T5_To_CanonicalReference_Bridge
plain-language theorem explainer
From uniqueness of the recognition cost J, one obtains a bridge certificate that every costed object space with positive complexity admits a mathematical symbol space, canonically realized by the zero-cost Unit backbone. The certificate also packages Wigner-style effectiveness: near-balanced symbols can mean any higher-cost object. Downstream, the complete forcing chain and the holding theorem for this bridge cite it. As a Prop structure it is a pure interface; the holding theorem fills the fields.
Claim. Given uniqueness of the recognition cost $J$ (reciprocity, normalization, the Recognition Composition Law, calibration, and continuity), the following hold as a single certificate: (i) the Unit costed space is mathematical (uniformly zero cost); (ii) every costed object space $P$ with some $o$ of positive cost admits a mathematical symbol space $S$ and a reference structure with a nonempty symbol relating them; (iii) the weaker existential (some symbol space, not necessarily mathematical) follows; (iv) for every $\varepsilon>0$ and every object of cost $>\varepsilon$, there exist a symbol space and a symbol of cost $<\varepsilon$ that means that object.
background
The Unified Forcing Chain module shows that T0–T8 are forced from the cost foundation (Recognition Composition Law, normalization, calibration). T5 is the uniqueness step: those axioms pin $J(x)=\frac12(x+1/x)-1$ on $(0,\infty)$, equivalently $\cosh(\log x)-1$.
Reference theory treats costed spaces (types equipped with a nonnegative cost $J$) and reference structures that let symbols in one space mean objects in another. A space is mathematical when cost is uniformly zero; the Unit costed space is the zero-parameter backbone. A symbol is a witness that a low-cost (or zero-cost) carrier refers to a chosen object.
The legacy theorem only asserted existence of some symbol space for any complex costed object space. The canonical strengthening fixes the backbone as Unit with the indicator reference at a chosen complex object, and adds effectiveness: symbols with cost below $\varepsilon$ can mean any object above $\varepsilon$.
proof idea
This declaration is a Prop-valued structure, not a proved theorem. It names four fields that a T5 instance is expected to supply: mathematicality of the Unit costed space; the canonical mathematical-backbone universal property; the legacy existential; and the effectiveness principle.
No proof body lives here. The companion theorem t5_to_canonical_reference_bridge_holds constructs an inhabitant by wiring Reference.unit_is_mathematical, Reference.mathematics_is_absolute_backbone, a derivation of the legacy existential from the canonical field, and the corresponding effectiveness lemma. Propositional uniqueness of certificates is immediate (Subsingleton by rfl).
why it matters
In the forcing chain, T5 (unique $J$) is the last pure cost-functional step before $\phi$, the eight-tick octave, and $D=3$. This bridge converts that uniqueness into a reference-theoretic backbone: mathematics as the zero-cost Unit carrier that can symbolize any complex object space, plus Wigner effectiveness at the cost-compression level.
CompleteForcingChain consumes the bridge as part of the full T-1 through T8 package (logic, MP, discreteness, ledger, recognition, unique $J$, $\phi$, eight-tick, $D=3$). The holding theorem shows T5 alone discharges every field, so the reference layer does not need extra axioms beyond the RCL bundle.
The doc-comment frames the move from a bare existential (reference_is_forced) to a named canonical witness. That is the step that makes "mathematics is absolute backbone" a chain-level fact rather than an external slogan.
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