equilibrium_entries_eq_uniform
plain-language theorem explainer
Any equilibrium configuration of N positive ledger entries equals the uniform configuration sharing its conserved log-charge. Ground-state and stability arguments cite this to collapse equilibria onto the explicit minimizer of the charge sector. The proof is a one-line uniqueness application: equilibrium means self-successor, and the uniform config is always the variational successor.
Claim. Let $N > 0$ and let $c$ be a configuration of $N$ positive real entries. If $c$ is an equilibrium (its own variational successor), then the entries of $c$ equal those of the uniform configuration with the same log-charge $\sigma = \sum_i \log(c_i)$, i.e. every entry equals $\exp(\sigma/N)$.
background
The module extracts a B4-style dynamic statement from the variational ledger update: equilibria coincide with variational minimizers; in a zero-charge sector the unique equilibrium is unity; stability of a one-channel ratio then forces $r = 1$.
A configuration is an $N$-tuple of positive real ratios. Its conserved charge is the total log-ratio $\sigma = \sum_i \log(\mathrm{entries}_i)$. The uniform configuration with charge $\sigma$ sets every entry to $\exp(\sigma/N)$. Equilibrium means the configuration is its own variational successor.
Upstream, the explicit equation of motion states that for any $c$ the uniform config of charge $\log\text{-charge}(c)$ is always a variational successor. Uniqueness of the variational step then pins the successor completely.
proof idea
Term-mode one-liner. Apply variational-step uniqueness to the triple $(c, c, u)$ where $u$ is the uniform config of charge $\log\text{-charge}(c)$: the equilibrium hypothesis says $c$ is a successor of itself, and the explicit-solution theorem says $u$ is a successor of $c$. Uniqueness forces $c = u$ on entries.
why it matters
This is the sector-wise identification of equilibria with uniform minimizers that the ground-state module needs. The immediate parent is zero-charge equilibrium is unity: specialize to $\sigma = 0$ and the uniform config collapses to the all-ones (unity) configuration. That in turn feeds the stable zero-charge ratio equals one statement for one-channel observables.
In the Recognition framework this is the dynamic half of cost-minima equals recognition: once the ledger update is variational, resting states cannot carry spatial inhomogeneity inside a fixed charge sector. It sits under the forcing chain's cost geometry (T5 J-uniqueness and the RCL) without reopening those uniqueness proofs; it only uses the already-established variational successor calculus.
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