First-Order Geometry, Spectral Compression, and Structural Compatibility under Bounded Computation

First-Order Geometry, Spectral Compression, and Structural Compatibility under Bounded Computation Changkai Li lck271828@gmail.com (Date:February,24,2026) Abstract Optimization under structural constraints is typically analyzed through projection or penalty methods, obscuring the geometric mechanism by which constraints shape admissible dynamics. We propose an operator-theoretic formulation in which computational or feasibility limitations are encoded by self-adjoint operators defining locally reachable subspaces. In this setting, the optimal first-order improvement direction emerges as a pseudoinverse-weighted gradient, revealing how constraints induce a distorted ascent geometry. We further demonstrate that effective dynamics concentrate along dominant spectral modes, yielding a principled notion of spectral compression, and establish a compatibility principle that characterizes the existence of common admissible directions across multiple objectives. The resulting framework unifies gradient projection, spectral truncation, and multi-objective feasibility within a single geometric structure. 1 Introduction Many optimization and dynamical frameworks implicitly assume unrestricted computational capacity. Under such assumptions, the gradient ∇J​(θ)∇ J(θ) directly determines the first-order admissible strategy direction. However, when computational resources are bounded, the set of admissible variations is constrained by structural limitations on accessible directions. The presence of computational constraints raises a structural question: how should first-order admissible strategy directions be characterized when only a restricted subspace of variations is available? While constrained optimization and related bounded models have been studied in various contexts, a unified geometric characterization of computationally constrained first-order structure remains incomplete. In particular, three aspects require clarification: (i) the intrinsic form of the constrained first-order admissible strategy direction, (i) the possibility of compressing this direction into a finite structural representation, and (i) the compatibility of multiple structural constraints. Related Work: Optimization on smooth manifolds has been extensively studied in the context of Riemannian optimization [1, 2]. Classical constrained first-order methods and projection-based approaches are treated in standard convex optimization literature [3]. In parallel, bounded rationality and computational constraint models have been discussed in economic and decision-theoretic contexts [5, 4]. The present work differs in that it isolates a structural first-order geometry induced by computational accessibility and studies its spectral compression and multi-constraint compatibility properties. Against this background, we now formalize the structural framework developed in this paper. The main contributions are as follows. First, Theorem 1 establishes that the unique first-order admissible strategy direction under computational constraints is given by the pseudoinverse-weighted gradient Δ​θ⋆∝HC†​(θ)​∇J​(θ). θ H_C (θ)∇ J(θ). Second, Theorem 2 shows that this direction admits a low-rank spectral approximation. The resulting rule kernel provides a compressed representation with explicit residual bounds. Third, Principle 3 introduces a structural compatibility threshold characterized by a critical coupling parameter governing the existence of a common admissible strategy tangent direction. Together, these results provide a three-layer structural decomposition of bounded computational systems: constrained first-order geometry, spectral rule compression, and multi-constraint compatibility. The remainder of the paper is organized as follows. Section 2 presents the problem statement and the minimal geometric setting. Section 3 develops the geometric structure induced by bounded computation. Section 4 establishes the first-order constrained optimal direction (Theorem 1). Section 5 proves the low-rank rule-kernel approximation result (Theorem 2). Section 6 formulates the structural compatibility threshold principle (Principle 3). Section 7 concludes with a minimal discussion summarizing the structural framework. 