T4_To_CanonicalUniversalForcing_Bridge
plain-language theorem explainer
Certificate that forced recognition (T4) yields the canonical Universal Forcing bridge: every Law-of-Logic realization extracts a canonically equivalent arithmetic surface. Citers of the complete T0–T8 inevitability chain use this interface. It packages pairwise arithmetic equivalence, reference equivalence to LogicNat, the Peano-surface property, and continuous-positive-ratio invariance. As a Prop structure it is definitional; the companion existence theorem discharges the fields.
Claim. Assume recognition is forced: a non-trivial discrete distinction supplies a recognition witness on the Boolean carrier. Then: (i) for any two Law-of-Logic realizations $R,S$, the carriers of their forced Peano arithmetic are equivalent; (ii) each such carrier is equivalent to the reference naturals $\mathrm{LogicNat}$ forced by the Law of Logic; (iii) every forced arithmetic satisfies the Peano-surface axioms (zero $\neq$ successor, injective successor, induction); (iv) the continuous-positive-ratio realization shares forced arithmetic with every other realization.
background
The Unified Forcing Chain module aims to show T-1 through T8 as inevitabilities from the cost foundation (Recognition Composition Law, normalization, calibration). T4 sits mid-chain: once a ledger and observables exist, recognition itself is forced on the Boolean floor.
A Law-of-Logic realization is any structure satisfying the logical laws extracted from cost. From each realization one builds an arithmetic object ArithmeticOf: a Peano carrier with an initiality witness. The reference carrier is LogicNat, the two-constructor inductive type (identity / step) mirroring the orbit ${1,\gamma,\gamma^2,\ldots}$. A Peano surface asserts the usual Peano axioms on that carrier (zero not a successor, injective step, induction).
Universal Forcing is the claim that this arithmetic content is realization-independent: different logical presentations yield canonically equivalent Peano surfaces. The continuous-positive-ratio comparison is the analytic realization that links the cost-uniqueness layer (unique $J$) back to the discrete arithmetic forced here.
proof idea
Definitional Prop structure, not a proved theorem. The four fields are certificate obligations parameterized by a T4 hypothesis (recognition forced on the Boolean floor). No tactic body; propositional uniqueness is immediate from Subsingleton via rfl.
Discharge lives in the companion theorem t4_to_canonical_universal_forcing_bridge_holds, which fills each field from Universal Forcing: pairwise orbit equivalences give arithmetic_invariant and to_reference; the Peano-surface lemmas on ArithmeticOf give peano_surface; specializing one side to the positive-ratio comparison realization gives continuous_positive_ratio_invariant.
why it matters
This is the Lean-facing form of the central Universal Forcing theorem at the T4 layer. Downstream, CompleteForcingChain requires the bridge so the full T-1..T8 package can cite a single arithmetic invariant rather than realization-dependent naturals. The companion t4_to_canonical_universal_forcing_bridge_holds is the existence proof that populates the certificate.
In the forcing narrative, T4 (recognition from ledger + observables) is the hinge between discrete ledger structure and the analytic cost layer. Without canonical arithmetic, T5 (unique $J$ via d'Alembert + calibration) and T6 ($\varphi$ as self-similar fixed point) would float free of the logical floor. The continuous-positive-ratio invariance field is exactly the glue to that cost-uniqueness layer. No open scaffold remains at this declaration; it is a pure interface.
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