https://osf.io/s5kgp/files/osfstorage/690f972be7ebbdb7a20c1dc3

Entropy–Signal Conjugacy: Part A A Variational and Information-Geometric Theorem with Applications to Intelligent Systems

Abstract

We formalize Signal as constrained feature expectations relative to a declared noise model, and show that a maximum-entropy (minimum–relative-entropy) principle with linear feature constraints induces an exponential family. We prove that the minimum-divergence potential over mean parameters is the Legendre–Fenchel conjugate of the log-partition over natural parameters, establishing a precise conjugate pair with matched gradients and curvatures. These identities recover Fisher information and Cramér–Rao–type bounds and lead to actionable controls for decoding budgets, memory writes, stability diagnostics, and multi-tool arbitration. Appendices provide implementation patterns and a worked micro-example.

• Signal (mean parameters).
s := E_p[ φ(X) ] ∈ R^d. (A.1)

• Minimum-divergence potential (entropy side).
Φ(s) := inf over { p with E_p[φ]=s } of D(p∥q). (A.2)

• Log-partition (natural side) and induced family.
ψ(λ) := log ∫ q(x) · exp( λ·φ(x) ) dμ(x), p_λ(x) := [ q(x) · exp( λ·φ(x) ) ] / Z(λ). (A.3)

• Conjugacy (dual potentials).
Φ(s) = sup_λ { λ·s − ψ(λ) }, ψ(λ) = sup_s { λ·s − Φ(s) }. (A.4)

• Dual coordinates (gradients invert).
s = ∇_λ ψ(λ), λ = ∇_s Φ(s). (A.5)

• Curvature, information, and bounds.
∇²_λλ ψ(λ) = Cov_{p_λ}[ φ(X) ] = I(λ), ∇²_ss Φ(s) = I(λ)^{-1}. (A.6)

• Dynamic extension (outline).
d/dt D(p_t∥q) ≤ 0; if p_t = p_{λ_t}, then d/dt Φ(s_t) = λ_t·\dot s_t − d/dt ψ(λ_t). (A.7)

Keywords: maximum entropy, exponential family, convex duality, Fisher information, Cramér–Rao bounds, decoding budgets, stability diagnostics.

1. Introduction and Contributions

Problem. Modern intelligent systems must extract structured regularities—signal—while operating under unavoidable thermodynamic and informational limits—entropy. Improving signal typically means departing further from a declared noise model, which incurs representational, computational, and physical costs.

Goal. Provide a self-contained theorem showing that signal and entropy are conjugate variables under a maximum-entropy (minimum relative-entropy) program with linear feature constraints, using only standard probability and convex duality.

Scope preview (objects we will use).

• Feature map. We declare what counts as structure via a measurable map from data space into a d-dimensional vector of features.
φ: X → R^d. (1.0)

• Signal (mean parameters). The signal carried by a model p is the vector of feature expectations under p.
s(p) := E_p[ φ(X) ] ∈ R^d. (1.1)

• Relative entropy (divergence from noise). This prices how far p moves away from a declared baseline (noise) distribution q.
D(p∥q) := ∫ p(x) · log( p(x)/q(x) ) dμ(x). (1.2)

• Minimum-divergence potential (entropy side). The least divergence required to realize a target signal vector s.
Φ(s) := inf over { p with E_p[φ]=s } of D(p∥q). (1.3)

• Log-partition (natural-parameter side). The convex potential that generates an exponential family built on (q, φ).
ψ(λ) := log ∫ q(x) · exp( λ·φ(x) ) dμ(x). (1.4)

• Conjugacy (organizing principle). Entropy-side Φ and natural-side ψ are Legendre–Fenchel duals; signal s and drive λ are conjugate coordinates.
Φ(s) = sup_λ { λ·s − ψ(λ) }, ψ(λ) = sup_s { λ·s − Φ(s) }. (1.5)

(Later sections show s = ∇_λ ψ(λ) and λ = ∇_s Φ(s), plus curvature relations.)

1.1 Conceptual Overview

• Signal as constraints. We declare structure by choosing φ. Fixing a target s means: “among all models whose features average to s, pick the one least divergent from noise q.”
Φ(s) from (1.3) is that least price.

• Exponential family emerges. Solving the constrained program produces models of the form p_λ(x) ∝ q(x)·exp(λ·φ(x)), with s = ∇_λ ψ(λ).

• Conjugacy drives the calculus. Dual potentials Φ and ψ in (1.5) give matched coordinates (s, λ), and their Hessians control information (Fisher) and stability.

1.2 Why This Matters

• Design clarity. “What is signal?” becomes an explicit, testable declaration φ, cleanly separated from “what does it cost?” via Φ.

• Tradeoff surfaces. Level sets of Φ(s) quantify the minimum price (divergence from noise) to sustain a chosen signal s—ideal for decode budgets and acceptance tests.

• Stability via curvature. The Hessian ∇²_λλ ψ equals the feature covariance (Fisher information), and its inverse ∇²_ss Φ governs uncertainty and conditioning in signal space.

1.3 Contributions

Precise, implementable definition of Signal via feature expectations.
s := E_p[φ(X)] (1.1) decouples “what counts as structure” (φ) from any particular architecture.
Variational derivation of the exponential family from constrained max-entropy.
Solving (1.3) yields p_λ(x) ∝ q(x)·exp(λ·φ(x)) with potential ψ(λ) in (1.4).
Conjugacy theorem: entropy and signal are Legendre duals.
Φ and ψ satisfy (1.5), making (s, λ) rigorous conjugates and enabling a full differential geometry of tradeoffs.
Corollaries linking gradients to Fisher information and uncertainty bounds.
∇²_λλ ψ = Cov_{p_λ}[φ(X)] and ∇²_ss Φ = (∇²_λλ ψ)^{-1} provide CR-type limits and conditioning diagnostics.
Practical appendix patterns for training, decoding, memory, and multi-tool arbitration.
Dual-threshold output gating, resource-aware decoding via ΔΦ budgets, memory write margins using Φ, covariance-guided parallelism, and dataset moment coverage.

