RTSG Axioms — v4 Entropy Update¶

April 2026 · Addendum to Axioms (0–9)

The Σ-reparameterization and Action-Entropy Identity (Pokrovskaia, April 2026) change the status of one axiom and add two new structural elements.

Axiom 0 — Relational Reality (ZFA)¶

Status: Unchanged. Only relational reality exists. ZFA with Aczel's Anti-Foundation Axiom.

Axiom 1 — Three Co-Primordial Spaces¶

Status: Strengthened. QS, PS, CS arise simultaneously. The entropy \(\Sigma = -\mathrm{Tr}(\rho_{PS}\ln\rho_{PS})\) now provides a metric on PS: it measures how much instantiated structure exists. The three-space picture gains a quantitative bridge.

Axiom 2 — Instantiation¶

Status: Strengthened. \(C\) is a BRST cohomological filter. Physical observables are \(H^0(s)\). NEW: the rate of instantiation IS the entropy production rate \(\dot\Sigma\). Instantiation doesn't merely produce entropy as a side effect — by the Action-Entropy Identity, the action governing \(C\) IS the entropy.

Arrow of Time — STATUS CHANGE¶

v3 status: Axiom. "The arrow of time is the arrow of complexification: the monotonic growth of instantiated structure in PS."

v4 status: Theorem-candidate. Demoted from axiom to near-theorem.

Argument: If \(S_E[W] = -\Sigma + \text{const}\) (Action-Entropy Identity), then:

1. The Will Field drift is \(\mu = +\delta\Sigma/\delta\bar{W}\) — entropy gradient ascent
2. Every non-equilibrium trajectory moves toward higher \(\Sigma\)
3. The GL ground state is a global attractor (known for standard \(\phi^4\) GL)
4. Therefore \(\dot\Sigma \geq 0\) along trajectories, with equality only at equilibrium

What remains to prove: That the GL global attractor property survives the BRST quotient structure. For standard scalar GL theory this is established. With bisimulation quotienting on top, it requires explicit verification.

Proposed restatement: If the axiom is retained as a safety net, restate as:

Axiom (Arrow): The GL action \(S[W]\) has a unique global minimum with \(\dot\Sigma > 0\) for all non-equilibrium configurations.

This is weaker than the current version (it's a statement about the GL potential shape, not a metaphysical claim) and should be provable from the \(\phi^4\) structure.

New Structural Element: Σ as Primary Variable¶

The Σ-reparameterization introduces a new structural principle:

Entropy primacy: The fundamental independent variable of RTSG dynamics is \(\Sigma\) (von Neumann entropy of the bisimulation quotient), not clock-time \(t\). Time is derived: \(dt = d\Sigma/\dot\Sigma\).

This is not an axiom — it follows from the Action-Entropy Identity. But it changes how every equation in the framework is read.

New Structural Element: Action-Entropy Duality¶

Every variational statement in RTSG is simultaneously a thermodynamic statement:

See Also¶

• Axioms (0–9) — canonical
• Master Reference v4 Addendum
• Action-Entropy Identity
• Σ-Reparameterization