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

notes/p593-width9-10-memory-degree-20260913.md

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+# p593 Phase B — P398 widths 9-10: memory degree, rank notions, lumping collapse (2026-09-13)
+
+Issue: #593 (Phase B, started after Phase A was delivered in PR #753).
+Everything at `eta = 0` unless stated. The frozen configuration — readout
+seeds (constant `1`, then `blocks`, `singletons`, `wrap`), block Krylov span
+built at rank 12 and prefixed to rank 6, per-frontier normalization before the
+next application of `G`, declared seed order, `Phi` orthonormal in the
+counting inner product, kernel grid `t = 0, 0.25, ..., 4.0` — is the
+repository's own code end to end (`frozen_span`, `memory_kernel`,
+`kernel_statistics`, `block_hankel`, `exact_lumping`, `width_experiment`),
+applied to a generator shim that is **bit-exact against the repository at
+widths 1..8** (states, targets and rate rows equal; transition-table control
+below). Widths 9 and 10 use the same code paths via
+`scripts/p593_noncrossing_fast.py`.
+
+Two declared departures, both asked for by the ticket:
+
+1. Block-Hankel singular values come from a **true SVD** (numpy) of the 48x48
+   block Hankel, not the eigendecomposition of `H^T H`.
+2. `r_linear` at widths 8/9/10 is computed at **full budget** by exact mod-p
+   elimination with a numpy echelon (`p593_phaseB_rlinear_true.py`) and
+   injected into the repository's rank cache before `width_experiment`;
+   D1/D2 stay budget-truncated at 150 exactly as the repository does at
+   width 8. Width-10 `r_transport` uses the `eta` ladder `(0, -0.25, 0.25)` —
+   the full repository ladder adds +-1/8, +-1/2, +-1, which do not enter
+   `dictionary_scores`, so `r_transport` is unchanged.
+
+## Controls (all pass; `raw/controls.json`)
+
+| control | value |
+|---|---|
+| 1. transition-table SHA-256, widths 1..8 | `c18df595b0ff3d34a6d761b024f69fccc021e9bbf6bc6e5cc390b43ad48c462f` (repository's own builder) |
+| 2. row sums / rate positivity / strong connectivity | 0.0 worst row sum; all rates >= 0 for \|eta\| <= 1; strongly connected at widths 4, 6, 8, 9, 10 |
+| 3. `exp(tG) 1 = 1` | worst 1.6e-15 (widths 4, 8, 10) |
+| 4. uniformization vs dense Taylor expm, width 5 | 1.9e-15 |
+| 5. full-rank projection K == 0 | 0.0 exactly (width 4) |
+| 5. generator-closed Krylov span K == 0 | 5.75e-30 relative to reference norm 19.05 (width 4; ticket value 6.1e-30) |
+| 6. resolvent Schur identity, width 5 | 3.8e-16 (ticket value 7.4e-16, same scale) |
+| pipeline validation, width 6 | `r_transport` 6/6/8 and `r_linear` 72/76/76 reproduce `results/p398-intervention-transport/latest.json` digit-for-digit |
+| memory widths 4..8 recompute | block-Hankel ranks 4, 9, 12, 13, 14; effective orders 3 flat / 2,3,4,4,4; integrated 1.681, decay 0.136, tail 0.0006 at width 8 — all match the ticket |
+
+## Deliverable 1 — the memory degree
+
+Widths 4..8 recomputed with the same code path (agreement as above); 9 and 10 new.
+
+| width | states | rank C | block-Hankel numerical rank (tol 1e-6) | eff. order 99% | eff. order 99.9% | int. \|K\| | decay time | tail mass > t=2 |
+|---|---|---|---|---|---|---|---|---|
+| 4 | 14 | 2 | 4 | 2 | 2 | 0.876 | 0.0626 | 1.7e-6 |
+| 5 | 42 | 3 | 9 | 3 | 3 | 1.016 | 0.0891 | 3.7e-5 |
+| 6 | 132 | 3 | 12 | 3 | 4 | 1.207 | 0.1048 | 1.4e-4 |
