T5_T3_To_Variational_Canonical_Bridge
plain-language theorem explainer
Certificate that T5 J-uniqueness plus T3 ledger conservation force a canonical variational successor: existence, uniqueness up to entry equality, defect monotonicity, and the universal argmin property. Cited by anyone assembling the complete forcing chain or the variational layer. Pure Prop structure; the companion theorem fills every field from variational-dynamics lemmas.
Claim. A Prop certificate asserting: for every configuration $c$ of size $N>0$, a variational successor $\mathrm{next}=\arg\min_{c'\in\mathrm{Feasible}(c)}\mathrm{total\_defect}(c')$ exists and is unique up to entry equality; defect is non-increasing along any such pair; feasibility is exactly conservation of $\log$-charge; the unity configuration is an equilibrium; and the bundled variational layer follows.
background
The Unified Forcing Chain module aims to show every level T-1 through T8 is forced from the cost foundation (Recognition Composition Law, normalization, calibration). T5 pins the unique cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$); its log-coordinate form is $J_{\log}(t)=(\mathrm{e}^{t}+\mathrm{e}^{-t})/2-1$, strictly convex. T3 forces the ledger from cost symmetry $J(x)=J(1/x)$.
Configurations carry a total defect and a $\log$-charge. Feasible moves are those that conserve $\log$-charge. A variational successor of $c$ is a feasible next state realizing the global minimum of total defect among feasible states. The sibling structure VariationalLayer_Forced packages existence of a successor, defect non-increase, and unity equilibrium without extra axioms.
This bridge is the named interface between those two forcing steps and the variational dynamics layer: it records that the argmin construction is the canonical universal object supplied by T5 strict convexity plus T3 conservation.
proof idea
Definitional Prop structure, not a proved theorem. Fields are named hypotheses (existence, uniqueness up to entries, universal packaging, defect monotonicity, feasible iff charge conservation, unity equilibrium, and the bundled variational layer). No tactics run here.
The companion theorem t5_t3_to_variational_canonical_bridge_holds discharges every field under hypotheses T5_J_Unique and T3_Ledger_Forced, wiring VariationalDynamics.variational_step_exists, variational_step_unique, and related lemmas. A Subsingleton instance records propositional uniqueness of certificates by rfl.
why it matters
Closes the T5+T3 to variational bridge inside the complete inevitability chain. Downstream, CompleteForcingChain requires the variational layer among the layers available in the main namespace (quarter-turn, Hamiltonian, projective, coupled-core, variational, measurement). The companion holds theorem is the discharge site.
Framework landmark: T5 J-uniqueness supplies strict convexity of $J_{\log}$, which forces uniqueness of the defect-minimizing successor; T3 ledger conservation identifies the feasible set with $\log$-charge conservation. Together they give a canonical dynamics step without extra axioms, matching the module claim that every level is forced rather than merely compatible.
Does not itself advance T6--T8 ($\varphi$, eight-tick, $D=3$), but supplies the variational substrate those later steps sit on once the chain is assembled.
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