2 Problem Statement and Minimal Setting 2.1 Multi-Agent Strategy Structure Let A denote a finite set of agents. Each agent selects strategies from a common strategy manifold Θ , assumed to be a smooth finite-dimensional manifold endowed with a Riemannian metric. The metric induces an inner product on each tangent space Tθ​ΘT_θ . A collective strategy profile is represented by a point on the strategy manifold θ∈Θ.θ∈ . Here θ denotes the strategy itself (in local coordinates on Θ ), rather than a preference-dependent parameterization. We assume the existence of a payoff functional J:Θ→ℝ,J: , which is continuously differentiable. 2.2 Complexity Constraint and Computational Budget Each strategy profile θ incurs a complexity cost measured by a functional C:Θ→ℝ≥0,C: _≥ 0, assumed to be continuously differentiable. The functional C encodes computational cost of representing or evaluating strategies and does not modify the payoff functional J. The system operates under a fixed computational budget C​(θ)≤κ,C(θ)≤κ, where κ>0κ>0 is a constant. The feasible set is defined as Θκ:=θ∈Θ∣C​(θ)≤κ. _κ:=\θ∈ C(θ)≤κ\. 2.3 Local Feasibility Structure We consider local perturbations Δ​θ∈Tθ​Θ θ∈ T_θ such that θ+Δ​θ∈Θκ.θ+ θ∈ _κ. For sufficiently small perturbations, first-order feasibility requires ∇C​(θ)⋅Δ​θ≤0,∇ C(θ)· θ≤ 0, where the gradient is defined with respect to the Riemannian metric. This condition defines the local feasible tangent cone at θ. 2.4 Problem Formulation Given the continuously differentiable functionals J and C under the fixed budget κ, we seek to characterize the first-order structure of first-order admissible strategy directions in Θκ _κ. In particular, the problem is to determine whether there exists a canonical first-order strategy variation direction determined jointly by the gradient structure of J and the computational accessibility geometry induced by bounded computation. Throughout this paper, all first-order variations are understood as variations of strategies on the strategy manifold, rather than updates of an external parameterization. 3 Geometric Structure of Bounded Computation 3.1 Reachable Direction Structure For each strategy profile θ∈Θθ∈ , we associate a computationally reachable subspace θ⊂Tθ​Θ.V_θ⊂ T_θ . We assume: (A1) θV_θ is a linear subspace of Tθ​ΘT_θ . (A2) θV_θ is closed with respect to the Riemannian inner product. (A3) The assignment θ↦θ _θ is measurable. 3.2 Computational Geometry Operator Bounded computation does not merely restrict admissible strategy tangent directions; it induces a geometric distortion within the reachable subspace. We therefore define a constraint-induced linear operator HC​(θ):Tθ​Θ→Tθ​ΘH_C(θ):T_θ → T_θ satisfying: (B1) HC​(θ)H_C(θ) is self-adjoint. (B2) HC​(θ)H_C(θ) is positive semidefinite. (B3) Im⁡(HC​(θ))=θIm(H_C(θ))=V_θ. (B4) ker⁡(HC​(θ))=θ⟂ (H_C(θ))=V_θ . Thus HC​(θ)H_C(θ) encodes both accessibility and directional weighting induced by bounded computation. When HC​(θ)H_C(θ) reduces to the orthogonal projection onto θV_θ, the geometry is isotropic within the reachable subspace. In general, however, HC​(θ)H_C(θ) may exhibit nontrivial spectral structure. 