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A Mathematical Theory of Interpretation: Rational Entropy, Spectral Readout, and Confusability as a Resource
Blake Reynolds
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 94%
Last extracted: 8/26/2026, 4:57:26 AM
Summary
The paper introduces the Mathematical Theory of Interpretation (MTI), a framework treating interpretation as an observer-relative spectral measurement problem. It defines Rational Entropy to measure residual uncertainty across knowledge, utility, and medium. Key contributions include a fiber-first method-design protocol, a classification of zero-error states via pairwise confusability, and a four-condition certificate for sharp, decodable, and order-independent readout. The theory establishes a theoretical basis for constructing interpretation methods with explicit guarantees and failure modes.
Entities (9)
Relation Signals (7)
Mathematical Theory of Interpretation → isauthoredby → Blake Reynolds
confidence 99% · A Mathematical Theory of Interpretation Blake Reynolds
Blake Reynolds → affiliatedwith → Conjecture Labs
confidence 95% · Affiliation: Conjecture Labs
Mathematical Theory of Interpretation → defines → Rational Entropy
confidence 95% · On a learning-invariant Hilbert realization, Rational Entropy measures residual uncertainty across knowledge, utility, and medium.
Mathematical Theory of Interpretation → proposes → Fiber-First Protocol
confidence 93% · The construction is target-relative... In operational form, the protocol is: write the readout, characterize its fiber...
Mathematical Theory of Interpretation → treats → Confusability
confidence 92% · Pairwise confusability is equivalent to uniform atomic collapse... This reverses the usual zero-error role of confusability
Rational Entropy → measures → residual uncertainty
confidence 90% · Rational Entropy measures residual uncertainty across knowledge, utility, and medium.
Access Structure → determines → observer capabilities
confidence 88% · access, query, utility, and medium determine what an observer can select, identify, communicate, or refuse.
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Abstract
Abstract:This article presents the abridged core of \emph{A Mathematical Theory of Interpretation} (MTI), which treats interpretation as observer-relative spectral measurement under an access structure. MTI makes interpretation a method-design problem: access, query, utility, and medium determine what an observer can select, identify, communicate, or refuse. On a learning-invariant Hilbert realization, Rational Entropy measures residual uncertainty across knowledge, utility, and medium. In the finite-effective regime, we classify its zero set. Pairwise confusability is equivalent to uniform atomic collapse, while a unique utility maximum can select one atom even when other zero-cost states remain non-atomic. This reverses the usual zero-error role of confusability: agreement in at least one observer direction excludes unresolved multi-atom readings, while the joint label preserves identification. The corresponding free-design capacity is the product of all but the smallest direction budget. A four-condition certificate characterizes sharp, decodable, medium-faithful, and order-independent readout on a finite commuting code sector and returns typed obstructions when those guarantees fail. Together, these results establish MTI as a theoretical basis for constructing interpretation methods with explicit access assumptions, guarantees, and failure modes.
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- Source: https://arxiv.org/abs/2608.23892v1
- Canonical: https://arxiv.org/abs/2608.23892v1
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A Mathematical Theory of Interpretation Blake Reynolds †thanks: Blake Reynolds holds a Ph.D. from The University of Wisconsin-Madison and is the founder of Conjecture Labs. Affiliation: Conjecture Labs Email: blake@conjecturelabs.com August 2026 Abstract This article presents the abridged core of A Mathematical Theory of Interpretation (MTI), which treats interpretation as observer-relative spectral measurement under an access structure. MTI makes interpretation a method-design problem: access, query, utility, and medium determine what an observer can select, identify, communicate, or refuse. On a learning-invariant Hilbert realization, Rational Entropy measures residual uncertainty across knowledge, utility, and medium. In the finite-effective regime, we classify its zero set. Pairwise confusability is equivalent to uniform atomic collapse, while a unique utility maximum can select one atom even when other zero-cost states remain non-atomic. This reverses the usual zero-error role of confusability: agreement in at least one observer direction excludes unresolved multi-atom readings, while the joint label preserves identification. The corresponding free-design capacity is the product of all but the smallest direction budget. A four-condition certificate characterizes sharp, decodable, medium-faithful, and order-independent readout on a finite commuting code sector and returns typed obstructions when those guarantees fail. Together, these results establish MTI as a theoretical basis for constructing interpretation methods with explicit access assumptions, guarantees, and failure modes. Rational Entropy, Spectral Readout, and Confusability as a Resource Abridged Core Theory Keywords: interpretation; Rational Entropy; information geometry; spectral measurement; observer-relative access; method design; zero-error coding; mechanistic interpretability; partial identification. Relation to the dissertation and article scope. This article abridges the core theory of the published dissertation A Mathematical Theory of Interpretation [20]. It retains the access-structured setup, fiber-first method-design protocol, static Rational Entropy law, compatibility boundary, and finite coding/readout results. The dissertation remains the source of record for theorem provenance and for the two-temporal, multi-observer, full-access, aggregation-mediated, and empirical developments omitted here. The theorem statements below carry their local hypotheses; in particular, finite-capacity and zero-error results use a finite atom-separated commuting realization. Full-access GEB and aggregation-mediated GAS–GNN/ECB realizations are reserved for companion papers and interactive materials. No empirical validation claim is made in this article. 1 Introduction Interpretation usually follows a model or institution producing an output. A method identifies a feature, traces a circuit, estimates a latent state, or narrates a reason. These procedures can be useful without resolving the prior mathematical question: what object has been selected when an observer reports one reading rather than another? The difficulty persists even under complete causal access. For example, a white-box learned system can expose weights, activations, gradients, and training records while leaving the relevant semantic partition, query, and readout unspecified. On the other hand, under systems with aggregation-mediated access, the problem is stricter: an institutional or statistical map may merge latent states before the observer reaches them. The theory developed here begins from that distinction. Interpretation is an observer-relative measurement problem. Its mathematical data are not only a system state but an access structure, a query, a utility used to select among admissible readings, and a medium through which the result must be expressed. The learning mechanism supplies an invariant sector on which stable observation is posed. Rational Entropy then measures residual uncertainty in three observer directions as knowledge, utility, and medium, on the spectral outcome law induced by the query. The access structure also supplies a practical method-design rule: write the readout, characterize its fibers, name (explicitly) and validate the information that narrows those fibers, and return the strongest typed result the construction supports. Figure 1: The Meeting toy model. The same knowledge–utility–medium structure appears first inside one presenter and then as a distributed reconciliation problem. The green throat is the access/readout boundary. The diagram is a dictionary for the construction and is pedagogical, not a claim that every organization is literally one observer or is structured in an observer-complementary manner. 1.1 The Meeting toy model A finite meeting provides the smallest useful dictionary for the theory. A presenter must recommend one action from =Launch,Pilot,Hold.A=\ Launch, Pilot, Hold\. The recommendation is constrained by three maps. Knowledge records which actions are supported by the source material. Utility ranks the supported actions under the declared objective. Medium records which distinctions can be stated faithfully in the presentation and converted into an executable decision. The rule is: Choose the highest-utility action that remains supported by the accessible evidence and expressible in the operating medium; otherwise return a typed refusal. Internally, one presenter maintains all three constraints. Externally, the presentation crosses an access boundary, and the same functions may be distributed among a leader, analyst, and operator. The object entering the room is thus fixed, while the observer realization changes. The toy model makes five distinctions easy to see and offers a mental map before the operator theory begins. First, source information and observed readout are different objects. Second, support and preference are different operations. Third, a selected internal state and a communicable label need not coincide. Fourth, a failure can be typed, such that the obstruction may be inaccessible evidence, unresolved selection, or an unfaithful medium. Fifth, the same target can occupy different access regimes for different observers. The following corresponding design questions form a checklist that is converted into a fiber-first protocol in Section 2: □ Who is the observer? □ What is the observer’s position? □ What target is being claimed? □ What map mediates access to that target? □ Which query defines the measurement? □ Which utility selects among admissible outputs? □ Which medium carries the interpretation? □ Which certificate or refusal is returned? 1.2 Interpretation as spectral measurement Let a learned system be represented by a regular statistical manifold ℓ(M) (M) with Fisher–Rao metric g and Hilbert lift ℋ=L2(ℓ(M),dμg)H=L^2( (M),d _g). An observer O supplies a hard access projector POP_O, a graded weight WOW_O, a utility structure, a medium, and a context. A self-adjoint query realization Q on the observer space induces the projection-valued spectral measure EQE_Q and the outcome law μI(B)=⟨I,EQ(B)I⟩. _I(B)= I,E_Q(B)I . (1) The observer realization supplies measurable coarse-grainings πK,πU,πM _K, _U, _M of that outcome. Rational Entropy is HR(I∣O,Q)=H(ΛQ∣ℱK)+H(ΛQ∣ℱU)+H(ΛQ∣ℱM).H_R(I O,Q)=H( _Q _K)+H( _Q _U)+H( _Q _M). (2) The theory evaluates stable observation on the learning-invariant sector and separates the following claims: (i) constrained minimizers satisfy a variational stationarity condition; (i) in a finite pure-point regime, the zero set of (2) can be classified exactly and utility selects among it; (i) in the general self-adjoint regime, a full-mass selected Borel component transfers to a Hilbert-space support statement; (iv) only a selected singleton pure-point component licenses eigenvector shorthand. This separation is essential, noting that a Lagrange multiplier in the stationarity equation is not a spectral value of Q. Likewise, the implication μI(A∗)=1⇒EQ(A∗)I=I _I(A_*)=1 E_Q(A_*)I=I does not derive the set A∗A_*, but rather it translates a supplied support statement into operator language. 