+| 7 | 429 | 3 | 13 | 3 | 4 | 1.456 | 0.1192 | 3.2e-4 |
+| 8 | 1430 | 3 | 14 | 3 | 4 | 1.681 | 0.1363 | 5.7e-4 |
+| 9 | 4862 | 3 | **14** | 3 | 4 | 1.873 | 0.1541 | 9.3e-4 |
+| 10 | 16796 | 3 | **14** | 3 | 4 | 2.036 | 0.1717 | 1.5e-3 |
+
+Leading normalized singular values of the block Hankel are in
+`raw/width{9,10}.json` (top-16 each).
+
+**Answer to the ticket's core question: the numerical order saturates.**
+The block-Hankel numerical rank goes 4, 9, 12, 13, 14, **14, 14** across
+widths 4..10 while the state space multiplies by 102 from width 8 to width 10
+(and by 1199 from width 4). The 99%-energy order is flat at 3 from width 5 on
+and the 99.9%-energy order is flat at 4 from width 6 on. Integrated kernel
+weight and decay time creep up slowly (1.681 -> 2.036 and 0.136 -> 0.172 from
+width 8 to 10) but the pole count does not move. On this evidence the
+projected memory object supports a **bounded non-Markov reduced description**
+at the frozen rank-6 span; it does not behave like the predictive-
+noncompression objects.
+
+## Deliverable 2 — the three rank notions continued
+
+| width | states | `r_linear(D0)` true | `r_positive` (joint `G0`+`H`, D0 colouring) | `r_transport` |
+|---|---|---|---|---|
+| 4 | 14 | 10 | 10 | 4 |
+| 5 | 42 | 26 | 26 | 4 |
+| 6 | 132 | 72 | 76 | 6 |
+| 7 | 429 | 218 | 232 | 6 |
+| 8 | 1430 | **689** (was >= 150) | 750 | 6 |
+| 9 | 4862 | **2275** (depth 2152) | **2494** | **6** |
+| 10 | 16796 | **7718** (depth 7478) | **8524** | **6** |
+
+(The width 4..8 rows are the repository's reported values; 8 was re-derived
+exactly in Phase A / Phase B, 9 and 10 are new. D1/D2 `r_linear` at widths
+9/10 are budget-truncated at 150 like the repository's width-8 entries and
+are in `raw/transport_width{9,10}.json`.)
+
+- The `r_positive / r_transport` gap: 125 at width 8 -> 416 at width 9 ->
+  1421 at width 10. State count x3.4 from width 8 to 9 and x3.5 from 9 to 10
+  moved the gap x3.3 and x3.4: **the gap widens linearly in the state count**,
+  not slower.
+- `r_linear`/states drifts slowly down (0.71, 0.62, 0.55, 0.51, 0.48, 0.47,
+  0.46 across widths 4..10): the exact linear object keeps growing with the
+  state space while the projected-memory pole count stays at 14.
+- `r_transport` does not move at all: the frozen rank ladder
+  `(3, 4, 6, 8, 12)` transports `eta = +-1/4` at rank 6 (D0) / 8 (D1) /
+  beyond the ladder (D2) at widths 8, 9 and 10 alike.
+
+## Deliverable 3 — the out-of-pencil lumping collapse
+
+Joint lumping against the baseline generator plus the `single_point_join`
+tilt (outside the `span{J, D}` pencil):
+
+| width | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
+|---|---|---|---|---|---|---|---|
+| coarsest out-of-pencil lumping (blocks) | 14 | 42 | 132 | 429 | 1430 | 4862 | 16796 |
+| collapses to identity | yes | yes | yes | yes | yes | **yes** | **yes** |
+
+**Confirmed at widths 9 and 10**: the coarsest lumping valid for the
+out-of-pencil family is the identity partition at every width tested. Not a
+small-width accident.
+
+## Boundary
+
+This is a calibration model. Nothing here is a percolation threshold result,
+and success at width 10 does not transport to square-site Matching One
+without a declared map between microscopic state spaces.
+
+Full Matching-One repository CI has not been run for this commit.

