Appendix E: Rigorous Proof Sketches for Ontological and Metabolic Laws

TLDR

• This appendix contains rigorous proof sketches backing the ontology/metabolism results in the later cognition chapters.

• Read it when you want the mathematical spine behind the narrative statements; otherwise treat it as a reference.

This appendix collects calculations for the cognition and multi-agent chapters. The multi-agent proofs track the operator, measure, and variational functional used in each result, so that changes of representation can be checked directly.

We operate on the latent Riemannian manifold \((\mathcal{Z}, G)\) with belief measures \(\rho \in \mathcal{P}(\mathcal{Z})\).

E.1 Proof of Theorem Theorem 21

Statement: The ontology should expand from \(N_c\) to \(N_c + 1\) charts if and only if \(\Xi > \Xi_{\text{crit}}\) and \(\Delta V_{\text{proj}} > \mathcal{C}_{\text{complexity}}\).

Hypothesis: Let \(\mathcal{S}[N_c] = \inf_{\theta} \mathcal{S}_{\text{onto}}(\theta, N_c)\) be the value function of the ontological action for \(N_c\) charts.

Proof

Consider the discrete variation \(\Delta \mathcal{S} = \mathcal{S}[N_c + 1] - \mathcal{S}[N_c]\). By the definition of the Ontological Action (Section 30.3):

\[ \mathcal{S}_{\text{onto}} = -\mathcal{S}_{\text{task}} + \mu_{\text{size}} \cdot N_c,\]

where \(\mathcal{S}_{\text{task}} = \mathbb{E}[\langle V \rangle]\) is the expected task value.

Expanding \(\mathcal{S}_{\text{task}}\) via a first-order Taylor approximation in the space of representations:

\[ \mathcal{S}_{\text{task}}[N_c + 1] \approx \mathcal{S}_{\text{task}}[N_c] + \frac{\partial \langle V \rangle}{\partial N_c}.\]

The marginal utility of a new chart is \(\frac{\partial \langle V \rangle}{\partial N_c} = \Delta V_{\text{proj}}\). The complexity cost is \(\mu_{\text{size}}\). Therefore:

\[ \Delta \mathcal{S} = -\Delta V_{\text{proj}} + \mu_{\text{size}}.\]

The transition \(N_c \to N_c + 1\) is the global minimizer iff \(\Delta \mathcal{S} < 0\), which yields:

\[ \Delta V_{\text{proj}} > \mu_{\text{size}} = \mathcal{C}_{\text{complexity}}.\]

The condition \(\Xi > \Xi_{\text{crit}}\) ensures that the second variation of the texture-entropy functional \(\delta^2 H(z_{\text{tex}})\) is negative-definite at the vacuum. This precludes the absorption of the signal into the existing noise floor: if \(\Xi \le \Xi_{\text{crit}}\), the texture residual \(z_{\text{tex}}\) is truly unpredictable noise, and adding a chart provides no informational benefit. \(\square\)

E.2 Proof of Theorem Theorem 22

Statement: The emergence of a new chart follows a supercritical pitchfork bifurcation with control parameter \(\mu = \Xi - \Xi_{\text{crit}}\).

Hypothesis: The potential \(\Phi_{\text{onto}}(r)\) is \(SO(n)\)-invariant near \(r=0\), where \(r = \|q_* - q_{\text{parent}}\|\) is the radial distance of the new query from the parent.

Proof

Let \(f(\Xi) = \Xi - \Xi_{\text{crit}}\) be the control parameter. By \(SO(n)\) symmetry, the Ontological Action can only depend on even powers of \(r\) near the origin. We expand in a power series:

\[ \mathcal{S}(r) = \mathcal{S}_0 - \frac{1}{2}f(\Xi)r^2 + \frac{1}{4}\beta r^4 + O(r^6),\]

where \(\beta > 0\) for stability (the quartic term must be positive for bounded energy).

The stationarity condition \(\frac{\partial \mathcal{S}}{\partial r} = 0\) yields:

\[ -f(\Xi)r + \beta r^3 = 0 \implies r(f(\Xi) - \beta r^2) = 0.\]

This has solutions:

1. \(r = 0\) (trivial, no new chart)

2. \(r^2 = f(\Xi)/\beta\) (symmetry-broken state)

Analysis of stability:

• For \(f(\Xi) < 0\) (i.e., \(\Xi < \Xi_{\text{crit}}\)): The Hessian at \(r=0\) is \(\frac{\partial^2 \mathcal{S}}{\partial r^2}|_{r=0} = -f(\Xi) > 0\). Thus \(r=0\) is a stable minimum.

• For \(f(\Xi) > 0\) (i.e., \(\Xi > \Xi_{\text{crit}}\)): The Hessian at \(r=0\) becomes \(-f(\Xi) < 0\) (unstable). New minima appear at \(r^* = \sqrt{f(\Xi)/\beta}\).

Since \(r \ge 0\) is a radial coordinate, this constitutes a supercritical pitchfork bifurcation where the symmetry-broken state \(r^* > 0\) becomes the unique stable equilibrium for \(\Xi > \Xi_{\text{crit}}\).

The bifurcation diagram: for \(\Xi < \Xi_{\text{crit}}\), the system has a single stable fixed point at \(r=0\); for \(\Xi > \Xi_{\text{crit}}\), the origin becomes unstable and two symmetric branches (in the full space, a sphere of radius \(r^*\)) emerge. \(\square\)

E.3 Proof of Theorem Theorem 29

Statement. Under the compact-domain, positive-density, no-flux, mass-preservation, and calibration hypotheses in the theorem, the conditional Landauer-form estimate is \(\dot{\mathcal M}(s)\ge T_c\lvert dH(\rho_s)/ds\rvert\).