3.3 Pseudoinverse Structure Since HC​(θ)H_C(θ) may be rank-deficient, we define its Moore–Penrose pseudoinverse HC†​(θ).H_C (θ). The operator HC†​(θ)H_C (θ) acts as the generalized inverse within the reachable subspace and vanishes on its orthogonal complement. The pair (HC​(θ),HC†​(θ))(H_C(θ),H_C (θ)) defines the first-order computational geometry at strategy profile θ. 3.4 Decomposition of Tangent Space The tangent space admits the orthogonal decomposition Tθ​Θ=θ⊕θ⟂,T_θ =V_θ _θ , with HC​(θ)H_C(θ) acting nontrivially only on θV_θ. This structure determines the admissible first-order variation geometry under bounded computation. 4 Theorem 1: First-Order Constrained Optimal Direction 4.1 Computational Effort Geometry Fix θ∈Θθ∈ . Let HC​(θ):Tθ​Θ→Tθ​ΘH_C(θ):T_θ → T_θ be the self-adjoint, positive semidefinite operator defined in Section 2, with Im⁡(HC​(θ))=θ,ker⁡(HC​(θ))=θ⟂.Im(H_C(θ))=V_θ, (H_C(θ))=V_θ . Define the computational effort functional ℰθ​(Δ​θ):=⟨Δ​θ,HC​(θ)​Δ​θ⟩θ.E_θ( θ):= θ,\,H_C(θ) θ _θ. (1) We restrict admissible first-order variations to the reachable subspace: θ:=Δ​θ∈θ|ℰθ​(Δ​θ)=1.D_θ:= \ θ _θ\; |\;E_θ( θ)=1 \. (2) 4.2 First-Order Admissible Strategy Direction Problem Assume J:Θ→ℝJ: is differentiable at θ. Let g:=∇J​(θ)g:=∇ J(θ). We consider the variational problem maxΔ​θ∈θ⟨g,Δθ⟩θ. _ θ _θ\; g, θ _θ. (3) 4.3 Main Result Theorem 1 (First-Order Constrained Optimal Direction). Fix θ∈Θθ∈ and assume J is differentiable at θ. Let HC​(θ)H_C(θ) satisfy the properties in Section 2 and let HC†​(θ)H_C (θ) denote its Moore–Penrose pseudoinverse. If HC​(θ)​g≠0H_C(θ)g≠ 0, then the maximizers of (3) are exactly the rays generated by Δ​θ⋆∝HC†​(θ)​g, θ H_C (θ)\,g, (4) normalized so that ℰθ​(Δ​θ⋆)=1E_θ( θ )=1. If HC​(θ)​g=0H_C(θ)g=0, then ⟨g,Δ​θ⟩θ=0for all ​Δ​θ∈θ, g, θ _θ=0 all θ _θ, and no admissible strategy tangent direction yields positive first-order payoff under the computational constraint. 4.4 Proof Proof. Fix θ and abbreviate HC:=HC​(θ)H_C:=H_C(θ), HC†:=HC†​(θ)H_C :=H_C (θ), g:=∇J​(θ)g:=∇ J(θ). Since θ⊂θ=Im⁡(HC)D_θ _θ=Im(H_C), we may restrict all variations to Im⁡(HC)Im(H_C). Because HCH_C is self-adjoint, the operator HC​HC†H_CH_C is the orthogonal projection onto Im⁡(HC)Im(H_C). Therefore for any Δ​θ∈θ θ _θ, ⟨g,Δ​θ⟩θ=⟨HC​HC†​g,Δ​θ⟩θ=⟨HC†​g,HC​Δ​θ⟩θ. g, θ _θ= H_CH_C g, θ _θ= H_C g,\;H_C θ _θ. (5) Define the semi-inner product ⟨u,v⟩HC:=⟨u,HC​v⟩θ. u,v _H_C:= u,\;H_Cv _θ. On Im⁡(HC)Im(H_C) this form is non-degenerate. Indeed, if v∈Im⁡(HC)v (H_C) and ⟨v,HC​v⟩θ=0 v,H_Cv _θ=0, then v∈ker⁡(HC)∩Im⁡(HC)v∈ (H_C) (H_C), which implies v=0v=0 since ker(HC)=Im(HC)⟂ (H_C)=Im(H_C) . Thus on Im⁡(HC)Im(H_C) the norm ‖v‖HC2:=⟨v,HC​v⟩θ\|v\|_H_C^2:= v,H_Cv _θ is well-defined and coincides with ℰθ​(v)E_θ(v). For Δ​θ∈θ θ _θ we have ‖Δ​θ‖HC=1\| θ\|_H_C=1. Applying Cauchy–Schwarz in ⟨⋅,⋅⟩HC ·,· _H_C, ⟨g,Δ​θ⟩θ=⟨HC†​g,Δ​θ⟩HC≤‖HC†​g‖HC​‖Δ​θ‖HC=‖HC†​g‖HC. g, θ _θ= H_C g, θ _H_C≤\|H_C g\|_H_C\,\| θ\|_H_C=\|H_C g\|_H_C. Equality holds if and only if Δ​θ θ is positively colinear with HC†​gH_C g in Im⁡(HC)Im(H_C). Therefore, if HC​g≠0H_Cg≠ 0 (equivalently HC†​g≠0H_C g≠ 0), the maximizers are exactly the rays generated by HC†​gH_C g, normalized to unit computational effort. If HC​g=0H_Cg=0, then g∈ker⁡(HC)g∈ (H_C), and since Δ​θ∈Im⁡(HC) θ (H_C), ⟨g,Δ​θ⟩θ=0 g, θ _θ=0 for all admissible strategy tangent directions, the first-order payoff variation is non-positive. ∎ 5 Theorem 2: Rule Kernel Approximation 5.1 Spectral Structure of the Computational Operator Fix θ∈Θθ∈ . Since HC​(θ)H_C(θ) is self-adjoint and positive semidefinite, it admits the spectral decomposition HC​(θ)=∑i=1rλi​ui⊗uiT,H_C(θ)= _i=1^r _i\,u_i u_i^T, where: • λ1≥λ2≥⋯≥λr>0 _1≥ _2≥…≥ _r>0 are the positive eigenvalues, • u1,…,ur\u_1,…,u_r\ form an orthonormal basis of Im⁡(HC​(θ))Im(H_C(θ)), • ker⁡(HC​(θ)) (H_C(θ)) corresponds to eigenvalue 0. The Moore–Penrose pseudoinverse is therefore HC†​(θ)=∑i=1r1λi​ui⊗uiT.H_C (θ)= _i=1^r 1 _i\,u_i u_i^T. 5.2 Definition of the Rule Kernel For any k<rk<r, define the rank-k truncated operator HC,k†​(θ):=∑i=1k1λi​ui⊗uiT.H_C,k (θ):= _i=1^k 1 _i\,u_i u_i^T. We refer to HC,k†​(θ)H_C,k (θ) as the rank-k rule kernel associated with HC​(θ)H_C(θ). 