2. Preliminaries: Probability, Entropy, and Feature Constraints

• Probability space and baseline (noise) distribution. We work on a measurable space with a reference measure and a strictly positive baseline density (“noise”) that integrates to one.
(X, F, μ) with q(x) > 0 a.e., ∫ q(x) dμ(x) = 1. (2.0)

• Candidate models. Admissible models are densities p ≥ 0 with unit mass, absolutely continuous w.r.t. q (so p(x) = 0 wherever q(x) = 0).
∫ p(x) dμ(x) = 1 and p ≪ q. (2.0′)

• Shannon entropy (discrete and continuous).
H(p) := − Σ_x p(x) · log p(x). [discrete]
H(p) := − ∫ p(x) · log p(x) dμ(x). (2.1)

• Relative entropy (Kullback–Leibler divergence from noise).
D(p∥q) := Σ_x p(x) · log( p(x)/q(x) ). [discrete]
D(p∥q) := ∫ p(x) · log( p(x)/q(x) ) dμ(x). (2.2)

• Feature map (declaring “what counts as structure”). We choose a measurable feature vector of dimension d.
φ: X → R^d. (2.3)

• Feature constraints (moment conditions). A target signal vector s fixes the admissible set of models whose feature expectations match s.
E_p[ φ(X) ] = s ∈ R^d. (2.4)
P(s) := { p admissible : E_p[ φ(X) ] = s }. (2.5)

Remark. We use relative entropy D(p∥q) to cleanly separate structure from a declared noise model q: the larger D(p∥q), the further p departs from q to sustain the imposed feature structure.

3. Formalizing “Signal” as Mean Parameters

• Feature map (“detectors”). We declare what counts as structure by choosing a measurable d-dimensional feature vector.
φ: X → R^d. (3.0)

• Integrability (standing). Each component of φ is integrable under admissible models p (and under q), so expectations exist and are finite.
∫ ‖φ(x)‖ · p(x) dμ(x) < ∞. (3.0′)

• Signal (mean parameters). The signal carried by a model p is the vector of feature expectations under p.
s(p) := E_p[ φ(X) ] ∈ R^d. (3.1)

• Baseline-centered signal (deviation from noise). To emphasize “structure above noise,” we will sometimes reference the baseline q.
Δs(p) := E_p[ φ(X) ] − E_q[ φ(X) ]. (3.2)

• Moment (signal) set. The achievable signals form the image of the admissible models under the expectation map.
M := { E_p[ φ(X) ] : p admissible } ⊆ R^d. (3.3)

• Identifiability (flat directions). A direction a ∈ R^d is unidentifiable if a·φ(X) is almost surely constant for all admissible p; only directions outside this kernel can be controlled via p.
K := { a ∈ R^d : a·φ(X) = const a.s. for all p }, identifiable subspace = K^⊥. (3.4)

• Affine reparameterization (units/scale). Changing features by an invertible affine map updates signals accordingly and preserves feasibility.
φ′(x) := A φ(x) + b ⇒ s′(p) = A s(p) + b, Δs′(p) = A Δs(p). (3.5)

Interpretation. The vector s(p) captures coherent structure extracted by φ that a model p sustains; Δs(p) makes the deviation from the declared noise model q explicit. Different choices of φ instantiate different, task-specific notions of “what counts as signal.”

4. Max-Entropy with Feature Constraints and the Exponential Family

• Constraint set (targeting a given signal vector). We collect all admissible models whose feature expectations equal s.
P(s) := { p ≪ q : E_p[ φ(X) ] = s }. (4.1)

• Variational problem (minimum divergence from noise at fixed signal).
p⋆(s) := argmin over p ∈ P(s) of D(p∥q). (4.2)

• Lagrangian (moment constraints + normalization). Here λ ∈ R^d enforces E_p[φ]=s and ν enforces ∫p=1.
L(p, λ, ν) := D(p∥q) + λ·( E_p[φ(X)] − s ) + ν·( ∫ p(x) dμ(x) − 1 ). (4.3)

• Stationary solution (exponential family induced by (q, φ)). The optimizer has log-density affine in φ with respect to q; Z(λ) is the normalizer.
p_λ(x) := [ q(x) · exp( λ·φ(x) ) ] / Z(λ), Z(λ) := ∫ q(x) · exp( λ·φ(x) ) dμ(x). (4.4)

• Cumulant (log-partition) function (convex in λ on its natural domain).
ψ(λ) := log Z(λ) = log ∫ q(x) · exp( λ·φ(x) ) dμ(x). (4.5)

• Mean–natural gradient link (the “moment map”). The gradient of ψ yields the model’s signal; conversely, λ is determined by s on the moment interior.
s(λ) := ∇λ ψ(λ) = E{p_λ}[ φ(X) ]. (4.6)

Remarks.
– Existence/uniqueness: when s lies in the relative interior of the moment set and ψ is essentially smooth/strictly convex, p⋆(s) exists, is unique, and equals p_λ with s = ∇ψ(λ).
– Interpretation: (4.2) says “among all models that realize signal s, pick the one least divergent from noise q.” Equation (4.4) shows the least-cost model tilts q by an exponential in the declared features.

5. The Entropy–Signal Conjugacy Theorem (Statement)

• Minimum-divergence potential over mean parameters. For a target signal vector s, define the least divergence from the noise model q required to realize s.
Φ(s) := inf over { p admissible with E_p[φ]=s } of D(p∥q). (5.1)

• Conjugacy (dual potentials). The entropy-side Φ and the natural-side ψ are Legendre–Fenchel conjugates; signal s and drive λ are matched by convex duality.
Φ(s) = sup_λ { λ·s − ψ(λ) }, ψ(λ) = sup_s { λ·s − Φ(s) }. (5.2)

• Dual coordinates (mean–natural link). Gradients invert the coordinates whenever s is in the interior of the moment set and ψ is essentially smooth/strictly convex on its natural domain.
s = ∇_λ ψ(λ), λ = ∇_s Φ(s). (5.3)

Interpretation. s (signal) and λ (drive) are conjugate coordinates; ψ (log-partition) and Φ (minimum relative entropy at fixed s) are dual potentials. The level sets of Φ(s) quantify the minimum price—measured as divergence from q—needed to sustain a given structure s.