Proof

Proof. The WFR equation is \(\partial_s\rho=-\nabla\!\cdot(\rho v)+\rho r\). For \(H(\rho)=-\int\rho\ln\rho\,d\mu_G\), differentiation gives

\[ \frac{dH}{ds}=-\int_{\mathcal Z}(1+\ln\rho)\partial_s\rho\,d\mu_G. \]

The no-flux condition removes the boundary contribution. Mass preservation removes the term \(\int\rho r\,d\mu_G\), so

\[ \frac{dH}{ds} =-\int_{\mathcal Z}\rho\langle\nabla\ln\rho,v\rangle_G\,d\mu_G -\int_{\mathcal Z}\rho r\ln\rho\,d\mu_G. \]

Set

\[ E_v=\int\rho\|v\|_G^2d\mu_G,\quad E_r=\int\rho r^2d\mu_G,\quad I_\rho=\int\rho\|\nabla\ln\rho\|_G^2d\mu_G,\quad J_\rho=\int\rho(\ln\rho)^2d\mu_G. \]

Cauchy–Schwarz in \(L^2(\rho d\mu_G)\) gives

\[ \left|\int\rho\langle\nabla\ln\rho,v\rangle_Gd\mu_G\right| \le\sqrt{I_\rho E_v},\qquad \left|\int\rho r\ln\rho\,d\mu_G\right| \le\sqrt{J_\rho E_r}. \]

Adding the estimates yields \(|dH/ds|\le\sqrt{I_\rho E_v}+\sqrt{J_\rho E_r}\). The theorem’s explicit calibration compares this right-hand side with \(\dot{\mathcal M}=\sigma_{\mathrm{met}}(E_v+\lambda^2E_r)\) and therefore gives the claimed Landauer-form inequality. No de Bruijn identity or universal physical identification is required. \(\square\)

E.4 Proof of Theorem Theorem 30

Statement: The optimal computation budget \(S^*\) satisfies \(\frac{d}{ds} \langle V \rangle_{\rho_s}|_{s=S^*} = \dot{\mathcal{M}}(S^*)\).

Hypothesis: \(S^*\) is an interior point of \([0, S_{\max}]\).

Proof

Define the deliberation functional:

\[ \mathcal{F}(S) = -\int_{\mathcal{Z}} V(z) \rho(S, z) \, d\mu_G + \int_0^S \dot{\mathcal{M}}(u) \, du.\]

The necessary condition for an extremum is \(\mathcal{F}'(S) = 0\). By the Leibniz integral rule:

\[ \mathcal{F}'(S) = -\int_{\mathcal{Z}} V(z) \partial_s \rho(S, z) \, d\mu_G + \dot{\mathcal{M}}(S).\]

Using the result that \(\partial_s \rho\) is governed by the WFR operator \(\mathcal{L}_{\text{WFR}}\):

\[ \mathcal{F}'(S) = -\int_{\mathcal{Z}} V \mathcal{L}_{\text{WFR}}\rho \, d\mu_G + \dot{\mathcal{M}}(S).\]

By the adjoint property of the WFR operator (the formal \(L^2(\rho)\) adjoint):

\[ \int V \mathcal{L}_{\text{WFR}}\rho \, d\mu_G = \int \rho \mathcal{L}_{\text{WFR}}^* V \, d\mu_G,\]

where \(\mathcal{L}_{\text{WFR}}^* V = -\langle \nabla V, v \rangle_G + Vr\) (transport-adjoint plus reaction).

For gradient flows in the covariant case, \(v = -G^{-1}\nabla_A V\) with \(\nabla_A V := \nabla V - A\):

\[ \mathcal{L}_{\text{WFR}}^* V = G^{-1}(\nabla V, \nabla_A V) + Vr.\]

Thus:

\[ \mathcal{F}'(S) = -\int \rho \left( G^{-1}(\nabla V, \nabla_A V) + Vr \right) d\mu_G + \dot{\mathcal{M}}(S).\]

In the conservative case (\(A=0\)), \(G^{-1}(\nabla V, \nabla_A V) = \|\nabla V\|_G^2\), the power dissipated by the value-gradient flow. The stationarity condition \(\mathcal{F}'(S^*) = 0\) gives:

\[ \frac{d}{ds} \langle V \rangle_{\rho_s}\bigg|_{s=S^*} = \dot{\mathcal{M}}(S^*).\]

This states that the optimal stopping time \(S^*\) is reached when the power dissipated by the value-gradient flow exactly matches the metabolic cost rate. \(\square\)

E.5 Proof of Theorem Definition 167

Statement: \(F_{\text{total}} = -G^{-1}\nabla_A V + \beta_{\text{exp}} G^{-1}\nabla\Psi_{\text{causal}}\).

Hypothesis: The agent’s path minimizes \(\mathcal{S} = \int L(z, \dot{z}) \, dt\) with Lagrangian \(L = \frac{1}{2}\|\dot{z}\|_G^2 - (V + \beta_{\text{exp}}\Psi_{\text{causal}})\).

Proof

The Euler-Lagrange equations for the functional are:

\[ \frac{d}{dt} \frac{\partial L}{\partial \dot{z}^k} - \frac{\partial L}{\partial z^k} = 0.\]

Computing the momentum:

\[ \frac{\partial L}{\partial \dot{z}^k} = \frac{\partial}{\partial \dot{z}^k}\left( \frac{1}{2}G_{ij}(z)\dot{z}^i \dot{z}^j \right) = G_{kj}\dot{z}^j = p_k.\]

Time derivative of momentum:

\[ \frac{d}{dt}(G_{kj}\dot{z}^j) = G_{kj}\ddot{z}^j + \frac{\partial G_{kj}}{\partial z^m}\dot{z}^m \dot{z}^j.\]

Potential gradient:

\[ \frac{\partial L}{\partial z^k} = \frac{1}{2}\frac{\partial G_{ij}}{\partial z^k}\dot{z}^i\dot{z}^j - \partial_k V - \beta_{\text{exp}}\partial_k \Psi_{\text{causal}}.\]

