t4_to_canonical_universal_forcing_bridge_holds
plain-language theorem explainer
Given that recognition is forced on the Boolean floor, every Law-of-Logic realization extracts a canonically equivalent arithmetic surface. The bridge records pairwise orbit equivalence, reference equivalence to LogicNat, the Peano-surface universal property, and continuous positive-ratio invariance. Citers assembling the complete T-1..T8 chain use it to wire T4 into universal arithmetic. The proof is a short term construction from orbit equivalences and the extracted Peano surface.
Claim. Assume recognition is forced: the normalized two-point Boolean floor carries a non-trivial distinction, a recognition witness, and a recognition structure. Then a bridge certificate holds: (i) for any two Law-of-Logic realizations $R,S$, the forced Peano carriers of their arithmetic surfaces are equivalent; (ii) each realization's arithmetic is equivalent to the reference $\mathrm{LogicNat}$; (iii) the Peano-surface universal property is available; (iv) continuous positive-ratio comparisons preserve orbit equivalence of arithmetic surfaces.
background
The Unified Forcing Chain module aims to show that every level T-1 through T8 is forced from the cost foundation (Recognition Composition Law, normalization $F(1)=0$, calibration $F''(1)=1$). T4 is the recognition step: once a ledger and observables exist, a non-trivial discrete distinction on the Boolean floor already supplies a recognition witness and relation; richer J-stability is deferred to analytic refinements.
Universal forcing packages each Law-of-Logic realization as an arithmetic surface with a Peano carrier. The bridge structure asks four things of that package: pairwise canonical equivalence of arithmetic carriers, a reference map into LogicNat, the extracted Peano-surface universal property, and invariance under continuous positive-ratio comparisons (via the LogicRealization of such a comparison).
Upstream cost infrastructure supplies the reciprocal automorphism and the closed-form recognition cost $J(x)=(x+x^{-1})/2-1$, but this particular bridge is purely about orbit equivalences of arithmetic surfaces, not yet about uniqueness of $J$ (that is T5).
proof idea
Term-mode construction of the four bridge fields under the T4 hypothesis (which is not further unpacked).
arithmetic_invariant: for realizations $R,S$, change the goal to $R.\mathrm{Orbit}\simeq S.\mathrm{Orbit}$ and compose $R$'s orbit-to-LogicNat equivalence with the inverse of $S$'s.to_reference: same pattern, mapping $R.\mathrm{Orbit}$ straight to LogicNat via $R$'s orbit equivalence.peano_surface: invoke the already-extracted Peano surface of the arithmetic-of-$R$ package.continuous_positive_ratio_invariant: form the LogicRealization of the positive-ratio comparison, then again compose its orbit-to-LogicNat map with the target realization's inverse.
No new arithmetic is built; the proof only reassembles existing orbit equivalences into the bridge certificate.
why it matters
In the complete inevitability chain, T4 is the hinge from ledger/observables to recognition. This bridge is the certificate that recognition already forces a unique (up to canonical equivalence) arithmetic surface, so later steps do not depend on a choice of realization.
The sole downstream consumer is complete_forcing_chain, which threads T-1 through T8 by successively applying each bridge. Without this certificate, the chain would stall between recognition and the universal-forcing arithmetic layer that T5-T8 refine.
Framework landmarks: this is the T4 slot in the forcing spine (before T5 J-uniqueness, T6 $\varphi$, T7 eight-tick, T8 $D=3$). It does not itself force $J$ or $\varphi$; it only guarantees that the arithmetic substrate is canonical once recognition is present.
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