t5_to_t6_forced_bridge_holds
plain-language theorem explainer
Given uniqueness of the recognition cost J on the positive reals, the T5-to-T6 producer bridge holds: self-similarity of the discrete ledger forces φ as the unique positive root of x² = x + 1. Chain assemblers cite this when wiring T5 into the φ-forcing step. The proof is a term-mode structure pack: the self-similarity bridge lemma plus the standard φ equation, positivity, and uniqueness facts.
Claim. If the Recognition Composition Law, reciprocity $J(x)=J(1/x)$, normalization $J(1)=0$, calibration, and continuity uniquely fix $J(x)=\frac12(x+x^{-1})-1$ on $(0,\infty)$, then the T5-to-T6 producer bridge holds: that uniqueness induces a self-similarity bridge whose downstream output is the T6 package that $\varphi>0$ uniquely solves $\varphi^2=\varphi+1$.
background
The module UnifiedForcingChain aims at a complete inevitability chain from an absolute floor through T0–T8, all forced from the cost foundation (Recognition Composition Law plus normalization and calibration). T5 is the uniqueness package for the cost: reciprocity, normalization, the composition law, log-coordinate calibration, and continuity pin $J(x)=\frac12(x+1/x)-1$ on $(0,\infty)$.
T6 is the claim that the golden ratio is forced, not chosen: $\varphi$ is the unique positive solution of the self-similarity fixed-point equation $\varphi^2=\varphi+1$. The producer bridge sits between them. Its doc states that the self-similarity bridge consumes the T5 uniqueness package and records the extra realized-hierarchy structure needed to force $\varphi$, then exposes the actual T6 theorem surface as downstream output so T6 is not inserted independently of the bridge.
Upstream algebra supplies the bare identity $\varphi^2=\varphi+1$ and positivity; the bridge’s job is to attach those facts to T5 rather than treat φ as a free constant.
proof idea
Term-mode structure construction for T5_To_T6_Forced_Bridge h5, not a long tactic script.
- Fill
self_similarityby applying the existing lemmat5_to_t6_bridge_holdsat the T5 hypothesish5. That lemma is the actual self-similarity content of the bridge. - Fill the
t6field as a T6 package with three components:PhiForcing.phi_equation(the identity $\varphi^2=\varphi+1$),PhiForcing.phi_pos(positivity), andt6_phi_unique_from_derived(uniqueness of the positive fixed point from the derived forcing path).
No new analytic work happens here; the declaration is the producer wiring that makes T6 a consequence of T5 plus the self-similarity bridge rather than a free insertion.
why it matters
In the forcing chain, T5 is J-uniqueness and T6 is φ forced as the self-similar fixed point. This declaration is the certified handoff: once J is unique, the ledger’s self-similarity forces φ, matching the primer landmarks T5 and T6 and the module’s claim that every level is forced from the cost foundation.
Downstream it is consumed by complete_forcing_chain in this module ("the unconditional mathematical forcing chain holds") and by complete_forcing_chain_t8 in TMinus1ToT8Bridge, which threads T−1 through T8 by successive bridges. A parallel bridge theorem in TMinus1ToT8Bridge reuses the same shape.
Without this producer, T6 could be smuggled in as an independent axiom. With it, φ enters only after T5, so later steps (eight-tick octave T7, D = 3 as T8, and RS-native constants built from φ) sit on a forced rather than postulated golden ratio.
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