pattern_forced
plain-language theorem explainer
Every eight-tick ladder with unit base, constant positive step ratio, and T6 self-similarity is exactly the geometric sequence φⁿ. Cited by anyone closing M2 pattern forcing or the α-genesis certificate, and by the identification of the GapWeight window pattern with every admissible ladder. Proof is induction on the tick index, feeding the already-forced ratio r=φ into the constant-ratio recurrence.
Claim. Let $L$ be an eight-tick ladder: a sequence $u:\mathbb{N}\to\mathbb{R}$ with $u_0=1$, constant positive step ratio $r>0$ satisfying the self-similarity equation $r^2=r+1$, and recurrence $u_{n+1}=r\,u_n$. Then $u_n=\varphi^n$ for every $n\in\mathbb{N}$, where $\varphi$ is the positive golden ratio.
background
Module Alpha Genesis M2 (Pattern Forcing) shows that the time-domain pattern $u_t=\varphi^t$ consumed by the $w_8$ spectral projection is not a free choice. The local carrier is the T7 eight-tick window; the algebraic constraint is T6 self-similarity.
An eight-tick ladder is a real sequence with unit base $u_0=1$, a single positive constant step ratio $r$, the recurrence $u_{n+1}=r,u_n$, and the quadratic $r^2=r+1$. Upstream, the sibling result that the ratio equals $\varphi$ identifies $r$ with the unique positive root of that quadratic (the T6 fixed point). The module pairs this growth display with the reciprocal spectral envelope $\varphi^{-k}$, which is the T9 forced measure term by term; reciprocity is the ledger J-symmetry $J(x)=J(1/x)$.
No CODATA enters: the pattern is forced from the Recognition chain alone.
proof idea
Fix an eight-tick ladder $L$ and induct on the tick index $n$.
Base case $n=0$: the ladder axiom $u_0=1$ matches $\varphi^0$, finished by simplification.
Successor: rewrite $u_{k+1}$ by the constant-ratio step axiom, substitute the inductive hypothesis $u_k=\varphi^k$, replace the step ratio by $\varphi$ via the upstream theorem that every such ladder has ratio equal to $\varphi$, and close with ring arithmetic on $\varphi^{k+1}=\varphi\cdot\varphi^k$.
why it matters
This is the M2 pattern-forcing theorem proper: every admissible ladder is the $\varphi$-pattern. It discharges discrete choice (ii) of the no-fit proposition (the "canonical $\varphi$-pattern" is forced by T6 given the T7 carrier, not selected).
Downstream, the GapWeight window pattern equals $L.u$ on every $\mathrm{Fin},8$ tick for every admissible ladder, so the spectral machinery cannot choose a different discrete shape. The Pattern Forcing certificate bundles this as clause 1 of M2 closure (pattern forced; GapWeight is that ladder; pattern and forced measure are reciprocal; envelope is the forced measure). The Alpha Genesis certificate lists the same fact as item 2 of the forward derivation of $\alpha$.
Framework landmarks: T6 ($\varphi$ as self-similar fixed point) and T7 (eight-tick octave). Together with the reciprocal forced-measure identity, it removes independent inputs from both the growth and weight sides of $w_8$.
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