N_e_arithmetic

The module records the baryon-asymmetry rung on the φ-ladder and the arithmetic identities that factor it through the eight-tick period and the passive-mode count. Core entry: rung 44 carries η_B (baryon asymmetry) as φ^{-44}, status theorem.

Here 44 is that baryon rung, 11 is the passive-mode count (appearing elsewhere as the eleven-times table and Z_cf-scale bookkeeping), and 55 is the conjectured number of inflation e-foldings N_e. The identity is pure ℕ arithmetic; the physical gloss is supplied by the doc-comment and sibling factorizations.

No upstream lemmas are required: the statement is self-contained natural-number equality inside Cosmology.PhiRungLadder.

One-line tactic proof: norm_num discharges the closed natural-number equation 44 + 11 = 55. No lemmas, no rewriting, no induction.

Places the inflation e-fold count on the same φ-ladder arithmetic that already hosts the baryon rung. Sibling declarations (N_e_factorization, baryon_rung_factorization, eleven_times_table, eta_B_rung_val) assemble the same integers into product and rung identities; this sum is the additive companion that lets N_e be read as baryon rung plus passive modes.

In the broader Recognition framework the eight-tick octave (T7) and the φ-ladder mass/yardstick bookkeeping make small integer rungs load-bearing. The module claims 0 sorry and 0 axioms; this line is part of that clean arithmetic spine. It does not itself derive inflation dynamics from the forcing chain, but it locks the integer relation those later arguments would use.