LightChainr · LightChainr · Sep 13, 2026
diff --git a/notes/p593-phaseA-rlinear-definition-20260913.md b/notes/p593-phaseA-rlinear-definition-20260913.md
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+# p593 Phase A — pinning down the definition of `r_linear` (2026-09-13)
+
+Issue: #593. Context: the #599 probe measured block-Hankel / McMillan orders of
+9/8, 4/5, 2/2 at widths 4..6 for the P398 projected-memory object, while the
+rank tables in #593/#588 report `r_linear` = 10, 26, 72.  Phase A asks which
+candidate definition of `r_linear` reproduces 10/26/72, and whether the probe
+numbers contradict the reported ones.
+
+## Method
+
+All quantities computed at `eta = 0` from the repository's own exact generator
+(`p398_intervention_transport.Generator`, widths 4..8), with integer-valued
+readouts and exact elimination over the Mersenne prime 2147483647 (the
+repository's `RANK_PRIME`; no pivot tolerance anywhere).  Script:
+`scripts/p593_phaseA_rlinear_candidates.py`; raw output:
+`results/p593-phaseA-rlinear/raw/candidates.json`.
+
+## Result table
+
+| quantity | w=4 | w=5 | w=6 | w=7 | w=8 |
+|---|---|---|---|---|---|
+| states | 14 | 42 | 132 | 429 | 1430 |
+| **C1: joint Krylov `dim span{G0^k f}`, D0 seeds** | **10** | **26** | **72** | **218** | **689** |
+| C1 variant: seeds + constant `1` | 10 | 26 | 72 | 218 | 689 |
+| C1 variant: D2 dictionary seeds (8 readouts) | 10 | 42 | 76 | 415 | 750 |
+| C1 variant: `k >= 1` only (drop the seeds) | 9 | 25 | 71 | 217 | 688 |
+| C1 variant: Krylov under `H` instead of `G0` | 10 | 22 | 76 | 214 | 750 |
+| C1 variant: floating-point Krylov | 10 | 27 | 81 | – | – |
+| C2: coarsest exact lumping vs `G0` only (block count) | 10 | 26 | 76 | 232 | 750 |
+| C2: coarsest exact lumping vs `G0` and `H` jointly | 10 | 26 | 76 | 232 | 750 |
+| C2: distinct D0 readout value-tuples | 8 | 12 | 18 | 24 | 32 |
+| C3: rank `G0` mod p | 13 | 41 | 131 | 428 | 1429 |
+| C3: rank `H` mod p | 13 | 32 | 130 | 392 | 1429 |
+| C3: rank `J` / rank `D` mod p | 13 | 41 | 131 | 428 | 1429 |
+| C3: distinct rows of `G0` | 14 | 42 | 132 | 429 | 1430 |
+| repository `observable_reachable_dimension` (cross-check) | 10 | 26 | 72 | 218 | not run (budget) |
+
+## Conclusion
+
+**Candidate 1 is pinned, exactly and only.**  `r_linear` is the joint Krylov
+dimension
+
+```text
+r_linear = dim span{ G0^k f : k >= 0, f in {blocks, singletons, wrap} }
+```
+
+computed by exact elimination over the prime 2147483647.  It reproduces
+10/26/72 at widths 4..6 and 218 at width 7, and agrees digit-for-digit with
+`rank_notions.D0_additive_local_counts.r_linear` stored in
+`results/p398-intervention-transport/latest.json`.
+
+- **Candidate 2 (lumping degrees of freedom) is falsified**: the coarsest
+  exact strong lumping gives 76/232/750 at widths 6..8 — that is `r_positive`,
+  not `r_linear`.  (It coincides with `r_linear` only at widths 4 and 5.)
+- **Candidate 3 (pencil ranks / combinatorial structures) is falsified**:
+  every pencil rank is `size - 1` or near it, and no combinatorial count
+  matches.
+- The #599 probe numbers 9/8, 4/5, 2/2 are **not** a competing measurement of
+  `r_linear`; they are block-Hankel / McMillan orders of the *projected
+  memory kernel* on the frozen rank-6 span — a strictly smaller and different
+  object (its `k >= 1`-flavoured count at width 4 is 9, which is the probe's
+  block-Hankel value, but the sequence diverges immediately at width 5).