1.3 From interpretability tools to a certified readout Modern mechanistic-interpretability methods make learned structure legible by choosing concrete measurement components. For example, a probe defines a restricted readout, a sparse dictionary proposes a representation basis, an activation patch specifies an intervention, and circuit analysis proposes a candidate causal support [11, 4, 6, 3]. These are substantive contributions to basis discovery and causal localization. However, each method has already chosen some combination of representation, query, intervention, and readout before it reports an explanation, which is why recent work emphasizes formal claim standards and causal support as well as tool construction [14, 16]. MTI treats these methods as components of an access-structured observation protocol rather than as competitors to one universal interpreter. In the MTI protocol: the target must be stated before the decoder; the readout must be typed before its fibers can be characterized; the query must admit a valid realization on the accessible invariant sector; and utility and medium must be declared before a selected state becomes a communicable reading. Therefore, while basis discovery is useful, the observation law begins only after the observer, query, admissible sector, selection rule, and medium have been specified. The full-access companion program instantiates this contract computationally, while this article provides the mathematical event-level guarantee that such implementations must satisfy to produce interpretations as certified readouts. 1.4 Confusability as a resource Shannon’s theory organizes reliable transmission around probability laws, entropy, conditioning, and channel capacity [21, 8]. In zero-error communication, a code avoids confusable messages [22]. The finite interpretation problem reverses that local requirement. Suppose a finite spectral atom set A is viewed through three label maps πK:A→YK,πU:A→YU,πM:A→YM. _K:A→ Y_K, _U:A→ Y_U, _M:A→ Y_M. For every pair of atoms, agreement in at least one direction prevents that pair from forming a zero-Rational-Entropy multi-atom support. At the same time, the joint label (a)=(πK(a),πU(a),πM(a)) s(a)=( _K(a), _U(a), _M(a)) must remain injective if the selected atom is to be identified after collapse. Controlled confusability therefore supplies sharpness, while joint distinguishability supplies readout. The finite coding result is precise. With d observer directions and finite label budgets y1,…,ydy_1,…,y_d, the largest injective family in which every pair agrees in at least one coordinate has size Nd∗(y1,…,yd)=max∏j≠i1≤i≤dyj.N_d^*(y_1,…,y_d)= _1≤ i≤ d _j≠ iy_j. (3) The extremal graph theorem underlying (3) is a standard Hoffman-ratio consequence for a tensor product of complete graphs [15, 10, 13]. However, here we give it a different role, such that it becomes the free-design envelope for collapse-compatible, jointly identifiable observer labels. 1.5 Results retained in the abridged theory Table 1: Result spine of the abridged article. Result family Conclusion Regime Access-structured method design Compiles a target–readout pair into its fibers, added sources of resolution, validation obligations, and strongest licensed output. General set-theoretic Invariant-sector construction The mean-ergodic projector selects the learning-invariant sector; L-covariance makes query restriction measurement-consistent. Unitary/covariant Finite collapse classification Classifies every zero minimizer; pairwise confusability is equivalent to uniform atomicity; unique utility maximization gives selected atomicity. Finite pure point Static observation law Returns selected spectral support; eigenvector language is reserved for a selected pure-point atom. Finite derived or general conditional Compatibility boundary Commuting selected projectors remove order effects; isolated-projector order error is bounded by the query commutator. Finite dimensional Interpretive coding capacity The free-design capacity is the product of all but the smallest direction budget. Finite labels Zero-error readout certificate Conditions (C1)–(C4) are necessary and sufficient on the declared finite code window; failures are typed as (O1)–(O4). Finite, separated, commuting Perturbation certificate Spectral and label margins preserve a realized code under bounded operator error. Finite empirical approximation The result families answer three different methodology questions. (1) The fiber criterion determines whether the readout identifies the target. (2) The static observation law determines whether the declared query and observer select supported spectral structure. (3) The zero-error certificate determines whether that selected structure survives query order and medium encoding as one decodable readout. Therefore, a method can stop at any earlier layer and still return a useful set, conditional distribution, sensitivity surface, or typed refusal. The method simply cannot borrow the guarantee of a later layer. 1.6 Article structure Section 2 develops the fiber-first protocol and access data. Sections 3 and 4 give the invariant spectral setup and static observation law. Section 5 isolates the order boundary required by the finite certificate. Sections 6 and 7 prove the finite capacity and readout results. The appendices collect the dependency ledgers, expanded proofs, and notation. The main text retains proof maps only where they expose the logical seam between results. 2 Access-structured method design The access structure is both a theorem boundary and a method-design grammar. It specifies where the observed object lives, which latent distinctions survive the readout, and which additional assumptions are responsible for any sharper claim. The construction is target-relative: the same observed system can provide full access to one functional and aggregation-mediated access to another. In operational form, the protocol is: write the readout, characterize its fiber, name and validate each added source of resolution, and return a typed result. 2.1 Readout fibers and identification Let X be a latent state space, Y an observed space, and R:→R:X a readout map. For y∈R()y∈ R(X), define the readout fiber ℱR(y):=R−1(y).F_R(y):=R^-1(y). (4) Let φ:→ :X be the target of inference or interpretation. Proposition 2.1 (Fiber criterion). The target φ is identified from R on a subset D⊆D if and only if φ is constant on every nonempty fiber ℱR(y)∩DF_R(y)∩ D. Equivalently, there exists a unique map φ~:R(D)→ :R(D) satisfying φ|D=φ~∘R|D |_D= R|_D (5) if and only if R(x)=R(x′)R(x)=R(x ) with x,x′∈Dx,x ∈ D implies φ(x)=φ(x′) (x)= (x ). Proof map. The forward implication follows from factorization. For the converse, define φ~(y) (y) by evaluating φ at any x∈ℱR(y)∩Dx _R(y)∩ D; fiber constancy makes the definition independent of the representative. The full proof appears in Appendix B.1.1. Let A denote declared assumptions or side information and let ⊆C_A be the compatible latent class. The assumption-indexed identified set is ℐφ(y,):=φ(x):x∈ℱR(y)∩.I_ (y;A):= \ (x):x _R(y) _A \. (6) The data enter through ℱR(y)F_R(y). Added restrictions enter through C_A. A probabilistic decoder introduces a conditional law or weighting on the intersection. An auxiliary channel may further restrict or calibrate the selected section. Equation (6) keeps these sources of resolution separate. This distinction applies at both ends of the access spectrum. Under a faithful internal medium, a learned model can preserve the relevant state while leaving the appropriate query or semantic basis unresolved. Under a many-to-one institutional medium, some distinctions have already been removed. In the latter case, no change of estimator can make φ a function of R unless the target is constant on the admissible fiber or additional information changes the admissible class. 2.2 Fiber-first protocol For a target-map pair (φ,R)( ,R), the method-design protocol is: 1. Declare the medium and readout. Specify X, Y, and the map R. 2. Declare the target. Specify φ before choosing the decoder or estimator. 3. Characterize the unassisted fiber. Compute or bound φ(ℱR(y)) (F_R(y)) under support and accounting constraints alone. 4. Classify each added source of resolution. Distinguish exact side information, substantive restrictions, priors, decoders, anchor channels, and calibration bridges. 5. Return the strongest licensed object. Return a point only when φ is constant on the admissible fiber; otherwise return a set, a decoder-conditional distribution or band, a sensitivity surface, or a typed refusal. The protocol does not prescribe one estimator. It compiles the observation regime into objects that any estimator must declare. In a learned-system audit, R may be a layer, probe, dictionary, or intervention readout. In aggregate inference, R is the institutional or statistical aggregation rule. In both cases, the readout defines the fiber on which later interpretation is attempted. Any restriction, decoder, auxiliary channel, or calibration bridge used in step 4 must be validated on the object it contributes before it can narrow the output in step 5. The report then pairs the returned object with the applicable measurement, identification, or readout certificate (or with the corresponding typed refusal). 2.3 Interpretability methods as protocol components The protocol places familiar interpretability methods in a common dependency order. A sparse dictionary or representation decomposition proposes coordinates in which candidate structure can be expressed. A probe supplies a readout and a prediction task. Activation patching and circuit analysis supply interventions and candidate causal supports. Each method can improve access to the target, but none alone determines whether the target is constant on the induced fiber, whether the query has selected supported spectral structure, or whether the selected label survives the medium. These are separate obligations. This ordering is especially useful under full causal access. Complete access to weights, activations, and trajectories can make the internal state addressable while leaving the semantic map and query unresolved. In that regime, the method-design problem is not to reconstruct a latent state that has already been lost; it is to specify which state functional is being measured, what observer distinctions define the outcome, and what certificate turns the selected support into a communicable reading. Under aggregation-mediated access, the same protocol begins one step earlier because the readout map has already merged latent states. The method contribution of MTI can therefore be summarized as a four-layer dependency. The declaration layer fixes a partial order for the observation problem: the observer and access regime are stated; the medium and readout map are fixed before the readout fiber is analyzed; and the target, query, and utility rule are declared before selection. The fiber criterion then licenses an identification claim from the declared readout. The static observation law licenses a query-conditioned support claim on the accessible invariant sector. The finite certificate licenses a zero-error readout claim by requiring spectral sharpness, unique selection, medium faithfulness, and order independence on the declared code sector. Methods that supply only one component or complete only part of this dependency remain useful; however, they return the object established at that stage rather than a stronger interpretation claim made possible by MTI. 2.4 Observer and access data Let M be an ambient space of informational configurations and let Lt:M→ML_t:M→ M be a measurable learning mechanism. Fix an observation horizon or realized training regime and write ℓ(M):=Lt∗(M). (M):=L_t_*(M). (7) Assume ℓ(M) (M) is a regular statistical manifold with model family pθ\p_θ\ and Fisher–Rao metric gij(θ)=θ[∂ilogpθ∂jlogpθ].g_ij(θ)=E_θ\! [ _i p_θ\, _j p_θ ]. (8) In a Hessian or dually flat chart, one may additionally write gij=∂i∂jΨg_ij= _i _j for a convex potential Ψ ; this local representation is not assumed globally [18, 5, 1, 2]. Let ℋ:=L2(ℓ(M),dμg)H:=L^2( (M),d _g) (9) be the Hilbert lift. Definition 2.2 (Observer). An observer is a tuple O=(ℓ(O),mO,UO,cO).O=( (O),m_O,U_O,c_O). (10) Here ℓ(O)⊆ℓ(M) (O) (M) is the observer’s learned state, mOm_O is the readout medium, and UOU_O is an observer-relative utility structure. The context cOc_O selects the active admissible query class. Let κO:ℓ(M)→[0,1] _O: (M)→[0,1] be a measurable accessibility kernel. Define the hard admissible cone and its orthogonal support projector by O=x∈ℓ(M):κO(x)>0,(POf)(x)=O(x)f(x),C_O=\x∈ (M): _O(x)>0\, (P_Of)(x)=1_C_O(x)f(x), (11) and define the soft accessibility weight (WOf)(x)=κO(x)f(x).