The Identity Ladder: One Inequality Certifies a Quantum Memory at Every Length

Version 2 (2026-08-20). One CORRECTION, inherited from the companion flagship's v3. Version 1 twice attributed the mediation rank cap to the cell being read: "each cell has exactly one outward-facing quadrature" in the discussion, and "multi-collision cells (where a cell's channel matrix can reach rank three)" in the named frontier. The cap is supplied by the PARTNERS that read a cell, not by that cell's own collision count. A cell's own re-collisions leave its own channel rank at machine zero on any axis (9.79e-17 same axis, 1.04e-16 rotated); re-colliding a partner on a non-commuting axis lifts the rank to three (5.17e-02 to 7.13e-01). Both attributions predict identical numbers in this note's single-collision architecture, so nothing computed here is affected and the frontier paragraph's substantive point is untouched, but the second parenthetical is false under its natural reading and it named the wrong cell for a follow-up experiment. Companion DOIs in this version cite the current flagship (v2, 10.5281/zenodo.22019475), superseding v1's citation of the flagship v1 record. A quantum memory that accumulates k successive records of a single system can audit itself: its memory-side statistics obey polynomial identities, and its physical realizability is a positivity question. The companion paper (Gleason from Inside, doi:10.5281/zenodo.22019475) proved the two-record certificate (identity ideal + one back-action-corrected moment inequality M*) and left the k >= 3 ladder open. This note closes it for the single-collision comb architecture at every k. The entire identity structure is generated by the transverse Bloch block of the per-record collision channel, read in its two operator orders: a route-damping chain law (every skip-pair correlation as a sum over sine-pair routes, each damped by the visibility of every record strictly between its sine vertices) and a response transfer law carrying odd higher-sine routes; both are census-exact at machine precision through five records at every span and start. The identity variety's free chart has dimension exactly 2k+1, one transport identity per added record. The completeness theorem: a chart point is realizable if and only if its outcome probabilities are valid and the FIRST pair's M* is positive semidefinite; every later adjacent certificate is a consequence (the transported effective state inherits the Bloch ball). The proof is a constructive, optimization-free rung inversion; machine confirmations include 200/200 exact round trips, a 1703-point redundancy test with zero violations, and exact inversion of all 1094 points of a persisted certification sample. The comb moment hierarchy collapses at level one: identities do all equality work (linear in k) and a single two-record inequality does all positivity work forever. The frontier is named: multi-collision cells, reused cells, and non-commuting rotation axes are outside the theorem. The ledger of failed predictions, including two wrong transfer compositions killed by census gates, is an appendix.