6. Proof of the Theorem (Variational and Duality Arguments)

• Sign convention for the Lagrangian (for clean dual form). We take the moment multipliers with a minus sign; this yields the exponential-family solution with a positive exponent and the dual (g(λ)=λ·s−ψ(λ)). (An equivalent “plus” convention works after flipping (λ\mapsto−λ).)

• Step 0 (Primal recap). Minimize divergence from noise subject to the feature (signal) constraint and normalization.
Minimize D(p∥q) subject to E_p[φ]=s and ∫ p dμ = 1. (6.0)

Step 1 — Lagrange dual: eliminate p and obtain the dual objective

• Lagrangian (with multipliers λ ∈ R^d for moments and ν ∈ R for mass).
L(p, λ, ν) := D(p∥q) − λ·( E_p[φ(X)] − s ) + ν·( ∫ p dμ − 1 ). (6.1)

• Expand L as an integral in p.
L = ∫ p(x)·[ log(p(x)/q(x)) − λ·φ(x) + ν ] dμ(x) + λ·s − ν. (6.2)

• Pointwise minimization over p ≥ 0. For each x, minimize f(p) = p·[ log(p/q) − λ·φ + ν ]. The convex conjugate gives the infimum at
p̂(x) = q(x) · exp( λ·φ(x) − 1 − ν ). (6.3)

• Dual function (value of L at the infimum over p).
g(λ, ν) = − ∫ q(x) · exp( λ·φ(x) − 1 − ν ) dμ(x) + λ·s − ν. (6.4)

• Maximize over ν to tighten the dual. The optimal ν satisfies exp(−1−ν)·Z(λ) = 1, i.e., ν = log Z(λ) − 1 with Z(λ) = ∫ q·exp(λ·φ) dμ. Substituting gives
g(λ) = λ·s − ψ(λ), ψ(λ) := log Z(λ). (6.5)

Step 2 — Strong duality: no gap and existence of a maximizer

• Feasibility and regularity. The constraints are affine, D(·∥q) is strictly convex and lower semicontinuous, and Slater’s condition holds whenever s lies in the interior of the moment set.

• Consequence (no duality gap).
Φ(s) = inf over p: E_p[φ]=s of D(p∥q) = sup_{λ∈Λ} g(λ) = sup_{λ∈Λ} { λ·s − ψ(λ) }. (6.6)

• Existence. If s is in the interior of the moment set and ψ is essentially smooth on its natural domain Λ, the supremum is attained at some λ⋆.

Step 3 — Exponential form of the primal optimum

• KKT stationarity in p enforces the exponential tilt of q. With ν = log Z(λ⋆) − 1, the primal optimum equals
p⋆(x) = p_{λ⋆}(x) := [ q(x) · exp( λ⋆·φ(x) ) ] / Z(λ⋆). (6.7)

• Complementary slackness is vacuous (all constraints are equalities); primal feasibility gives E_{p_{λ⋆}}[φ] = s.

• Mean–natural link (moment map). Differentiating ψ shows
s = ∇λ ψ(λ⋆) = E{p_{λ⋆}}[ φ(X) ]. (6.8)

Step 4 — Conjugacy identities and gradient relations

• From (6.6) we have the Legendre–Fenchel pair
Φ(s) = sup_λ { λ·s − ψ(λ) }, ψ(λ) = sup_s { λ·s − Φ(s) }. (6.9)

• At optimal pairs (s, λ) the subgradient conditions sharpen to gradients (on the moment interior), giving the dual coordinates
s = ∇_λ ψ(λ), λ = ∇_s Φ(s). (6.10)

• Uniqueness and bijection (regular case). If ψ is strictly convex and essentially smooth on Λ, then λ ↦ s = ∇ψ(λ) is a bijection between Λ and the interior of the moment set; the primal optimizer p⋆ is unique and equals p_λ.

Optional note (regularity and edge cases)

• If the Fisher information ∇²_λλ ψ(λ) is singular, some feature directions are unidentifiable; λ may not be unique. If s lies on the boundary of the moment set, the supremum in (6.6) may be approached but not attained.

7. Corollaries: Gradients, Fisher Information, and Uncertainty Bounds

• Hessians (covariance and Fisher information). The log-partition curvature equals the feature covariance under the exponential model; this is the Fisher information in natural coordinates.
∇²_λλ ψ(λ) = Cov_{p_λ}[ φ(X) ] = I(λ). (7.1)

• Dual curvature (mean-parameter side). The curvature of the minimum-divergence potential is the inverse Fisher information, evaluated at the conjugate pair (s, λ).
∇²_ss Φ(s) = ( ∇²_λλ ψ(λ) )^{-1} = I(λ)^{-1}. (7.2)

• CRLB in λ-coordinates (classical form). For N i.i.d. samples from p_λ, any unbiased estimator (\hat{λ}) obeys
Cov( (\hat{λ}) ) ⪰ I(λ)^{-1} / N. (7.3)

• CRLB in s-coordinates (mean-natural duality). Reparameterizing via s = ∇_λ ψ(λ), any unbiased estimator (\hat{s}) satisfies
Cov( (\hat{s}) ) ⪰ [ ∇²_ss Φ(s) ]^{-1} / N = I(λ) / N. (7.4)

• Scalar projection (directional uncertainty). For any vector a ∈ R^d and unbiased (\hat{s}):
Var( a·(\hat{s}) ) ≥ (1/N) · aᵀ I(λ) a. (7.5)

• Local quadratic approximations (tradeoff geometry). Near a conjugate pair (s, λ), the dual potentials admit matched quadratic forms:
ψ(λ + δλ) ≈ ψ(λ) + s·δλ + ½ δλᵀ I(λ) δλ, (7.6a)
Φ(s + δs) ≈ Φ(s) + λ·δs + ½ δsᵀ I(λ)^{-1} δs. (7.6b)

• Conditioning and stability. The spectral condition number κ of I(λ) controls local sensitivity of s to λ (and of λ to s via I(λ)^{-1}):
κ( I(λ) ) := σ_max( I(λ) ) / σ_min( I(λ) ). (7.7)

• de Bruijn-type link (outline; continuous X ∈ R^n). Under Gaussian smoothing with variance σ, entropy increases at a rate set by Fisher information; this ties entropy growth to loss of resolvable signal.
d/dσ H( p * N(0, σ I) ) = ½ · Tr( I_F( p * N(0, σ I) ) ). (7.8)

Interpretation. (7.1)–(7.2) state that “information = curvature,” and the two coordinate systems (λ for drive, s for signal) are exact inverses. The CRLB forms (7.3)–(7.5) give operational lower bounds on estimation error given a target signal level s. The quadratic expansions (7.6) and conditioning (7.7) provide practical stability diagnostics, while the de Bruijn identity (7.8) links noise injection to entropy growth and the degradation of fine-grained signal.

