variational_layer_holds
plain-language theorem explainer
The variational ledger layer is forced with no extra axioms: every positive-size configuration admits a variational successor that never raises total defect, unity is an equilibrium, and the update is global. Downstream T5/T3-to-variational bridges cite it as the packaged layer certificate. Proof is a two-field structure packing of existing VariationalDynamics results.
Claim. The variational ledger-dynamics layer holds: for every configuration $c$ of size $N>0$ there exists a variational successor; every variational successor $c'$ satisfies $\mathrm{total\_defect}(c')\le\mathrm{total\_defect}(c)$; the unity configuration is an equilibrium; and the variational update is a global map on configurations.
background
In the Unified Forcing Chain, T0–T8 are claimed as inevitabilities from the Recognition Composition Law plus normalization and calibration. The local layer here sits after the ledger (T3) and unique analytic cost $J$ (T5): configurations carry a non-negative total defect built from the defect functional (equal to $J$ on positive ratios), and the conserved ledger quantity is log-charge.
Feasible successors are those that preserve that charge. Variational dynamics then select, among feasible successors, global minimizers of total defect. Several cost aliases in the dependency cone (observer cost, multiplicative-recognizer cost, PRC bridge cost, rung-coarsen cost) all ultimately route through $J$ or a doubled $J$-weight; the conserved multiset functionals under rung coarsening justify treating totals and spectra as ledger invariants.
VariationalLayer_Forced packages existence of a variational successor, monotone non-increase of total defect along those successors, equilibrium of the unity configuration, and globality of the update, without adding axioms beyond the dynamics already developed.
proof idea
Term-mode structure construction with two fields. The certificate field is the lambda that, given $N>0$ and a configuration $c$, applies VariationalDynamics.variational_dynamics_certificate to obtain existence of a variational successor, defect non-increase along every such successor, and equilibrium of the unity configuration. The globality field is the direct citation of VariationalDynamics.update_is_global. No new algebra is performed; the theorem only assembles the pre-proved dynamics facts into the layer Prop.
why it matters
This declaration is the layer seal that the T5/T3 variational bridges consume. Downstream, t5_t3_to_variational_bridge_holds states that the analytic $J$-cost plus ledger conservation surface supplies variational dynamics (non-negative total defect, feasible set = charge conservation). The canonical twin t5_t3_to_variational_canonical_bridge_holds upgrades that to existence, uniqueness, and universality of the variational step.
In the forcing chain this is the dynamics interface between T5 (unique $J(x)=(x+x^{-1})/2-1$) and T3 (ledger from $J(x)=J(1/x)$ symmetry): once those are in hand, evolution is forced to be global defect minimization on the charge shell. It does not itself force $\varphi$, the eight-tick octave, or $D=3$; those remain T6–T8. It closes the scaffolding gap that would otherwise leave variational update as an extra postulate rather than a forced layer.
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