(W_Of)(x)= _O(x)f(x). (12) The observer space is ℋO=POℋ=L2(O,dμg).H_O=P_OH=L^2(C_O,d _g). (13) Hard support and graded accessibility are not interchangeable. POP_O decides which states are admissible. WOW_O changes weighting or quality inside that domain but cannot create support where PO=0P_O=0. Definition 2.3 (Access regime). An access regime is the specialization of POP_O, WOW_O, and any readout map through which the observer receives the system. The canonical cases are: Regime Signature Consequence Full hard access PO=IP_O=I on the target domain The full admissible sector is measurable. Partial hard access 0≠PO≠I0≠ P_O≠ I Observation is posed on a proper compressed domain. No hard access PO=0P_O=0 on the target region No nonzero normalized observed state is supported there. Graded access WOW_O nontrivial on RanPORanP_O Resolvability changes without enlarging hard support. Aggregation-mediated observation factors through R:→R:X The primary observed law lives on the codomain of R; latent pullback requires additional structure. Proposition 2.4 (Access specialization). The static observation problem is posed on POℋP_OH. If PO=0P_O=0 on a target region, no normalized observed state can be supported there. The soft operator WOW_O changes weighting only inside POℋP_OH. If observation factors through R, then the observer’s primary query and outcome law are defined on the observed space; a latent-space interpretation requires a justified pullback. The proof is a direct consequence of POP_O being multiplication by an indicator and of the type of the observed object. It is given in Appendix B.1.2. 2.5 Queries and primitive axioms Definition 2.5 (Admissible query realization). An admissible query realization is a tuple (q,Λ(q),Dq,QO,q),(q, (q),D_q,Q_O,q), (14) where Λ(q) (q) is a symmetric pre-operator or closed quadratic form on ℋH, DqD_q is its domain, and QO,qQ_O,q is a specified self-adjoint operator on ℋOH_O. In the bounded case, one may take QO,q=POΛ(q)PO|ℋO.Q_O,q=P_O (q)P_O|_H_O. (15) In unbounded or nonreducing cases, the self-adjoint realization is part of the query data. The article uses four primitive axioms. Axiom 2.6 (Admissible domain). Interpretation is defined only on OC_O. Information outside the hard cone is inadmissible for observer O. Axiom 2.7 (Observable realization). Queries are evaluated through the Hilbert lift of accessible information and require a specified self-adjoint realization on the relevant observer or invariant-sector domain. Axiom 2.8 (Learning invariance). Stable interpretation is insensitive to nuisance variation along the declared learning symmetry. It is evaluated on the L-invariant, recoverable sector. Axiom 2.9 (Variational selection). Observed interpretation is selected by minimizing Rational Entropy relative to the observer and query, followed by the declared utility and medium rules. These axioms do not assert that every query has a unique answer. They type the domain, operator, invariant restriction, and selection functional. Existence, atomicity, uniqueness, and decodability enter through separate hypotheses and theorems. Appendix A records the complete article-level dependency ledger. 3 Learning-invariant spectral setup The static theory distinguishes the state produced by learning from variation along the learning path. It does so operator-theoretically: a Koopman representation of the declared learning flow determines a fixed-point sector, and admissible queries must be covariant with that representation. This section states the minimal construction needed by the observation law. 3.1 Koopman lift and mean-ergodic projection Assume the learning mechanism induces a measurable flow Φt:ℓ(M)→ℓ(M), _t: (M)→ (M), (16) with associated Koopman operators (Utf)(x)=f(Φt(x)).(U_tf)(x)=f( _t(x)). (17) For the static regime, assume Φt _t preserves OC_O modulo null sets, is invertible and measure preserving on that domain, and induces a strongly continuous unitary representation on ℋOH_O. The invariant sector is ℋV:=Fix(U)=f∈ℋO:Utf=f for all t.H_V:=Fix(U)=\f _O:U_tf=f for all t\. (18) Proposition 3.1 (Mean-ergodic invariant projection). Under the standing unitary hypothesis, ℋVH_V is closed and the strong limit ΠHLf=limT→∞1T∫0TUtft _H^Lf= _T→∞ 1T _0^TU_tf\,dt (19) exists as the orthogonal projector from ℋOH_O onto ℋVH_V. This is the standard mean-ergodic construction for unitary representations on Hilbert space [7, 19]. We write ΠhkL=IℋO−ΠHL,ℋO=ℋV⊕ℋV⟂. _hk^L=I_H_O- _H^L, _O=H_V _V . (20) The term housekeeping refers only to the orthogonal complement of the declared learning-invariant structure. It does not imply that entropy minimization by itself removes that complement. Stable observation is restricted to ℋVH_V by Axiom 2.8; the finite theorem below shows that this restriction is variationally lossless when the invariant sector contains an admissible atom. 3.2 L-covariant queries and safe compression Definition 3.2 (L-covariant query). An admissible self-adjoint query Q on ℋOH_O is L-covariant if its spectral measure commutes with the learning representation: UtEQ(B)=EQ(B)UtU_tE_Q(B)=E_Q(B)U_t (21) for every Borel B⊆σ(Q)B σ(Q) and every t. The covariance condition ensures that the query concerns learned invariant structure rather than an arbitrary learning-path coordinate. It also supplies the operator consistency needed to restrict the query. Lemma 3.3 (Safe self-adjoint compression). Let T be self-adjoint on a Hilbert space ℋH and let P be an orthogonal projector. (i) If T is bounded, then PTP|PℋPTP|_PH is bounded self-adjoint on PℋPH. (i) If P reduces T, equivalently P commutes with every spectral projection of T, then T|Pℋ∩Dom(T)T|_PH (T) is self-adjoint on PℋPH and ET|Pℋ(B)=ET(B)|Pℋ.E_T|_PH(B)=E_T(B)|_PH. (22) (i) In the unbounded nonreducing case, a self-adjoint compression must be supplied by a closed quadratic form or another explicit realization and is part of the query data. Case (i) guarantees self-adjointness but not equality of outcome laws. For example, take ℋO=ℂ2H_O=C^2, Ut=diag(1,eit),ℋV=span(e1),Q=(0110).U_t=diag(1,e^it), _V=span(e_1), Q= pmatrix0&1\\ 1&0 pmatrix. The compression ΠHLQΠHL|ℋV=0 _H^LQ _H^L|_H_V=0 is self-adjoint. Nevertheless, the Q-law of e1e_1 is 12δ1+12δ−1 12 _1+ 12 _-1, while the compressed-law is δ0 _0. The seam is not self-adjointness but failure of reduction. Under L-covariance, each spectral projector of Q commutes with every UtU_t and hence with ΠHL _H^L; therefore ΠHL _H^L reduces Q, and the restriction QL:=Q|ℋV∩Dom(Q)Q_L:=Q|_H_V (Q) (23) is self-adjoint with the restricted spectral measure. The proof is in Appendix B.1.3. 3.3 Spectral outcome law Let QLQ_L be a self-adjoint admissible invariant-sector query with projection-valued spectral measure EQLE_Q_L. For a normalized state I∈V:=I∈ℋV:‖I‖=1I _V:=\I _V: I =1\, define μI(B)=⟨I,EQL(B)I⟩,B∈ℬ(σ(QL)). _I(B)= I,E_Q_L(B)I , B (σ(Q_L)). (24) The canonical outcome random variable is the identity map ΛO,Q(λ)=λ _O,Q(λ)=λ on the spectral probability space (σ(QL),ℬ(σ(QL)),μI).(σ(Q_L),B(σ(Q_L)), _I). Lemma 3.4 (Spectral law and support projection). For every Borel set A⊆σ(QL)A σ(Q_L), μI(A)=1⟺EQL(A)I=I. _I(A)=1 E_Q_L(A)I=I. (25) If A=λA=\λ\ is a pure-point atom, then EQL(λ)I=IE_Q_L(\λ\)I=I is equivalent to QLI=λIQ_LI=λ I. The proof uses only the identity ‖(I−EQL(A))I‖2=1−⟨I,EQL(A)I⟩. (I-E_Q_L(A))I ^2=1- I,E_Q_L(A)I . This lemma is the operator-theoretic support transfer used by the general observation law. It does not supply A. Figure 2: Exact spectral outcome law. A self-adjoint query induces a projection-valued spectral measure and hence a probability law. The observer maps that one outcome law into knowledge, utility, and medium coarse-grainings. 3.4 Observer realization on the spectral outcome Definition 3.5 (Observer realization). An observer realization of (O,Q)(O,Q) is a triple of measurable maps πK:σ(QL)→YK,πU:σ(QL)→YU,πM:σ(QL)→YM, _K:σ(Q_L)→ Y_K, _U:σ(Q_L)→ Y_U, _M:σ(Q_L)→ Y_M, (26) with generated sub-σ-algebras ℱK=σ(πK∘ΛO,Q),ℱU=σ(πU∘ΛO,Q),ℱM=σ(πM∘ΛO,Q).F_K=σ( _K _O,Q), _U=σ( _U _O,Q), _M=σ( _M _O,Q). (27) The maps represent distinctions resolved by knowledge/context, utility indifference classes, and the readout medium. The joint semantic map is O,Q(λ)=(πK(λ),πU(λ),πM(λ)). s_O,Q(λ)=( _K(λ), _U(λ), _M(λ)). (28) Two spectral values are iso-semantic when their joint labels agree. A unique spectral atom and a unique communicable label are therefore separate claims: spectral simplicity is an operator property, while medium faithfulness and joint injectivity are properties of the observer realization. On a finite pure-point window S⊆σp(QL)S _p(Q_L), call the realization pairwise confusable if every distinct λ,λ′∈Sλ,λ ∈ S agrees in at least one direction: πK(λ)=πK(λ′)orπU(λ)=πU(λ′)orπM(λ)=πM(λ′). _K(λ)= _K(λ ) _U(λ)= _U(λ ) _M(λ)= _M(λ ). (29) Pairwise confusability controls the entire zero-minimizer class. The utility rule, by contrast, may select one atom without pairwise confusability if one atom uniquely maximizes the declared utility. This distinction is the central content of Theorem 4.5. 4 Rational Entropy and the static observation law Rational Entropy is defined on the spectral probability space induced by the query. Its three terms charge unresolved outcome uncertainty separately under knowledge, utility, and medium. This per-direction redundancy is deliberate: zero cost requires the outcome to be resolved by each observer direction, not merely by their joint refinement. 4.1 Definition and zero set Definition 4.1 (Rational Entropy). Provided the conditional entropies are defined, HR(I∣O,Q)=H(ΛO,Q∣ℱK)+H(ΛO,Q∣ℱU)+H(ΛO,Q∣ℱM). H_R(I O,Q)=H( _O,Q _K)+H( _O,Q _U)+H( _O,Q _M). (30) In the finite or pure-point regime, the terms in (30) are ordinary Shannon conditional entropies [21, 8]. For a finite law μI=∑j=1Npjδλj,pj=‖EQL(λj)I‖2, _I= _j=1^Np_j _ _j, p_j= E_Q_L(\ _j\)I ^2, and a partition Bk\B_k\ induced by one observer map π, the corresponding term is H(Λ∣σ(π∘Λ))=−∑k∑λj∈Bkpjlog2(pj∑λi∈Bkpi).H( σ(π ))=- _k _ _j∈ B_kp_j _2\! ( p_j _ _i∈ B_kp_i ). (31) Each term is nonnegative. Under the standard continuity hypotheses on the induced law, HRH_R is lower semicontinuous. If the admissible class is compact in the corresponding topology, a minimizer exists. Strictly positive weights may be inserted in front of the three terms without changing the zero set. They alter the landscape away from zero but not the static minimizer class whenever a zero state is available. Proposition 4.2 (Finite zero criterion). Let μI _I have finite support S. Then HR(I∣O,Q)=0⟺πK,πU,πM are each injective on suppμI.H_R(I O,Q)=0 _K, _U, _M are each injective on _I. (32) Proof map. For one partition, conditional entropy vanishes exactly when every positive-mass cell contains one supported outcome. The three-term criterion follows by nonnegativity. Appendix B.2.1 gives the cellwise proof. In a finite pure-point sector, every probability vector on the admissible atoms is realized by a Hilbert state. If vjv_j is a unit vector in the λj _j eigenspace, then Ip=∑j=1NpjvjI_p= _j=1^N p_j\,v_j (33) has spectral law p. Thus point masses are available and the minimum value of HRH_R is zero. 4.2 Invariant restriction and utility selection L-covariance makes the spectral law invariant along the learning representation: μUtI=μI,HR(UtI∣O,Q)=HR(I∣O,Q). _U_tI= _I, H_R(U_tI O,Q)=H_R(I O,Q). (34) The full minimizer set on OS_O is therefore invariant under UtU_t. Stable observation is evaluated on V=I∈ℋV:‖I‖=1.S_V=\I _V: I =1\. (35) Proposition 4.3 (Variational losslessness of invariant restriction). Assume a finite-dimensional or atom-separated pure-point L-covariant regime and suppose QLQ_L has at least one admissible eigenvalue. Then minI∈VHR(I∣O,Q)=minI∈OHR(I∣O,Q)=0. _I _VH_R(I O,Q)= _I _OH_R(I O,Q)=0. (36) Hence the invariant-sector restriction required by Axiom 2.8 loses no variational value. Entropy minimization does not itself exclude housekeeping eigenvectors of an ambient covariant query. The architecture assigns distinct roles to the assumptions: learning invariance defines the stable domain; Rational Entropy classifies zero-cost states inside that domain; utility selects among the minimizers; and the medium determines whether the selected result is faithfully decoded. Let ℳ0=argminI∈VHR(I∣O,Q).M_0= _I _VH_R(I O,Q). (37) The utility selection rule returns a maximizer of expected query utility: IOobs∈argmaxI∈ℳ0μI[uQ∘ΛO,Q].I_O^obs∈ _I _0E_ _I[u_Q _O,Q]. (38) 4.3 Variational stationarity Theorem 4.4 (Variational stationarity on the invariant sector). Assume HR(⋅∣O,Q)H_R(· O,Q) is Fréchet differentiable on a neighborhood of VS_V and attains a minimum at IOobs∈VI_O^obs _V. Then there exists ηOnorm∈ℝ _O^norm such that ∇IHR(IOobs∣O,Q)=ηOnormIOobs. _IH_R(I_O^obs O,Q)= _O^normI_O^obs. (39) The scalar ηOnorm _O^norm is the multiplier for the normalization constraint. It is not the observed spectral readout λOobs _O^obs of QLQ_L. Equation (39) is a first-order condition. The gradient map is generally nonlinear; it becomes an ordinary eigenvalue equation only when that map is represented by a linear self-adjoint Euler–Lagrange operator. The finite collapse theorem does not require Fréchet differentiability. It follows directly from the spectral-law zero criterion and utility selection. 