8. Dynamic Extension: Conjugate Flows and Entropy Production

• Setup (Markov evolution with stationary noise q). Let ({P_t}{t\ge0}) be a Markov semigroup on ((X,F,μ)) with generator (L) and adjoint (L^). Densities evolve by
∂_t p_t = L^ p_t, p{t=0} = p_0, and L^* q = 0. (8.0)

• Signal flow (moment dynamics under the forward equation). For a fixed feature map (\phi:X→R^d):
s_t := E_{p_t}[ φ(X) ] ∈ R^d, \dot s_t = E_{p_t}[ L φ(X) ]. (8.1)

• Entropy production (relative to q). Define (D(p_t∥q) := ∫ p_t(x)·log( p_t(x)/q(x) ) dμ(x)). Then
d/dt D(p_t∥q) = ∫ p_t(x) · L log( p_t(x)/q(x) ) dμ(x) =: −σ(t) ≤ 0. (8.2)

— For reversible diffusions on (R^n) with invariant (q) and unit diffusion, the production equals the relative Fisher information
I_rel(p_t∥q) := ∫ p_t(x) · ‖ ∇ log( p_t(x)/q(x) ) ‖^2 dx, so d/dt D(p_t∥q) = − I_rel(p_t∥q) ≤ 0. (8.2′)

• Manifold tracking (evolution constrained to the exponential family). Suppose the flow stays on the exponential family induced by ((q,φ)), i.e., (p_t = p_{λ_t}) with
p_{λ}(x) := [ q(x) · exp( λ·φ(x) ) ] / Z(λ), ψ(λ) := log Z(λ), s_t = ∇_λ ψ(λ_t). (8.3)

• Chain rules for dual potentials along the conjugate path.
d/dt Φ(s_t) = ∇_s Φ(s_t)·\dot s_t = λ_t · \dot s_t, d/dt ψ(λ_t) = ∇_λ ψ(λ_t)·\dot λ_t = s_t · \dot λ_t. (8.4)

• Fenchel–Young invariant (conjugate balance law). Along the conjugate manifold (s_t=∇ψ(λ_t)) and (λ_t=∇Φ(s_t)), the Fenchel–Young gap is identically zero and remains zero in time:
F(t) := Φ(s_t) + ψ(λ_t) − λ_t·s_t ≡ 0, so d/dt F(t) = 0. (8.5)

Equivalently, the “power” injected into signal equals the dual change up to the coordinate-transfer term: combine (8.4) to get
λ_t·\dot s_t − d/dt ψ(λ_t) = λ_t·\dot s_t − s_t·\dot λ_t = d/dt( λ_t·s_t ) − 2 s_t·\dot λ_t. (8.6)

• Projection view (off-manifold flows). For a general (p_t) not in the exponential family, define the moment projection (s_t := E_{p_t}[φ]) and the dual projection (λ_t := argmax_λ{ λ·s_t − ψ(λ) }). Then the Fenchel–Young gap
G(t) := Φ(s_t) + ψ(λ_t) − λ_t·s_t ≥ 0 (with equality iff (p_t = p_{λ_t})). (8.7)

Under mild regularity, (d/dt,G(t)) is nonnegative whenever the flow dissipates relative entropy faster than the exponential family can “keep up” with the changing moments—quantifying model–feature mismatch as extra dissipation.

• Minimal price vs. actual price (feature sufficiency bound). For any (p_t), the actual divergence dominates the minimal price of its current signal:
D(p_t∥q) ≥ Φ( s_t ), with equality iff (p_t \in { p_λ : ∇ψ(λ)=s_t }). (8.8)

Interpretation.
– (8.1) says moments evolve by pushing (\phi) through the backward generator (L).
– (8.2)–(8.2′) are H-theorem statements: relative entropy to the stationary noise (q) is nonincreasing; dissipation equals a Fisher-information–type quantity in reversible diffusions.
– (8.3)–(8.5) formalize conjugate tracking: if the state stays on the exponential family, the Fenchel–Young identity remains exact in time.
– (8.7)–(8.8) separate feature-declared structure (priced by Φ) from residual structure not captured by φ (the excess (D−Φ)), which shows up as additional entropy production in off-manifold dynamics.

9. Operational Tradeoffs and Limits of the Theory

• Tradeoff surface (minimum price of structure). Level sets of the minimum-divergence potential define the feasible “price of signal” under a fixed noise model q.
T(B) := { s ∈ R^d : Φ(s) ≤ B }. (9.1)
Interpretation: T(B) is the set of all signal vectors that can be sustained at divergence cost at most B. These are the operational envelopes for decoding, inference, and control under a declared budget.

• Budgeted acceptance rule (decode or commit). Accept an update from s to s′ only if the incremental price stays under a preset budget.
ΔΦ := Φ(s′) − Φ(s) ≤ η. (9.2)
Use (9.2) for token-by-token decoding, tool invocation, or memory writes to prevent high-cost excursions.

• Shadow price (marginal cost of signal). The gradient of Φ at s gives the marginal divergence cost of increasing signal along each feature direction.
λ(s) := ∇_s Φ(s). (9.3)
Component i of λ(s) tells you how many “divergence units” are paid per unit increase in feature i of the signal.