+
+## New value produced
+
+- **True `r_linear` at width 8 = 689** (Krylov depth 621).  The repository
+  entry was budget-truncated (`>= 150`, `LINEAR_RANK_BUDGET_ABOVE`); the
+  exact elimination at full budget is now affordable and the number is a
+  certified lower bound on the rational rank.
+
+## Notes
+
+- The floating-point Krylov variant overestimates (27 at width 5, 81 at
+  width 6 against true 26/72), reproducing the failure mode the ticket
+  documents for float Gram-Schmidt.
+- `dim span{G0^k f}` with the constant function added to the seeds is
+  unchanged at every width: `1` is already in the Krylov span of the D0
+  readouts.
+- Full-budget `r_linear` at widths 9 and 10 is computed in Phase B
+  (`results/p593-width9-10-memory-degree/`).
+
+Full Matching-One repository CI has not been run for this commit.
diff --git a/results/p593-phaseA-rlinear/REPORT.md b/results/p593-phaseA-rlinear/REPORT.md
@@ -0,0 +1,30 @@
+# p593 Phase A — r_linear definition pinned (raw: candidates.json)
+
+- date: 2026-09-13
+- issue: 593 (Phase A)
+- question: which candidate definition of `r_linear` reproduces 10/26/72 at widths 4..6?
+- answer: the joint Krylov dimension `dim span{G0^k f : k >= 0, f in D0 readouts}` under exact elimination mod 2147483647; candidates 2 (lumping) and 3 (pencil ranks) are falsified.
+- new value: true `r_linear` at width 8 = 689 (repository had budget-truncated `>= 150`).
+
+## Commands
+
+```bash
+cd scripts
+python p593_phaseA_rlinear_candidates.py
+```
+
+## Environment
+
+- Python 3.13 (managed env), numpy 2.5.2, scipy 1.18.1
+- repository generator/codec code unchanged; all arithmetic exact over Z/2147483647
+- no frozen configuration was substituted: seeds, readouts, dictionary nesting and the prime are the repository's own declarations
+
+## Verdict per candidate
+
+| candidate | definition | verdict |
+|---|---|---|
+| 1 | joint Krylov over D0 seeds, exact mod-p | **matches 10/26/72/218 exactly** |
+| 2 | linear lumping (coarsest exact strong lumping block count) | gives 76/232/750 at widths 6..8 = `r_positive`, not `r_linear` |
+| 3 | pencil ranks (G0, H, J, D mod p; row patterns; value tuples) | no match at any width |
+
+Full Matching-One repository CI has not been run for this commit.
diff --git a/results/p593-phaseA-rlinear/commands.txt b/results/p593-phaseA-rlinear/commands.txt
@@ -0,0 +1,2 @@
+cd scripts && python p593_phaseA_rlinear_candidates.py
+# output -> raw/candidates.json (stdout lines, one JSON object per width)
diff --git a/results/p593-phaseA-rlinear/metadata.json b/results/p593-phaseA-rlinear/metadata.json
@@ -0,0 +1,12 @@
+{
+  "issue": 593,
+  "phase": "A",
+  "date": "2026-09-13",
+  "topic": "r_linear definition",
+  "conclusion": "r_linear = dim span{G0^k f : k>=0, f in D0 readouts}, exact elimination mod 2147483647; candidates 2 and 3 falsified; true r_linear(8) = 689",
+  "widths": [4, 5, 6, 7, 8],
+  "prime": 2147483647,
+  "artifacts": ["raw/candidates.json"],
+  "scripts": ["scripts/p593_phaseA_rlinear_candidates.py"],
+  "ci_status": "Full Matching-One repository CI has not been run for this commit."
