←(x,y)Path(x,y) (x,y) and ℎ(x,z)←(x,y),ℎ(y,z)Path(x,z) (x,y),\,Path(y,z). Agents. The sender (agent 1) stores |SO(1)|=8|S_O^(1)|=8 facts: four Edge facts (a,b),(a,c),(b,c),(c,d)\Edge(a,b),Edge(a,c),Edge(b,c),Edge(c,d)\ and four ℎPath shortcuts ℎ(a,b),ℎ(b,c),ℎ(c,d),ℎ(b,d);\Path(a,b),Path(b,c),Path(c,d),Path(b,d)\; irredundantization yields A(1)=(a,b),(a,c),(b,c),(c,d)A^(1)=\Edge(a,b),Edge(a,c),Edge(b,c),Edge(c,d)\ with |A(1)|=4|A^(1)|=4, so all four ℎPath facts are stored shortcuts. Three receivers are defined to illustrate distinct overlap regimes: receiver 2 (core loss: |A−12|=1|A_-^12|=1, non-derivable surplus: |S+,n12|=1|S_+,n^12|=1); receiver 2′ (augmented: A−12′=∅A_-^12 = , S+,n12′=∅S_+,n^12 = ); receiver 3 (broadcast: A−13=∅A_-^13= , S+,n13=∅S_+,n^13= ). Carrier channel. A q-ary symmetric channel with q=10q=10 and crossover probability p=0.1p=0.1, giving C(W)≈2.536C(W)≈ 2.536 bits. Results. Table I reports the overlap decomposition and set-level invariants. Table I lists all six invariant families of Theorem 3.3. Table IV compares the minimum blocklength under Hamming and closure reliability. Key observations: TABLE I: Overlap decomposition and set-level invariants for sender agent 1 paired with each receiver. Quantity Recv. 2 Recv. 2′ Recv. 3 |A−ij||A_-^ij| 1 0 0 |S+,nij||S_+,n^ij| 1 0 0 ρAtom _Atom 3/4 1 1 Cn F_Cn 3/73/7 1 1 Definition 5.6 (Experimental semantic capacity estimate). In the numerical validation, we report an experimental semantic-capacity estimate under a fixed encoding and a restricted decoder class. Fix an encoding kernel κencid _enc^id (the identity-injection encoding used in the experiments) and a decoder family (j)⊆(ℐdec(j))D^(j) (I_dec^(j)). Define C^semij:=maxPO∈Δ(SO(i)),D∈(j)I(o;^o), C_sem^ij\;:=\; _P_O∈ (S_O^(i)),\;D ^(j)I\! ( S_o; S_o ), where the joint law is induced by the end-to-end kernel κsemij=D∘Wij∘κencid _sem^ij=D W_ij _enc^id. For each fixed D, the maximization over POP_O is a standard finite-alphabet channel-capacity computation and is carried out via the Blahut–Arimoto algorithm. We also report the achieved mutual information I^semij:=I(o;^o) I_sem^ij\;:=\;I\! ( S_o; S_o ) under κencid _enc^id and a decoder D chosen to maximize I(o;^o)I( S_o; S_o) within the same decoder family (j)D^(j). TABLE I: Semantic channel invariants for the Datalog instance in Example 5.1. Family Invariant Pair (1,2)(1,2) Pair (1,2′)(1,2 ) Pair (1,3)(1,3) I. Source A 4 4 4 D_d 2 2 2 I. Set-level ρAtom _Atom 0.750 1.000 1.000 Cn F_Cn 0.429 1.000 1.000 I. Noise-pair ΦAtom _Atom 0 0.900 0.900 Ψ+ _+ 0.900 0 0.911 IV. Quality F 0.900 0.980 0.981 E 0.078 0.078 0.494 V. Comparison Δ A 0 0 0 Δ D_d +1+1 0 0 VI. Info-th. C(W)C(W) 2.536 2.536 2.536 C^semij C_sem^ij 2.280 2.280 1.958 I^semij I_sem^ij 2.273 2.273 1.808 Configuration: Carrier: q-ary symmetric channel, q=10q=10, p=0.1p=0.1; source: POP_O uniform on SO(1)S_O^(1). Families I–I and V are determined by the knowledge-base pair alone and are independent of the channel configuration. Information-theoretic quantities. The quantities C^semij C_sem^ij and I^semij I_sem^ij are experimental and follow Definition 5.6. In particular, C^semij C_sem^ij is obtained by maximizing I(o;^o)I( S_o; S_o) over POP_O (via the Blahut–Arimoto algorithm) and over the chosen decoder family, under the fixed identity-injection encoding. The reported I^semij I_sem^ij is evaluated under the same encoding and a mutual-information-maximizing decoder within the same decoder family. Sanity check. The inequalities I^semij≤C^semij≤C(W) I_sem^ij≤ C_sem^ij≤ C(W) are verified for all three pairs. TABLE IV: Minimum blocklength estimates (ϵ→0ε→ 0). Criterion (1,2)(1,2) (1,2′)(1,2 ) (1,3)(1,3) Hamming nH∗n^*_H N/A† 1.183 N/A† Closure nCn∗n^*_Cn ∄ ‡ 0.789 0.789 Ratio — 2/3 — †S−ij≠∅S_-^ij≠ (Hamming reconstruction undefined). ‡A−ij≠∅A_-^ij≠ , so Cn<1 F_Cn<1 by Corollary 5.3. Data processing sanity check. The experimental quantities satisfy the chain I^semij≤C^semij≤C(W) I_sem^ij≤ C_sem^ij≤ C(W) for every pair: 2.273≤2.280≤2.5362.273≤ 2.280≤ 2.536 for (1,2)(1,2), 2.273≤2.280≤2.5362.273≤ 2.280≤ 2.536 for (1,2′)(1,2 ), and 1.808≤1.958≤2.5361.808≤ 1.958≤ 2.536 for (1,3)(1,3). The equality C^sem12=C^sem12′=2.280 C_sem^12= C_sem^12 =2.280 indicates that, under the fixed encoding and decoder family of Definition 5.6, the semantic-capacity estimate is driven primarily by the carrier channel and the effective source/receiver alphabet sizes, and is insensitive to the particular logical content of the receiver vocabulary in these two cases. Under mutual-information-maximizing decoding, I^sem12=I^sem12′=2.273 I_sem^12= I_sem^12 =2.273: the vocabulary mismatch in pair (1,2)(1,2) (core loss and non-derivable surplus) is invisible at the information-theoretic level. The discrepancy is instead captured by the set-level and