4.4 Finite collapse classification Theorem 4.5 (Finite collapse classification and selected readout). Assume the discrete effective regime on ℋVH_V. Let QLQ_L be self-adjoint and let S=λ1,…,λN⊆σp(QL)S=\ _1,…, _N\ _p(Q_L) (40) be a nonempty finite atom window exhaustive on the effective invariant sector: EQL(S)=IℋV.E_Q_L(S)=I_H_V. (41) Let the observer maps and utility valuation be defined on S, and let ℳ0M_0 be given by (37). Then: (i) The minimum of HRH_R on VS_V is zero, and ℳ0=I∈V:πK,πU,πM are each injective on suppμI.M_0= \I _V: _K, _U, _M are each injective on _I \. (42) (i) The following are equivalent: (a) the realization is pairwise confusable on S; (b) every state in ℳ0M_0 has singleton spectral support; (c) ℳ0=⋃λ∈SI∈RanEQL(λ):‖I‖=1.M_0= _λ∈ S\I _Q_L(\λ\): I =1\. (43) (i) If Λmax=argmaxλ∈SuQ(λ), _ = _λ∈ Su_Q(λ), (44) then the utility-selected set is ℳsel=I∈ℳ0:suppμI⊆Λmax.M_sel= \I _0: _I _ \. (45) (iv) If Λmax=λ∗ _ =\λ^*\, every selected state satisfies EQL(λ∗)IOobs=IOobs,QLIOobs=λ∗IOobs,E_Q_L(\λ^*\)I_O^obs=I_O^obs, Q_LI_O^obs=λ^*I_O^obs, (46) whether or not the realization is pairwise confusable on all of S. If, additionally, dimRanEQL(λ∗)=1 _Q_L(\λ^*\)=1, the selected interpretation is unique up to phase. Proof map. Point masses give zero cost. Proposition 4.2 yields part (i). Pairwise confusability excludes every two-atom zero state. Conversely, failure of confusability produces a two-atom superposition whose support is injective in all three directions. Expected utility is a convex average of atom utilities, so equality with the maximum occurs exactly on laws supported in Λmax _ . Appendix B.2.3 gives the full argument. The theorem distinguishes uniform atomic collapse from selected atomic readout. Pairwise confusability is necessary and sufficient for every zero minimizer to be atom-supported. It is not necessary for the final selected state to be atomic: a unique utility maximum selects one atom from the zero set even when other zero states are multi-atom. The coding theory in Section 6 studies the stronger uniform condition because capacity concerns the whole declared code window. 4.5 General support transfer and the observation law Proposition 4.6 (Selected-support transfer). Let QLQ_L be self-adjoint on ℋVH_V and let I∈VI _V. If a theorem-local selection rule or support bridge supplies a Borel set A∗⊆σ(QL)A_* σ(Q_L) satisfying μI(A∗)=1, _I(A_*)=1, (47) then EQL(A∗)I=I.E_Q_L(A_*)I=I. (48) If A∗=λ∗A_*=\ _*\ is a pure-point atom, then QLI=λ∗IQ_LI= _*I. This proposition is exactly Lemma 3.4. It translates a full-mass component into Hilbert-space support; it does not derive the component. Theorem 4.5 supplies it in the finite effective regime. Continuous-spectrum and noncommuting generalizations require additional mathematics at the point where A∗A_* is produced. Theorem 4.7 (Static observation law). Assume the invariant-sector construction and a self-adjoint admissible query QLQ_L on ℋVH_V. Let IOobs∈argminI∈VHR(I∣O,Q)I_O^obs∈ _I _VH_R(I O,Q) (49) be the state returned by the declared selection rule. Suppose either: (a) the finite hypotheses of Theorem 4.5 hold and IOobs∈ℳselI_O^obs _sel, in which case set A∗=ΛmaxA_*= _ ; or (b) a theorem-local support bridge supplies A∗A_* with μIOobs(A∗)=1 _I_O^obs(A_*)=1. Then IOobs∈argminI∈VHR(I∣O,Q)⟹EQL(A∗)IOobs=IOobs. I_O^obs∈ _I _VH_R(I O,Q) E_Q_L(A_*)I_O^obs=I_O^obs. (50) If A∗=λOobsA_*=\ _O^obs\ is a singleton pure-point atom, the output may be written QLIOobs=λOobsIOobs.Q_LI_O^obs= _O^obsI_O^obs. (51) If A∗A_* is a finite set of pure-point eigenvalues, the selected range is the finite orthogonal sum of the corresponding eigenspaces. Figure 3: Static observation law. The finite route derives the selected support from the Rational Entropy zero set and utility. The general route is conditional on a theorem-local support bridge. Output is projection-valued, with eigenvector shorthand only in the singleton pure-point case. 4.6 Discrete and perturbative consequences Corollary 4.8 (Discrete selected support). If ℋVH_V is finite dimensional, every selected spectral range is finite dimensional. If QLQ_L has compact resolvent, its spectrum is a finite or countably infinite set of isolated real eigenvalues of finite multiplicity with no finite accumulation point. In that regime, every isolated bounded selected component has finite-dimensional range. Compact resolvent does not imply that the full spectrum is finite. Proposition 4.9 (Query perturbation bound). Let Q1Q_1 and Q2Q_2 be self-adjoint on the same invariant sector with a common domain, and suppose V=Q2−Q1V=Q_2-Q_1 extends to a bounded self-adjoint operator. Let Σ1 _1 be a closed selected component of Q1Q_1 separated from the complementary spectrum by δ=dist(Σ1,σ(Q1)∖Σ1)>0.δ= ( _1,σ(Q_1) _1)>0. If ε=‖V‖<δ/4 = V <δ/4, the corresponding component Σ2 _2 of Q2Q_2 remains isolated. If EjE_j is the spectral projector onto Σj _j, then ‖E1−E2‖≤4‖Q1−Q2‖δ. E_1-E_2 ≤ 4 Q_1-Q_2 δ. (52) The estimate is a Davis–Kahan/Riesz-projector stability bound [9, 17, 23]. It certifies query dependence only while a no-crossing margin persists. When the selected gap closes, the theorem no longer permits a stable one-component correspondence. 5 Compatibility and the order boundary The static law applies to one self-adjoint query on one invariant sector. However, a finite readout certificate may involve several queries, and then order independence requires a common measurement structure. In this section, we retain only the compatibility results needed by the coding layer. It does not reproduce the dissertation’s two-temporal flow, moving Hilbert bundle, path action, or aggregate observer construction. Let A and B be finite-dimensional self-adjoint compressed queries on a common effective sector. Let PA∗P_A^* and PB∗P_B^* be the selected orthogonal spectral projectors supplied by isolated spectral windows. For an input state h, the unnormalized sequential outputs are hAB=PB∗PA∗h,hBA=PA∗PB∗h.h_AB=P_B^*P_A^*h, h_BA=P_A^*P_B^*h. (53) When nonzero, the observed rays are obtained by normalization. Proposition 5.1 (No order effects under common simple joint selection). Suppose the selected projectors arise from a common joint spectral measure and both sequential procedures land in the same one-dimensional joint spectral component. Then their normalized outputs agree up to phase. The proposition is immediate because both orders terminate in the same one-dimensional range. More generally, pairwise commuting orthogonal projectors have order-independent products for every finite subfamily. This hereditary property becomes condition (C1) of the zero-error certificate. Proposition 5.2 (Commutator-controlled order effect). Let A and B be finite-dimensional self-adjoint queries. Assume the selected projectors PA∗P_A^* and PB∗P_B^* are defined by isolated spectral windows whose Riesz contours remain a positive distance from the complementary spectra. Then there is a finite constant C, depending only on those contours and resolvent bounds, such that ‖PB∗PA∗−PA∗PB∗‖≤C‖[A,B]‖. P_B^*P_A^*-P_A^*P_B^* ≤ C [A,B] . (54) Consequently, for every normalized h, ‖PB∗PA∗h−PA∗PB∗h‖≤C‖[A,B]‖. P_B^*P_A^*h-P_A^*P_B^*h ≤ C [A,B] . (55) Proof map. Represent each selected projector by a Riesz contour integral. The commutator of the projectors is a double contour integral with one factor [A,B][A,B] between resolvents. Uniform contour-resolvent bounds yield (54). Appendix B.3 gives the calculation. Order effects occur in the smallest nontrivial example. On ℝ2R^2, let P project onto span(e1)span(e_1) and let R project onto span(u)span(u) with u=(e1+e2)/2u=(e_1+e_2)/ 2. Then RPe1=12(e1+e2),PRe1=12e1.RPe_1= 12(e_1+e_2), PRe_1= 12e_1. (56) After normalization, the two outputs are the distinct rays span(u)span(u) and span(e1)span(e_1). Thus, one-shot zero Rational Entropy at each stage does not imply protocol-level order independence. We can now cleanly state the boundary between commuting and noncommuting. In a commuting finite code sector, the queries admit a common joint spectral calculus and the selected projector family can be checked as one classical refinement. In a noncommuting protocol, candidate generation, selection, and readout may change after each query. A single static agreement graph no longer represents the entire protocol. Therefore, in this article, we use noncommutation only as a typed obstruction to the finite certificate, and we do not assign a universal capacity formula to the noncommuting regime. 6 A finite coding theory of interpretation We now have a projection-valued static law. The following coding layer uses a finite effective specialization in which selected spectral cells can be treated as atoms and observer maps have finite ranges. Outside that regime, finite graph capacities require an additional discretization or atom-separation assumption and do not follow from the general support-transfer theorem. 6.1 Finite atom model Assumption 6.1 (Finite coding regime). The compressed query QLQ_L is self-adjoint on a finite effective invariant sector; the coding window is a finite family of pairwise disjoint non-null spectral cells; every invoked gap is positive at the declared scale; the observer maps have finite ranges; and Rational Entropy is the finite conditional-entropy functional on the induced atom law. Let δ(O,Q)=Ba:a∈AA_δ(O,Q)=\B_a:a∈ A\ (57) be a finite family of non-null Borel spectral cells with pairwise separation at least δ when a gap statement is used. Write Ea=EQL(Ba),pI(a)=‖EaI‖2.E_a=E_Q_L(B_a), p_I(a)= E_aI ^2. (58) The atom-level support is suppA(I)=a:pI(a)>0 _A(I)=\a:p_I(a)>0\. The observer realization is a triple of finite maps πK:A→YK,πU:A→YU,πM:A→YM. _K:A→ Y_K, _U:A→ Y_U, _M:A→ Y_M. (59) For the A-valued random variable AIA_I with law pIp_I, HR(I∣O,Q)=H(AI∣σ(πK))+H(AI∣σ(πU))+H(AI∣σ(πM)).H_R(I O,Q)=H(A_I σ( _K))+H(A_I σ( _U))+H(A_I σ( _M)). (60) By Proposition 4.2, HR=0H_R=0 exactly when each label map is injective on suppA(I) _A(I). Definition 6.2 (Pairwise confusability, identifiability, and selectivity). A subset S⊆AS A is: (i) pairwise confusable if every distinct a,b∈Sa,b∈ S satisfies πK(a)=πK(b)orπU(a)=πU(b)orπM(a)=πM(b); _K(a)= _K(b) _U(a)= _U(b) _M(a)= _M(b); (61) (i) identifiable if the joint map (a)=(πK(a),πU(a),πM(a)) s(a)=( _K(a), _U(a), _M(a)) (62) is injective on S; (i) selectable if πU _U is injective on S. Pairwise confusability forces uniform atomicity on S, such that any support containing two atoms has a direction that lumps them and thus has positive Rational Entropy. Identification performs the opposite function after selection, in that it preserves the atom in the joint observer label. Figure 4: Interpretive-code inversion. Collapse admissibility requires every pair to agree in at least one direction; readout identification requires that no pair agree in all directions. 6.2 Agreement graphs and realized capacities Definition 6.3 (Interpretive code). An interpretive code is a subset S⊆AS A with the restricted joint map |S s|_S. It is class-admissible if it is pairwise confusable and identifiable. It is selective-admissible if it is additionally selectable. For i∈K,U,Mi∈\K,U,M\, define the agreement graph Gi=(S,Ei)G_i=(S,E_i) by a,b∈Ei⟺a≠b and πi(a)=πi(b).\a,b\∈ E_i a≠ b and _i(a)= _i(b). (63) Let G∪=(S,EK∪EU∪EM)G_∪=(S,E_K∪ E_U∪ E_M) and G∩=(S,EK∩EU∩EM)G_∩=(S,E_K∩ E_U∩ E_M). Lemma 6.4 (Graph dictionary). S is pairwise confusable if and only if it is a clique of G∪G_∪. It is identifiable if and only if it is independent in G∩G_∩. If S is selectable, GUG_U has no edges on S, so pairwise confusability must be supplied by GK∪GMG_K∪ G_M. Define the realized direction budgets yi(S)=|πi(S)|,capi(S)=log2yi(S).y_i(S)= _i(S) , _i(S)= _2y_i(S). (64) The realized class capacity is Nclass(O,Q;δ)=max|S|:S⊆Aδ(O,Q),S class-admissible,N_class(O,Q;δ)= \ S :S A_δ(O,Q),\ S class-admissible\, (65) with NselN_sel defined analogously. The free-design class capacity retains only budgets: Nclass∗(yK,yU,yM)=max|F|:F⊆YK×YU×YM is injective,and every distinct pair agrees in at least one coordinate.N^*_class(y_K,y_U,y_M)= \ F : array[]lF Y_K× Y_U× Y_M is injective,\\ and every distinct pair agrees in at least one coordinate array \. (66) Every realized code is a feasible free-design family after relabeling the used direction ranges. Hence |S|≤Nclass∗(yK(S),yU(S),yM(S)). S ≤ N^*_class(y_K(S),y_U(S),y_M(S)). (67) This is an upper envelope for realized capacity. It does not assert that arbitrary abstract labels can be realized by the access geometry, utility, medium, or spectrum of a particular observer. 