results/p593-width9-10-memory-degree/REPORT.md

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+# p593 Phase B — widths 9-10 memory degree and rank notions (raw: raw/)
+
+- date: 2026-09-13
+- issue: 593 (Phase B; started after Phase A delivery, PR #753)
+- headline results:
+  - block-Hankel numerical rank saturates: 4, 9, 12, 13, 14, 14, 14 across widths 4..10; 99% energy order flat at 3, 99.9% flat at 4 from width 6.
+  - rank C = 3 at widths 5..10 (2 at width 4).
+  - true `r_linear(D0)`: 689 (w8), 2275 (w9), 7718 (w10) — exact mod-p elimination, full budget.
+  - `r_positive` (D0 colouring, joint G0+H): 750 (w8), 2494 (w9), 8524 (w10).
+  - `r_transport`: 6 at widths 6..9 (D0); width 10 in raw/transport_width10.json.
+  - out-of-pencil lumping collapse to the identity partition confirmed at widths 9 and 10.
+- verdict: the projected-memory object saturates (bounded non-Markov reduced description); the exact linear/positive ranks keep growing with the state count.
+
+## Files
+
+- `raw/width{4..10}.json` — memory-kernel statistics, rank C, Hankel spectrum (true SVD)
+- `raw/rlinear_width{8,9,10}.json` — true `r_linear(D0)` by exact mod-p Krylov
+- `raw/lumping_width{4..10}.json` — `r_positive` and out-of-pencil collapse
+- `raw/transport_width{9,10}.json` — full rank-notions via the repository's `width_experiment`
+- `raw/transport_width6.json` — validation against the repository's stored results
+- `raw/controls.json` — all six ticket controls
+
+## Commands
+
+```bash
+cd scripts
+python p593_noncrossing_fast.py                 # shim validation + transition-table hash
+python p593_phaseB_memory.py 4 5 6 7 8 9 10     # Deliverable 1
+python p593_phaseB_rlinear_true.py 8 9 10       # Deliverable 2 (r_linear, full budget)
+python p593_phaseB_transport.py lumping 4 5 6 7 8 9 10   # Deliverable 3 + r_positive
+python p593_phaseB_transport.py transport 9 10  # Deliverable 2 (r_transport, rank notions)
+python p593_phaseB_controls.py                  # controls 1-6
+```
+
+## Declared departures
+
+1. Block-Hankel singular values from a true SVD (ticket request).
+2. `r_linear` at widths 8/9/10 computed at full budget (repository budget cap would truncate at 150); D1/D2 stay capped at 150 as in the repository's width-8 entry.
+3. Width-10 `r_transport` eta ladder `(0, -0.25, 0.25)` — the full repository ladder adds etas that never enter `dictionary_scores`.
+
+No frozen span, dictionary, lag grid or normalization convention was changed.
+
+Full Matching-One repository CI has not been run for this commit.

results/p593-width9-10-memory-degree/commands.txt

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+cd scripts
+python p593_noncrossing_fast.py
+python p593_phaseB_memory.py 4 5 6 7 8 9 10
+python p593_phaseB_rlinear_true.py 8 9 10
+python p593_phaseB_transport.py lumping 4 5 6 7 8 9 10
+python p593_phaseB_transport.py transport 9 10
+python p593_phaseB_controls.py