noise-pair invariants: Cn12=3/7≠1=Cn12′ F_Cn^12=3/7≠ 1= F_Cn^12 , ΦAtom12=0≠0.9=ΦAtom12′ _Atom^12=0≠ 0.9= _Atom^12 , and Ψ+12=0.900≠0=Ψ+12′ _+^12=0.900≠ 0= _+^12 . Here Ψ+12=0.900 _+^12=0.900 is attained at the lost-core input so=(a,c)s_o=Edge(a,c): when this message is sent, the optimized decoder outputs the surplus atom (d,a)Edge(d,a) with probability 0.90.9, hence the maximal spurious-output probability equals 0.90.9 by Definition 3.15. The core preservation index detects the loss of (a,c)Edge(a,c) from the receiver’s vocabulary, while the spurious probability index captures the complementary effect—the decoder outputs the surplus element (d,a)Edge(d,a) with probability 0.90.9 whenever the lost core element is sent. This illustrates the diagnostic value of the multi-family invariant architecture of Theorem 3.3: families I–I detect a semantic impairment that family VI cannot distinguish. For pair (1,3)(1,3), the smaller receiver vocabulary (|SO(3)|=6|S_O^(3)|=6) constrains C^sem13=1.958<2.280 C_sem^13=1.958<2.280, demonstrating a capacity-level vocabulary bottleneck that complements the set-level bottleneck of Proposition 5.9. Deductive compression. The ratio nCn∗/nH∗=2/3=log|A(1)|/log|SO(1)|n^*_Cn/n^*_H=2/3= |A^(1)|/ |S_O^(1)| for pair (1,2′)(1,2 ) matches Theorem 5.2(i) exactly. Vocabulary design. Augmenting receiver 2 to receiver 2′ (adding the single lost core element (a,c)Edge(a,c) and removing the non-derivable surplus (d,a)Edge(d,a)) raises Cn F_Cn from 3/73/7 to 11 and ΦAtom _Atom from 0 to 0.90.9, while eliminating the spurious output probability (Ψ+ _+ from 0.9000.900 to 0), confirming Proposition 5.8. Broadcast bottleneck. Agent 1 broadcasts to receivers 2 and 3. Receiver 3 satisfies (BH1)–(BH2) and achieves Cn=1 F_Cn=1 with nbc∗=⌈log4/C(W)⌉=1n^*_bc= 4/C(W) =1. Receiver 2 violates (BH1) and is a semantic bottleneck (Cn<1 F_Cn<1 regardless of W), confirming Proposition 5.9. Medium-scale Experimental Metrics The following metrics are used in the medium-scale experiments. Let ℱF denote the set of stored base-predicate facts and Cl(ℱ):=Cn(ℱ)∩OCl(F):=Cn(F) _O the closure restricted to the ambient universe. Definition 5.7 (Deductive amplification factor). γamp(ℱ):=|Cl(ℱ)|/|ℱ| _amp(F):=|Cl(F)|/|F|. Definition 5.8 (Syntactic and semantic Jaccard indices). For agents i,ji,j with fact sets ℱ(i),ℱ(j)F^(i),F^(j): Ωijsyn:=|ℱ(i)∩ℱ(j)||ℱ(i)∪ℱ(j)|,Ωijsem:=|Cl(ℱ(i))∩Cl(ℱ(j))||Cl(ℱ(i))∪Cl(ℱ(j))|. ^syn_ij:= |F^(i) ^(j)||F^(i) ^(j)|, ^sem_ij:= |Cl(F^(i)) (F^(j))||Cl(F^(i)) (F^(j))|. The pairwise averages are Ω¯syn ^syn and Ω¯sem ^sem. Note that Ωijsem=Cn(ℱ(i),ℱ(j)) ^sem_ij= F_Cn(F^(i),F^(j)) (Definition 2.9). Definition 5.9 (Closure fidelity curve). For a base fact set ℱF and a randomly selected subset ℱ^⊆ℱ F of fraction R: Φ(R):=|Cl(ℱ^)∩Cl(ℱ)|/|Cl(ℱ)| (R):=|Cl( F) (F)|/|Cl(F)|. Definition 5.10 (Deductive compression ratio). For a knowledge base SOS_O containing both base facts and materialized derived facts: ρcomp:=log|Atom(SO)|/log|SO| _comp:= |Atom(S_O)|/ |S_O| (the single-shot ratio of Corollary 3.3). The entropy-based ratio under uniform source is ρent:=PAH(πA)/H(PO)=(klogk)/(|SO|log|SO|) _ent:=P_AH( _A)/H(P_O)=(k k)/(|S_O| |S_O|) where k=|Atom(SO)|k=|Atom(S_O)|. Note that ρcomp=Λ1−1 _comp= _1^-1 and ρent=Λ∞−1 _ent= _∞^-1; the two coincide only when PA=1P_A=1 (irredundant source). When |J|>0|J|>0 and the source is uniform, ρent≪ρcomp _ent _comp because the additional factor PA=k/|SO|P_A=k/|S_O| makes the entropic gain much larger than the log-scale gain. Medium-Scale Supply-Chain Experiments We test scalability on synthetic supply-chain knowledge graphs with three base predicates ( connected, supplies, produces) and four Datalog rules computing transitive reachability and item availability (see (5.27)–(5.30) below). (X,Y) reachable(X,Y) ←(X,Y), ← connected(X,Y), (5.27) (X,Z) reachable(X,Z) ←(X,Y),(Y,Z), ← reachable(X,Y),\; connected(Y,Z), (5.28) (I,L) available(I,L) ←(S,I),(S,L), ← produces(S,I),\; supplies(S,L), (5.29) (I,L) available(I,L) ←(S,I),(S,L0), ← produces(S,I),\; supplies(S,L_0), (L0,L). reachable(L_0,L). (5.30) The stored knowledge base consists exclusively of base-predicate facts, so Atom(ℱ)=ℱAtom(F)=F and the deductive compression ratio is 11 (Corollary 3.2). All experiments are implemented in Python 3 using NetworkX [13]. TABLE V: Deductive amplification across knowledge-base scales. γamp=|Cl|/|ℱ| _amp=|Cl|/|F|. |||V| p |ℱ||F| |Cl||Cl| γamp _amp d 50 0.060 188 2 727 14.5 10 200 0.040 1 705 45 105 26.5 5 500 0.020 5 179 268 679 51.9 5 1 000 0.012 12 305 1 050 305 85.4 5 2 000 0.006 24 304 4 118 304 169.4 6 Experiment 1: Deductive amplification Table V reports the closure amplification γamp=|Cl(ℱ)|/|ℱ| _amp=|Cl(F)|/|F| for configurations spanning 50 to 2 000 locations. At ||=2,000|V|=2,000, the closure exceeds the base fact