6.3 Elementary regimes and Fano bound Proposition 6.5 (Elementary regimes). For positive finite budgets: (i) one direction gives N∗=1N^*=1; (i) two directions give N∗(y1,y2)=max(y1,y2)N^*(y_1,y_2)= (y_1,y_2); (i) the three-direction selective capacity is Nsel∗(yK,yU,yM)=yU.N^*_sel(y_K,y_U,y_M)=y_U. (68) The selective bound states a direct utility bottleneck, wherein an observer cannot uniquely select more readings than it can assign distinct utility labels. Knowledge and medium may supply the agreement needed for atomicity, but they cannot enlarge an injective utility coordinate. Proposition 6.6 (Interpretive Fano inequality). Let Λ be uniform on an admissible atom set S with |S|=N≥2 S =N≥ 2, and let Λ^=g((Λ)) =g( s( )) be any decoder from the joint labels. If Pe=ℙ(Λ^≠Λ)P_e=P( ≠ ), then h2(Pe)+Pelog2(N−1)≥log2N−∑i∈K,U,Mcapi(S),h_2(P_e)+P_e _2(N-1)≥ _2N- _i∈\K,U,M\cap_i(S), (69) and therefore Pe≥max0,1−capK(S)+capU(S)+capM(S)+1log2N.P_e≥ \0,1- cap_K(S)+cap_U(S)+cap_M(S)+1 _2N \. (70) The proof is the standard Fano inequality after bounding I(Λ,(Λ))I( ; s( )) by the entropy of the three finite labels. It appears in Appendix B.4.2. 6.4 Three-direction cylinder theorem Theorem 6.7 (Cylinder optimality). For all positive three-direction budgets, Nclass∗(yK,yU,yM)=maxyKyU,yKyM,yUyM.N^*_class(y_K,y_U,y_M)= \y_Ky_U,y_Ky_M,y_Uy_M\. (71) Equivalently, the free-design log-capacity is the sum of the two largest direction log-budgets. Proof map. Achievability holds by fixing a smallest-budget coordinate and enumerating the other two. For the converse, either some two-coordinate projection is injective, immediately giving a pair-product bound, or a projection collision forces the family into two crossing cylinders and yields |F|≤y1+y2+y3−2 F ≤ y_1+y_2+y_3-2, which is bounded by the largest pair product. Appendix B.4.3 gives the complete combinatorial proof. Figure 5: Free-design cylinder capacity. A smallest direction is held constant to supply pairwise confusability, while the other two directions carry the identifiable grid. Cylinders are canonical optimizers but not the only optimizers. For example, with YK=YU=YM=0,1Y_K=Y_U=Y_M=\0,1\, the parity family 000,011,101,110\000,011,101,110\ (72) has size four and every pair agrees in exactly one coordinate. It attains the cylinder value without any constant coordinate. The theorem yields the one-, two-, and three-direction sequence 1,max(y1,y2),max(yKyU,yKyM,yUyM).1, (y_1,y_2), (y_Ky_U,y_Ky_M,y_Uy_M). (73) The third direction is the first regime in which class capacity can be multiplicative in two independent observer budgets. Every realized class-admissible code satisfies the converse |S|≤maxyK(S)yU(S),yK(S)yM(S),yU(S)yM(S), S ≤ \y_K(S)y_U(S),y_K(S)y_M(S),y_U(S)y_M(S)\, (74) and every selective code satisfies |S|≤yU(S) S ≤ y_U(S). Therefore, equality is a realized-achievability question, not a consequence of the abstract envelope. 6.5 General number of directions For d≥1d≥ 1 and positive budgets y=(y1,…,yd)y=(y_1,…,y_d), let Nd∗(y)N_d^*(y) be the maximum size of an injective family F⊆Y1×⋯×Yd,|Yj|=yj,F Y_1×·s× Y_d, Y_j =y_j, such that every pair of distinct words agrees in at least one coordinate. Theorem 6.8 (General cylinder optimality). For every d≥1d≥ 1, Nd∗(y1,…,yd)=max∏j≠i1≤i≤dyj.N_d^*(y_1,…,y_d)= _1≤ i≤ d _j≠ iy_j. (75) Proof map. A cylinder gives the lower bound. For the upper bound, form the avoidance graph on the full product, connecting words that differ in every coordinate. It is the tensor product ⨂jKyj _jK_y_j. Its least eigenvalue is obtained by taking −1-1 in a smallest-budget coordinate and yj−1y_j-1 elsewhere. The Hoffman ratio bound gives exactly the largest (d−1)(d-1)-coordinate product. Appendix B.4.4 records the spectral calculation. The extremal graph calculation is classical [15, 10, 12, 13]. The interpretation-theoretic contribution is the correspondence between its independent sets and finite observer labels that simultaneously enforce pairwise lumping and joint separation. Theorem 6.9 (Sectional identity). Fix d≥2d≥ 2 and slice along coordinate 11. Let Q′=Y2×⋯×YdQ =Y_2×·s× Y_d and let G′G be the avoidance graph on Q′Q . Then Nd∗(y)=|Q′|+maxI⊆Q′I independent in G′((y1−1)|I|−|NG′(I)|),N_d^*(y)= Q + _ subarraycI Q \\ I independent in G subarray ((y_1-1) I - N_G (I) ), (76) where NG′(I)N_G (I) is the open neighborhood of I. The identity separates the gain from reusing suffixes across first-coordinate slices from the avoidance neighborhood that such reuse excludes. It is proved constructively in Appendix B.4.5. When y1y_1 is a smallest budget, Theorem 6.8 is equivalent to |NG′(I)|≥(y1−1)|I| N_G (I) ≥(y_1-1) I (77) for every independent I⊆Q′I Q . 7 From generated candidates to zero-error readout The finite capacity theorem counts admissible labels on one joint atom structure. However, a protocol can generate more candidate distinctions than it can collapse to one stable, communicable reading. The distinction between generation and readout is therefore typed as a pair rather than compressed into one scalar. 7.1 Generative and readout coordinates Definition 7.1 (Measurement protocol). A measurement protocol of length m is an ordered tuple π=(Q1,…,Qm)π=(Q_1,…,Q_m) (78) of admissible queries, where Qk+1Q_k+1 is applied to the post-readout state from QkQ_k whenever that readout exists. The protocol is commuting on a finite atom window if the relevant spectral projections commute pairwise on every cell used by the model. Let AτA_τ be a finite active candidate set. A generative coordinate is a finite map γτ:Aτ→Zτ _τ:A_τ→ Z_τ (79) through which the observer distinguishes candidates before readout. Its generative count is Gγ(O,τ,Aτ)=|γτ(Aτ)|.G_γ(O,τ;A_τ)= _τ(A_τ) . (80) The knowledge-only and full-semantic specializations are |πK(Aτ)| _K(A_τ) and |(Aτ)| s(A_τ) , respectively. Suppose a protocol-induced finite agreement structure and selection rule have been specified. Define the readout count by the realized class capacity R(O,π,δ)=Nclass(O,π,δ).R(O,π;δ)=N_class(O,π;δ). (81) If no finite agreement structure or selected readout rule has been supplied, the static coding theorem does not define R. When every admissible readout code is γ-injective, define the compatible deficit Δγ(O,π,Aτ,δ)=Gγ(O,τ,Aτ)−R(O,π,δ)≥0. _γ(O,π;A_τ,δ)=G_γ(O,τ;A_τ)-R(O,π;δ)≥ 0. (82) The safe object is the pair (Gγ,R)(G_γ,R). GγG_γ is a finite label count and does not require a joint spectral law. R uses the collapse-and-readout structure and is therefore regime-dependent. Proposition 7.2 (Fixed-support generative monotonicity). Let A be fixed and suppose γ1:A→Z1 _1:A→ Z_1 refines γ0:A→Z0 _0:A→ Z_0, so γ0=r∘γ1 _0=r _1 for some map r. Then Gγ0(O,τ,A)≤Gγ1(O,τ,A),G_ _0(O,τ;A)≤ G_ _1(O,τ;A), (83) and, for every law on A, H(A∣σ(γ1))≤H(A∣σ(γ0)).H(A σ( _1))≤ H(A σ( _0)). (84) The statement is fixed-support. Along a moving observer trajectory, support transport must be specified before monotonicity can be inferred. Refining knowledge can enlarge the generated partition without improving readout when utility does not select one candidate, the medium merges selected labels, or the query family is order-sensitive. 7.2 Code-level zero-error interpretation To start, work on a finite effective invariant sector ℋVH_V with an observer realization (πK,πU,πM)( _K, _U, _M). Let =Q1,…,QmQ=\Q_1,…,Q_m\ be a finite family of admissible L-covariant queries with selected orthogonal spectral projectors Pi∗P_i^* on a declared finite candidate sector HSH_S. Definition 7.3 (Code-level zero-error readout). A code-level zero-error interpretive readout is an observed interpretation IOobsI_O^obs satisfying: (Z1) Uniform atomic sharpness. The candidate set is a finite family of isolated non-null joint pure-point atoms with positive gaps; Rational Entropy minimization and utility selection choose one maximizing atom λ∗ _*; and the corresponding joint eigenspace is one-dimensional. (Z2) Decodable code. The medium map is injective on the declared code, so the selected atom is determined by the communicated label. (Z3) Hereditary order independence. For every nonempty query subfamily J and every two orderings σ,τσ,τ of J, ∏j∈JσPj∗|HS=∏j∈JτPj∗|HS. _j∈ J^σP_j^*|_H_S= _j∈ J^τP_j^*|_H_S. (85) Theorem 7.4 (Zero-error interpretive readout certificate). Under the standing finite code regime, a code-level zero-error interpretive readout exists if and only if the following four conditions hold: (C1) Joint measure on the code sector. The selected projectors commute pairwise on HSH_S. Equivalently, the selected family admits a joint projection-valued measure there. Strong commutation of the compressed queries is sufficient. (C2) Atomic separated code. The declared selected spectrum is a finite family of isolated non-null pure-point atoms with positive separating gaps, with no continuous component included in the code sector. (C3) Unique utility selection. On the atom-supported zero set, the utility objective has one maximizing atom λ∗ _* and the corresponding joint eigenspace is one-dimensional. (C4) Faithful medium. The medium label is injective on the declared code. When (C1)–(C4) hold, E(λ∗)IOobs=IOobs,HR(IOobs∣O,)=0,E(\ _*\)I_O^obs=I_O^obs, H_R(I_O^obs O,Q)=0, (86) IOobs∈Fix(U)I_O^obs (U), and (Z1)–(Z3) hold. For one query, (C1) is vacuous and hereditary order independence is automatic. Proof map. Commuting orthogonal projectors give order-independent products. Atomic separation and Theorem 4.5 reduce the zero set to the declared atom window. Expected utility is a convex average, and the unique maximum selects λ∗ _*. One-dimensionality fixes a ray; medium injectivity decodes it. Necessity follows by applying hereditary order independence to every two-query subfamily and unpacking the definitions of uniform sharpness and decodability. Appendix B.5 gives both directions. Pairwise confusability is not required by condition (C3) to select the final atom. It remains essential for the capacity layer because it is the exact condition that prevents larger multi-atom supports from being zero-cost readings before utility selection. Figure 6: The zero-error readout boundary. The static code sector is certified when commutation, atomic separation, unique selection, and medium faithfulness hold. Spectral, selection, and medium failures appear at the observer aperture; noncommutativity is a property of the protocol chain. Corollary 7.5 (Complete obstruction list on the finite code window). Failure of the zero-error certificate is equivalent to at least one of: (O1) Noncommutativity. Some selected projector pair fails to commute, so a two-query subprotocol is order-sensitive. (O2) Spectral-type failure. The candidate window is not a finite isolated atom-separated pure-point code; the safe output is then selected Borel support rather than a uniform singleton-atom certificate. (O3) Selection degeneracy. More than one atom maximizes utility, the selected joint eigenspace has dimension greater than one, or an order-sensitive preference protocol fails to return a maximal candidate. (O4) Medium identification failure. Distinct candidate atoms share a readout label, so communication cannot identify the selected atom within the code. No fifth obstruction occurs for the declared finite attained code-level object. On the one-shot code window itself, taking Aτ=SA_τ=S and γ=O,Q|Sγ= s_O,Q|_S makes the generated alphabet exactly the declared code alphabet, so no separate compatible-deficit obstruction is present. Other choices of active alphabet or generative coordinate are not covered by the four-obstruction equivalence. Beyond the declared regime, failure of minimizer attainment, moving support, or protocol-level nonconvergence can become additional problems. 