Euler-Lagrange equation:

\[ G_{kj}\ddot{z}^j + \frac{\partial G_{kj}}{\partial z^m}\dot{z}^m \dot{z}^j - \frac{1}{2}\frac{\partial G_{ij}}{\partial z^k}\dot{z}^i\dot{z}^j = -\partial_k V - \beta_{\text{exp}}\partial_k \Psi_{\text{causal}}.\]

Recognizing the Christoffel symbols of the first kind \([ij, k] = \frac{1}{2}(\partial_i G_{jk} + \partial_j G_{ik} - \partial_k G_{ij})\):

\[ G_{kj}\ddot{z}^j + [ij, k]\dot{z}^i\dot{z}^j = -\partial_k V - \beta_{\text{exp}}\partial_k \Psi_{\text{causal}}.\]

Contracting with \(G^{mk}\) and using \(\Gamma^m_{ij} = G^{mk}[ij, k]\):

\[ \ddot{z}^m + \Gamma^m_{ij}\dot{z}^i\dot{z}^j = -G^{mk}\partial_k V - \beta_{\text{exp}} G^{mk}\partial_k \Psi_{\text{causal}}.\]

This is the geodesic equation with forcing terms. In the overdamped limit (Section 22.3), inertia is negligible and the acceleration term vanishes, leaving:

\[ \dot{z}^m = -G^{mk}\partial_k V + \beta_{\text{exp}} G^{mk}\partial_k \Psi_{\text{causal}} = F^m_{\text{total}}.\]

The drift field \(F_{\text{total}}\) is the first-order velocity approximation, proving the additive force of curiosity. \(\square\)

E.6 Proof of Theorem Definition 168

Statement: The macro-ontology \(K\) is interventionally closed iff \(I(K_{t+1}; Z_{\text{micro}, t} | K_t, do(K^{\text{act}}_t)) = 0\).

Hypothesis: Let \(\mathcal{M}\) be a Markov Blanket for \(K\).

Proof

We compare the mutual information under the observational measure \(P\) and the interventional measure \(P_{do(K^{\text{act}})}\).

Observational case: By the Causal Enclosure condition (Section 2.8):

\[ I(K_{t+1}; Z_{\text{micro}, t} | K_t, K^{\text{act}}_t) = 0 \quad \text{under } P.\]

This states that the macro-state \(K_{t+1}\) is conditionally independent of the micro-texture \(Z_{\text{micro}, t}\) given the current macro-state and action.

Interventional case: The \(do(K^{\text{act}}_t)\) operator performs a graph surgery that removes all incoming edges to \(K^{\text{act}}_t\) while preserving all other mechanisms. By Pearl’s Causal Markov Condition [Pearl, 2009]:

\[ P(K_{t+1} | K_t, K^{\text{act}}_t, Z_{\text{micro}, t}) \text{ remains invariant under } do(K^{\text{act}}_t).\]

This is because the mechanism \(P(K_{t+1} | \text{parents}(K_{t+1}))\) is a structural equation that does not depend on how \(K^{\text{act}}_t\) was generated.

Combining the conditions: If the observational distribution satisfies \(I = 0\), then:

\[ P(K_{t+1} | K_t, K^{\text{act}}_t) = P(K_{t+1} | K_t, K^{\text{act}}_t, Z_{\text{micro}, t}) \quad \forall Z_{\text{micro}, t}.\]

Since the mechanism is invariant under intervention:

\[ P(K_{t+1} | K_t, do(K^{\text{act}}_t)) = P(K_{t+1} | K_t, K^{\text{act}}_t) = P(K_{t+1} | K_t, K^{\text{act}}_t, Z_{\text{micro}, t}).\]

Therefore, \(I(K_{t+1}; Z_{\text{micro}, t} | K_t, do(K^{\text{act}}_t)) = 0\).

Contrapositive (violation): If \(I > 0\) under \(do(K^{\text{act}}_t)\), there exists a back-door path through \(Z_{\text{micro}, t}\):

\[ K_t \leftarrow Z_{\text{micro}, t} \to K_{t+1}.\]

This path was confounded in observational data (the correlation between \(Z_{\text{micro}}\) and \(K_{t+1}\) was screened by the policy generating \(K^{\text{act}}_t\)). The intervention breaks this screening, exposing the hidden variable. The remedy is Ontological Expansion (Section 30): promote the relevant component of \(Z_{\text{micro}}\) to a new macro-variable in \(K\). \(\square\)

E.7 Scalar spectral and barrier calculations

Definition 325 (The scalar metric realization)

Use the compact connected smooth scalar manifold and its positive strategic metric from the existing Appendix E.7 setup. The same spectral margin is \(\|G^{-1/2}hG^{-1/2}\|_{\mathrm{op}}<1\) for sign-indefinite perturbations; positive perturbations preserve positivity directly. The joint volume is \(w=\sqrt{\det\widetilde G}\), as in Theorem 63. Compactness makes a fixed smooth positive metric uniformly elliptic. This statement concerns the compact realization already used here, not a replacement of unbounded confinement by an artificial compact domain.

Definition 326 (Scalar form realization)

For the existing real \(C^2\) potential \(U\) bounded below, set \(q_\sigma[u]=\tfrac{\sigma^2}{2}\int|\nabla u|^2+\int U|u|^2\). The form domain is \(H^1\) on a closed manifold or for the Neumann realization, and \(H_0^1\) for the Dirichlet realization. Its self-adjoint operator is \(H_\sigma=-\sigma^2\Delta_{\widetilde G}/2+U\) with the selected boundary condition. The domain is the operator domain associated to this form, not unrestricted \(H^2\) on a domain with boundary. Compact embedding gives compact resolvent for this fixed model.

Definition 327 (Forbidden and allowed regions)

For an energy \(E\) define \(A_E=\{U\le E\}\) and \(K_E=\{U>E\}\). These are sets of a scalar potential. Their relation to payoff basins is tested separately by the unilateral inequalities of Theorem 48.

Theorem 125 (Fixed scalar ground state)

The compact connected nonmagnetic scalar realization has a simple ground eigenvalue \(E_0\) and an eigenfunction positive in the interior.

