T5_To_T6_Forced_Bridge
plain-language theorem explainer
Packages the T5→T6 step of the forcing chain: given J-uniqueness, it records both the self-similarity bridge and the resulting φ-forced surface. Anyone assembling CompleteForcingChain cites this so T6 is not inserted independently of T5. It is a Prop structure with two fields, not a proved theorem.
Claim. Given a T5 package (unique cost $J(x)=\frac12(x+1/x)-1$ from the Recognition Composition Law, reciprocity, normalization, and calibration), a T5-to-T6 producer bridge consists of (i) a self-similarity bridge certificate built from that T5 package and (ii) a T6 surface asserting that $\varphi=(1+\sqrt5)/2$ is the unique positive solution of $x^2=x+1$.
background
The Unified Forcing Chain module aims to show every level T−1 through T8 is forced from the cost foundation (Recognition Composition Law plus normalization and calibration), not merely compatible with it. T5 is the uniqueness package for the cost $J$: reciprocity $J(x)=J(1/x)$, normalization $J(1)=0$, the composition law, calibration, and continuity pin $J(x)=\frac12(x+1/x)-1$ on $(0,\infty)$.
T6 asserts that in a discrete ledger with self-similar cost structure the only scaling ratio is the golden ratio $\varphi$, characterized by $\varphi^2=\varphi+1$ and $\varphi>0$, with uniqueness among positive solutions. The intermediate self-similarity bridge routes through hierarchy-dynamics: a closed observable framework with a realized hierarchy forces the scale ratio to be $\varphi$, and it records that bare closed-framework fields alone do not force the hierarchy fields, so nothing is smuggled past T5.
proof idea
No proof body: this is a Prop-valued structure definition. It takes a T5 uniqueness hypothesis and exposes two fields: self_similarity of type T5-to-T6 self-similarity bridge (indexed by that same T5 package), and t6 of type T6 φ-forced. Inhabitation is supplied downstream by t5_to_t6_forced_bridge_holds, which fills the first field via t5_to_t6_bridge_holds and the second from the PhiForcing equation, positivity, and uniqueness lemmas.
why it matters
In the primer forcing chain, T5 is J-uniqueness and T6 is φ forced as the self-similar fixed point. This producer bridge is the explicit link that makes T6 a downstream output of T5 rather than an independent insertion. Downstream, CompleteForcingChain and CompleteForcingChainT8 assemble the full T−1..T8 inevitability record; the companion theorem t5_to_t6_forced_bridge_holds is the witness that the bridge is inhabited. Without this packaging, the chain could claim φ while skipping the self-similarity certificate that ties φ to the unique cost $J$. It sits on the path to derived constants ($c$, $\hbar$, $G$, $\alpha$) that all flow from φ once T6 is locked.
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