T4_To_T5_Realization_Bridge
plain-language theorem explainer
Certificate that the T4 Boolean recognition floor and any continuous positive-ratio Law-of-Logic comparison are admissible realizations of the same interface, with Universal Forcing identifying their extracted Peano arithmetic. On the continuous side the derived cost obeys multiplicative consistency with the bilinear RCL combiner. Downstream complete-chain assemblies and the T4-to-T5 cost bridge cite it. As a Prop structure it packages the interface obligations rather than proving them.
Claim. Given a T4 recognition-forced hypothesis (normalized two-point Boolean floor, a recognition witness, and a nontrivial Boolean distinction), the following hold as a bridge certificate: the Boolean floor and its normalized form each yield a Law-of-Logic realization; every continuous comparison operator $C$ satisfying the laws of logic likewise yields a realization; Universal Forcing extracts isomorphic Peano carriers from the floor realization and from any such continuous realization; and on each continuous realization the derived cost $F$ admits a combiner $P(u,v)=2u+2v+c\,uv$ with multiplicative consistency $F(xy)+F(x/y)=P(F(x),F(y))$.
background
The Unified Forcing Chain module aims to show T0–T8 as forced inevitabilities from the cost foundation (Recognition Composition Law, normalization, calibration). T4 asserts that a nontrivial discrete distinction on the Boolean carrier already supplies a recognition witness and relation; its analytic J-stability refinement is separate.
A normalized two-point recognition floor is the abstract Boolean floor: one empty/consistent point, one marked inconsistent point, unit-normalized recognition-work cost, and an equivalence showing Bool is only the canonical representative. A comparison operator is a map $\mathbb{R}{>0}\times\mathbb{R}{>0}\to\mathbb{R}$ whose Aristotelian constraints encode well-posed comparison; the derived cost fixes the second argument at the multiplicative identity.
Multiplicative consistency (d'Alembert inevitability) is $F(xy)+F(x/y)=P(F(x),F(y))$ for positive $x,y$. The Recognition Composition Law is the special case with bilinear combiner $P(u,v)=2u+2v+c,uv$, the surface on which T5 uniqueness of $J$ is proved.
proof idea
No proof body: this is a Prop-valued structure (interface certificate), not a theorem. Inhabiting instances are built elsewhere by packing existing constructions fieldwise.
Typical fillers (see t4_to_t5_bridge_holds in the T−1–T8 bridge module): copy floor_recognition and floor_distinction from the T4 hypothesis; supply floorRealization and positiveRatioRealization as the Boolean and continuous Law-of-Logic realizations; discharge arithmetic invariance via Universal Forcing's extracted Peano carriers; and obtain rcl_surface from the continuous derived cost plus the d'Alembert multiplicative-consistency lemmas that force the bilinear combiner family.
why it matters
This is the T4→T5 realization bridge in the complete inevitability chain: it records that the pre-analytic recognition floor and the continuous positive-ratio surface used by T5 are co-realizations of one Law-of-Logic interface, so RCL can be stated on the continuous side while arithmetic is identified across settings (primer T5: unique $J$ via d'Alembert + normalization + calibration; RCL as the functional equation).
Parents include CompleteForcingChain and CompleteForcingChainT8, the cost-bridge structure T4_To_T5_Cost_Bridge, and the inhabiting defs t4_to_t5_bridge_holds / t4_to_t5_cost_bridge_holds. Downstream honesty notes stress a real gap: T5 uniqueness is proved from CostUniqueness and law_of_logic_forces_jcost and does not consume the floor beyond re-exporting rcl_surface; deleting T−1..T4 would not break T5 proofs. The certificate therefore certifies interface compatibility and RCL availability, not a causal feed of the Boolean floor cost into J-uniqueness.
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