Proof. Compact resolvent and the lower form bound give a minimizer of the normalized Rayleigh quotient. Replacing it by its modulus does not increase its gradient energy, so a nonnegative minimizer exists. Interior elliptic regularity and the strong maximum principle make it strictly positive there. The scalar heat kernel is positivity improving on a connected domain with the selected Dirichlet or Neumann realization; a bounded real potential preserves this through its positive Feynman–Kac weight. For compact positive time the leading eigenspace of this positivity-improving self-adjoint semigroup is one-dimensional, hence so is the ground eigenspace. In the Dirichlet case the eigenfunction vanishes on the boundary. Every nonempty interior open set contains a relatively compact ball on which its continuous square has a positive minimum, giving positive mass in that open set. This proves stationary positivity, not a dynamical crossing rate. Compact resolvent and simplicity also give \(E_1>E_0\) for this fixed operator. \(\square\)

Definition 328 (Barrier metric at a fixed energy)

Define \(g_E=2(U-E)_+\widetilde G\) and \(d_E(x,A)=\inf_{\gamma:A\to x}\int\sqrt{2(U-E)_+}\,|\dot\gamma|_{\widetilde G}\). The factor two matches the kinetic normalization \(-\sigma^2\Delta/2\). Distances may vanish within an allowed connected region.

Theorem 126 (Exact weighted eigenfunction identity)

For an eigenfunction \((H_\sigma-E)u=0\) in the scalar realization and a bounded smooth real weight \(f\), put \(v=e^{f/\sigma}u\). Then

\[ \frac{\sigma^2}{2}\int|\nabla v|^2 +\int\left(U-E-\frac12|\nabla f|^2\right)|v|^2=0. \]

All integrals use \(d\mu_{\widetilde G}\); the weight is taken constant in the normal direction for the Neumann case. Compactly supported cutoffs give the local form, with their explicit derivative terms retained.

Proof. Test the weak eigenvalue equation against \(e^{2f/\sigma}u\) and take real parts. The gradient product is

\[ \operatorname{Re}\langle\nabla u,\nabla(e^{2f/\sigma}u)\rangle =|\nabla(e^{f/\sigma}u)|^2 -\sigma^{-2}|\nabla f|^2e^{2f/\sigma}|u|^2. \]

Substitution proves the identity. For \(0<\epsilon<1\) and a bounded Lipschitz approximation to \(f=(1-\epsilon)d_E(\cdot,A_E)\), its gradient satisfies \(|\nabla f|^2\le2(1-\epsilon)^2(U-E)_+\) almost everywhere. Weak approximation preserves the resulting energy inequality. On the forbidden region the remaining potential coefficient is at least \((2\epsilon-\epsilon^2)(U-E)\); the allowed-region negative term controls the weighted integral. For a region where \(U-E\ge\delta>0\) and \(f\ge a\),

\[ \int_{\mathrm{region}}|u|^2 \le\frac{e^{-2a/\sigma}}{(2\epsilon-\epsilon^2)\delta} \int_{A_E}(E-U)|u|^2. \]

This is a weighted stationary estimate, including its constants. No dimension-independent \(H^1\to L^\infty\) embedding or uniform pointwise prefactor is used. \(\square\)

Corollary 48 (Metric comparison at fixed potential and energy)

For \(g_1\succeq g_0\) and the same \(U,E\), every path obeys \(\int\sqrt{2(U-E)_+}|\dot\gamma|_{g_1} \ge\int\sqrt{2(U-E)_+}|\dot\gamma|_{g_0}\). Taking infima gives \(d_E^{g_1}\ge d_E^{g_0}\). This compares barrier actions at the same energy. Changing a Hamiltonian generally changes its ground energy as well, so one cannot substitute two different ground energies into this fixed-energy inequality or deduce an ordering of transition probabilities from two upper bounds. \(\square\)

Theorem 127 (Feynman–Kac clock and normalization)

Let \(X_s\) have generator \(\Delta_{\widetilde G}/2\), killed at a Dirichlet boundary or reflected for the Neumann realization. Then

\[ (e^{-tH_\sigma/\sigma^2}\phi)(x) =\mathbb E_x\left[e^{-\sigma^{-2}\int_0^tU(X_s)ds}\phi(X_t)\right], \]

with the survival indicator in the killed case. To obtain the ground vector,

\[ e^{tE_0/\sigma^2}e^{-tH_\sigma/\sigma^2}\phi \longrightarrow\langle u_0,\phi\rangle u_0 \quad\text{in }L^2. \]

Proof. The diffusion generator and multiplication weight give \(\partial_tu=(\Delta/2-U/\sigma^2)u=-H_\sigma u/\sigma^2\) with the same initial and boundary data, which is the Feynman–Kac semigroup. The spectral expansion multiplies each eigencomponent by \(e^{-t(E_n-E_0)/\sigma^2}\); dominated convergence leaves exactly its ground projection. Divide by the nonzero overlap to recover \(u_0\). The original WFR generator is compared separately; Brownian motion here is the process associated to this displayed scalar semigroup. \(\square\)

Corollary 49 (Action-length inequality)

For any absolutely continuous path in the forbidden region, set \(a=|\dot\gamma|_{\widetilde G}\), \(b=\sqrt{2(U-E)}\). Then \(a^2/2+U-E-ab=(a-b)^2/2\ge0\). Integration gives \(\int(|\dot\gamma|^2/2+U-E)dt \ge\int\sqrt{2(U-E)}|\dot\gamma|dt\). Equality is attained on a parametrization with \(a=b\) wherever that parametrization is defined. This relates an action to barrier length. An asymptotic probability additionally belongs to a specified stochastic law and event; it is not furnished by this algebraic inequality.

E.8 Proof of Corollary Remark 18

Statement: \(V_H(z) = T_c^2 \frac{\partial H(\pi)}{\partial T_c}\).

