The Action-Entropy Identity¶

Theorem Candidate · Veronika Pokrovskaia (@B_Veronika) · April 2026

Statement¶

Let \(W\) be the Will Field on the RTSG configuration space, \(S_E[W]\) its Euclidean Ginzburg-Landau action, and \(\Sigma = -\mathrm{Tr}(\rho_{PS}\ln\rho_{PS})\) the von Neumann entropy of the bisimulation quotient \(PS = QS/\!\sim_{\text{bisim}}\). Then:

\[\boxed{S_E[W] = -\Sigma + \text{const}}\]

up to a topological constant independent of field configuration.

The Wick Rotation Bridge¶

In the Lorentzian path integral, field configurations are weighted by \(e^{iS}\) — oscillatory, quantum. Wick rotation (\(t \to i\tau\)) transforms this to:

\[e^{iS[W]} \;\longrightarrow\; e^{-S_E[W]}\]

The Euclidean GL action after Wick rotation:

\[S_E[W] = \int\left(|\partial_\tau W|^2 + |\nabla W|^2 + \alpha|W|^2 + \tfrac{\beta}{2}|W|^4\right)d\tau\,d^3x\]

All plus signs. This is exactly a Ginzburg-Landau free energy functional — and free energy minimization IS entropy maximization. Therefore \(S_E = -\Sigma\).

Consequences¶

1. Drift = Entropy Gradient Ascent¶

The Will Field drift was \(\mu = -\delta S/\delta\bar{W}\). With \(S = -\Sigma\):

\[\mu = +\frac{\delta\Sigma}{\delta\bar{W}}\]

The Will Field drifts up the entropy gradient. Arrow of time falls out of the equation of motion, not added by hand.

2. Path Integral = Entropy Maximization¶

\[Z = \int e^{-S_E}\,\mathcal{D}W = \int e^{\,\Sigma[W]}\,\mathcal{D}W\]

Configurations with more entropy dominate exponentially. The classical path maximizes entropy.

3. Decoherence = Entropy Selection¶

Quantum interference (real time: \(e^{iS}\)) becomes entropy selection (Euclidean: \(e^{\Sigma}\)). Decoherence is the transition from entropy-superposition to entropy-selection.

4. Σ-Reparameterization Is Natural¶

The substitution \(d/dt \to \dot\Sigma\,d/d\Sigma\) isn't an external coordinate change — it reveals the parameterization the physics was already in. \(\dot\Sigma\) is not a conversion factor; it's the Jacobian of an identity hidden by clock-time.

Verification Checklist¶

Condition	Status	Notes
Wick rotation clean (no boundary terms)	✅ Expected	Standard scalar GL theory
BRST cohomology \(H^0(s)\) survives \(t \to i\tau\)	⚠️ Open	2,500 COG bounty
\(\rho_{PS}\) density matrix well-defined under Euclidean continuation	⚠️ Open	Depends on BRST survival
Topological constant field-independent	⚠️ Open	Expected from GL structure

COG Bounties¶

Problem	Reward	Status
Verify BRST survival under Wick rotation for Will Field GL theory	2,500 COG	Open
Prove \(\dot\Sigma > 0\) monotonicity (arrow of time as theorem)	1,000 COG	Open

Connection to RTSG Framework¶

The Identity unifies three previously separate structures:

The GL action (governing Will Field dynamics, mass gaps, phase transitions)
The von Neumann entropy (measuring accumulated structure in PS)
The Σ-reparameterization (making entropy the independent variable)

into a single statement: the action IS the entropy, sign-flipped. Minimizing action = maximizing entropy = complexification = the arrow of time.