set by a factor of 169169: approximately 24 000 base facts generate over 4.1 million derived consequences. The maximum derivation depth stabilizes at d=5d=5–1010, reflecting the short diameter of dense random directed graphs. This confirms substantial deductive amplification (formalized by the closure operator Cn) and quantifies the semantic leverage available to a receiver possessing the shared rules. Experiment 2: Multi-agent overlap We partition the base facts of a 300-location universe (|ℱ|=2,946|F|=2,946) among K=8K=8 agents, each retaining each fact independently with probability 0.40.4. The average pairwise semantic Jaccard index is Ω¯sem=0.795 ^sem=0.795, compared with the syntactic index Ω¯syn=0.246 ^syn=0.246—a factor of 3.2×3.2×. Thus agents sharing roughly 25%25\% of base facts share nearly 80%80\% of semantic content after closure, confirming that condition (F2) of Proposition 5.3 is substantially easier to satisfy at the semantic level. Combining any two agents’ knowledge bases produces novel derivations constituting 1.8%1.8\%–4.5%4.5\% of the combined closure, exhibiting non-trivial deductive synergy. Experiment 3: Rate–fidelity trade-off Using the 200-location base (|ℱ|=1,705|F|=1,705), we vary the fraction R of base facts transmitted and measure closure fidelity Φ(R)=|Cl(ℱ^)∩Cl(ℱ)|/|Cl(ℱ)| (R)=|Cl( F) (F)|/|Cl(F)|. Under random selection: Φ(0.25)=0.53 (0.25)=0.53, Φ(0.50)=0.89 (0.50)=0.89, Φ(0.75)=0.97 (0.75)=0.97. The ratio Φ(R)/R>1 (R)/R>1 for all R<1R<1, demonstrating a semantic leverage effect: inference rules allow the receiver to reconstruct a disproportionately large fraction of the closure from a partial base. Since the knowledge base is irredundant, perfect recovery Φ=1 =1 requires R=1R=1, consistent with Corollary 3.3. A connectivity-first strategy outperforms random at low rates (R≲0.35R 0.35) but underperforms at higher rates due to delayed transmission of supplier/product facts, illustrating a predicate-balance trade-off. Experiment 4: Deductive compression with materialized shortcuts The preceding experiments store only base-predicate facts, so Atom(ℱ)=ℱAtom(F)=F and ρcomp=1 _comp=1. To demonstrate the deductive compression gain predicted by Theorem 4.1 at scale, we augment the stored knowledge base by materializing a fraction μ of the derived facts (reachable and available) as stored shortcuts, simulating a common scenario in which an agent caches query results or materialized views. For the 200-location supply-chain universe (|ℱbase|=1,705|F_base|=1,705, |Cl|=45,105|Cl|=45,105), we set SO:=ℱbase∪ℱmatS_O:=F_base _mat where ℱmatF_mat is a uniformly random subset of Cl∖ℱbaseCl _base of size ⌊μ⋅|Cl∖ℱbase|⌋ μ·|Cl _base| . Under the given Datalog rules, no rule derives a base-predicate fact, so every materialized IDB fact is redundant and Atom(SO)=ℱbaseAtom(S_O)=F_base; hence |A|=|ℱbase||A|=|F_base| and |J|=|ℱmat||J|=|F_mat|. This property—that no Datalog rule derives a base-predicate (EDB) fact—is specific to the rule set (5.27)–(5.30), in which all rule heads are IDB predicates. In knowledge bases with integrity constraints or recursive rules whose heads include EDB predicates, some base facts could become derivable, altering the core; the above identification Atom(SO)=ℱbaseAtom(S_O)=F_base would then require verification via the core-extraction procedure of Definition 2.1. Table VI reports the compression ratios for varying materialization fractions μ. TABLE VI: Deductive compression gain for the 200-location supply-chain knowledge base with materialized shortcuts. |A|=|ℱbase|=1,705|A|=|F_base|=1,705 throughout; POP_O uniform on SOS_O. μ (%) |J||J| |SO||S_O| ρcomp _comp ρent _ent Λ1 _1 0 0 1 705 1.000 1.000 1.00 10 4 340 6 045 0.855 0.241 1.17 20 8 680 10 385 0.805 0.132 1.24 30 13 020 14 725 0.775 0.090 1.29 50 21 700 23 405 0.740 0.054 1.35 80 34 720 36 425 0.709 0.033 1.41 100 43 400 45 105 0.694 0.026 1.44 We also verify cross-scale consistency by repeating the experiment at μ=0.3μ=0.3 across five graph sizes (Table VII). TABLE VII: Deductive compression at μ=0.3μ=0.3 across scales. |||V| |A||A| |SO||S_O| ρcomp _comp ρent _ent dmaxd_ 50 188 949 0.764 0.151 10 200 1 705 14 725 0.775 0.090 5 500 5 179 84 229 0.754 0.046 5 1 000 12 305 323 705 0.742 0.028 5 2 000 24 304 1 252 504 0.719 0.014 6 The compression gain increases with scale. At ||=2,000|V|=2,000, the single-shot ratio is ρcomp=0.719 _comp=0.719 (a 28%28\% blocklength reduction), while the entropic ratio drops to ρent=0.014 _ent=0.014: in the i.i.d. regime, the semantic rate is less than 1.5%1.5\% of the classical rate. The entropic gains are amplified at larger scales because the deductive amplification factor γamp _amp grows super-linearly (Table V): materializing 30%30\% of a larger closure produces a proportionally larger shortcut set relative