(sec-appendix-e-rigorous-proof-sketches-for-ontological-and-metabolic-laws)= # {ref}`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. :::{div} feynman-prose 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 $\rho \in \mathcal{P}(\mathcal{Z})$. (sec-appendix-e-proof-of-theorem-prf-ref)= ## E.1 Proof of Theorem {prf:ref}`thm-fission-criterion` **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-thm-the-fission-trigger)= :::{prf:proof} Consider the discrete variation $\Delta \mathcal{S} = \mathcal{S}[N_c + 1] - \mathcal{S}[N_c]$. By the definition of the Ontological Action ({ref}`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. $\square$ ::: (sec-appendix-e-proof-of-theorem-prf-ref-a)= ## E.2 Proof of Theorem {prf:ref}`thm-supercritical-pitchfork-bifurcation-for-charts` **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-thm-supercritical-pitchfork-bifurcation)= :::{prf: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. $\square$ ::: (sec-appendix-e-proof-of-theorem-prf-ref-b)= ## E.3 Proof of Theorem {prf:ref}`thm-generalized-landauer-bound` **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-thm-generalized-landauer-bound)= :::{prf: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 {prf:ref}`thm-deliberation-optimality-condition` **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-thm-deliberation-optimality)= :::{prf: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. $\square$ ::: (sec-appendix-e-proof-of-theorem-prf-ref-d)= ## E.5 Proof of Theorem {prf:ref}`thm-augmented-drift-law` **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-thm-the-augmented-drift-law)= :::{prf: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** ({ref}`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. $\square$ ::: (sec-appendix-e-proof-of-theorem-prf-ref-e)= ## E.6 Proof of Theorem {prf:ref}`thm-interventional-closure` **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-thm-interventional-closure)= :::{prf: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 ({ref}`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 {cite}`pearl2009causality`: $$ P(K_{t+1} | K_t, K^{\text{act}}_t, Z_{\text{micro}, t}) \text{ remains invariant under } do(K^{\text{act}}_t). $$ 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** ({ref}`Section 30 `): promote the relevant component of $Z_{\text{micro}}$ to a new macro-variable in $K$. $\square$ ::: (sec-appendix-e-rigorous-proof-of-multi-agent-strategic-tunneling)= (proof-thm-e7-ground-state-positivity)= (proof-thm-e7-agmon-decay-bound)= (proof-cor-e7-adversarial-suppression)= (proof-thm-e7-feynman-kac)= (sec-appendix-e-ground-state-existence)= (pi-spectral-gap)= (insight-tunneling-inevitable)= (pi-agmon-estimates)= (interpretation-adversarial-barrier)= (pi-feynman-kac)= (pi-large-deviation)= ## E.7 Scalar spectral and barrier calculations :::{prf:definition} The scalar metric realization :label: def-e7-strategic-metric 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 {prf:ref}`thm-game-augmented-laplacian`. 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. ::: :::{prf:definition} Scalar form realization :label: def-e7-strategic-hamiltonian 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. ::: :::{prf:definition} Forbidden and allowed regions :label: def-e7-forbidden-region 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 {prf:ref}`thm-nash-equilibrium-as-geometric-stasis`. ::: :::{prf:theorem} Fixed scalar ground state :label: thm-e7-ground-state-positivity 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. $\square$ ::: :::{prf:definition} Barrier metric at a fixed energy :label: def-e7-agmon-metric 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. ::: :::{prf:theorem} Exact weighted eigenfunction identity :label: thm-e7-agmon-decay-bound 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. $\square$ ::: :::{prf:corollary} Metric comparison at fixed potential and energy :label: cor-e7-adversarial-suppression 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. $\square$ ::: :::{prf:theorem} Feynman--Kac clock and normalization :label: thm-e7-feynman-kac 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. $\square$ ::: :::{prf:corollary} Action-length inequality :label: cor-e7-large-deviations 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. ::: (sec-appendix-e-proof-of-corollary-varentropy-stability)= ## E.8 Proof of Corollary {prf:ref}`cor-varentropy-stability` **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 {prf:ref}`def-cognitive-temperature`. (proof-cor-varentropy-stability)= :::{prf: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. $\square$ ::: (sec-appendix-e-proof-of-corollary-bimodal-instability)= ## E.9 Proof of Corollary {prf:ref}`cor-bimodal-instability` **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-cor-bimodal-instability)= :::{prf: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. $\square$ ::: (sec-appendix-e-proof-of-corollary-varentropy-brake)= ## E.10 Proof of Corollary {prf:ref}`cor-varentropy-brake` **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-cor-varentropy-brake)= :::{prf: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. $\square$ ::: (sec-appendix-e-proof-of-corollary-epistemic-curiosity-filter)= ## E.11 Proof of Corollary {prf:ref}`cor-epistemic-curiosity-filter` **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-cor-epistemic-curiosity-filter)= :::{prf: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). $\square$ ::: (proof-hjb-klein-gordon)= (proof-madelung-transform)= (proof-markov-restoration)= (proof-nash-standing-wave)= (proof-game-tensor-derivation)= (proof-bianchi-identity)= (proof-higgs-mechanism)= (proof-nash-ground-state)= (sec-references)= ## E.12 Bellman generator and scalar wave variation :::{prf:proof} For the diffusion and discount already used in {prf:ref}`thm-the-hjb-helmholtz-correspondence`, 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 :::{prf: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 :::{prf: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 :::{prf: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 {prf:ref}`thm-nash-equilibrium-as-geometric-stasis`. $\square$ ::: ## E.16 Strategic Hessian and response composition :::{prf:proof} Use the smooth local best-response branch and Strategic Jacobian already specified in {prf:ref}`def-strategic-jacobian`. 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 {prf:ref}`def-e7-strategic-metric`. The curvature equation {prf:ref}`thm-capacity-constrained-metric-law` 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 $\widetilde G^{-1}-G^{-1}=-G^{-1}h\widetilde G^{-1}$. ::: :::{prf:definition} Strategic Jacobian on the established response branch :label: def-strategic-jacobian 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 {prf:ref}`def-the-game-tensor`. ::: ## E.17 Curvature and Bianchi identity :::{prf: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 :::{prf: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 :::{prf: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. $\square$ ::: ## References ```{bibliography} ```