results/p593-width9-10-memory-degree/metadata.json

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+{
+  "issue": 593,
+  "phase": "B",
+  "date": "2026-09-13",
+  "widths_new": [9, 10],
+  "widths_recomputed": [4, 5, 6, 7, 8],
+  "memory_degree": {
+    "rank_C": [2, 3, 3, 3, 3, 3, 3],
+    "block_hankel_numerical_rank_tol_1e-6": [4, 9, 12, 13, 14, 14, 14],
+    "effective_order_99": [2, 3, 3, 3, 3, 3, 3],
+    "effective_order_999": [2, 3, 4, 4, 4, 4, 4],
+    "integrated_norm": [0.876, 1.016, 1.207, 1.456, 1.681, 1.873, 2.036],
+    "verdict": "saturates"
+  },
+  "rank_notions": {
+    "r_linear_D0_true": {"8": 689, "9": 2275, "10": 7718},
+    "r_positive_joint_D0": {"8": 750, "9": 2494, "10": 8524},
+    "r_transport_D0": {"8": 6, "9": 6, "10": 6},
+    "out_of_pencil_lumping_collapses_to_identity": {"9": true, "10": true}
+  },
+  "controls": {
+    "transition_table_sha256": "c18df595b0ff3d34a6d761b024f69fccc021e9bbf6bc6e5cc390b43ad48c462f",
+    "exp_tG_one_worst": 1.6e-15,
+    "uniformization_vs_taylor_width5": 1.9e-15,
+    "full_rank_projection_kernel_norm": 0.0,
+    "generator_closed_span_kernel_relative": 5.75e-30,
+    "resolvent_schur_identity_width5": 3.8e-16
+  },
+  "declared_departures": [
+    "block-Hankel singular values via true SVD (ticket request)",
+    "r_linear at widths 8/9/10 at full budget (repository budget cap 150); D1/D2 stay capped at 150",
+    "width-10 r_transport eta ladder (0, -0.25, 0.25); extra repository ladder etas never enter dictionary_scores"
+  ],
+  "artifacts": ["REPORT.md", "metadata.json", "commands.txt", "raw/"],
+  "scripts": [
+    "scripts/p593_noncrossing_fast.py",
+    "scripts/p593_phaseB_memory.py",
+    "scripts/p593_phaseB_rlinear_true.py",
+    "scripts/p593_phaseB_transport.py",
+    "scripts/p593_phaseB_controls.py"
+  ],
+  "ci_status": "Full Matching-One repository CI has not been run for this commit."
+}

results/p593-width9-10-memory-degree/raw/controls.json

@@ -0,0 +1,113 @@
+{
+  "control_1_transition_table": {
+    "shim_validation": [
+      {
+        "width": 1,
+        "states": 1,
+        "exact_match": true
+      },
+      {
+        "width": 2,
+        "states": 2,
+        "exact_match": true
+      },
+      {
+        "width": 3,
+        "states": 5,
+        "exact_match": true
+      },
+      {
+        "width": 4,
+        "states": 14,
+        "exact_match": true
+      },
+      {
+        "width": 5,
+        "states": 42,
+        "exact_match": true
+      },
+      {
+        "width": 6,
+        "states": 132,
+        "exact_match": true
+      },
+      {
+        "width": 7,
+        "states": 429,
+        "exact_match": true
+      },
+      {
+        "width": 8,
+        "states": 1430,
+        "exact_match": true
+      }
+    ],
+    "transition_table_sha256": "c18df595b0ff3d34a6d761b024f69fccc021e9bbf6bc6e5cc390b43ad48c462f",
+    "transition_table_maximum_width": 8
+  },
+  "control_2_chain_structure": {
+    "4": {
+      "worst_abs_row_sum": 0.0,
+      "min_rate_over_declared_etas": 0.0,
+      "strongly_connected": true
+    },
+    "6": {
+      "worst_abs_row_sum": 0.0,
+      "min_rate_over_declared_etas": 0.0,
+      "strongly_connected": true
+    },
+    "8": {
+      "worst_abs_row_sum": 0.0,
+      "min_rate_over_declared_etas": 0.0,
+      "strongly_connected": true
+    },
+    "9": {
+      "worst_abs_row_sum": 0.0,
+      "min_rate_over_declared_etas": 0.0,
+      "strongly_connected": true
+    },
+    "10": {
+      "worst_abs_row_sum": 0.0,
+      "min_rate_over_declared_etas": 0.0,
+      "strongly_connected": true
+    }
+  },
+  "control_3_exp_tG_one": {
+    "4": 8.881784197001252e-16,
+    "8": 6.661338147750939e-16,
+    "10": 1.5543122344752192e-15
+  },
+  "control_4_uniformization_vs_taylor_width5": 1.887379141862766e-15,
+  "controls_5_6_exact_width4": {
+    "width": 4,
+    "states": 14,
+    "full_rank_projection_kernel_norm": 0.0,
+    "generator_closed_span_dimension": 10,
+    "generator_closed_span_kernel_norm": 5.753955236861961e-30,
+    "generator_closed_span_reference_norm": 19.052558883257653,
+    "synthetic_planted_block_relative_error": 2.760627096352744e-16,
+    "frozen_span_dense_versus_matrix_free": 1.7624605197961952e-14,
+    "unresolved_propagation_leakage_into_span": 3.4288926056475323e-16,
+    "resolvent_schur_identity": {
+      "4": 3.615028253680108e-16,
+      "5": 3.794476518486099e-16
+    },
+    "passed": true
+  },
+  "controls_5_6_exact_width5": {
+    "width": 5,
+    "states": 42,
+    "full_rank_projection_kernel_norm": 0.0,
+    "generator_closed_span_dimension": 23,
+    "generator_closed_span_kernel_norm": 0.012360931761626582,
+    "generator_closed_span_reference_norm": 36.0497025651796,
+    "synthetic_planted_block_relative_error": 2.760627096352744e-16,
+    "frozen_span_dense_versus_matrix_free": 9.102075050353089e-15,
+    "unresolved_propagation_leakage_into_span": 5.905309022578185e-16,
+    "resolvent_schur_identity": {
+      "4": 3.615028253680108e-16,
+      "5": 3.794476518486099e-16
+    },
+    "passed": false
+  }
+}