Hypothesis: Let \(\pi(a|z) = \frac{1}{Z} \exp\left(\frac{Q(z,a)}{T_c}\right)\) be the policy, where \(Z = \sum_a \exp(Q/T_c)\) is the partition function. Let \(\beta_{\text{ent}} = 1/T_c\) be the inverse Definition 73.

Proof

Step 1: Express Entropy in terms of \(\beta_{\text{ent}}\). The entropy of the policy is:

\[ H(\pi) = -\sum_a \pi(a) \ln \pi(a).\]

Substituting \(\ln \pi(a) = \beta_{\text{ent}} Q(a) - \ln Z\):

\[ H(\pi) = -\sum_a \pi(a) [\beta_{\text{ent}} Q(a) - \ln Z] = \ln Z - \beta_{\text{ent}} \mathbb{E}_\pi[Q].\]

Step 2: Derivative of Entropy w.r.t. \(\beta_{\text{ent}}\). Differentiating with respect to \(\beta_{\text{ent}}\):

\[ \frac{\partial H}{\partial \beta_{\text{ent}}} = \frac{\partial \ln Z}{\partial \beta_{\text{ent}}} - \mathbb{E}_\pi[Q] - \beta_{\text{ent}} \frac{\partial \mathbb{E}_\pi[Q]}{\partial \beta_{\text{ent}}}.\]

Using the identity \(\frac{\partial \ln Z}{\partial \beta_{\text{ent}}} = \mathbb{E}_\pi[Q]\):

\[ \frac{\partial H}{\partial \beta_{\text{ent}}} = -\beta_{\text{ent}} \frac{\partial \mathbb{E}_\pi[Q]}{\partial \beta_{\text{ent}}} = -\beta_{\text{ent}} \mathrm{Var}_\pi(Q),\]

where we used \(\frac{\partial \mathbb{E}[Q]}{\partial \beta_{\text{ent}}} = \mathrm{Var}(Q)\) (standard fluctuation-response relation).

Step 3: Relate \(\mathrm{Var}(Q)\) to Varentropy. Recall \(\mathcal{I}(a) = -\ln \pi(a) = -\beta_{\text{ent}} Q(a) + \ln Z\). The variance of the surprisal is:

\[ V_H(\pi) = \mathrm{Var}(\mathcal{I}) = \mathrm{Var}(-\beta_{\text{ent}} Q + \ln Z) = \beta_{\text{ent}}^2 \mathrm{Var}(Q).\]

Step 4: Change of variables to \(T_c\). We have \(V_H = \beta_{\text{ent}}^2 \mathrm{Var}(Q)\) and \(\frac{\partial H}{\partial \beta_{\text{ent}}} = -\beta_{\text{ent}} \mathrm{Var}(Q)\). Therefore \(V_H = -\beta_{\text{ent}} \frac{\partial H}{\partial \beta_{\text{ent}}}\).

Using the chain rule \(\frac{\partial}{\partial T_c} = -\frac{1}{T_c^2} \frac{\partial}{\partial \beta_{\text{ent}}}\):

\[ \frac{\partial H}{\partial T_c} = -\frac{1}{T_c^2} \frac{\partial H}{\partial \beta_{\text{ent}}} = \frac{1}{T_c^2} \cdot \beta_{\text{ent}} \mathrm{Var}(Q) = \frac{V_H}{T_c^2 \cdot \beta_{\text{ent}}} = \frac{V_H}{T_c}.\]

Final Result: Rearranging yields:

\[ V_H(z) = T_c \frac{\partial H(\pi)}{\partial T_c} = \beta_{\text{ent}}^2 \mathrm{Var}(Q) = C_v.\]

This proves that Varentropy equals the heat capacity and measures the sensitivity of the entropy to temperature fluctuations. \(\square\)

E.9 Proof of Corollary Corollary 13

Statement: For a bimodal policy on a value ridge, \(V_H\) is significant, distinguishing it from uniform noise.

Hypothesis: Let \(\pi\) be a mixture of two dominant modes with values \(Q_1, Q_2\) and a background of \(N-2\) negligible modes.

Proof

Step 1: Variance of Surprisal Form.

\[ V_H = \mathbb{E}[\mathcal{I}^2] - (\mathbb{E}[\mathcal{I}])^2.\]

Since \(\mathcal{I} = -\beta Q + \ln Z\), we have \(V_H = \beta^2 \mathrm{Var}(Q)\).

Step 2: Two-Point Statistics. Consider two actions \(a_1, a_2\) with probabilities \(p, 1-p\). The variance of a Bernoulli variable taking values \(Q_1, Q_2\) is:

\[ \mathrm{Var}(Q) = p(1-p)(Q_1 - Q_2)^2.\]

Thus:

\[ V_H = \beta^2 p(1-p) (\Delta Q)^2 = p(1-p) \left( \frac{\Delta Q}{T_c} \right)^2.\]

For equally weighted modes (\(p = 1/2\)), this simplifies to:

\[ V_H = \frac{1}{4} \left( \frac{\Delta Q}{T_c} \right)^2.\]

Step 3: Interpretation of \(\Delta Q\). \(\Delta Q\) is the value gap between the two modes.

• Perfect Symmetry (The Ridge): If \(Q_1 = Q_2\) exactly, then \(\Delta Q = 0 \implies V_H = 0\).

• Structural Instability: When the agent is slightly off-center or when sampling includes the tails, the effective \(\Delta Q > 0\).

Step 4: Distinguishing Structure from Noise. For a distribution with structure (peaks and valleys), \(\mathrm{Var}(Q) > 0\). For a flat distribution (noise), \(\mathrm{Var}(Q) = 0\).

Specifically, on a ridge, the agent samples \(a_{\text{left}}\) and \(a_{\text{right}}\) (high \(Q\)) but also transitively samples the separating region (lower \(Q\)) during exploration. The variance of \(Q\) along the trajectory corresponds to \(V_H\):

\[ V_H \propto (\Delta Q_{\text{peak-valley}})^2.\]

This proves that \(V_H\) detects the topological feature (the valley) that distinguishes a fork from a flat plane. \(\square\)

E.10 Proof of Corollary Corollary 12

Statement: To maintain stability, the cooling rate must satisfy \(|\dot{T}_c| \ll T_c / \sqrt{V_H}\).

