t5_to_canonical_reference_bridge_holds
plain-language theorem explainer
Given uniqueness of the J-cost (T5), the canonical reference bridge holds: the Unit costed space is mathematical, every costed object space with a complex object admits a canonical mathematical symbol, the legacy reference-forcing statement follows, and the effectiveness principle applies. Cited by the complete forcing chain and the spine-to-extras bridge. Proof is a structure constructor wiring four existing Reference lemmas.
Claim. Assume $J$ is the unique cost on $(0,\infty)$ fixed by reciprocity, normalization, the Recognition Composition Law, calibration, and continuity (so $J(x)=\frac12(x+1/x)-1$). Then the T5-to-canonical-reference bridge holds: the Unit costed space is mathematical (uniformly zero cost); for every costed object space $P$ with some $o$ of positive cost, the canonical mathematical symbol space refers to a complex object of $P$; the legacy existential reference-forcing theorem follows; and the effectiveness principle holds for admissible tolerances.
background
The Unified Forcing Chain module shows T-1 through T8 as inevitabilities from the cost foundation (Recognition Composition Law, normalization $F(1)=0$, calibration $F''(1)=1$). T5 is the uniqueness step: those axioms force $J(x)=\frac12(x+1/x)-1$ on $(0,\infty)$, equivalently $\cosh(\log x)-1$.
The bridge structure packages how T5 feeds the Reference layer. The canonical construction fixes the symbol space as Unit (zero-parameter mathematical backbone) with the unit costed space and an indicator reference at a chosen complex object. The legacy theorem reference_is_forced was an existential over symbol space; the bridge names the canonical witness and the universal property that any costed object space with positive-cost objects admits such a reference.
Upstream Reference results supply the four fields: unit space is mathematical, mathematics is an absolute backbone for any costed space with a complex object, reference is forced, and the effectiveness principle holds for small tolerances.
proof idea
Term-mode structure construction under hypothesis h5 : T5_J_Unique. Each field is filled by a direct application of an existing Reference lemma; h5 is carried only as the bridge parameter and is not reopened.
unit_costed_mathematicalisReference.unit_is_mathematical.canonical_mathematical_symbolisReference.mathematics_is_absolute_backboneon any costed space with a positive-cost object.reference_forced_legacyisReference.reference_is_forced(the legacy existential).effectivenessisReference.effectiveness_principle.
No new analytic work: the proof is a certificate assembler that re-exports the Reference backbone under the T5 hypothesis interface.
why it matters
In the forcing chain, T5 (unique $J$) is the hinge between ledger/recognition structure and the $\phi$-forced, eight-tick, $D=3$ tail (T6–T8). This bridge is how T5 licenses the canonical mathematical reference: physics and measurement sit on a zero-parameter Unit backbone rather than an ad hoc symbol space.
Downstream, spine_to_extras_bridge_holds instantiates ref_canonical := t5_to_canonical_reference_bridge_holds t5 among the extras sourced from the spine. complete_forcing_chain threads the same bridge into the unconditional CompleteForcingChain assembly. The doc-comment states the design intent: the legacy existential is derived from the canonical mathematical-backbone theorem, so callers need not choose a symbol space.
Framework landmark: T5 J-uniqueness from RCL plus normalization and calibration. The bridge does not itself derive $\phi$ or constants; it only certifies that unique $J$ yields a forced reference layer for later constant and measurement work.
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