to the fixed core. Summary The small-scale instance verifies the deductive compression ratio (nCn∗/nH∗=2/3n^*_Cn/n^*_H=2/3), the vocabulary design criterion (Proposition 5.8), and the broadcast bottleneck (Proposition 5.9). The medium-scale experiments confirm deductive amplification exceeding two orders of magnitude (Experiment 1), semantic overlap amplification by a factor of 3.23.2 (Experiment 2), the semantic leverage effect Φ(R)>R (R)>R for all R<1R<1 (Experiment 3), and—crucially—the deductive compression gain ρent<1 _ent<1 at scale whenever materialized shortcuts are present (Experiment 4), with entropic compression exceeding an order of magnitude at moderate materialization levels. Together, the four experiments bracket the theoretical range: irredundant-source deductive amplification (Experiments 1–3) versus redundant-source deductive compression (Experiment 4), closing the loop with Theorem 4.1 and Corollary 3.3. In particular, the noise-pair indices for pair (1,2)(1,2) (ΦAtom=0 _Atom=0, Ψ+=0.900 _+=0.900) quantify the vocabulary-mismatch impairment that is invisible to the information-theoretic invariants (Isem12=Isem12′=2.273I_sem^12=I_sem^12 =2.273), underscoring the diagnostic complementarity of the six invariant families. The semantic capacity CsemijC_sem^ij, computed via Blahut–Arimoto for all three pairs, verifies the data processing chain Isemij≤Csemij≤C(W)I_sem^ij≤ C_sem^ij≤ C(W) and reveals a capacity-level vocabulary bottleneck: when the receiver’s vocabulary is smaller (|SO(3)|=6<8=|SO(1)||S_O^(3)|=6<8=|S_O^(1)|), Csem13=1.958<2.280=Csem12′C_sem^13=1.958<2.280=C_sem^12 , demonstrating that vocabulary mismatch constrains not only closure fidelity (Proposition 5.9) but also the maximum achievable mutual information. VI Conclusion This paper has developed a rate–distortion theory for semantic communication grounded in formal proof systems. The framework rests on three pillars: an axiomatic information model with computable enabling maps (Section I); a semantic channel built as a composition of enabling kernels (Section I); and an overlap-based heterogeneous multi-agent theory (Section V). The central quantitative finding is the deductive compression gain. Under a closure-based fidelity criterion that accepts any reconstruction preserving the deductive closure, the minimum blocklength drops from nH∗≈log|SO|/C(W)n_H^*≈ |S_O|/C(W) to nCn∗≈log|Atom(SO)|/C(W)n_Cn^*≈ |Atom(S_O)|/C(W), yielding a compression ratio log|Atom(SO)|/log|SO|<1 |Atom(S_O)|/ |S_O|<1 that is invariant under receiver vocabulary heterogeneity. This gain arises because the receiver’s inference engine reconstructs all redundant states from the irredundant core at zero additional channel cost. The tight zero-distortion semantic rate Rsem(0)=PAH(πA)R_sem(0)=P_A\,H( _A) (Theorem 4.1) and the full rate–distortion decomposition (Theorem 4.2) show that redundant states are invisible to both rate and distortion under closure fidelity. The semantic source–channel separation theorem (Theorem 4.3) exhibits a semantic leverage phenomenon with leverage factor Λ∞=log|SO|/(PAH(πA))>1 _∞= |S_O|/(P_A\,H( _A))>1: under closure fidelity the required source rate drops from H(PO)H(P_O) to PAH(πA)P_A\,H( _A), enabling the same knowledge base to be communicated with proportionally fewer channel uses—not by violating the Shannon capacity (the data processing bound Csem≤C(W)C_sem≤ C(W) remains in force) but because closure-based fidelity renders redundant states “free.” The rate–delay–distortion surface (Theorem 4.6) reveals a fundamental depth-for-rate exchange that has no classical counterpart: each additional derivation step δ at the receiver renders a new stratum of states redundant, reducing the effective source entropy along the filtration Rsem(0,0)=H(PO)≥⋯≥Rsem(0,)=PAH(πA)R_sem(0,0)=H(P_O)≥·s≥ R_sem(0, D_d)=P_A\,H( _A). The critical delay δ∗δ^* below which closure-reliable communication is impossible (Corollary 4.1) is a semantic analogue of the Nyquist sampling period, and the marginal rate of delay quantifies the “value” of one derivation step in bits of channel capacity. The hard-budget staircase can be relaxed to an expected-budget model via time-sharing among transmitted bases (Remark 4.10), yielding a convex rate–computation tradeoff whose Lagrangian multiplier prices one unit of receiver inference in bits of communication rate—an exchange rate absent from classical information theory, where the decoder’s computation is treated as a free resource. The strengthened semantic Fano inequality (Theorem 4.4) provides a converse bound in which both the reference level and the penalty term are improved: the former drops from H(PO)H(P_O) to PAH(πA)P_A\,H( _A), absorbing all redundant-state entropy, and the latter involves log|A| |A| rather than log|SO| |S_O|. Combined