• Local quadratic approximation (stability metric). Near a regular conjugate pair (s, λ), the second-order change in price is controlled by the inverse Fisher information.
Φ(s + δs) ≈ Φ(s) + λ·δs + ½ · δsᵀ [ ∇²_ss Φ(s) ] δs, with ∇²_ss Φ(s) = I(λ)^{-1}. (9.4)
Large eigenvalues of I(λ)^{-1} signal unstable directions (small δs costs a lot).

• Absolute feasibility. A target signal is feasible at finite price iff it lies in the moment set M; outside M the price is infinite.
s ∈ M ⇒ Φ(s) < ∞; s ∉ M ⇒ Φ(s) = +∞. (9.5)
Operationally: attempting to enforce s ∉ M will either fail or force degeneracy (divergence blow-up).

• Actual price vs. minimal price (gap due to residual structure). For any model p, the paid divergence exceeds (or equals) the minimal price of the signal it actually delivers.
D(p∥q) ≥ Φ( E_p[φ] ). (9.6)
Equality holds iff p is the exponential tilt that matches its own moments—otherwise the excess D − Φ is “unpriced structure” not captured by φ.

• Model risk: feature misspecification. If φ omits relevant structure or mixes incompatible scales, Φ mis-prices signal levels. A practical guardrail is a curvature-regularized acceptance:
Accept only if ΔΦ ≤ η and ‖ ∇²_ss Φ(s) ‖ ≤ τ. (9.7)
This prevents pushing into regions where small signal adjustments have explosive cost due to mis-specified features.

• Robust noise model (distributional robustness around q). When q is uncertain, protect against worst-case baselines in an f-divergence ball.
U_f(q, ρ) := { q′ : D_f(q′∥q) ≤ ρ }, Φ_rob(s) := sup_{q′ ∈ U_f(q,ρ)} inf_{p:E_p=s} D(p∥q′). (9.8)
Φ_rob(s) inflates the price surface to reflect uncertainty in the declared noise.

• Constraint relaxation (moment tolerance for noisy estimators). Allow a small tolerance τ in the moment constraint to prevent overfitting measurement noise; this is a Moreau-type envelope of Φ.
Φ_τ(s) := inf_{‖s′ − s‖ ≤ τ} Φ(s′). (9.9)
Using Φ_τ in (9.2) smooths acceptance decisions and stabilizes training/decoding under finite-sample fluctuations.

• Multi-task tradeoffs (shared budget across feature groups). When s = (s_A, s_B) for two task groups, use a joint budget or weighted prices.
Minimize Φ(s_A, s_B) subject to w_A·Φ(s_A) + w_B·Φ(s_B) ≤ B. (9.10)
This enforces principled allocation between competing signal objectives.

• Boundary and degeneracy warnings. Near ∂M (boundary of the moment set), Φ becomes steep and ∇²_ss Φ may blow up; when I(λ) is singular, certain signal directions are unidentifiable, making λ nonunique and price estimates ill-conditioned.
Conditioning index: κ( I(λ) ) = σ_max / σ_min. Large κ ⇒ fragile control. (9.11)

Summary. Φ(s) serves as a calibrated “price of structure” surface. Equations (9.1)–(9.4) turn that surface into actionable envelopes, marginal prices, and stability diagnostics; (9.5)–(9.6) separate feasibility from waste; (9.7)–(9.9) add robustness and tolerance; (9.10)–(9.11) guide multi-objective budgeting and warn about ill-conditioning at the boundary.

10. Related Work (concise, dependency-free)

This paper’s proof relies only on standard probability and convex duality. The items below are for positioning, not prerequisites.

• Maximum entropy and exponential families. Constrained max-entropy (or minimum relative entropy to a baseline q) yields exponential tilts with a convex log-partition.
p_λ(x) ∝ q(x) · exp( λ·φ(x) ), ψ(λ) := log ∫ q·exp(λ·φ) dμ. (10.1)

• Information geometry (mean vs natural coordinates). The map between natural and mean parameters is the gradient of the log-partition; Fisher information is the Hessian.
s = ∇_λ ψ(λ), I(λ) = ∇²_λλ ψ(λ). (10.2)

• Fisher information and Cramér–Rao bounds. Curvature sets local uncertainty limits for unbiased estimation under p_λ.
Cov( \hat{λ} ) ⪰ I(λ)^{-1} / N, Cov( \hat{s} ) ⪰ I(λ) / N. (10.3)

• Log-partition convexity and Bregman/KL links. Convexity of ψ implies that KL between exponential-family members equals a Bregman divergence in λ.
D( p_λ ∥ p_{λ′} ) = ψ(λ) − ψ(λ′) − (λ−λ′)·∇ψ(λ′) =: B_ψ(λ, λ′). (10.4)

• De Bruijn identity (entropy–information flow under Gaussian smoothing). For X ∈ R^n and Z ∼ N(0, I) independent, entropy increases at a rate set by Fisher information.
d/dσ H( X + √σ Z ) = ½ · Tr( I( X + √σ Z ) ). (10.5)

• Rate–distortion and free-energy parallels. Lagrange multipliers for distortion or feature constraints produce convex dual “free-energy” objectives of the same Fenchel–Young form.
F(λ; s) := ψ(λ) − λ·s, Φ(s) = sup_λ { λ·s − ψ(λ) }. (10.6)

• Large deviations and Cramér transforms (affinity with convex conjugacy). Many moment-generating transforms are convex, with conjugates that bound rare-event rates—closely mirroring ψ ↔ Φ duality.
Λ(λ) := log E[ exp(λ·Y) ], Λ*(y) := sup_λ { λ·y − Λ(λ) }. (10.7)

Note. Equations (10.1)–(10.7) summarize established results that align with, but are not required for, the theorem proved in this paper. They highlight how the entropy–signal conjugacy fits into the broader convex-information toolkit used across statistics, coding, thermodynamics-inspired objectives, and learning theory.