5.3 Main Result Theorem 2 (Low-Rank Approximation of First-Order Direction). Let g=∇J​(θ)g=∇ J(θ). The first-order constrained optimal direction Δ​θ⋆=HC†​(θ)​g θ =H_C (θ)\,g admits the decomposition Δ​θ⋆=HC,k†​(θ)​g+Rk​(g), θ =H_C,k (θ)\,g+R_k(g), where the residual satisfies ‖Rk​(g)‖2=∑i=k+1r1λi2​|⟨g,ui⟩|2.\|R_k(g)\|^2= _i=k+1^r 1 _i^2| g,u_i |^2. (6) In particular, the operator-norm error satisfies ‖HC†​(θ)−HC,k†​(θ)‖op=1λk+1.\|H_C (θ)-H_C,k (θ)\|_op= 1 _k+1. 5.4 Proof Proof. Fix θ and abbreviate HC:=HC​(θ)H_C:=H_C(θ). Let g=∇J​(θ)g=∇ J(θ). Using the spectral representation of HC†H_C , HC†​g=∑i=1r1λi​⟨g,ui⟩​ui.H_C g= _i=1^r 1 _i g,u_i u_i. Similarly, HC,k†​g=∑i=1k1λi​⟨g,ui⟩​ui.H_C,k g= _i=1^k 1 _i g,u_i u_i. Therefore the residual equals Rk​(g)=∑i=k+1r1λi​⟨g,ui⟩​ui,R_k(g)= _i=k+1^r 1 _i g,u_i u_i, and its squared norm is ‖Rk​(g)‖2=∑i=k+1r1λi2​|⟨g,ui⟩|2.\|R_k(g)\|^2= _i=k+1^r 1 _i^2| g,u_i |^2. Since the largest singular value of HC†−HC,k†H_C -H_C,k is 1λk+1 1 _k+1, the operator-norm identity follows immediately. ∎ 6 Principle 3: Structural Compatibility Threshold 6.1 Family of Admissible Rule Cones Fix θ∈Θθ∈ . Let ii=1m\K_i\_i=1^m be a finite family of closed convex cones in θ⊂Tθ​ΘV_θ⊂ T_θ . Assume: (C1) Each iK_i is closed and convex. (C2) i⊂θK_i _θ. (C3) iK_i is nonempty. 6.2 Coupling Family Let γ≥0γ≥ 0. A coupling family is a collection of maps Lγ:θ→θ.L_γ:V_θ _θ. For any set A⊆θA _θ, we use the convention Lγ​(A):=Lγ​(x):x∈A.L_γ(A):=\L_γ(x):x∈ A\. Assume: (D1) L0=IdL_0=Id. (D2) If γ1≤γ2 _1≤ _2, then Lγ1​(i)⊆Lγ2​(i)for all ​i.L_ _1(K_i) L_ _2(K_i) all i. (D3) For each i and γ, Lγ​(i)L_γ(K_i) is a closed convex cone. Define the γ-coupled intersection: int​(γ):=⋂i=1mLγ​(i).K_int(γ):= _i=1^mL_γ(K_i). 6.3 Compatibility Functional Define Φ​(γ):=S​(int​(γ)∩), (γ):=S (K_int(γ) ), where S is the unit sphere in θV_θ and S​(⋅)S(·) is the spherical measure (with S​(∅)=0S( )=0). 6.4 Monotonicity Proposition 1. If γ1≤γ2 _1≤ _2, then Φ​(γ1)≤Φ​(γ2). ( _1)≤ ( _2). Proof. By (D2), int​(γ1)⊆int​(γ2).K_int( _1) _int( _2). Intersecting with S preserves inclusion. Since S is monotone under inclusion, the claim follows. ∎ 6.5 Compatibility Threshold Define γ∗:=infγ≥0:int​(γ)≠∅.γ^*:= \γ≥ 0:K_int(γ)≠ \. Principle 3 (Structural Compatibility Threshold). For γ<γ∗γ<γ^*, one has int​(γ)=∅K_int(γ)= . For γ>γ∗γ>γ^*, one has int​(γ)≠∅K_int(γ)≠ . Remark. If the set γ≥0:int​(γ)≠∅\γ≥ 0:K_int(γ)≠ \ is closed, then int​(γ∗)≠∅K_int(γ^*)≠ and the second statement can be strengthened to γ≥γ∗γ≥γ^*. 7 Minimal Discussion We summarize the structural results established in this paper. First, under bounded computational geometry, Theorem 1 characterizes the unique first-order admissible strategy direction as the pseudoinverse-weighted gradient Δ​θ⋆∝HC†​(θ)​∇J​(θ). θ H_C (θ)∇ J(θ). Second, Theorem 2 shows that this direction admits a low-rank spectral approximation. The rule kernel HC,k†H_C,k provides a compressed representation of the constrained first-order strategy structure, with explicit residual bounds. Third, Principle 3 introduces a structural compatibility threshold for families of admissible rule cones. The parameter γ governs the enlargement of constraint sets, and the threshold γ∗γ^* characterizes the minimal structural coupling required for the existence of a common admissible strategy tangent direction. Taken together, these results define a three-layer structural decomposition: • constrained first-order geometry, • spectral rule compression, • multi-constraint compatibility. 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