t6_holds
plain-language theorem explainer
The golden ratio is forced as the unique positive solution of x² = x + 1, completing step T6 of the Recognition Science forcing spine. Anyone citing the self-similar hierarchy scale or deriving RS constants from that scale will reference this witness. The proof is a three-field structure constructor packaging the algebraic identity, positivity, and uniqueness lemmas already proved upstream.
Claim. The T6 forcing claim holds: the canonical positive real $\varphi$ satisfies $\varphi^2 = \varphi + 1$, and it is unique among positive reals, i.e. every $r > 0$ with $r^2 = r + 1$ equals $\varphi$.
background
The module publishes the theory-only T-1 through T8 forcing spine. T6 is the step that forces the golden ratio as the unique scaling ratio once a discrete ledger carries a self-similar cost structure.
In the framework, $\varphi$ is the positive root of the golden equation $x^2 = x + 1$. The structure packaged here records three facts: the equation holds for the canonical $\varphi$, $\varphi > 0$, and no other positive real solves the same quadratic. Upstream doc-comment on the chain form states: "In a discrete ledger with self-similar cost structure, the only scaling ratio is $\varphi = (1 + \sqrt{5})/2$. $\varphi$ is not chosen; it's the unique solution to $x^2 = x + 1$ with $x > 0$."
The algebraic identity is already proved in PhiForcing (and PhiRing) by expanding the closed form. Uniqueness is supplied by the local derived lemma that excludes the negative root and the degenerate $r = 1$ case.
proof idea
Term-mode structure construction with no tactics. The three fields of the T6 claim are filled directly:
PhiForcing.phi_equationsupplies $\varphi^2 = \varphi + 1$,PhiForcing.phi_possupplies positivity,t6_phi_unique_from_derivedsupplies the uniqueness quantifier over positive reals.
It is a pure record assembly of already-established lemmas; the same pattern appears in the UnifiedForcingChain twin.
why it matters
This is the public bridge witness that T6 holds on the T-1..T8 spine. Downstream it is consumed by t5_to_t6_forced_bridge_holds (linking T5 J-uniqueness to $\varphi$-forcing), by constants_from_phi and constants_from_phi_canonical (which fix $c = 1$, $\hbar = \varphi^{-5}$, $G = \varphi^5/\pi$), by the classical-negation-plus-unique-minimizer theorem, and by the ultimate-inevitability root theorems.
In the framework landmarks this is exactly T6: $\varphi$ forced as the self-similar fixed point after T5's uniqueness of the reciprocal cost $J$. Without this witness the mass ladder, the eight-tick octave (T7), and $D = 3$ (T8) have no forced scale. The module deliberately stops before private operator and measurement layers.
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