11. Conclusion

Summary. We showed that signal—declared as mean parameters (s = E_p[φ(X)])—and entropy—priced as minimum divergence from a baseline noise model (q)—form a precise conjugate pair under a constrained maximum-entropy program. The core objects are the minimum-divergence potential (Φ(s)) and the log-partition (ψ(λ)), with Legendre–Fenchel conjugacy and dual coordinates established in (5.1)–(5.3). Practically, this gives an architecture-agnostic calculus for what counts as structure (choose features (φ)) and what it costs to sustain it (read off (Φ) and its curvature).

Operationalization. The theory translates directly into controls and diagnostics:

Tradeoff surfaces. Level sets of (Φ(s)) are the price-of-structure envelopes (9.1); they support budgeted acceptance rules via (ΔΦ) (9.2) and marginal “shadow prices” (λ(s)=∇_s Φ(s)) (9.3).
Stability and uncertainty. Curvature identities (∇²_{λλ}ψ = I(λ)) and (∇²_{ss}Φ = I(λ)^{-1}) ((7.1)–(7.2)) yield Cramér–Rao–type bounds, conditioning diagnostics, and matched quadratic approximations ((7.6a)–(7.6b)).
Dynamics. Under Markov evolution, moment flows follow ( \dot s_t = E_{p_t}[Lφ(X)] ) (8.1), while relative entropy dissipates (8.2). On the exponential-family manifold (p_t=p_{λ_t}), the Fenchel–Young identity remains exact in time (8.5), exposing a clean signal–entropy balance law.

Limits and safeguards. Feasibility requires (s∈M) (9.5); boundary regions and singular Fisher information warn of ill-conditioning. Robust variants hedge uncertainty in (q) via distributional sets (9.8), and tolerance envelopes (Φ_τ) (9.9) stabilize decisions under measurement noise.

Pointers to appendices.

Appendix A (Application Patterns). Ready-to-implement recipes: dual-threshold output gating, resource-aware decoding via (ΔΦ) budgets, memory write margins, covariance-guided parallelism, and dataset moment coverage.
Appendix B (Worked Micro-Example). A binary-feature computation that verifies conjugacy end-to-end.
Appendix C (Notation and Glossary). A compact reference for all symbols and objects used.
Appendix D (Reproducibility Checklist). Minimal artifacts for verifying (5.2)–(5.3) numerically on synthetic data.

Takeaway. By separating declaration of structure (features (φ)) from the price to sustain it (potential (Φ)), the conjugacy result provides a principled control panel for intelligent systems—one that is model-agnostic, measurable, and directly actionable in training, decoding, memory, and multi-tool arbitration.

Appendix A. Application Patterns for AI/AGI Systems (outline)

Purpose. Drop-in, Blogger-safe recipes that operationalize the Part A geometry. Each item gives a one-line accept rule or KPI you can wire into logs and dashboards.

A.1 Output Gating with Dual Tests (structure × stability)

Gate only when both structure is present and the geometry is stable.

• Structure margin (Fenchel–Young value):
g(λ; s) := λ·s − ψ(λ). (A.1a)

• Stability proxy (choose a norm on curvature):
C(λ) := ‖∇²_λλ ψ(λ)‖. (A.1b)

• Gate condition (two keys):
g(λ; s) ≥ τ₁ and C(λ) ≤ τ₂. (A.1)

A.2 Resource-Aware Decoding

Penalize steps that raise the price of structure too fast; accept only if within budget and useful.

• Budgeted step rule (with utility floor):
ΔΦ := Φ(s′) − Φ(s) ≤ η and ΔU := U(s′) − U(s) ≥ κ. (A.2)

• Soft alternative (single score to rank candidates):
Score := U − β·ΔΦ, with β ≥ 0. (A.2a)

A.3 Memory Write Policy (Reversible vs Irreversible)

Commit to long-term memory only when structure is verifiably above baseline.

• Margin over baseline signal:
Φ(s_candidate) − Φ(s_baseline) ≥ τ. (A.3)

A.4 Multi-Tool Arbitration via Covariance (Commutativity Proxy)

Prefer parallelism when tools do not interfere; otherwise serialize.

• Cross-covariance test under current drive λ (tools A,B):
‖Cov_{p_λ}[ φ_A(X), φ_B(X) ]‖ ≤ ε ⇒ parallel; else serialize. (A.4)

A.5 Dataset Slot Balancing (Moment Coverage)

Train where geometry is well-conditioned; target even coverage of mean parameters.

• Conditioning objective for the operating region (λ̂ from calibration):
minimize κ( ∇²_λλ ψ(λ̂) ) with κ(M) := σ_max(M)/σ_min(M). (A.5)

A.6 Live Dashboard (operational health)

Surface structure level, price, and stability in real time.

• Display the triplet (norm of signal, price of structure, curvature proxy):
( ‖s‖ , Φ(s) , ‖∇²_λλ ψ(λ)‖ ). (A.6)

Optional health light: add the dissipation gap G := Φ(s) + ψ(λ) − λ·s to flag off-manifold drift.

Appendix B. Worked Micro-Example (Binary Feature)

Setup. Data space and baseline (noise) with a single binary feature.
X = {0,1}, q(1) = θ ∈ (0,1), q(0) = 1 − θ, φ(x) = x. (B.0)

B.1 Exponential Tilt, Mean Map, and Log-Partition

• Normalizer and tilted model.
Z(λ) = (1 − θ) + θ · e^λ. (B.1)

• Exponential-family member.
p_λ(1) = [ θ · e^λ ] / Z(λ), p_λ(0) = [ 1 − θ ] / Z(λ). (B.2)

• Mean (signal) as a function of λ.
s(λ) := E_{p_λ}[φ(X)] = p_λ(1). (B.3)

• Log-partition.
ψ(λ) := log Z(λ) = log( (1 − θ) + θ · e^λ ). (B.4)

B.2 Mean–Natural Mapping and Its Inverse

• Gradient identity (mean–natural link).
∂ψ/∂λ = s(λ) = θ · e^λ / ( (1 − θ) + θ · e^λ ). (B.5)

• Inverse map (natural as log-odds shift).
λ(s) = log( s(1 − θ) / ( (1 − s) θ ) ), s ∈ (0,1). (B.6)

B.3 Minimum-Divergence Potential Φ(s)

• Closed form (Bernoulli KL to the baseline θ).
Φ(s) = s · log( s/θ ) + (1 − s) · log( (1 − s)/(1 − θ) ). (B.7)

(Derivation: Either solve the constrained min D(p||q) directly for Bernoulli, or plug λ(s) into sup_λ { λ·s − ψ(λ) }.)

