rung44_is_lattice_weight

Module T9 closes the missing weighting rule on recognition states after the T0–T8 shape chain. On the lattice layer, any admissible weight factorizes over independent steps and obeys the single-step self-similar balance ρ = 1/(1+ρ), which forces ρ = φ^{-1} and thus w(n) = φ^{-n}.

latticeWeight n is defined as (1/φ)^n, the forced geometric measure after n recognition steps. Independently, phi_neg_44 is the cosmology-side constant φ^{-44}, the same scale that appears as the baryogenesis prefactor η_B = φ^{-44} via the phi-rung ladder.

This lemma is the pure arithmetic identification of those two writings of the same real number. It does not re-derive the forcing of the weight rule; it only equates the named constants so the cosmology scale can be read as a lattice measure.

Unfold both sides to φ^{-44} and (1/φ)^{44}. Rewrite with zpow_neg to turn the integer negative power into a reciprocal, cast 44 : ℤ as a natural via norm_num, apply zpow_natCast, then div_pow, one_pow, and one_div to match (1/φ)^{44}. Pure power-arithmetic; no measure or cosmology lemmas.

Feeds the master certificate measureForcingCert in the same module, which packages lattice forcing, uniqueness, continuum Gibbs form, and nonvacuity as the T9 closure. The identification lets the cosmology rung-44 scale (dark-energy structural constants and η_B = φ^{-44}) be read literally as the forced lattice measure at 44 steps, rather than an ad-hoc exponent.

In the broader framework this is the concrete link between the geometric φ-measure (weight φ^{-1} per recognition step) and a named physical scale already used downstream. It supports the module claim that BIT kernel rung dilution and the forced measure are the same object, with rung 44 as the canonical high-rung witness.