7.3 Perturbation-stable realized codes A numerical implementation estimates QLQ_L or its spectral data. A realized code is therefore certified only relative to operator and label margins. Assumption 7.6 (Stable labelled atoms). Let QLQ_L be self-adjoint with finite atom cells Ba:a∈S\B_a:a∈ S\ and let Q Q be a self-adjoint approximation on the same finite-dimensional sector. Assume: (a) the cells are separated from one another and the remaining relevant spectrum by at least δ>0δ>0; (b) ‖Q^−QL‖≤ε<δ/4 Q-Q_L ≤ <δ/4; (c) the observer labels are constant on the 2ε2 spectral neighborhood of each BaB_a. Proposition 7.7 (Perturbation stability of a realized code). Under Assumption 7.6, every atom BaB_a has a corresponding empirical spectral cell B^a B_a with the same label triple. Every realized class-admissible or selective-admissible code therefore remains admissible in the empirical model with the same capacity count. The proof combines the no-crossing and projector stability of Proposition 4.9 with the label-margin assumption. If an estimator satisfies ℙ‖Q^−QL‖>εn≤ηn,P\ Q-Q_L > _n\≤ _n, (87) with εn<δ/4 _n<δ/4, then the realized code is capacity-stable with probability at least 1−ηn1- _n. Matrix concentration can supply exponential templates under bounded or sub-exponential hypotheses [24]; without those hypotheses, no universal rate is asserted. Figure 7: Realized capacity pipeline. The free-design envelope is an outer bound. A realized certificate additionally depends on actual spectral atoms, observer labels, separating gaps, and empirical perturbation margins. 7.4 Readout-only converse The fiber criterion has an immediate coding form. Proposition 7.8 (Readout-only capacity converse). Let S be a finite latent candidate family and let R:S→YR:S→ Y be the observed readout. Any code or decoder using only R(s)R(s) can distinguish at most |R(S)| R(S) latent classes. If R(s)=R(s′)R(s)=R(s ) for distinct candidates, no readout-only procedure can identify which of s,s′s,s occurred. A model can add a conditional section, prior, auxiliary channel, or calibration law. It can thereby produce useful and testable latent pullbacks. It does not change the fact that the distinctions absent from R(S)R(S) enter through the additional structure rather than through the aggregate readout itself. 8 Conclusion The article develops one reusable dependency chain: target and access⟶readout fiber⟶invariant query sector⟶spectral outcome law and access fiber query sector outcome law (88) ⟶Rational Entropy zero set⟶utility selection⟶finite readout certificate. Entropy zero set selection readout certificate. Herein, each arrow has a separate proof obligation. The readout fiber determines which target distinctions are supplied by observation and which enter through restrictions, decoders, auxiliary channels, or calibration. The learning representation determines which structure is stable under the declared learning symmetry. L-covariance makes the query restriction measurement-consistent. The spectral theorem supplies the outcome law. Rational Entropy measures residual uncertainty separately under knowledge, utility, and medium. The finite collapse theorem classifies the zero set and separates uniform atomicity from selected atomicity. The coding layer then counts finite label systems that preserve both collapse and readout. This chain is likewise MTI’s method-design contribution. An applied practitioner begins by declaring the target and readout rather than by choosing an estimator, characterizes what the readout leaves unresolved, records which additional information narrows the fiber, validates that information on the object it contributes, and returns the strongest licensed result. The fiber criterion supplies the identification guarantee. The static observation law supplies the query-conditioned support guarantee. Conditions (C1)–(C4) supply the finite zero-error readout guarantee. When one of those layers fails, the method returns the corresponding set, conditional object, sensitivity statement, or typed refusal instead of silently borrowing a stronger claim. Herein, the main finite inversion is exact. In zero-error communication, confusability is an obstruction. In finite interpretation, pairwise agreement in at least one observer direction excludes multi-atom zero-cost supports, while joint injectivity preserves the selected atom. The corresponding free-design capacity with d directions is the product of all but the smallest direction budget. The combinatorial theorem is classical in its graph-theoretic form, while in this theory, it quantifies the observer-label architecture required by the Rational Entropy zero criterion. Additionally, we derive a projection-valued static observation law. Here, a finite atom theorem derives the selected support from the zero set and utility. Then, a general self-adjoint theorem transfers a supplied full-mass Borel component into Hilbert-space support. Accordingly, eigenvector language is valid only for a selected pure-point atom, and unique-ray language requires simplicity or a stated gauge convention. The zero-error certificate then adds the readout boundary: an internally sharp interpretation is not yet a decodable, order-independent result. The zero-error certificate’s four obstruction classes identify whether the failure lies in commutation, spectral type, selection, or medium faithfulness, while the perturbation result states the margins required for numerical certification. The companion program develops the two assembled ends of this access spectrum without changing the present theorem chain. The forthcoming full-access GEB-Lite and GEB-Workshop work will instantiate the trace, query, Rational Entropy selection, certificate, and refusal interface for learned systems, with interactive materials supplying executable demonstrations. The GAS–GNN and ECB work will extend the dissertation’s results to further instantiate the aggregation-mediated branch through validated aggregate readouts, model-conditional pullbacks, and explicit uncertainty accounting. Those papers and open-source packages can cite this article for the event-level law and method-design protocol, while the published dissertation remains the archival source for the extended dynamic, multi-observer, and application development. This abridgment of MTI’s core theory proves the access-structured static law and finite coding/readout theory carried by the displayed chain. We note that continuous-spectrum collapse requires a theorem that produces selected support; genuinely noncommuting protocols require event- and order-dependent readout mathematics; and aggregate latent inference requires restrictions or auxiliary channels that remain visible in the returned claim, each of which is a result outside the scope of this abridgment. However, within the finite compatible window, the method shown herein is complete enough to use the mental model introduced through The Meeting to understand how the idea of interpretation is shifted from a silent source artifact to a construction where we get either one rule-faithful action or an explicit reason that the action has not yet been earned. Acknowledgments and disclosures. The underlying dissertation and related research program were supported by the author, Conjecture Labs, and Conjecture Labs investors. Some algorithmic programs referenced in this article are derived from patented or proprietary intellectual property of the author and Conjecture Labs; those references point to companion implementations and do not provide evidence for the mathematical results proved here. Frontier artificial-intelligence systems were used, under the author’s supervision and direction, to assist in preparing and editing this abridgment. The author is solely responsible for the manuscript, its mathematical claims, and any remaining errors. Appendix A Assumption and claim ledgers The tables below collect the recurring hypotheses of the abridged article. They do not replace theorem-local assumptions; they identify the dependency groups that recur across results. A.1 Assumption ledger Table 2: Assumption ledger for the abridged theory. ID Group Content controlled A1 Statistical geometry ℓ(M) (M) is a regular statistical manifold on the admissible region; the Fisher–Rao metric exists and is positive on the relevant tangent bundle; Hessian-potential notation is chart-local. A2 Hilbert realization dμgd _g is the measure used for L2(ℓ(M),dμg)L^2( (M),d _g); Riesz, compactness, and density-operator language are invoked only under their stated hypotheses. A3 Access structure POP_O is an orthogonal hard-support projector; WOW_O is a positive multiplication weight on the hard domain; soft weighting cannot enlarge hard support; aggregation-mediated observation factors through its readout map. A4 Learning representation The declared learning flow preserves the observer domain and induces a strongly continuous unitary representation when the mean-ergodic theorem is used. Semigroup or non-preserving regimes require separate hypotheses. A5 Invariant projection ΠHL _H^L exists as the mean-ergodic projector onto ℋV=Fix(U)H_V=Fix(U). Stable observation is posed on ℋVH_V by Axiom 2.8; this is a domain restriction, not an entropy consequence. A6 Self-adjoint query Every spectral statement uses a specified self-adjoint realization. Unless stated otherwise, the query is L-covariant, so ΠHL _H^L reduces it and the invariant-sector spectral measure is the restriction of the ambient one. A7 Spectral law and observer maps μI(B)=⟨I,EQ(B)I⟩ _I(B)= I,E_Q(B)I is well defined; the maps πK,πU,πM _K, _U, _M are measurable; conditional entropies are defined; finite zero criteria use finite ranges and atom separation where required. A8 Variational problem The admissible set, topology, lower semicontinuity, compactness/coercivity, and differentiability are stated locally. The unit-sphere multiplier is ηOnorm _O^norm and is not a query spectral value. A9 Collapse route In the finite regime, the zero set and selected support are derived from finite conditional entropy and utility. In the general regime, a full-mass Borel component is a theorem-local bridge before support transfer. Eigenvector shorthand requires a selected pure-point atom. A10 Spectral stability Perturbation results require a common-domain bounded self-adjoint difference, a positive selected gap, and the no-crossing condition ε<δ/4 <δ/4. Realized-code stability additionally requires label margins. A11 Compatibility Multi-query order-independence is asserted only for selected projectors admitting a common joint measure on the code sector. Commutator bounds require isolated spectral windows and controlled resolvents. A12 Finite coding Capacity theorems use a finite atom-separated model and finite observer-label ranges. Free-design capacity is an abstract envelope; realized capacity may be smaller because of geometry, spectrum, utility, or medium constraints. A13 Zero-error readout The code-level certificate requires commuting selected projectors, an isolated finite pure-point code, one utility-maximizing atom with a one-dimensional joint eigenspace, and a medium injective on the code. Hereditary order independence is required for every subprotocol. A.2 Theorem dependency map Table 3: Principal theorem dependencies. Result Assumptions Dependency note Fiber criterion Set-theoretic map data No probabilistic or spectral hypothesis. Mean-ergodic sector A2, A4–A5 Provides the stable observation domain. Safe query restriction A5–A6 Reduction preserves the spectral outcome law. Rational Entropy zero set A7–A8 Finite conditional entropy yields exact injectivity criterion. Finite collapse classification A5–A9 Uses full finite spectral-law realization, zero criterion, and utility selection. Static observation law A5–A9 Finite derived route or conditional general support route. Query/order stability A10–A11 Requires positive gaps and no spectral crossing. Cylinder capacity A12 Exact abstract label envelope; realized converse follows by relabeling. Zero-error certificate A7–A13 Adds protocol commutation, unique selection, and medium faithfulness. Empirical code stability A10, A12–A13 Exact code persists under operator and label margins. A.3 Claim ledger Table 4: Claim classes used in the abridged article. Class Use Boundary Definitions, axioms, and assumptions Constructs the object, primitive rule, or local hypothesis. Does not prove a downstream consequence without a stated result. Formal results Formal statement with explicit regime and proof route. Carries only the hypotheses written locally or imported through a cited assumption. Finite realization Specializes the projection-valued law to atoms and finite labels. Does not imply a global finite spectrum or a universal discretization. Dictionary/diagram Translates the construction into