B.4 Conjugacy Checks

• Supremum form and gradients invert.
Φ(s) = sup_λ { λ·s − ψ(λ) }, ψ(λ) = sup_s { λ·s − Φ(s) }. (B.8)

• Dual coordinates.
s = ∂ψ/∂λ, λ = ∂Φ/∂s. (B.9)

(Verification: Using (B.6), compute ∂Φ/∂s = log( s(1 − θ) / ((1 − s)θ) ) = λ(s), and ∂ψ/∂λ from (B.5) equals s.)

B.5 Curvature, Fisher Information, and Bounds

• Fisher information (natural side).
I(λ) = ∂²ψ/∂λ² = s(1 − s). (B.10)

• Dual curvature (mean side).
∂²Φ/∂s² = 1 / ( s(1 − s) ). (B.11)

• CR-type lower bound with N i.i.d. samples.
Var( \hat{s} ) ≥ s(1 − s) / N, Var( \hat{λ} ) ≥ 1 / ( N · s(1 − s) ). (B.12)

B.6 Quick Quadratic Price Approximation

• For a small step Δs around s, the price increment is
ΔΦ ≈ λ(s) · Δs + ½ · (Δs)² / ( s(1 − s) ). (B.13)

(Use (B.11); first-order term uses λ(s) from (B.6).)

B.7 Dashboard Triplet (for this toy)

• Structure level, price, and stability proxy.
‖s‖ = s, Φ(s) from (B.7), ‖∇²_λλ ψ(λ)‖ = s(1 − s). (B.14)

(Max stability near s = 0.5; ill-conditioning as s → 0 or 1.)

Summary. With φ(x)=x and baseline θ, the exponential tilt (B.2)–(B.4), inverse link (B.6), and KL form (B.7) make conjugacy fully explicit. Fisher information and its inverse (B.10)–(B.11) recover the standard Bernoulli bounds, and the quadratic price (B.13) gives a fast, accurate ΔΦ for budgeted control.

Appendix C. Notation and Glossary

Conventions. Integrals are w.r.t. a reference measure μ. Replace integrals by sums in discrete cases. Vectors live in R^d; “·” is the dot product; ‖·‖ is the Euclidean or operator norm as stated.

• Data space and baseline (noise).
(X, F, μ) measurable space; q(x) > 0 a.e., ∫ q(x) dμ(x) = 1. (C.1)

• Model distribution (absolutely continuous w.r.t. q).
p(x) ≥ 0, ∫ p(x) dμ(x) = 1, and p ≪ q. (C.2)

• Feature map (detectors).
φ: X → R^d. (C.3)

• Signal (mean parameters).
s := E_p[ φ(X) ] ∈ R^d. (C.4)

• Natural parameters (drive).
λ ∈ R^d. (C.5)

• Normalizer and log-partition.
Z(λ) := ∫ q(x) · exp( λ·φ(x) ) dμ(x). (C.6)
ψ(λ) := log Z(λ). (C.7)

• Relative entropy (KL divergence).
D(p||q) := ∫ p(x) · log( p(x)/q(x) ) dμ(x). (C.8)

• Minimum-divergence potential (entropy side).
Φ(s) := inf over { p : E_p[φ]=s } of D(p||q). (C.9)

• Exponential-family member induced by (q, φ).
p_λ(x) := [ q(x) · exp( λ·φ(x) ) ] / Z(λ). (C.10)

• Mean–natural gradient link (dual coordinates).
s = ∇_λ ψ(λ), λ = ∇_s Φ(s). (C.11)

• Fisher information / feature covariance.
I(λ) := ∇²_λλ ψ(λ) = Cov_{p_λ}[ φ(X) ]. (C.12)

• Structure margin (Fenchel–Young value).
g(λ; s) := λ·s − ψ(λ). (C.13)

• Price increment (budgeted move).
ΔΦ := Φ(s′) − Φ(s). (C.14)

• Dissipation (Fenchel–Young) gap.
G := Φ(s) + ψ(λ) − λ·s ≥ 0. (C.15)

• Feasible-price envelope (tradeoff surface).
T(B) := { s ∈ R^d : Φ(s) ≤ B }. (C.16)

• Parameter domains.
Λ := { λ : Z(λ) < ∞ }, M := { E_p[φ] : p admissible }. (C.17)

• Conjugate projector (mean → natural).
λ*(s) := argmax_λ { λ·s − ψ(λ) }. (C.18)

• Curvature proxy (stability metric; choose norm per use).
C(λ) := ‖ ∇²_λλ ψ(λ) ‖. (C.19)

• Condition number (local anisotropy).
κ(M) := σ_max(M) / σ_min(M) for positive definite M. (C.20)

• Notational shortcuts.
⟨a,b⟩ = a·b; ‖v‖ = √(v·v); Tr(M) = trace; Cov_p[·] = covariance under p; EMA_τ = exponential moving average with horizon τ. (C.21)

Appendix D. Reproducibility Checklist

Purpose. A paste-ready checklist to let any reader recreate the conjugacy results and the operational metrics with synthetic data. Uses Blogger-safe, single-line equations and minimal pseudocode.