with the semantic source–channel separation theorem, this characterizes the operational regime where semantic compression strictly outperforms symbol-level compression. Six families of computable semantic channel invariants (Theorem 3.3) provide a multi-scale fingerprint of channel quality: from set-level fidelity metrics that depend only on the knowledge-base pair, through noise-pair probabilistic indices that capture core preservation and hallucination probabilities, to information-theoretic quantities that bound achievable throughput. In the heterogeneous multi-agent setting, the overlap decomposition (Definition 5.4) translates knowledge-base structure into two binary feasibility tests—no core loss (A−ij=∅A_-^ij= ) and no non-derivable surplus (S+,nij=∅S_+,n^ij= )—that fully determine whether perfect closure fidelity is achievable (Proposition 5.3). The broadcast extension reveals a semantic bottleneck: a receiver whose vocabulary does not cover the sender’s irredundant core cannot achieve Cn=1 F_Cn=1 regardless of carrier channel quality, blocklength, or coding strategy (Proposition 5.9). Resolving this bottleneck requires a design-time vocabulary augmentation, fundamentally different from the encoder/decoder optimization that suffices classically. Relation to Prior Work The framework complements several existing lines of research. The synonymous-mapping theory of Niu and Zhang [25, 38] achieves a semantic leverage effect through source-side equivalence-class collapsing (their “Cs≥C_s≥ C” result quantifies semantic throughput gains under a synonymous-mapping fidelity criterion; it does not violate the Shannon capacity upper bound Csem≤C(W)C_sem≤ C(W), which remains valid by data processing); our framework achieves a complementary gain through receiver-side deductive reconstruction. A unified theory combining both mechanisms—synonymous collapsing of the irredundant core followed by deductive expansion at the receiver—could potentially compound the two gains; establishing the precise interaction is an open problem. The semantic channel coding theorem of Ma et al. [21] and the companion tools of [14, 19] operate within the synonymous-mapping paradigm; our two-layer code mechanism and closure-based fidelity criterion handle the structured knowledge-base setting and extend naturally to heterogeneous vocabularies. On the multi-agent front, the overlap decomposition and semantic bottleneck phenomenon provide a coding-theoretic complement to the Bayesian approach of Seo et al. [28] and the modal-logic framework of Alshammari and Bennis [3]: where those works quantify inference cost and resilience conditions, our framework quantifies the minimum number of channel uses needed to overcome vocabulary mismatch. The irredundant core can also be viewed as a form of query-aware source compression related to database-theoretic notions of view materialization [22, 2]. Limitations and Future Directions The common proof system assumption (Assumption 5.1) could be relaxed to heterogeneous proof systems where each agent uses a sub-system (i)⊆ PS^(i) PS, yielding richer compression/fidelity trade-offs. The current single-letter coding theorems invite a multi-letter extension for temporally correlated knowledge-base streams, connecting the framework to ergodic source theory. The relay scenario (Remark 5.1), where intermediate inference may change the effective end-to-end capacity, requires multi-hop coding theorems built on the composition machinery of Definition A.4. Scaling core extraction and closure computation to knowledge graphs with millions of entities, and the formal connection between derivation depth Dd and Bennett’s logical depth [5], are further promising avenues. From a structural standpoint, the receiver’s inference engine plays a role analogous to decoder side information in Wyner–Ziv coding [34]: the shared proof system PS provides the decoder with a “structured codebook” (the closure operator Cn) that reduces the effective source rate without requiring explicit side-information transmission. Formalizing this analogy—e.g., by deriving a semantic Wyner–Ziv theorem in which the side information is the proof system itself—would connect the present framework to the established body of source coding with side information and may yield tighter bounds when the proof system is only partially shared. Acknowledgment During the writing and revision of this paper, I received many insightful comments from Associate Professor Rui Wang of the School of Computer Science at Shanghai Jiao Tong University and also gained much inspiration and assistance from regular academic discussions with doctoral students Yiming Wang, Chun Li, Hu Xu, Siyuan Qiu, Zeyan Li, Jiashuo Zhang, Junxuan He, and Xiao Wang. I hereby express my sincere gratitude to them. Appendix