T5_To_CanonicalExistent_Bridge
plain-language theorem explainer
Certificate that T5 (unique J-cost) pins the unique RS-existent real at 1 and the unique zero of consistent cost at ratio 1. Anyone assembling the complete T0–T8 forcing chain cites it as the T5-to-ontology bridge. It is a Prop-valued structure bundling the iff uniqueness, the nothing-boundary, and the zero-cost consistency facts under a T5 hypothesis.
Claim. Given uniqueness of the recognition cost $J$ (reciprocity, normalization, the Recognition Composition Law, calibration, and continuity), the following hold as a single certificate: $1$ is RS-existent; for all real $x$, $x$ is RS-existent if and only if $x=1$; there is a unique RS-existent real; some $\varepsilon>0$ excludes all $(0,\varepsilon)$ from RS-existence; a consistent configuration has cost zero iff its ratio is $1$; consistent cost is nonnegative; and some consistent configuration has cost zero.
background
The module UnifiedForcingChain aims at a complete inevitability chain: every level from the absolute floor through T0–T8 is forced from the cost foundation (Recognition Composition Law plus normalization and calibration), not merely shown compatible.
T5 asserts that those axioms determine the unique cost $J(x)=\frac12(x+1/x)-1$ on $(0,\infty)$. The present bridge packages the ontological reading of that uniqueness: RS-existence collapses to the single point $x=1$, and the consistent-configuration cost (the cost attached to logic-from-cost configurations) vanishes exactly on ratio $1$.
Legacy surfaces used bare $\exists!$ uniqueness and a bare zero-cost existence claim. The bridge keeps those as derived fields while preferring the iff characterizations and the boundary statement that arbitrarily small positive values are not RS-existent.
proof idea
This declaration is a Prop-valued structure (a certificate type), not a proved theorem. It names seven fields under a fixed T5 hypothesis and carries a Subsingleton instance (any two certificates for the same T5 are definitionally equal by rfl).
The inhabiting theorem t5_to_canonical_existent_bridge_holds fills the fields by direct appeal to ontology lemmas: rs_exists_one, rs_exists_unique_one, rs_exists_unique, nothing_not_rs_exists, and the corresponding consistent-cost facts from the logic-from-cost layer. No new analytic work happens here; the structure only freezes the interface the complete chain expects after T5.
why it matters
In the forcing ladder, T5 is the step that unique-ifies $J$ from d'Alembert-type composition, reciprocity, normalization, and calibration. Downstream physics (T6 $\varphi$ as self-similar fixed point, the eight-tick octave, $D=3$) needs a canonical unit of existence and a zero-cost reference configuration; this bridge is that handoff.
CompleteForcingChain consumes the certificate as part of the full T-1 through T8 package. The companion theorem t5_to_canonical_existent_bridge_holds shows every T5 instance supplies the bridge, so the chain does not re-prove ontology uniqueness at each assembly site.
Framework landmark: T5 J-uniqueness in the primer, with $J(x)=\cosh(\log x)-1$ as the closed form. The bridge turns that analytic uniqueness into the ontological claim "the unique existent is 1" and the cost claim "consistent cost vanishes only at ratio 1."
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