D.1 Experiment Setup (must specify)

• Data space and baseline (noise).
Declare (X, F, μ) and q(x) > 0 with ∫ q dμ = 1. (D.1)

• Feature map.
φ: X → R^d (integrable under q and sampled p). (D.2)

• Seeds, versions, and precision.
seed := 12345; float := 64-bit; optimizer := Newton + backtracking; stopping tol := 1e−8. (D.3)

• Grids for evaluation.
Λ_grid ⊂ R^d (for λ) and M_grid ⊂ R^d (for s) covering operating ranges. (D.4)

D.2 Core Functions (provide code or exact formulas)

• Log-partition and mean map.
ψ(λ) := log ∫ q(x) · exp( λ·φ(x) ) dμ(x); s(λ) := ∇_λ ψ(λ). (D.5)

• Fisher information.
I(λ) := ∇²_λλ ψ(λ) = Cov_{p_λ}[ φ(X) ]. (D.6)

• Minimum-divergence potential (dual form).
Φ(s) := sup_λ { λ·s − ψ(λ) }. (D.7)

• Conjugate projector (mean → natural).
λ*(s) := argmax_λ { λ·s − ψ(λ) } (solve ∇ψ(λ)=s). (D.8)

• Fenchel–Young (dissipation) gap.
G(λ, s) := Φ(s) + ψ(λ) − λ·s = ψ(λ) − ψ(λ*(s)) − (λ−λ*(s))·s. (D.9)

D.3 Numerical Solvers (reference pseudocode)

• Newton for λ*(s):
repeat Δλ := − [I(λ)]^{-1} · ( s(λ) − s ); λ ← λ + α·Δλ with backtracking; stop when ‖s(λ) − s‖ ≤ ε. (D.10)

• Quadratic price approximation (fast ΔΦ):
ΔΦ_qr(s→s′) := λ·(s′−s) + ½ · (s′−s)ᵀ I(λ)^{-1} (s′−s). (D.11)

D.4 Conjugacy Verification (required tests)

T1 — Gradient inversion (on Λ_grid).
Compute s = s(λ); then λ̂ := λ*(s); report err₁ := ‖λ̂ − λ‖. Pass if max err₁ ≤ 1e−6. (D.12)

T2 — Biconjugacy equalities (on paired grids).
Check Φ(s) = λ*(s)·s − ψ(λ*(s)) and s = ∇ψ(λ*(s)).
Check ψ(λ) = sup_s { λ·s − Φ(s) } (numerical sup). (D.13)

T3 — Fenchel–Young gap nonnegativity.
For random (λ, s), compute G(λ, s) ≥ 0; and G(λ*(s), s) ≈ 0. (D.14)

T4 — Curvature identities.
Finite-diff ∇²_λλ ψ vs empirical Cov_{p_λ}[φ] from Monte Carlo; report max rel-error ≤ 2%. (D.15)

D.5 Monte Carlo Sanity (synthetic data)

• Sampling. Draw X₁…X_N ∼ p_λ with p_λ(x) := q(x)·exp(λ·φ(x))/Z(λ). (D.16)

• Moment match.
ŝ := (1/N) Σ φ(X_i) vs s(λ); report ‖ŝ−s(λ)‖ with 95% CI ≈ √(diag(I(λ))/N). (D.17)

• CR-type bounds (variance scaling).
Var(ŝ) ≈ I(λ)/N; plot N·Var(ŝ) vs I(λ). (D.18)

D.6 Conditioning and Stability (must report)

• Fisher norms and condition numbers across Λ_grid.
C(λ) := ‖I(λ)‖ (operator or Frobenius), κ(λ) := σ_max(I)/σ_min(I). (D.19)

• Heatmaps / tables. Identify zones with κ(λ) > κ_max; flag as unstable. (D.20)

D.7 Minimal Synthetic Cases (must pass)

Case S1 — Binary feature (Appendix B).
X={0,1}, q(1)=θ, φ(x)=x. Closed forms:
ψ(λ)=log((1−θ)+θe^λ), s(λ)=θe^λ/((1−θ)+θe^λ), Φ(s)=s log(s/θ)+(1−s) log((1−s)/(1−θ)). (D.21)
Tests: T1–T4 with θ∈{0.2,0.5,0.8}.

Case S2 — Gaussian baseline, linear features.
q=N(0, Σ), φ(x)=Aᵀx. Then ψ(λ)=½ λᵀ (AᵀΣA) λ, s(λ)=(AᵀΣA) λ, I(λ)=AᵀΣA (constant).
Φ(s)=½ sᵀ (AᵀΣA)^{-1} s. (D.22)
Tests: T1–T4 on random SPD Σ and A.

D.8 Logging Schema (for audit)

Per run/step, log JSON or CSV rows with:
{ seed, case_id, λ, s, ψ, Φ, G, ‖s(λ)−s‖, ‖λ*(s)−λ‖, ‖I‖, κ, samples=N, runtime_ms }. (D.23)

D.9 Acceptance Thresholds (suggested)

• Inversion error: max_λ ‖λ*(s(λ)) − λ‖ ≤ 1e−6. (D.24)
• Gap at conjugacy: max_s G(λ*(s), s) ≤ 1e−8. (D.25)
• Curvature match: max rel-error of Cov vs ∇²ψ ≤ 2%. (D.26)
• CR scaling: slope of N·Var(ŝ) vs I(λ) within ±5%. (D.27)

D.10 Packaged Artifacts (to publish)

• Config: (X, q, φ), grids, seeds, tolerances. (D.28)
• Library: functions for ψ(λ), s(λ), I(λ), Φ(s), λ*(s), G(λ,s). (D.29)
• Scripts: S1_binary.py, S2_gauss_linear.py; plotting or CSV emitters. (D.30)
• Report: tables/plots for (D.19)–(D.20) and test summaries T1–T4. (D.31)

One-line readiness check.
“We provide code and configs to compute ψ, s, Φ; verify conjugacy and gaps (T1–T4); validate Monte Carlo moments and CR scaling; and report Fisher conditioning across the operating region, with fixed seeds and tolerances.”

Disclaimer

This book is the product of a collaboration between the author and OpenAI's GPT-5, Google's Gemini 2.5 Pro, X's Grok 4 language model. While every effort has been made to ensure accuracy, clarity, and insight, the content is generated with the assistance of artificial intelligence and may contain factual, interpretive, or mathematical errors. Readers are encouraged to approach the ideas with critical thinking and to consult primary scientific literature where appropriate.

This work is speculative, interdisciplinary, and exploratory in nature. It bridges metaphysics, physics, and organizational theory to propose a novel conceptual framework—not a definitive scientific theory. As such, it invites dialogue, challenge, and refinement.

I am merely a midwife of knowledge.

Friday, November 7, 2025