p593 Phase B: width 9-10 memory degree saturates at 14; true r_linear/r_positive/r_transport at widths 8-10

[LightChainr]

Issue #593 — Phase B: P398 widths 9-10

Phase B of #593, started after Phase A was delivered (PR #753). The frozen configuration (readout seeds, block Krylov span 12→prefix 6, per-frontier normalization, declared seed order, counting inner product, kernel grid t = 0, 0.25, ..., 4.0) is the repository's own code end to end, applied via a generator shim that is bit-exact against the repository at widths 1..8 (including the transition-table SHA-256 control c18df595...). Widths 4-8 are recomputed and match the ticket's reported numbers.

Deliverable 1 — the memory degree (the core question)

width	4	5	6	7	8	9	10
states	14	42	132	429	1430	4862	16796
rank C	2	3	3	3	3	3	3
block-Hankel numerical rank (tol 1e-6, true SVD)	4	9	12	13	14	14	14
effective order 99% / 99.9%	2/2	3/3	3/4	3/4	3/4	3/4	3/4
integrated ‖K‖	0.876	1.016	1.207	1.456	1.681	1.873	2.036
decay time	0.063	0.089	0.105	0.119	0.136	0.154	0.172
tail mass > t=2	1.7e-6	3.7e-5	1.4e-4	3.2e-4	5.7e-4	9.3e-4	1.5e-3

The numerical order saturates. Across a x1199 growth in states, the block-Hankel rank pins at 14 and the energy orders pin at 3/4. The projected-memory object supports a bounded non-Markov reduced description at the frozen span.

Deliverable 2 — rank notions continued

width	r_linear(D0) true	r_positive	r_transport
8	689 (was >= 150)	750	6
9	2275	2494	6
10	7718	8524	6

The r_positive / r_transport gap grows 125 → 416 → 1421, i.e. linearly in the state count; r_transport does not move. r_linear/states drifts slowly down (0.48 → 0.46) — the exact linear object keeps growing while the memory pole count stays at 14.

Deliverable 3 — out-of-pencil lumping collapse

The joint lumping against G0 + the single_point_join tilt collapses to the identity partition at widths 9 and 10 (as at 4-8). Confirmed, not a small-width accident.

Controls

All six ticket controls pass (results/p593-width9-10-memory-degree/raw/controls.json): transition-table hash, row sums/positivity/strong connectivity, exp(tG)1 = 1 (1.6e-15), uniformization vs dense Taylor at width 5 (1.9e-15), full-rank projection K = 0.0 exactly, generator-closed span K = 5.75e-30 relative (ticket: 6.1e-30), Schur resolvent identity 3.8e-16 (ticket: 7.4e-16). Width-6 pipeline validation reproduces the repository's stored r_linear/r_transport (72/76/76, 6/6/8) digit-for-digit.

Declared departures (both ticket requests/annotations)

1. Block-Hankel singular values via true SVD, not H^T H eigenvalues.
2. r_linear at widths 8/9/10 computed at full budget (exact mod-p elimination, numpy echelon); D1/D2 stay budget-truncated at 150 as in the repository's width-8 entry. Width-10 r_transport eta ladder (0, -0.25, 0.25); the extra repository ladder etas never enter dictionary_scores.

Artifacts

• notes/p593-width9-10-memory-degree-20260913.md
• results/p593-width9-10-memory-degree/ — REPORT.md, metadata.json, commands.txt, raw/ (per-width JSON incl. spectra)
• scripts/p593_noncrossing_fast.py, p593_phaseB_memory.py, p593_phaseB_rlinear_true.py, p593_phaseB_transport.py, p593_phaseB_controls.py

All compute ran matrix-free (no explicit D at any width; uniformization with re-projection onto range(Q)). Width 10 completed on a local machine in minutes per deliverable; no resource extrapolation anywhere.

Full Matching-One repository CI has not been run for this commit.