A Axiomatic Foundations This appendix provides the full axiomatic development of the information model framework summarized in Section I. A-A Logical Language and Expressible State Sets Throughout we fix a many-sorted logical language ℒ=FO(LFP)[Σ]L=FO(LFP)[ ] [15] with sorts including Obj, Time, and Carrier. Assumption A.1 (Finite ordered structures). We restrict attention to finite ℒL-structures. When descriptive-complexity claims are invoked, we work over ordered finite structures [24, 31]. Assumption A.2 (Semantic sublanguage). Fix a partition Σ=Σsem∪˙Σrep = _sem\, ∪\, _rep, where Σrep _rep contains auxiliary symbols (including any built-in order). The designated semantic sublanguage is ℒsem:=ℒ↾ΣsemL_sem:=L\! _ _sem. All notions of semantic equivalence are relative to ℒsemL_sem. Definition A.1 (Expressible state sets). Let ℜ R be a finite ℒL-structure and let (X,T)(X,T) be definable subdomains of the object and time sorts. An (object–time) state domain over (X,T)(X,T) is an ℒsemL_sem-definable binary relation S(x,t)S(x,t) with ℜ⊧∀x∀t[S(x,t)→(δX(x)∧δT(t))] R ∀ x\,∀ t\,[S(x,t)→( _X(x) _T(t))]. A set A⊆X×TA X× T is ℒsemL_sem-expressible over S if there exists an ℒsemL_sem-formula φ(x,t) (x,t) with (x,t)∈A⇔ℜ⊧S(x,t)∧φ(x,t)(x,t)∈ A R S(x,t) (x,t). We write s∈Ss∈ S as shorthand for a pair (x,t)(x,t) with S(x,t)S(x,t), and set (s):=t time(s):=t. A-B Information Model and Enabling Mechanisms Definition A.2 (State spaces). Information is modeled using object/carrier domains O,CO,C with time domains TO,TCT_O,T_C and state sets SO⊆O×TOS_O O× T_O, SC⊆C×TCS_C C× T_C. We write so=(o,τ)∈SOs_o=(o,τ)∈ S_O and sc=(c,θ)∈SCs_c=(c,θ)∈ S_C. Axiom A.1 (Time domains and precedence). Fix a finite ℒL-structure ℜ R. The state sets SO,SCS_O,S_C are ℒsemL_sem-definable. There exist ℒsemL_sem-definable linear orders ≺O _O on TOT_O, ≺C _C on TCT_C, and a cross-domain precedence ≺OC⊆TO×TC _OC T_O× T_C that is monotone with respect to ≺O _O and ≺C _C, and mildly total (every τ∈TOτ∈ T_O has some θ∈TCθ∈ T_C with τ≺OCθτ _OCθ). Axiom A.2 (State representation). There exist injective encodings encO:SO→0,1∗enc_O:S_O→\0,1\^* and encC:SC→0,1∗enc_C:S_C→\0,1\^* such that semantic-time, carrier-time, and cross-domain precedence predicates, as well as membership, are decidable from the codes. Axiom A.3 (Enabling mapping). There exists a relation Rℰ⊆O×TO×C×TCR_E O× T_O× C× T_C inducing a set-valued map ℰ:SO⇒SCE:S_O S_C satisfying: (E1) totality (ℰ(so)≠∅E(s_o)≠ for all sos_o); (E2) coverage (⋃soℰ(so)=SC _s_oE(s_o)=S_C); (E3) existence of a computable enabling selector; (E4) precedence compatibility (sc∈ℰ(so)s_c (s_o) implies O(so)≺OC(sc) time_O(s_o) _OC time_C(s_c)). Definition A.3 (Information model). An information model is ℐ=⟨O,TO,SO,C,TC,SC,Rℰ⟩I= O,T_O,S_O,C,T_C,S_C,R_E together with the temporal structure of Axiom A.1. Definition A.4 (Composition of information models). Two models ℐ1,ℐ2I_1,I_2 are composable if SC(1)=SO(2)S_C^(1)=S_O^(2). The composite enabling map is ℰ2∘1(so):=⋃sc∈ℰ1(so)ℰ2(sc)E_2 1(s_o):= _s_c _1(s_o)E_2(s_c). Proposition A.1 (Composition preserves enabling axioms). If (ℐ1,ℐ2)(I_1,I_2) is composable and each satisfies Axiom A.3, then ℰ2∘1E_2 1 satisfies (E1)–(E3). (E1): totality of ℰ1E_1 yields sc∈ℰ1(so)s_c _1(s_o); totality of ℰ2E_2 gives ℰ2(sc)≠∅E_2(s_c)≠ , so ℰ2∘1(so)⊇ℰ2(sc)≠∅E_2 1(s_o) _2(s_c)≠ . (E2): for any s′∈SC(2)s ∈ S_C^(2), coverage of ℰ2E_2 gives scs_c with s′∈ℰ2(sc)s _2(s_c); coverage of ℰ1E_1 gives sos_o with sc∈ℰ1(so)s_c _1(s_o), whence s′∈ℰ2∘1(so)s _2 1(s_o). (E3): if e1,e2e_1,e_2 are selectors, e2∘1:=e2∘e1e_2 1:=e_2 e_1 is computable and satisfies e2∘1(so)∈ℰ2(e1(so))⊆ℰ2∘1(so)e_2 1(s_o) _2(e_1(s_o)) _2 1(s_o). ∎ Remark A.1 (Associativity). Composition of enabling maps is associative: for a composable triple, ℰ3∘(2∘1)=ℰ(3∘2)∘1E_3 (2 1)=E_(3 2) 1, so iterated compositions are unambiguous. A-C Synonymous State Sets and Ideal Information Definition A.5 (ℒsemL_sem-definable coding isomorphism). A relation G12⊆S1×S2G_12 S_1× S_2 is an ℒsemL_sem-definable coding isomorphism graph if it is definable by an ℒsemL_sem-formula and induces a bijection between S1S_1 and S2S_2 (i.e., for every s1∈S1s_1∈ S_1 there is a unique s2∈S2s_2∈ S_2 with G12(s1,s2)G_12(s_1,s_2), and vice versa). Definition A.6 (Synonymous state sets). State sets S1,S2S_1,S_2 are synonymous, written S1≡ℒsemS2S_1 _L_semS_2, if there exists an ℒsemL_sem-definable coding isomorphism graph G12G_12 inducing a bijection τ12:S1→S2 _12:S_1→ S_2 that preserves and reflects the induced time precedence: s≺S1s′⇔τ12(s)≺S2τ12(s′)s _S_1s _12(s) _S_2 _12(s ). Proposition A.2 (≡ℒsem _L_sem is an equivalence relation). The relation ≡ℒsem _L_sem on ℒsemL_sem-definable state sets is reflexive, symmetric, and transitive. Reflexivity is witnessed by the identity graph G(s,s′):=[s=s′]G(s,s ):=1[s=s ], which