T5_T3_To_Variational_Bridge
plain-language theorem explainer
Interface bundling the passage from unique analytic J-cost plus ledger conservation to variational ledger dynamics: total defect is nonnegative, feasible moves preserve log-charge, defect-nonincreasing successors exist, unity is equilibrium, and the dynamics is not purely local. The complete forcing chain cites it as the variational layer certificate. As a Prop structure it only states the bridge; the inhabitant is built elsewhere from T5 and T3.
Claim. The T5/T3-to-variational bridge is the conjunction of: (1) for every $N$ and every positive-ratio configuration $c$, the total defect satisfies $\mathrm{total\_defect}(c)\ge 0$; (2) $c'\in\mathrm{Feasible}(c)$ if and only if $\log\mathrm{charge}(c')=\log\mathrm{charge}(c)$; (3) if $N>0$ then a variational successor of $c$ exists; (4) every variational successor weakly decreases total defect; (5) the unity (all-ones) configuration is an equilibrium; (6) some successor is not realizable by a single-entry local update; (7) the bundled variational layer follows.
background
In the Unified Forcing Chain, T0–T8 are forced from the Recognition Composition Law with normalization and calibration. T5 pins the unique analytic cost $J(x)=(x+x^{-1})/2-1$; T3 forces the ledger from the symmetry $J(x)=J(1/x)$.
A configuration of size $N$ is an $N$-tuple of positive real ratios. Its total defect is the sum of individual defects (each the $J$-cost of an entry); upstream, that sum is nonnegative termwise. Feasibility in the variational dynamics is conservation of total log-charge on the ledger. A variational successor is a global minimizer of total defect on the charge-preserving set; the unity configuration (all entries $1$) is the zero-defect reference.
The sibling bundle VariationalLayer_Forced packages existence of successors, defect non-increase, and unity equilibrium without extra axioms. This bridge structure records the finer surface that justifies that layer once T5 and T3 are available.
proof idea
No proof body: the declaration is a Prop-valued structure (seven fields). It is an interface, not a derivation.
The concrete inhabitant is supplied by the companion theorem that assumes unique $J$ (T5) and the forced ledger (T3). That proof fills the fields by direct appeal to upstream facts: nonnegativity of total defect from the configuration module; feasibility rewritten as equality of log-charge; existence and defect-monotonicity of variational successors; unity as equilibrium; an explicit witness that some successor is nonlocal; and finally the old variational-layer certificate as a consequence of the preceding clauses.
why it matters
Closes the step from T5 (unique $J$) and T3 (ledger) to a working variational dynamics on configurations, as the doc-comment states: once the analytic $J$ surface is in hand, feasible successors are charge-preserving global defect minimizers.
Downstream, CompleteForcingChain requires this bridge among the layers (variational, Hamiltonian, measurement, etc.) that sit inside the main inevitability package after T0–T8. The companion theorem t5_t3_to_variational_bridge_holds is the discharge site that turns the interface into a proved link in that chain.
Framework role: it is the variational hinge between the cost/ledger foundation and dynamics that can relax toward the unity equilibrium, without smuggling a separate dynamical axiom. It does not itself force $\varphi$, the eight-tick octave, or $D=3$; those remain T6–T8.
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