Hypothesis: We require the probability distribution \(\pi_t\) to remain close to the equilibrium Boltzmann distribution \(\pi^*_{T_c(t)}\) during annealing. This is the Adiabatic Condition.

Proof

Step 1: Thermodynamic Speed. The rate of change of the policy distribution with respect to temperature is measured by the Fisher Information metric \(g_{TT}\) on the statistical manifold parameterized by \(T_c\):

\[ g_{TT} = \mathbb{E}\left[ \left( \frac{\partial \ln \pi}{\partial T_c} \right)^2 \right].\]

Step 2: Relate Fisher Metric to Varentropy. Recall \(\ln \pi = \frac{Q}{T_c} - \ln Z\). Then:

\[ \frac{\partial \ln \pi}{\partial T_c} = -\frac{Q}{T_c^2} + \frac{\mathbb{E}[Q]}{T_c^2} = -\frac{1}{T_c^2}(Q - \mathbb{E}[Q]).\]

Substituting into the Fisher definition:

\[ g_{TT} = \frac{1}{T_c^4} \mathbb{E}\left[ (Q - \mathbb{E}[Q])^2 \right] = \frac{\mathrm{Var}(Q)}{T_c^4}.\]

Using \(V_H = \frac{\mathrm{Var}(Q)}{T_c^2}\) (from Proof E.8):

\[ g_{TT} = \frac{V_H}{T_c^2}.\]

Step 3: Thermodynamic Length. The “distance” traversed in probability space for a small temperature change \(dT_c\) is \(ds^2 = g_{TT} dT_c^2\):

\[ ds = \sqrt{g_{TT}} |dT_c| = \frac{\sqrt{V_H}}{T_c} |dT_c|.\]

Step 4: Adiabatic Condition. For the system to relax to equilibrium (stay in the basin of attraction), the speed of change in distribution space must be bounded:

\[ \left| \frac{ds}{dt} \right| \leq C \cdot \tau_{\text{relax}}^{-1}.\]

Substituting \(ds/dt\):

\[ \frac{\sqrt{V_H}}{T_c} \left| \frac{dT_c}{dt} \right| \leq C.\]

Solving for the cooling rate:

\[ \left| \frac{dT_c}{dt} \right| \leq C \frac{T_c}{\sqrt{V_H}}.\]

Conclusion: When Varentropy \(V_H\) is large (phase transition/critical point), the permissible cooling rate goes to zero. The Governor must apply the “Varentropy Brake” to prevent quenching the system into a suboptimal metastable state. \(\square\)

E.11 Proof of Corollary Remark 36

Statement: \(\nabla \Psi_{\text{causal}} \propto \nabla \mathbb{E}_{z'} [ V_H[P(\theta_W | z, a, z')] ]\).

Hypothesis: We define \(\Psi_{\text{causal}}\) as the Expected Information Gain (EIG) about model parameters \(\theta\) given a transition \((z, a) \to z'\).

Proof

Step 1: Definition of EIG.

\[ \text{EIG}(z, a) = I(\theta; z' | z, a) = H(z' | z, a) - \mathbb{E}_{\theta} [ H(z' | z, a, \theta) ].\]

This is the Total Predictive Entropy minus the Expected Aleatoric Entropy.

Step 2: Decomposition of Uncertainty. For the “noisy TV” case (outcomes are stochastic noise independent of \(\theta\)):

\[ H(z' | z, a, \theta) \approx H(z' | z, a) \implies \text{EIG} \approx 0.\]

Step 3: Varentropy as Structure Detector. The varentropy \(V_H(z' | z, a)\) measures the variance of log-probabilities.

• Uniform noise: \(V_H^{\text{noise}} \to 0\) (all outcomes equally likely).

• Structured uncertainty: \(V_H^{\text{structured}} > 0\) (some outcomes much more likely).

Step 4: Connection to Multimodality. If the model is uncertain about structure (\(\theta\)), the predictive distribution \(p(z')\) is a mixture of distinct hypotheses \(p(z'|\theta_1), p(z'|\theta_2)\). As established in Proof E.9, a mixture of distinct modes has high Varentropy compared to a broad unimodal distribution (noise).

Step 5: Operational Equivalence. Thus, maximizing EIG is functionally equivalent to maximizing the Varentropy of the expected outcome, provided the aleatoric noise floor is constant:

\[ \nabla \Psi_{\text{causal}} \propto \nabla \mathrm{Var}_{z' \sim p(z'|z,a)} [ -\ln p(z'|z,a) ].\]

Conclusion: The agent should seek states where the World Model’s prediction has high Varentropy (conflicting hypotheses), as these offer the maximum potential for falsification (reduction of parameter variance). \(\square\)

E.12 Bellman generator and scalar wave variation

Proof

For the diffusion and discount already used in Theorem 12, write \(\mathcal L=b\cdot\nabla+T_c\Delta_G\) and \(\gamma_h=e^{-\lambda h}\). The smooth Bellman equation has continuous-time form

\[ \partial_tV+\mathcal LV-\lambda V+r=0. \]

In its stationary zero-drift sector, \((-\Delta_G+\lambda/T_c)V=r/T_c\); denote this screening coefficient by \(\kappa_B^2=\lambda/T_c\). The scalar field action used in this chapter instead defines the wave operator

\[ \Box_g=-|g|^{-1/2}\partial_\mu(|g|^{1/2}g^{\mu\nu}\partial_\nu), \qquad(\Box_g+\kappa^2)V=\rho_r. \]

The stationary operators coincide under the coefficient identification \(\kappa^2=\kappa_B^2\) and the same sources and boundary realization.

