RecognitionWeightRule
plain-language theorem explainer
A recognition weight rule packages the two lattice premises that force the measure on recognition states: a strictly positive weight on step count that multiplies under independent composition, with the single-step weight fixed by reciprocal self-similarity ρ = 1/(1+ρ). Anyone citing T9 lattice forcing or uniqueness of the φ-measure uses this interface. As a structure it is definitional packaging of those axioms, not a proved claim.
Claim. A recognition weight rule is a map $w:\mathbb{N}\to\mathbb{R}$ such that $w(n)>0$ for every $n$, $w(m+n)=w(m)\,w(n)$ for all $m,n\in\mathbb{N}$, and the single-step weight satisfies the reciprocal self-similarity balance $w(1)=1/(1+w(1))$.
background
The module closes T9. The T0–T8 chain forces the shape of the law: unique J-cost, φ as self-similar scale, eight-tick period, and D=3. What remained open was the weighting of allowed recognition states. Born weights, chirality selection, rung occupancy, and related instance-selection problems are projections of that single missing primitive.
On the lattice layer recognition is discrete, so a weight rule needs only two premises. Factorization under independent composition is the multiplicative shadow of ledger cost additivity: a non-factorizing weight would carry unpaid correlation. Per-step self-similar balance pins the unit weight by ρ = 1/(1+ρ), the only balance equation expressible with the ledger's forced reciprocal-shift generator; BIT kernel shape forcing then yields ρ = φ⁻¹, consistent with T6 uniqueness of scale.
The fields match those of a rung dilution (the BIT kernel object) one-for-one, so the two interfaces are the same premises under different names.
proof idea
Definitional structure, no proof body. Four fields package the lattice axioms: a weight map $w:\mathbb{N}\to\mathbb{R}$, strict positivity of every $w(n)$, multiplicativity $w(m+n)=w(m)w(n)$, and the fixed-point equation $w(1)=1/(1+w(1))$. Mathematical force lives in the field types and in the identification of factorization with cost-additivity and of the balance equation with reciprocal self-similarity. Downstream conversion to a rung dilution is field-for-field identity, after which existing kernel lemmas apply unchanged.
why it matters
Lattice-layer interface for T9, the forced measure on recognition states. Conversion to a rung dilution lets the lattice forcing theorem conclude $w(n)=\varphi^{-n}$ by the same argument that forces BIT kernel rung dilution; uniqueness then says any two rules agree pointwise. Both feed the master certificate (lattice forced, unique, continuum Gibbs form $\exp(-(ln\varphi)\cdot t)$, non-vacuity) and the one-statement theorem packaging weight $\varphi^{-1}$ per step, partition function $\varphi^2$, and mean rung $\varphi$. Closes the instance-selection gap left open by T0–T8 without importing a second scale.
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