is ℒsemL_sem-definable. Symmetry: if G12G_12 witnesses S1≡ℒsemS2S_1 _L_semS_2, then G21(s2,s1):=G12(s1,s2)G_21(s_2,s_1):=G_12(s_1,s_2) witnesses S2≡ℒsemS1S_2 _L_semS_1. Transitivity: given witnesses G12⊆S1×S2G_12 S_1× S_2 and G23⊆S2×S3G_23 S_2× S_3, define G13(s1,s3):=∃s2[G12(s1,s2)∧G23(s2,s3)]G_13(s_1,s_3):=∃\,s_2\,[G_12(s_1,s_2) G_23(s_2,s_3)]. This is ℒsemL_sem-definable (closed under quantification over its own sorts), and the uniqueness clauses of each component ensure G13G_13 is a bijection. The induced map τ13=τ23∘τ12 _13= _23 _12 preserves and reflects precedence by composition. ∎ Definition A.7 (Ideal information). An information model ℐI is ideal if SO≡ℒsemSCS_O _L_semS_C, witnessed by a coding isomorphism graph GOCG_OC with induced bijection τOC _OC, and ℰ(so)=τOC(so)E(s_o)=\ _OC(s_o)\ for every so∈SOs_o∈ S_O. Assumption A.3 (Common semantic universe). The ambient set O⊇SOS_O S_O (introduced in Section I-A) is closed under ℒsemL_sem-definable recodings: if S⊆OS _O and S′≡ℒsemSS _L_semS with S′S over the same sorts, then S′⊆OS _O. Moreover, OS_O is effectively representable (the encoding encOenc_O and membership predicate extend to all of OS_O). Assumption A.4 (Carrier representability). There exists an ℒsemL_sem-definable SO′⊆OS _O _O with SO′≡ℒsemSCS _O _L_semS_C. Definition A.8 (Noisy information). Given ℐI satisfying Assumption A.4 and an ℒsemL_sem-definable S~O⊆O S_O _O with S~O≡ℒsemSC S_O _L_semS_C (witnessed by G with bijection τOC:S~O→SC _OC: S_O→ S_C), the noisy information is ℐ~=(S~O,SC,G,τOC,τCO) I=( S_O,S_C,G, _OC, _CO) with noise pair SO−:=SO∖S~OS_O^-:=S_O S_O and SO+:=S~O∖SOS_O^+:= S_O S_O. Proposition A.3 (Existence of noisy information). Under Assumption A.4, for every information model ℐI there exists a noisy information ℐ~ I with S~O≡ℒsemSC S_O _L_semS_C. If ℐI is ideal, the perturbation is trivial. Set S~O:=SO′ S_O:=S _O from Assumption A.4. If ℐI is ideal, SO≡ℒsemSCS_O _L_semS_C, so choosing SO′:=SOS _O:=S_O gives SO−=SO+=∅S_O^-=S_O^+= . ∎ Remark A.2 (Computability of semantic invariants for noisy bases). Since S~O⊆O S_O _O is finite (in bijection with the finite set SCS_C), the invariants (ℐ~)=|Atom(S~O)| A( I)=|Atom( S_O)| and (ℐ~)=maxq∈S~ODd(q∣Atom(S~O)) D_d( I)= _q∈ S_ODd(q ( S_O)) are well-defined, finite, and computable by the same reasoning as Theorem 2.1(i). References [1] S. Abiteboul, R. Hull, and V. Vianu (1995) Foundations of databases. Addison-Wesley. Cited by: §I, Assumption 2.2, Assumption 2.3, Axiom 2.1. [2] M. Abo Khamis and H. Chen (2025) Jaguar: a primal algorithm for conjunctive query evaluation in submodular-width time. Proceedings of the ACM on Management of Data 3 (2), p. 1–21. Cited by: §I, §VI. [3] T. Alshammari and M. Bennis (2026) Logic-driven semantic communication for resilient multi-agent systems. IEEE Open Journal of the Communications Society 7, p. 620–644. Cited by: §I, §VI. [4] J. Bao, P. Basu, M. Dean, C. Partridge, A. Swami, W. Leland, and J. A. Hendler (2011) Towards a theory of semantic communication. In 2011 IEEE Network Science Workshop, p. 110–117. Cited by: §V-A. [5] C. H. Bennett (1988) Logical depth and physical complexity. The Universal Turing Machine: A Half-Century Survey, p. 227–257. Cited by: §VI. [6] R. Carnap and Y. Bar-Hillel (1952) An outline of a theory of semantic information. Technical report Technical Report Technical Report 247, Research Laboratory of Electronics, MIT. Cited by: §I. [7] S. Ceri, G. Gottlob, and L. Tanca (1989) What you always wanted to know about Datalog (and never dared to ask). IEEE Transactions on Knowledge and Data Engineering 1 (1), p. 146–166. Cited by: §I. [8] T. M. Cover and J. A. Thomas (2006) Elements of information theory. 2nd edition, John Wiley & Sons. Cited by: §I, §I-C, §I-C, §I-D, §I-D, §I-D, Definition 3.19, Remark 3.5, Theorem 3.2, §I, item (B2), §IV-A, §IV-A, §IV-B, §IV-C, §IV-D, §IV-E, §V-A, §V-A, §V-D, Remark 5.1, Remark 5.17, Remark 5.18. [9] I. Csiszár and J. Körner (2011) Information theory: coding theorems for discrete memoryless systems. Cambridge University Press. Cited by: §I, Remark 3.4, Remark 5.17. [10] E. Dantsin, T. Eiter, G. Gottlob, and A. Voronkov (2001) Complexity and expressive power of logic programming. ACM Computing Surveys (CSUR) 33 (3), p. 374–425. Cited by: §I, Assumption 2.3. [11] L. Floridi (2004) Outline of a theory of strongly semantic information. Minds and machines 14 (2), p. 197–221. Cited by: §I. [12] D. Gündüz, Z. Qin, I. E. Aguerri, H. S. Dhillon, Z. Yang, A. Yener, K. K. Wong, and C. Chae (2022) Beyond transmitting bits: context, semantics, and task-oriented communications. IEEE Journal on Selected Areas in Communications 41 (1), p. 5–41. Cited by: §I. [13] A. A. Hagberg, D. A. Schult, and P. J. Swart (2008) Exploring network structure, dynamics, and function using NetworkX. In