Proof. Generator consistency gives \(\mathbb E[V(Z_h,t+h)]=V+h(\partial_t+\mathcal L)V+o(h)\). Insert this and \(e^{-\lambda h}=1-\lambda h+o(h)\) into \(V=rh+e^{-\lambda h}\mathbb E[V(Z_h,t+h)]\), cancel \(V\), and divide by \(h\). The \(\partial_t^2V\) Taylor term has coefficient \(h/2\) after division and vanishes. Finite signal speed does not change that coefficient. For the wave model vary

\[ S[V]=\int\left[-\tfrac12g^{\mu\nu}\partial_\mu V\partial_\nu V -\tfrac12\kappa^2V^2+\rho_rV\right]\sqrt{|g|}\,dx. \]

Integration by parts against a compactly supported variation \(\eta\) gives \(\delta S=\int\eta[-\Box_gV-\kappa^2V+\rho_r]\sqrt{|g|}\,dx\). Stationarity proves the field equation. For a fixed product metric \(g=\operatorname{diag}(-c^2,G)\), \(\Box_g=c^{-2}\partial_t^2-\Delta_G\). These are explicit equations for two defined evolutions; equality of their stationary operators is the comparison established here. \(\square\)

E.13 Polar WFR calculation with exact signs

Proof

On a smooth positive-density chart with the fixed spatial metric of the WFR equations, put \(R=\sqrt\rho\), \(p=dV-B\), \(v=G^{-1}p\), \(D=\nabla-iB/\sigma\), and \(Q_B=-\sigma^2\Delta_GR/(2R)\). For the equations already stated, \(\partial_s\rho+\operatorname{div}_G(\rho v)=r\rho\) and \(\partial_sV+|p|_G^2/2+\Phi_{\mathrm{eff}}=0\), the exact amplitude equation is

\[ i\sigma\partial_s\psi= \left[-\tfrac{\sigma^2}{2}\Delta_B+\Phi_{\mathrm{eff}}-Q_B +\tfrac{i\sigma}{2}r\right]\psi, \qquad\psi=Re^{iV/\sigma}. \]

Proof. The product rule gives

\[ \frac{\Delta_B\psi}{\psi}=\frac{\Delta_GR}{R} -\frac{|p|_G^2}{\sigma^2} +\frac{i}{\sigma}\left(2\frac{\langle dR,p\rangle_G}{R} +\operatorname{div}_Gv\right). \]

Thus the kinetic real part is \(Q_B+|p|_G^2/2\). The \(-Q_B\) term cancels it to the given classical HJB expression. The imaginary part is \(-\sigma\operatorname{div}_G(\rho v)/(2\rho)+\sigma r/2\), equal to \(\sigma\partial_s\rho/(2\rho)\) by continuity. The time derivative is \(i\sigma\partial_s\psi/\psi=i\sigma\partial_s\rho/(2\rho)-\partial_sV\), so both parts agree. Reading these two parts backwards proves the local equivalence. At zeros use the density/current equations without division by \(R\); a global phase additionally retains its existing circulation data.

The compensating \(Q_B\) depends on \(|\psi|\), making this amplitude equation nonlinear. Omitting the compensation gives the distinct linear Schrödinger model with a \(+Q_B\) term in its Hamilton–Jacobi equation. A curl-modified mobility must be substituted into its own continuity equation; the above Laplacian yields precisely the canonical velocity \(G^{-1}p\). For time-dependent volume density \(w_s\), conservation reads \(\partial_s(w_s\rho)+\partial_i(w_s\rho v^i)=w_sr\rho\); the corresponding amplitude equation acquires \(-i\sigma\partial_s\log w_s/2\). \(\square\)

E.14 Complete-history conditional kernels

Proof

On the standard measurable path spaces of the specified process, the complete history state \((t,\mathsf H_t)\) is Markov with its conditional extension kernel. Neither a positive delay alone nor the reward occupation screen determines whether a smaller state is Markov.

Proof. The sigma-algebra generated by \((t,\mathsf H_t)\) contains the entire modeled history through \(t\). Let \(K_{t,u}(h,\cdot)\) be the regular conditional law of the extended history through \(u\) given \(\mathsf H_t=h\). For a bounded history functional \(F\),

\[ \mathbb E[F(\mathsf H_u)\mid\sigma(\mathsf H_s:s\le t)] =\mathbb E[F(\mathsf H_u)\mid\mathsf H_t] =K_{t,u}F(\mathsf H_t). \]

The tower property gives \(K_{t,u}=K_{t,v}K_{v,u}\) on realized histories. Including \(t\) in the state accounts for time-inhomogeneous coefficients. This proves the representation without identifying a finite compression. For the occupation screen take \(\alpha=0\): every history has screen zero, while histories with the same current position can have different \(z_{t-\tau}\) and therefore different delayed drifts. Conversely a delayed signal with zero coupling leaves a Markov local process Markov. These examples prove both limitations of the smaller-state claims. \(\square\)

E.15 Time averages and unilateral variations

Proof

For a bounded differentiable density trajectory, \(T^{-1}\int_0^T\partial_t\rho\,dt=(\rho(T)-\rho(0))/T\to0\). This holds for many nonequilibrium trajectories and does not test unilateral payoff improvements. Moreover \(\langle\rho v\rangle\) need not vanish when \(\langle v\rangle=0\): on a periodic clock take \(v=\sin t\) and \(\rho=1+\epsilon\sin t\), \(0<\epsilon<1\), giving \(\langle\rho v\rangle=\epsilon/2\). Standing-wave expansions describe solutions of the defined wave operator; Nash conditions are the payoff inequalities in Theorem 48. \(\square\)

E.16 Strategic Hessian and response composition

Proof

Use the smooth local best-response branch and Strategic Jacobian already specified in Definition 329. With the intrinsic connection on agent \(j\)’s manifold define the covariant tensor

\[ H^{(i)}_{jj,mn}=\nabla^{(j)}_m\nabla^{(j)}_nV^{(i)},\qquad \mathcal G^{(i)}_{ij,ab}=\mathcal J_{ji}^{m}{}_{a} H^{(i)}_{jj,mn}\mathcal J_{ji}^{n}{}_{b}. \]

No additional lowering of the Hessian indices is applied. The strategic metric prescription is \(\widetilde G^{(i)}=G^{(i)}+h^{(i)}\) with \(h^{(i)}=\sum_{j\ne i}\beta_{ij}\mathcal G^{(i)}_{ij}\). Its positive-definite domain is checked using the spectral margin already specified in Definition 325. The curvature equation Theorem 6 remains a separate differential identity; this algebraic prescription is not its solution.