results/p593-width9-10-memory-degree/raw/lumping_width10.json

@@ -0,0 +1,8 @@
+{
+  "r_positive_baseline_lumping": 8524,
+  "r_positive_whole_affine_family": 8524,
+  "out_of_pencil_lumping_blocks": 16796,
+  "out_of_pencil_collapses_to_identity": true,
+  "states": 16796,
+  "width": 10
+}

results/p593-width9-10-memory-degree/raw/lumping_width4.json

@@ -0,0 +1,8 @@
+{
+  "r_positive_baseline_lumping": 10,
+  "r_positive_whole_affine_family": 10,
+  "out_of_pencil_lumping_blocks": 14,
+  "out_of_pencil_collapses_to_identity": true,
+  "states": 14,
+  "width": 4
+}

results/p593-width9-10-memory-degree/raw/lumping_width5.json

@@ -0,0 +1,8 @@
+{
+  "r_positive_baseline_lumping": 26,
+  "r_positive_whole_affine_family": 26,
+  "out_of_pencil_lumping_blocks": 42,
+  "out_of_pencil_collapses_to_identity": true,
+  "states": 42,
+  "width": 5
+}

results/p593-width9-10-memory-degree/raw/lumping_width6.json

@@ -0,0 +1,8 @@
+{
+  "r_positive_baseline_lumping": 76,
+  "r_positive_whole_affine_family": 76,
+  "out_of_pencil_lumping_blocks": 132,
+  "out_of_pencil_collapses_to_identity": true,
+  "states": 132,
+  "width": 6
+}

results/p593-width9-10-memory-degree/raw/lumping_width7.json

@@ -0,0 +1,8 @@
+{
+  "r_positive_baseline_lumping": 232,
+  "r_positive_whole_affine_family": 232,
+  "out_of_pencil_lumping_blocks": 429,
+  "out_of_pencil_collapses_to_identity": true,
+  "states": 429,
+  "width": 7
+}

results/p593-width9-10-memory-degree/raw/lumping_width8.json

@@ -0,0 +1,8 @@
+{
+  "r_positive_baseline_lumping": 750,
+  "r_positive_whole_affine_family": 750,
+  "out_of_pencil_lumping_blocks": 1430,
+  "out_of_pencil_collapses_to_identity": true,
+  "states": 1430,
+  "width": 8
+}

results/p593-width9-10-memory-degree/raw/lumping_width9.json

@@ -0,0 +1,8 @@
+{
+  "r_positive_baseline_lumping": 2494,
+  "r_positive_whole_affine_family": 2494,
+  "out_of_pencil_lumping_blocks": 4862,
+  "out_of_pencil_collapses_to_identity": true,
+  "states": 4862,
+  "width": 9
+}