latticeWeight
plain-language theorem explainer
Defines the lattice weight on recognition states: after n discrete recognition steps the weight is φ to the power −n. This is the geometric φ-measure forced by T9 factorization and per-step self-similarity. Alpha-genesis pattern and spectral certificates cite it as the unique decay envelope factor. The body is a one-line definition in terms of the golden ratio constant.
Claim. For each natural number $n$, the lattice weight is $w(n) = \varphi^{-n}$, where $\varphi$ is the golden ratio (the unique self-similar scale from the forcing chain).
background
Module T9 closes the missing weighting rule on recognition states after the T0–T8 chain has fixed the shape of the law (unique cost $J$, scale $\varphi$, eight-tick period, $D=3$). The open instance-selection problems (Born weights, chirality, rung occupancy, and related prefactors) are projections of that single primitive.
On the lattice layer, recognition is discrete. An admissible weight rule must factorize over independent composition and obey per-step self-similar balance $\rho = 1/(1+\rho)$. The latter forces $\rho = \varphi^{-1}$ by the same reciprocal fixed-point uniqueness that pins $\varphi$ in T6. Iterating then yields $w(n) = \varphi^{-n}$.
Equivalently, the continuum form is the Gibbs rule with rate fixed by the self-similar ledger: probability proportional to $\exp(-( \ln \varphi) \cdot \mathrm{cost})$. This definition is the discrete geometric factor that those theorems name.
proof idea
Pure definition: $w(n)$ is written as $(1/\varphi)^n$. No tactics or lemmas. Downstream results rewrite it as $\varphi^{-n}$ via ordinary field arithmetic and $\varphi \neq 0$. The forcing that this is the only admissible weight lives in sibling theorems (RecognitionWeightRule.weight_forced and the conversion to rung dilution), not in this declaration.
why it matters
T9's geometric measure is the unique weighting of recognition states; this def is its discrete form. Downstream, pattern forcing proves reciprocity $\varphi^{t} \cdot w(t) = 1$ at every eight-tick index, and the envelope identity $w_{\mathrm{geom}}(k) = \sin^{2}(k\pi/8) \cdot w(k)$ for nonzero modes: the $\varphi^{-k}$ factor in the gap weight is not an $\alpha$-specific choice but the forced T9 measure.
Spectral forcing rewrites the same identity as (difference-operator spectrum)/4 times the lattice weight, so both factors are theorems. The Alpha Genesis certificate bundles this as the spectral envelope being the forced measure (M2). Framework landmarks: T6 uniqueness of $\varphi$, T7 eight-tick octave, and the RS-native $\hbar = \varphi^{-5}$ link via the sibling theorem that $\hbar$ is a lattice weight.
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