← CSM / 04 · Closure Dynamics, Reopening, Hysteresis, and Fixed-Point Evolution

CSM · 04 v0.1 · Dynamics Core Paper 2026-08-27

Closure Dynamics, Reopening, Hysteresis, and Fixed-Point Evolution

Advances CSM from a static graph to a time-indexed dynamical system, evolving through event-driven updates. Core claim: evidence accumulation can be monotonic, but the closure state usually is not — old evidence is never deleted, but BLOCKED/CLOSED/EXHAUSTED states can be downgraded to STALE/REOPENED. Defines closure events, closure scheduling, event commutation/non-commutation, closure hysteresis (the same final evidence arriving in a different order can produce path-dependent, different outcomes), reopening waves, debt discharge, frontier drift, local closure fixed points, relative equilibrium, closure attractors/cycles, metastability, and closure oscillation. Central non-collapse principle: $\mathfrak{C}_{t+1}=\mathfrak{C}_t \not\Rightarrow \mathfrak{C}_t=\Omega^{math}$ — reaching a fixed point under a fixed theorem library/grammar/representation/bridging policy is only a relative closure fixed point, and never means “the mathematical space is complete.”

A closure fixed point under a fixed policy is only a relative fixed point, never equal to a complete mathematical space; closure states are generally non-monotonic and can be downgraded by reopening. A dedicated NS Fixed Point Non-Claim: even if the NS closure graph reaches a complete relative fixed point, this does not mean Navier–Stokes has been solved — $\mathfrak{C}_{NS}^{\star,rel} \not\Rightarrow$ NS is solved.

Connections

Relationship to the rest of the series, stated as closely as possible in the document's own words, not my interpretation.

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