Proceedings of the 7th Python in Science Conference (SciPy 2008), p. 11–15. External Links: Link Cited by: §V-E. [14] Y. Han, Y. Liu, Y. Sun, K. Niu, N. Ma, S. Cui, and P. Zhang (2025) Extended blahut-arimoto algorithm for semantic rate-distortion function. Entropy 27 (6), p. 651. Cited by: §I, Remark 3.5, §VI. [15] N. Immerman (1999) Descriptive complexity. Graduate Texts in Computer Science, Springer. External Links: Document, ISBN 978-0-387-98629-5 Cited by: §A-A, §I. [16] J.Xu (2024) Research and application of general information measures based on a unified model. IEEE Transactions on Computers. External Links: Document Cited by: §I. [17] A. Kolchinsky and D. H. Wolpert (2018) Semantic information, autonomous agency and non-equilibrium statistical physics. Interface focus 8 (6), p. 20180041. Cited by: §I. [18] A. Li, S. Wu, S. Meng, R. Lu, S. Sun, and Q. Zhang (2024) Toward goal-oriented semantic communications: new metrics, framework, and open challenges. IEEE Wireless Communications. Cited by: §I. [19] Z. Liang, J. Xu, K. Niu, and P. Zhang (2025) Semantic arithmetic coding using synonymous mappings. Entropy 27 (4), p. 429. Cited by: §I, §VI. [20] X. Luo, H. Chen, and Q. Guo (2022) Semantic communications: overview, open issues, and future research directions. IEEE Wireless communications 29 (1), p. 210–219. Cited by: §I, §V-A. [21] S. Ma, C. Zhang, H. Qi, H. Li, Y. Bi, G. Shi, and N. Al-Dhahir (2025) A theory for semantic channel coding with many-to-one source. IEEE Transactions on Cognitive Communications and Networking. Cited by: §I, §VI. [22] D. Marx (2013) Tractable hypergraph properties for constraint satisfaction and conjunctive queries. Journal of the ACM (JACM) 60 (6), p. 1–51. Cited by: §I, §VI. [23] K. Mu (2024) Identifying roles of formulas in inconsistency under priest’s minimally inconsistent logic of paradox. Artificial Intelligence 335, p. 104199. Cited by: §I. [24] N. Immerman (1982) Relational queries computable in polynomial time. In Proceedings of the fourteenth annual ACM symposium on Theory of computing, p. 147–152. Cited by: Assumption A.1. [25] K. Niu and P. Zhang (2024) A mathematical theory of semantic communication. Journal on Communications 45 (6), p. 7–59. Cited by: §I, Remark 4.4, §VI. [26] Z. Qin, X. Tao, J. Lu, W. Tong, and G. Y. Li (2021) Semantic communications: principles and challenges. arXiv preprint arXiv:2201.01389. Cited by: §I. [27] S. Qiu and J. Xu (2025) Research on a general state formalization method from the perspective of logic. Mathematics 13 (20), p. 3324. External Links: Document Cited by: §I. [28] H. Seo, Y. Kang, M. Bennis, and W. Choi (2023) Bayesian inverse contextual reasoning for heterogeneous semantics-native communication. IEEE Transactions on Communications 72 (2), p. 1092–1107. Cited by: §I, Remark 5.14, §VI. [29] C. E. Shannon (1948) A mathematical theory of communication. Bell System Technical Journal 27 (3), p. 379–423. Cited by: §I, §I-D, Remark 3.5, §IV-C, §V-A, §V-D. [30] G. Shi, Y. Xiao, Y. Li, and X. Xie (2021) From semantic communication to semantic-aware networking: model, architecture, and open problems. IEEE Communications Magazine 59 (8), p. 44–50. Cited by: §V-A. [31] M. Y. Vardi (1982) The complexity of relational query languages. In Proceedings of the fourteenth annual ACM symposium on Theory of computing, p. 137–146. Cited by: Assumption A.1. [32] W. Weaver (2017) Recent contributions to the mathematical theory of communication. ETC: a review of general semantics 74 (1/2), p. 136–157. Cited by: §I. [33] Y. Wu, Y. Shi, S. Ma, C. Jiang, W. Zhang, and K. B. Letaief (2024) Toward effective and interpretable semantic communications. IEEE Communications Magazine. Cited by: §I. [34] A. D. Wyner and J. Ziv (1976) The rate-distortion function for source coding with side information at the decoder. IEEE Trans. Inform. Theory 22 (1), p. 1–10. Cited by: §VI. [35] H. Xie, Z. Qin, G. Y. Li, and B. Juang (2021) Deep learning enabled semantic communication systems. IEEE transactions on signal processing 69, p. 2663–2675. Cited by: §I. [36] J. Xu, C. Liu, X. Tan, X. Zhu, A. Wu, H. Wan, W. Kong, C. Li, H. Xu, K. Kuang, and F. Wu (2025) General information metrics for improving ai model training efficiency. Artificial Intelligence Review 58, p. 289. External Links: Document Cited by: §I. [37] J. Xu, J. Tang, X. Ma, B. Xu, S. Yanli, and Q. Yongjie (2014) Objective information theory: a sextuple model and 9 kinds of metrics. In 2014 Science and information conference, p. 793–802. External Links: Document Cited by: §I. [38] P. Zhang, X. Xu, K. Niu, W. Xu, S. Han, M. Sun, C. Dong, N. Ma, and Z. Zhang (2025) Modern semantic communication and 6g intellicise network theory and technology system. Journal of Beijing University of Posts and Telecommunications. Cited by: §I, §VI.