+}
diff --git a/results/p593-phaseA-rlinear/raw/candidates.json b/results/p593-phaseA-rlinear/raw/candidates.json
@@ -0,0 +1,104 @@
+[
+  {
+    "width": 4,
+    "size": 14,
+    "c1_krylov_D0": 10,
+    "c1_krylov_D0_depth": 5,
+    "c1_krylov_D0_plus_constant": 10,
+    "c1_krylov_D2": 10,
+    "c1_krylov_D0_kge1": 9,
+    "c1_krylov_D0_with_H": 10,
+    "c1_float_krylov_D0": 10,
+    "repo_observable_reachable_dimension": 10,
+    "c2_lumping_G0_only": 10,
+    "c2_lumping_G0_and_H_joint": 10,
+    "c2_distinct_readout_tuples": 8,
+    "c3_rank_G0_mod_p": 13,
+    "c3_rank_H_mod_p": 13,
+    "c3_rank_J_mod_p": 13,
+    "c3_rank_D_mod_p": 13,
+    "c3_distinct_rows_G0": 14,
+    "seconds": 0.0
+  },
+  {
+    "width": 5,
+    "size": 42,
+    "c1_krylov_D0": 26,
+    "c1_krylov_D0_depth": 17,
+    "c1_krylov_D0_plus_constant": 26,
+    "c1_krylov_D2": 42,
+    "c1_krylov_D0_kge1": 25,
+    "c1_krylov_D0_with_H": 22,
+    "c1_float_krylov_D0": 27,
+    "repo_observable_reachable_dimension": 26,
+    "c2_lumping_G0_only": 26,
+    "c2_lumping_G0_and_H_joint": 26,
+    "c2_distinct_readout_tuples": 12,
+    "c3_rank_G0_mod_p": 41,
+    "c3_rank_H_mod_p": 32,
+    "c3_rank_J_mod_p": 41,
+    "c3_rank_D_mod_p": 41,
+    "c3_distinct_rows_G0": 42,
+    "seconds": 0.0
+  },
+  {
+    "width": 6,
+    "size": 132,
+    "c1_krylov_D0": 72,
+    "c1_krylov_D0_depth": 53,
+    "c1_krylov_D0_plus_constant": 72,
+    "c1_krylov_D2": 76,
+    "c1_krylov_D0_kge1": 71,
+    "c1_krylov_D0_with_H": 76,
+    "c1_float_krylov_D0": 81,
+    "repo_observable_reachable_dimension": 72,
+    "c2_lumping_G0_only": 76,
+    "c2_lumping_G0_and_H_joint": 76,
+    "c2_distinct_readout_tuples": 18,
+    "c3_rank_G0_mod_p": 131,
+    "c3_rank_H_mod_p": 130,
+    "c3_rank_J_mod_p": 131,
+    "c3_rank_D_mod_p": 131,
+    "c3_distinct_rows_G0": 132,
+    "seconds": 0.3
+  },
+  {
+    "width": 7,
+    "size": 429,
+    "c1_krylov_D0": 218,
+    "c1_krylov_D0_depth": 184,
+    "c1_krylov_D0_plus_constant": 218,
+    "c1_krylov_D2": 415,
+    "c1_krylov_D0_kge1": 217,
+    "c1_krylov_D0_with_H": 214,
+    "repo_observable_reachable_dimension": 218,
+    "c2_lumping_G0_only": 232,
+    "c2_lumping_G0_and_H_joint": 232,
+    "c2_distinct_readout_tuples": 24,
+    "c3_rank_G0_mod_p": 428,
+    "c3_rank_H_mod_p": 392,
+    "c3_rank_J_mod_p": 428,
+    "c3_rank_D_mod_p": 428,
+    "c3_distinct_rows_G0": 429,
+    "seconds": 7.7
+  },
+  {
+    "width": 8,
+    "size": 1430,
+    "c1_krylov_D0": 689,
+    "c1_krylov_D0_depth": 621,
+    "c1_krylov_D0_plus_constant": 689,
+    "c1_krylov_D2": 750,
+    "c1_krylov_D0_kge1": 688,
+    "c1_krylov_D0_with_H": 750,
+    "c2_lumping_G0_only": 750,
+    "c2_lumping_G0_and_H_joint": 750,
+    "c2_distinct_readout_tuples": 32,
+    "c3_rank_G0_mod_p": 1429,
+    "c3_rank_H_mod_p": 1429,
+    "c3_rank_J_mod_p": 1429,
+    "c3_rank_D_mod_p": 1429,
+    "c3_distinct_rows_G0": 1430,
+    "seconds": 167.8
+  }
+]
Original file line number	Diff line number	Diff line change
		@@ -0,0 +1,2 @@
		cd scripts && python p593_phaseA_rlinear_candidates.py
		# output -> raw/candidates.json (stdout lines, one JSON object per width)