communication, measurement, or meeting language. Cannot carry proof burden. Free-design capacity Optimizes over abstract finite labels with fixed budgets. Is an outer envelope for realized observers, not an automatic achievability theorem. Numerical certificate Transfers exact finite structure through operator and label margins. Conditional on the estimator, gap, label stability, and concentration assumptions. Open boundary Continuous-spectrum selection, noncommuting protocol capacity, and dynamic return. Not inferred from the static support-transfer identity or finite agreement graph. Table 5: Major claim ledger. Claim Class Boundary Static observation law Theorem family Finite support is derived; general Borel support is conditional on a full-mass bridge. Finite collapse classification Theorem Pairwise confusability characterizes uniform atomicity; unique utility maximization characterizes selected atomicity. Finite coding capacity Theorem family Exact on abstract finite label products; realized codes obey the converse but need not attain it. Zero-error interpretive readout Finite iff certificate Complete only on the declared attained finite atom-separated code window. Compatibility boundary Proposition family Commuting selected projectors remove order effects; noncommuting protocols need additional dynamic structure. Access-structured method design Proposition/protocol Separates readout information from assumptions and decoder-supplied resolution; it is not a universal estimator. Appendix B Expanded proofs This appendix collects the proof details removed from the main text. Standard functional-analytic and spectral perturbation theorems are invoked with their native hypotheses rather than reproved from first principles. B.1 Access and operator setup B.1.1 Proof of the fiber criterion Proof of Proposition 2.1. Suppose φ|D=φ~∘R|D |_D= R|_D. If x,x′∈Dx,x ∈ D and R(x)=R(x′)R(x)=R(x ), then φ(x)=φ~(R(x))=φ~(R(x′))=φ(x′), (x)= (R(x))= (R(x ))= (x ), so φ is constant on each nonempty fiber intersected with D. Conversely, assume fiber constancy. For y∈R(D)y∈ R(D), choose any xy∈Dx_y∈ D with R(xy)=yR(x_y)=y and define φ~(y)=φ(xy). (y)= (x_y). If xy′x _y is another representative, then R(xy)=R(xy′)R(x_y)=R(x _y) and fiber constancy gives φ(xy)=φ(xy′) (x_y)= (x _y), so the definition is independent of the choice. For every x∈Dx∈ D, choosing y=R(x)y=R(x) gives φ~(R(x))=φ(x) (R(x))= (x). Uniqueness follows because any factor map must take y to the common value of φ on ℱR(y)∩DF_R(y)∩ D. ∎ B.1.2 Proof of access specialization Proof of Proposition 2.4. Since POP_O is multiplication by O1_C_O on L2(ℓ(M),dμg)L^2( (M),d _g), it is self-adjoint and idempotent. Its range consists exactly of equivalence classes supported on OC_O, and that range is closed. Thus the hard-admissible observer space is POℋP_OH. If PO=0P_O=0 on a target region, every vector supported there is mapped to zero and no unit vector can lie in the corresponding range. The soft operator WOW_O is multiplication by κO _O on the same ambient space. Wherever PO=0P_O=0, κO=0 _O=0 by definition of OC_O, so WOW_O cannot create nonzero support outside RanPORanP_O. If the observed object is y=R(x)y=R(x) or an observed law is R#μR_\#μ, then its measurable structure is defined on the codomain of R. A query on the latent domain requires an additional map or model that pulls the observed object back to X. This is a type requirement and does not follow from the existence of R alone. ∎ B.1.3 Mean-ergodic projection and safe compression Proof of Proposition 3.1. For a strongly continuous unitary representation UtU_t on a Hilbert space, the continuous-time mean ergodic theorem states that the Cesàro averages ATf=1T∫0TUtftA_Tf= 1T _0^TU_tf\,dt converge strongly to the orthogonal projection onto the fixed-point subspace. The fixed-point subspace is closed because it is the intersection of the kernels of the bounded operators Ut−IU_t-I. Hence the limit is an orthogonal projector and its range is Fix(U)Fix(U) [7, 19]. ∎ Proof of Lemma 3.3. For (i), if T is bounded and self-adjoint, then PTPPTP is bounded and (PTP)∗=PT∗P=PTP.(PTP)^*=PT^*P=PTP. It maps PℋPH into itself, so its restriction is bounded self-adjoint there. For (i), if P reduces T, both PℋPH and (I−P)ℋ(I-P)H are reducing subspaces and T=T1⊕T2T=T_1 T_2 with T1=T|Pℋ∩Dom(T)T_1=T|_PH (T) and T2=T|(I−P)ℋ∩Dom(T)T_2=T|_(I-P)H (T). A direct sum of closed symmetric restrictions is self-adjoint precisely when each summand is self-adjoint; reduction of a self-adjoint operator supplies that property. Equivalently, the functional calculus commutes with P, and for every Borel set B, ET1(B)=ET(B)|Pℋ.E_T_1(B)=E_T(B)|_PH. Part (i) records the absence of an automatic theorem. In the unbounded nonreducing case, PTPPTP may fail to be densely defined or self-adjoint on PℋPH. A closed semibounded form can define a self-adjoint operator through the representation theorem, but that realization is additional data. ∎ Covariance implies reduction. Let B be a bounded operator commuting with every UtU_t. Boundedness allows B to pass through the Bochner integral and the strong limit: BΠHLf=limT→∞1T∫0TBUtft=limT→∞1T∫0TUtBft=ΠHLBf.B _H^Lf= _T→∞ 1T _0^TBU_tf\,dt= _T→∞ 1T _0^TU_tBf\,dt= _H^LBf. For an L-covariant query, every spectral projection EQ(C)E_Q(C) is bounded and commutes with every UtU_t, hence with ΠHL _H^L. Thus ΠHL _H^L reduces Q, and Lemma 3.3(i) gives the self-adjoint restriction and restricted spectral measure. ∎ B.1.4 Spectral support transfer Proof of Lemma 3.4. Let P=EQL(A)P=E_Q_L(A). Since P is an orthogonal projector and ‖I‖=1 I =1, ‖(I−P)I‖2=⟨I,(I−P)I⟩=1−⟨I,PI⟩=1−μI(A). (I-P)I ^2= I,(I-P)I =1- I,PI =1- _I(A). Therefore μI(A)=1 _I(A)=1 if and only if (I−P)I=0(I-P)I=0, equivalently PI=IPI=I. If A=λA=\λ\ is a pure-point atom, RanEQL(λ)=ker(QL−λI)RanE_Q_L(\λ\)= (Q_L-λ I), so the projector identity is equivalent to QLI=λIQ_LI=λ I. ∎ B.2 Rational Entropy and the static theorem chain B.2.1 Zero criterion, existence, and stationarity Proof of Proposition 4.2. Fix one finite coarse-graining π with cells C\C\. The conditional entropy decomposes as H(Λ∣σ(π∘Λ))=∑CμI(C)H(μI,C),H( σ(π ))= _C _I(C)H( _I,C), where μI,C _I,C is the normalized within-cell conditional law when μI(C)>0 _I(C)>0. Every term is nonnegative. The sum is zero if and only if every positive-mass conditional law is a point mass, which is equivalent to each cell containing at most one supported outcome. This is exactly injectivity of π on suppμI _I. Applying the argument to πK,πU,πM _K, _U, _M and summing the three nonnegative terms proves the result. ∎ Existence of minimizers. Let (In)(I_n) be a minimizing sequence in a compact admissible set. A subsequence converges to some admissible I∗I_*. Lower semicontinuity gives HR(I∗∣O,Q)≤lim infnHR(In∣O,Q)=infHR,H_R(I_* O,Q)≤ _nH_R(I_n O,Q)= H_R, so I∗I_* is a minimizer. ∎ Proof of Theorem 4.4. Treat the complex Hilbert space as a real Hilbert space with inner product Re⟨⋅,⋅⟩Re ·,· . The tangent space of the unit sphere at I is TIV=h:Re⟨I,h⟩=0.T_IS_V=\h:Re I,h =0\. For every C1C^1 curve I(s)I(s) in VS_V with I(0)=IOobsI(0)=I_O^obs and I˙(0)=h∈TIOobsV I(0)=h∈ T_I_O^obsS_V, minimality implies 0=ds|s=0HR(I(s)∣O,Q)=Re⟨∇IHR(IOobs∣O,Q),h⟩.0= . dds |_s=0H_R(I(s) O,Q)=Re _IH_R(I_O^obs O,Q),h . Hence the gradient lies in the real orthogonal complement of TIOobsVT_I_O^obsS_V, which is spanℝIOobsspan_R\I_O^obs\. Therefore ∇IHR=ηOnormIOobs _IH_R= _O^normI_O^obs for a real scalar ηOnorm _O^norm. ∎ B.2.2 Invariant-sector losslessness Proof of Proposition 4.3. Under L-covariance, ΠHL _H^L reduces the query. Let λ be an admissible eigenvalue of QLQ_L and choose a unit vector v∈RanEQL(λ)⊆ℋVv _Q_L(\λ\) _V. The spectral law of v is δλ _λ. Each observer coarse-graining of a point mass is deterministic, so all three conditional entropies vanish and HR(v)=0H_R(v)=0. Since HR≥0H_R≥ 0, the minima over both VS_V and OS_O are zero, and the invariant-sector minimum is attained by v. ∎ The law invariance in (34) follows directly from covariance: μUtI(B)=⟨UtI,EQ(B)UtI⟩=⟨I,Ut∗EQ(B)UtI⟩=⟨I,EQ(B)I⟩. _U_tI(B)= U_tI,E_Q(B)U_tI = I,U_t^*E_Q(B)U_tI = I,E_Q(B)I . Every Rational Entropy term is a functional of this law through fixed observer maps, hence is invariant. B.2.3 Finite collapse classification Proof of Theorem 4.5. For every atom λj∈S _j∈ S, choose a unit eigenvector vj∈RanEQL(λj)v_j _Q_L(\ _j\). More generally, for every probability vector p=(p1,…,pN)p=(p_1,…,p_N), the state Ip=∑j=1NpjvjI_p= _j=1^N p_j\,v_j is normalized and has spectral law p, because the eigenspaces of distinct eigenvalues are orthogonal. In particular, every point mass is realized and has Rational Entropy zero. Proposition 4.2 therefore implies that the minimum is zero and gives the exact zero-set description in part (i). Assume pairwise confusability. If I∈ℳ0I _0 had two distinct atoms λa,λb _a, _b in its support, at least one observer map would assign them the same label. That map would fail to be injective on the support, contradicting Proposition 4.2. Thus every zero minimizer has singleton support, and it lies on one of the eigenspheres in (43). Conversely, suppose pairwise confusability fails for distinct atoms λa,λb _a, _b. Then all three maps separate that pair. Choose orthogonal unit eigenvectors va,vbv_a,v_b and set Iab=va+vb2.I_ab= v_a+v_b 2. Its spectral support is λa,λb\ _a, _b\, and each observer map is injective on that support. Proposition 4.2 gives HR(Iab)=0H_R(I_ab)=0. Hence not every minimizer is atom-supported. This proves the equivalences in part (i). For every I∈ℳ0I _0, expected utility is μI[uQ∘Λ]=∑λ∈SuQ(λ)μI(λ)≤maxλ∈SuQ(λ).E_ _I[u_Q ]= _λ∈ Su_Q(λ) _I(\λ\)≤ _λ∈ Su_Q(λ). Equality holds exactly when all mass is supported on Λmax _ . Since point masses on maximizing atoms belong to ℳ0M_0, the bound is attained and part (i) follows. If Λmax=λ∗ _ =\λ^*\, every selected law is δλ∗ _λ^*. Lemma 3.4 gives EQL(λ∗)IOobs=IOobs,E_Q_L(\λ^*\)I_O^obs=I_O^obs, and the pure-point spectral theorem gives QLIOobs=λ∗IOobsQ_LI_O^obs=λ^*I_O^obs. If the eigenspace is one-dimensional, all normalized vectors in it differ only by phase. Since selection is performed on V⊆ℋV=Fix(U)S_V _V=Fix(U), the selected state is learning-invariant. ∎ B.2.4 Static observation law and consequences Proof of Proposition 4.6. Apply Lemma 3.4 to the supplied set A∗A_*. The singleton conclusion is its pure-point specialization. ∎ Proof of Theorem 4.7. In the finite route, Theorem 4.5 gives suppμIOobs⊆Λmax _I_O^obs _ , so μIOobs(A∗)=1 _I_O^obs(A_*)=1 for A∗=ΛmaxA_*= _ . In the general route, the same full-mass statement is assumed theorem-locally. Proposition 4.6 gives EQL(A∗)IOobs=IOobsE_Q_L(A_*)I_O^obs=I_O^obs in either case. If A∗A_* is a singleton pure-point atom, the range is the corresponding eigenspace and (51) follows. ∎ Proof of Corollary 4.8. In finite dimension, every projector has finite-dimensional range. If QLQ_L has compact resolvent, the self-adjoint spectral theorem gives isolated real eigenvalues of finite multiplicity and no finite accumulation point. A bounded isolated component contains finitely many eigenvalues, so its spectral projector is a finite orthogonal sum of finite-dimensional eigenspaces [19, 17]. ∎ Proof of Proposition 4.9. A bounded self-adjoint perturbation of norm ε moves the spectrum by at most ε in Hausdorff distance. Because ε<δ/4 <δ/4, the spectral portion arising from Σ1 _1 remains inside the open δ/2δ/2 neighborhood of Σ1 _1, while the complementary spectral portion remains outside. Thus the corresponding component Σ2 _2 persists without crossing. The Davis–Kahan sin-Θ theorem for separated self-adjoint spectral sets gives ‖E1−E2‖≤C‖Q1−Q2‖δ. E_1-E_2 ≤ C Q_1-Q_2 δ. Under the stated no-crossing normalization one may take C=4C=4 [9, 17, 23]. ∎ B.3 Compatibility and order Proof of Proposition 5.1. Both sequential procedures terminate in the same one-dimensional joint spectral range. Any two normalized nonzero vectors in that range differ by a scalar of modulus one, so the observed rays agree. ∎ Proof of Proposition 5.2. Choose positively oriented Riesz contours ΓA _A and ΓB _B around the selected spectral windows. Then PA∗=12πi∫ΓA(zI−A)−1z,PB∗=12πi∫ΓB(wI−B)−1w.P_A^*= 12π i _ _A(zI-A)^-1\,dz, P_B^*= 12π i _ _B(wI-B)^-1\,dw. Let RA(z)=(zI−A)−1R_A(z)=(zI-A)^-1 and RB(w)=(wI−B)−1R_B(w)=(wI-B)^-1. The inverse-commutator identity gives [RA(z),RB(w)]=RA(z)RB(w)[A,B]RB(w)RA(z).[R_A(z),R_B(w)]=R_A(z)R_B(w)[A,B]R_B(w)R_A(z). Therefore [PA∗,PB∗]=1(2πi)2∫ΓA∫ΓBRA(z)RB(w)[A,B]RB(w)RA(z)wz.