For a \(C^2\) response \(y=b(x)\), direct differentiation gives

\[ \partial_{ab}V(x,b(x))=V_{ab}+V_{am}b^m_b+V_{bm}b^m_a +V_{mn}b^m_ab^n_b+V_m\partial_{ab}b^m. \]

Thus the pulled-back \(H_{jj}\) is one contribution, not the full Hessian of the composed value. The final term vanishes at a stationary point in \(y\). For the positive metric \(\widetilde G=G+h\), subtraction of the two metric-compatible torsion-free connections gives the exact identity

\[ \widetilde\Gamma^a_{bc}-\Gamma^a_{bc} =\tfrac12\widetilde G^{ad} (\nabla_bh_{dc}+\nabla_ch_{db}-\nabla_dh_{bc}). \]

Replacing \(\widetilde G^{-1}\) by \(G^{-1}\) gives its first-order expansion, with a remainder controlled by the inverse-metric identity \(\widetilde G^{-1}-G^{-1}=-G^{-1}h\widetilde G^{-1}\).

Definition 329 (Strategic Jacobian on the established response branch)

On the smooth, nondegenerate local best-response branch specified in the original strategic construction, differentiate \(\nabla_jV_j(b_j(z_{-j}),z_{-j})=0\). The chain rule gives \(H_{jj}^{(j)}\mathcal J_{ji}+H_{ji}^{(j)}=0\), hence \(\mathcal J_{ji}=-(H_{jj}^{(j)})^{-1}H_{ji}^{(j)}\). This identifies the branch derivative on the domain where the declared inverse exists. It is not a global selection of a multivalued best-response correspondence. Its pullback action on covariant Hessians is exactly that used in Definition 208.

E.17 Curvature and Bianchi identity

Proof

For the defined connection, the adjoint covariant derivative is \(\mathcal D_\rho F_{\mu\nu}=\partial_\rho F_{\mu\nu}-ig[A_\rho,F_{\mu\nu}]\). On a test section \(u\), expansion gives \([D_\rho,F_{\mu\nu}]u=(\mathcal D_\rho F_{\mu\nu})u\). Insert \([D_\mu,D_\nu]=-igF_{\mu\nu}\) into \([D_\rho,[D_\mu,D_\nu]]+\mathrm{cyclic}=0\). Dividing by \(-ig\) gives

\[ \mathcal D_\rho F_{\mu\nu}+\mathcal D_\mu F_{\nu\rho} +\mathcal D_\nu F_{\rho\mu}=0. \]

For \(g=0\) the same identity is \(d(dA)=0\). The geometric Christoffel terms cancel under cyclic antisymmetrization for the torsion-free connection. The identity holds in each smooth gauge chart and is preserved under transition functions by conjugation; nontrivial bundle topology does not violate it. \(\square\)

E.18 Vacuum expansion in the declared representation

Proof

For the stable quartic potential already specified, \(\lambda>0\) and \(\mu^2<0\), write \(\Phi_0=(v/\sqrt2)n\), \(n^\dagger n=1\) and \(v^2=-\mu^2/\lambda\). The radial mass is \(m_h^2=2\lambda v^2\). The gauge quadratic term is \(-\tfrac12A_\mu^a(M^2)_{ab}A^{\mu b}\) with

\[ (M^2)_{ab}=g^2\Phi_0^\dagger\{T_a,T_b\}\Phi_0. \]

Proof. Put \(q=\Phi^\dagger\Phi\). The minimum solves \(\mu^2+2\lambda q=0\). Substitute \(q=(v+h)^2/2\); the coefficient of \(h^2\) in \(U\) is \(\lambda v^2=m_h^2/2\). For a constant vacuum, \(D_\mu\Phi_0=-igA_\mu^aT_a\Phi_0\); the symmetric product of the commuting coefficients \(A^aA^b\) gives the displayed anticommutator. For any real \(u^a\), \(u^a(M^2)_{ab}u^b=2g^2\|(u^aT_a)\Phi_0\|^2\ge0\). Its kernel is the stabilizer Lie algebra of the vacuum. For a single \(SU(2)\) doublet with \(T_a=\sigma_a/2\), this evaluates to \((M^2)_{ab}=g^2v^2\delta_{ab}/4\). It is this representation that gives \(m_A=gv/2\). Other declared representations are evaluated by the same matrix formula. If \(\lambda\le0\), the stated stable quartic expansion does not apply; the potential itself reveals the failure. \(\square\)

E.19 Spectral minimization and the Nash comparison

Proof

The Rayleigh quotient of the scalar Hamiltonian minimizes its single joint energy. The Nash test instead compares each agent’s own payoff under a unilateral change. Their equality must be checked by differentiating the actual objectives and by evaluating their global inequalities.

Proof by explicit comparison. Let \(V_1(x,y)=-(x-y)^2\) and \(V_2(x,y)=-(y-1)^2-Kx\) with \(K>0\). Both are strictly concave in their own coordinate; their best responses are \(x=y\) and \(y=1\), hence the unique Nash profile is \((1,1)\). The sum of costs is \(U=(x-y)^2+(y-1)^2+Kx\). Its \(x\) derivative at \((1,1)\) is \(K\), so Nash is not a stationary point of this joint energy. This example meets the smooth nondegenerate best-response structure of the Strategic Jacobian. Thus that machinery cannot justify the former general identification of Nash with a joint ground state. The ground-state and variational calculations retain their meaning for the scalar operator actually defined. \(\square\)

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