RTSG Equations — Entropy-Time Duals
April 2026 · Companion to RTSG Equations
Every RTSG equation that contains \(\partial/\partial t\) has an entropy-time dual obtained by the master substitution \(\partial_t = \dot\Sigma\,\partial_\Sigma\). This page collects all duals in one reference.
The Master Substitution
\[\frac{\partial}{\partial t} = \dot\Sigma\,\frac{\partial}{\partial\Sigma}, \qquad \dot\Sigma = -\mathrm{Tr}(\dot\rho_{PS}\ln\rho_{PS}), \qquad \Sigma = -\mathrm{Tr}(\rho_{PS}\ln\rho_{PS})\]
Foundation
| Name | Clock-Time | Entropy-Time |
| Arrow of time | Axiom: \(\dot\Sigma > 0\) | Theorem-candidate: follows from \(\mu = +\delta\Sigma/\delta\bar{W}\) |
| Collapse | \(PS = QS/\!\sim_{\text{bisim}}\) | Same — \(\Sigma\) measures quotient diversity |
| Unitarity | \(\pi \circ U_t = \bar{U}_t \circ \pi\) | \(\pi \circ U_\Sigma = \bar{U}_\Sigma \circ \pi\) |
Will Field
| Name | Clock-Time | Entropy-Time |
| SDE | \(dw = \mu\,dt + \sigma\,dW_t\) | \(dw = (\mu/\dot\Sigma)\,d\Sigma + (\sigma/\sqrt{\dot\Sigma})\,dW_\Sigma\) |
| GL Action | $S = \int( | \partial W |
| EOM | $\Box W - \alpha W - \beta | W |
| d'Alembertian | \(\Box = -\partial_t^2 + \nabla^2\) | \(\Box_\Sigma = -(1/\dot\Sigma)\partial_\Sigma(\dot\Sigma\,\partial_\Sigma) + \nabla^2\) |
| Energy density | $\rho_W = | \partial W |
| Drift | \(\mu = -\delta S/\delta\bar{W}\) | \(\mu = +\delta\Sigma/\delta\bar{W}\) (via Action-Entropy Identity) |
| Utility | \(U = \text{value}/(\text{energy} \times \text{time})\) | \(U = \text{value}/(\text{energy} \times \Delta\Sigma / \dot\Sigma)\) |
Lyapunov Classification
| Regime | Clock-Time \(\lambda\) | Entropy-Time \(\lambda_\Sigma\) | \(\dot\Sigma\) | Meaning |
| Stable attractor | \(< 0\) | \(< 0\) | Moderate | GL ground state |
| Flow / critical | \(\approx 0\) | \(\approx 0\) | High | Phase transition |
| Dissolution | \(> 0\) | \(> 0\) | \(\to \infty\) or \(0\) | Structural explosion or stasis |
Interaction Matrices
| Name | Clock-Time | Entropy-Time |
| K-matrix | \(K_{ij}\) = coupling (spectral gap = coherence) | Unchanged — \(K\) is structural, not temporal |
| R-matrix | \(R_{AB} = \cos\theta(\mathbf{I}_A, \mathbf{I}_B)\) | Unchanged |
| J-matrix | Idea interaction | Unchanged |
| Interface | \(\mathbf{I}_{\text{eff}} = \mathcal{I} \cdot K \cdot \mathbf{I}\) | Reports \(\dot\Sigma\) per cognitive dimension |
Cosmology
| Name | Clock-Time | Entropy-Time |
| Gravity | \(\kappa_{\text{grav}} = \lim_{\dim(CS)\to 1}\kappa\) | Same — gravity = Stage 0 entropy production |
| Dark matter | Uncondensed \(W\) (\(\alpha > 0\)) | Maximum entropy phase |
| \(\Lambda_{\text{eff}}\) | \(\sim\langle\rho_W\rangle_{PS}\) | $\sim\langle\dot\Sigma^2 |
| Dark energy EOS | \(w = -1\) (constant) | \(w = -1 + d\ln\dot\Sigma/d\ln a\) — evolving |
Millennium Problems
| Problem | Clock-Time Formulation | Entropy-Time Formulation |
| Yang-Mills gap | \(\Delta = \sqrt{2\alpha} = 1/\xi_W\) | Same + entropic: \(\Delta\) = minimum entropy cost of excitation |
| Riemann Hypothesis | \(K_\theta\) on \(L^2(\Gamma\backslash\mathbb{H})\) | Spectral parameter as entropy variable (speculative) |
| Navier-Stokes | \(\sup_K\int_0^T \mathcal{D}_K^+\,dt < \infty\) | \(\sup_K\int_0^{\Sigma_{\max}} \mathcal{D}_K^+\,d\Sigma < \infty\) — bounded domain |
The Action-Entropy Identity
\[S_E[W] = -\Sigma + \text{const}\]
This single equation relates the left column to the right column throughout this page. It is the bridge between the variational (action) and thermodynamic (entropy) formulations of RTSG.
Moral Framework
| Clock-Time | Entropy-Time |
| Maximize \(\text{Id}_{\text{extended}}\) (total life force) | Maximize aggregate \(\dot\Sigma\) across all agents |
| Moral = life-force-producing | Moral = entropy-producing |
| Immoral = life-force-suppressing | Immoral = entropy-suppressing |
See Also