[P_A^*,P_B^*]= 1(2π i)^2 _ _A _ _BR_A(z)R_B(w)[A,B]R_B(w)R_A(z)\,dw\,dz. Taking norms yields ‖[PA∗,PB∗]‖≤len(ΓA)len(ΓB)(2π)2(supz∈ΓA‖RA(z)‖2)(supw∈ΓB‖RB(w)‖2)‖[A,B]‖. [P_A^*,P_B^*] ≤ len( _A)len( _B)(2π)^2 ( _z∈ _A R_A(z) ^2 ) ( _w∈ _B R_B(w) ^2 ) [A,B] . The prefactor is the finite constant C. Applying the operator to a normalized state gives (55). ∎ B.4 Finite coding proofs B.4.1 Graph dictionary and elementary regimes Proof of Lemma 6.4. A pair is adjacent in G∪G_∪ exactly when it agrees in at least one observer direction. Thus every pair in S is adjacent in G∪G_∪ exactly when S is pairwise confusable. A pair is adjacent in G∩G_∩ exactly when all three labels agree, which is exactly a collision of the joint map. Hence joint injectivity is equivalent to independence in G∩G_∩. If πU _U is injective, no distinct pair has a utility-agreement edge. ∎ Proof of Proposition 6.5. With one direction, two distinct codewords must agree to be pairwise confusable, but then they are not jointly identifiable. Thus N∗=1N^*=1. With two directions, let F⊆Y1×Y2F Y_1× Y_2 be pairwise agreeing and injective. If two words agree in coordinate 11 and differ in coordinate 22, every third word must have the same coordinate-11 value; otherwise it would have to equal both distinct coordinate-22 values to agree with both words. Hence F lies in a coordinate-11 cylinder and has size at most y2y_2. The symmetric case gives at most y1y_1. A full constant-coordinate cylinder attains max(y1,y2) (y_1,y_2). In the selective three-direction model, injectivity of the utility coordinate gives |F|≤yU F ≤ y_U. Conversely, hold the knowledge coordinate constant and choose yUy_U distinct utility labels. The family is pairwise confusable through knowledge and identifiable through utility, attaining yUy_U. ∎ B.4.2 Fano inequality Proof of Proposition 6.6. Because Λ is a function of the joint label, I(Λ,Λ^)≤I(Λ,(Λ))≤H((Λ))≤∑i∈K,U,Mlog2|πi(S)|.I( ; )≤ I( ; s( ))≤ H( s( ))≤ _i∈\K,U,M\ _2 _i(S) . Since Λ is uniform on N atoms, H(Λ∣Λ^)=log2N−I(Λ,Λ^)≥log2N−∑icapi(S).H( )= _2N-I( ; )≥ _2N- _icap_i(S). Fano’s inequality gives H(Λ∣Λ^)≤h2(Pe)+Pelog2(N−1),H( )≤ h_2(P_e)+P_e _2(N-1), which proves (69). Using h2(Pe)≤1h_2(P_e)≤ 1 and log2(N−1)≤log2N _2(N-1)≤ _2N yields (70). ∎ B.4.3 Three-direction cylinder theorem Proof of Theorem 6.7. For the lower bound, hold a smallest-budget coordinate constant and enumerate all pairs in the other two coordinates. Every pair agrees in the fixed coordinate, and the remaining pair identifies the word. This attains the largest pair product. For the upper bound, relabel coordinates as 1,2,31,2,3 and let F⊆Y1×Y2×Y3F Y_1× Y_2× Y_3 be injective and pairwise agreeing. If any two-coordinate projection is injective, |F| F is at most the corresponding budget product. Otherwise, after permuting coordinates, the (2,3)(2,3) projection has a collision: w=(a,b,c),w′=(a′,b,c),a≠a′.w=(a,b,c), w =(a ,b,c), a≠ a . No v∈Fv∈ F can satisfy both v2≠bv_2≠ b and v3≠cv_3≠ c, because to agree with both w and w′w it would then need v1=a=a′v_1=a=a . Hence F=R∪C,R=v:v2=b,C=v:v3=c.F=R∪ C, R=\v:v_2=b\, C=\v:v_3=c\. If F=RF=R or F=CF=C, it is a cylinder and obeys a pair-product bound. Otherwise choose r∈R∖Cr∈ R C and s∈C∖Rs∈ C R. They differ in coordinates 22 and 33, so they agree in coordinate 11. Comparing every element across the two sides shows that all elements of R∖CR C and C∖RC R share one first-coordinate value. Therefore |R∩C|≤y1,|R∖C|≤y3−1,|C∖R|≤y2−1, R∩ C ≤ y_1, R C ≤ y_3-1, C R ≤ y_2-1, and |F|≤y1+y2+y3−2. F ≤ y_1+y_2+y_3-2. Sort the budgets as m≥s≥nm≥ s≥ n. Then ms−(m+s+n−2)=(m−1)(s−1)−(n−1)≥0,ms-(m+s+n-2)=(m-1)(s-1)-(n-1)≥ 0, because m≥2m≥ 2 and s≥ns≥ n unless all budgets equal one, a trivial case. Thus |F|≤ms F ≤ ms, the largest pair product. ∎ B.4.4 General cylinder theorem Proof of Theorem 6.8. Fixing coordinate i and enumerating all remaining coordinates gives a feasible family of size ∏j≠iyj _j≠ iy_j, so Nd∗(y)≥max∏j≠iyj.N_d^*(y)≥ _i _j≠ iy_j. If some yj=1y_j=1, every pair automatically agrees in coordinate j, so the full product is feasible and the formula follows. Assume every yj≥2y_j≥ 2. Let G be the avoidance graph on Y1×⋯×YdY_1×·s× Y_d, with two words adjacent when they differ in every coordinate. A feasible interpretive family is exactly an independent set of G. The graph is the tensor product G=Ky1⊗⋯⊗Kyd.G=K_y_1 ·s K_y_d. It has n=∏j=1dyj,r=∏j=1d(yj−1)n= _j=1^dy_j, r= _j=1^d(y_j-1) vertices and degree. The adjacency eigenvalues are products of choices from yj−1,−1\y_j-1,-1\. Let j0j_0 be a smallest-budget coordinate. The least eigenvalue is λmin=−∏j≠j0(yj−1). _ =- _j≠ j_0(y_j-1). Indeed, a negative product uses an odd number of −1-1 choices; its magnitude is maximized by omitting only one positive factor, and omitting a smallest factor yj0−1y_j_0-1 gives the largest magnitude. The Hoffman ratio bound for an independent set I in an r-regular graph gives |I|≤n(−λmin)r−λmin. I ≤ n(- _ )r- _ . Now r−λmin=∏j≠j0(yj−1)((yj0−1)+1)=yj0∏j≠j0(yj−1),r- _ = _j≠ j_0(y_j-1) ((y_j_0-1)+1 )=y_j_0 _j≠ j_0(y_j-1), so |I|≤nyj0=∏j≠j0yj, I ≤ ny_j_0= _j≠ j_0y_j, which is the largest (d−1)(d-1)-coordinate product. This matches the cylinder lower bound. ∎ B.4.5 Sectional identity Proof of Theorem 6.9. For a feasible family F, decompose by first-coordinate slices: Sa=q∈Q′:(a,q)∈F,a∈Y1.S_a=\q∈ Q :(a,q)∈ F\, a∈ Y_1. Words in one slice already agree in coordinate 11. Cross-slice pairs must therefore avoid adjacency in G′G . Let M⊆Q′M Q be the suffixes used in at least two slices. If two suffixes in M were adjacent in G′G , choose occurrences in different first-coordinate slices; the resulting words would differ in coordinate 11 and in every suffix coordinate, contradicting feasibility. Hence M is independent. No neighbor of M can occur in any slice. To see this, let q∈Mq∈ M occur in two distinct first-coordinate slices and let q′∈NG′(q)q ∈ N_G (q). Whatever first-coordinate label is assigned to q′q , one of the two occurrences of q has a different first coordinate; because q and q′q differ in every suffix coordinate, those two words would disagree everywhere. Every suffix outside M∪NG′(M)M∪ N_G (M) can occur in at most one slice by definition of M. Therefore |F|≤y1|M|+|Q′|−|M|−|NG′(M)|=|Q′|+(y1−1)|M|−|NG′(M)|. F ≤ y_1 M + Q - M - N_G (M) = Q +(y_1-1) M - N_G (M) . Maximizing over independent M gives the upper bound. Conversely, fix any independent I⊆Q′I Q . Place every suffix in I in all y1y_1 slices. Place every suffix in Q′∖(I∪NG′(I))Q (I∪ N_G (I)) in one chosen slice. No cross-slice all-coordinate disagreement is possible: reused suffixes form an independent set, and all other used suffixes avoid its neighborhood. The construction attains |Q′|+(y1−1)|I|−|NG′(I)|, Q +(y_1-1) I - N_G (I) , proving equality. ∎ B.5 Generative and zero-error readout proofs Proof of Proposition 7.2. If γ0=r∘γ1 _0=r _1, every fiber of γ1 _1 is contained in a fiber of γ0 _0. Hence the partition induced by γ1 _1 refines that induced by γ0 _0 and has at least as many nonempty cells, proving (83). Conditional entropy is monotone under refinement of the conditioning σ-algebra, proving (84). ∎ Proof of Theorem 7.4: sufficiency. Assume (C1)–(C4). Pairwise commuting orthogonal projectors have order-independent products for every finite subfamily, so (C1) gives hereditary order independence (Z3). By (C2), the declared sector is a finite atomic model. Every one-atom state has zero Rational Entropy. On the atom-supported zero set, expected utility is ∑λ∈S‖E(λ)I‖2uQ(λ), _λ∈ S E(\λ\)I ^2u_Q(λ), a convex combination of the atom utilities. By (C3), only λ∗ _* maximizes this quantity, and its one-dimensional joint eigenspace leaves one projective ray. Thus (Z1) holds and (86) follows. The entire construction lies in ℋVH_V, so IOobs∈Fix(U)I_O^obs (U). Condition (C4) is exactly decodability (Z2). ∎ Proof of Theorem 7.4: necessity. Assume (Z1)–(Z3). Apply (Z3) to every two-query subfamily J=i,jJ=\i,j\. Equality of the two orderings gives Pi∗Pj∗=Pj∗Pi∗on HS,P_i^*P_j^*=P_j^*P_i^* H_S, so (C1) holds. Condition (Z2) is precisely injectivity of the medium label on the code, giving (C4). Condition (Z1) states that the code window is finite, atom-separated, pure point, uniquely utility-selected, and one-dimensional at the selected atom; these are (C2) and (C3). Therefore (C1)–(C4) are necessary. ∎ Proof of Corollary 7.5. Theorem 7.4 gives an equivalence between zero-error readout and the conjunction of (C1)–(C4). Negating the conjunction gives the disjunction of the four failures. The descriptions (O1)–(O4) are exactly the negations of commutation, atomic separated type, unique one-ray selection, and medium injectivity. They may co-occur. ∎ Proof of Proposition 7.7. By Assumption 7.6, each exact atom is isolated from every other atom and the complementary spectrum by at least δ. The operator perturbation has norm less than δ/4δ/4, so Proposition 4.9 supplies a unique corresponding empirical cluster and a close Riesz projector. The 2ε2 label-margin condition prevents any empirical cluster from crossing a knowledge, utility, or medium label boundary. Thus all pairwise agreement relations, joint collisions, and utility-label relations are unchanged. Every class-admissible or selective-admissible code remains so, with the same number of codewords. ∎ Proof of Proposition 7.8. Every procedure using only the observed value factors through the finite image R(S)R(S). Hence it can return at most one distinct deterministic label for each element of R(S)R(S), so no more than |R(S)| R(S) latent classes can be distinguished. If R(s)=R(s′)R(s)=R(s ), every readout-only function has the same input for s and s′s and therefore cannot identify which occurred. This is Proposition 2.1 with D=SD=S and target equal to the latent identity. ∎ Appendix C Notation glossary Table 6: Principal notation. Symbol Meaning M Ambient space of informational configurations. LtL_t, Φt _t Learning mechanism and the induced flow on the learned realization. ℓ(M) (M) Learned information manifold fixed at the observation regime. g, dμgd _g Fisher–Rao metric and its induced measure. ℋ=L2(ℓ(M),dμg)H=L^2( (M),d _g) Information Hilbert realization. O=(ℓ(O),mO,UO,cO)O=( (O),m_O,U_O,c_O) Observer: learned state, medium, utility, and active context. κO _O Measurable accessibility kernel. OC_O Hard admissible cone x:κO(x)>0\x: _O(x)>0\. POP_O Orthogonal hard-support projector, multiplication by O1_C_O. WOW_O Soft accessibility weight, multiplication by κO _O. ℋO=POℋH_O=P_OH Observer-accessible Hilbert space. UtU_t Koopman representation of the declared learning flow on observables. ℋV=Fix(U)H_V=Fix(U) Learning-invariant sector. ΠHL _H^L Mean-ergodic projector from ℋOH_O onto ℋVH_V. ΠhkL _hk^L Orthogonal complement I−ΠHLI- _H^L; the housekeeping projector. q, Q, QLQ_L Symbolic query, specified observer-space self-adjoint realization, and learning-invariant restriction or specified compression. EQE_Q, EQLE_Q_L Projection-valued spectral measures. μI(B)=⟨I,EQL(B)I⟩ _I(B)= I,E_Q_L(B)I Query-induced spectral outcome law for state I. ΛO,Q _O,Q Canonical spectral outcome random variable. πK,πU,πM _K, _U, _M Observer coarse-grainings for knowledge, utility, and medium. ℱK,ℱU,ℱMF_K,F_U,F_M Sub-σ-algebras generated by the three coarse-grainings. O,Q s_O,Q Joint semantic label (πK,πU,πM)( _K, _U, _M). HRH_R Rational Entropy, the sum of the three conditional entropies. OS_O, VS_V Unit spheres in ℋOH_O and ℋVH_V. ηOnorm _O^norm Normalization-constraint multiplier in the variational stationarity equation. It is not a query spectral value. IOobsI_O^obs State selected by Rational Entropy minimization and the declared selection rule. λOobs _O^obs Observed spectral readout, used only for a selected pure-point value of QLQ_L. A∗A_* Selected Borel spectral component in the projection-valued observation law. Λmax _ Set of utility-maximizing atoms in a finite spectral window. R:→R:X Readout or aggregation map. ℱR(y)F_R(y) Readout fiber R−1(y)R^-1(y). φ Target functional on the latent space. ℐφ(y,)I_ (y;A) Assumption-indexed identified set on the admissible fiber. A, BaB_a, EaE_a Finite atom index set, spectral cells, and their projectors. pI(a)=‖EaI‖2p_I(a)= E_aI ^2 Finite atom law. GK,GU,GMG_K,G_U,G_M Agreement graphs for the three observer directions. G∪,G∩G_∪,G_∩ Union and intersection agreement graphs. yi(S)y_i(S) Number of realized labels in observer direction i. NclassN_class, NselN_sel Realized class and selective capacities. 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