p593 Phase A: r_linear is the D0 joint Krylov dimension (10/26/72/218/689)

[LightChainr]

Issue #593 — Phase A: pinning down r_linear

Closes the Phase A question of #593 (definition audit only; no frozen configuration touched).

Conclusion

r_linear is pinned, exactly and only, as the joint Krylov dimension

r_linear = dim span{ G0^k f : k >= 0, f in {blocks, singletons, wrap} }

computed by exact elimination over the prime 2147483647. It reproduces the reported 10 / 26 / 72 at widths 4..6 and 218 at width 7, digit-for-digit against results/p398-intervention-transport/latest.json.

• Candidate 2 (linear 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.
• Candidate 3 (pencil ranks: G0, H, J, D mod p; row patterns; readout value tuples) is falsified at every width.
• The [P2 long-horizon probe] Experiment-relative state complexity atlas: symmetry, Hankel, memory, predictive language, and finite-size flow #599 probe values 9/8, 4/5, 2/2 are block-Hankel / McMillan orders of the projected memory kernel — a different, smaller object — not a competing measurement of r_linear.

New value

True r_linear at width 8 = 689 (Krylov depth 621). The repository entry was budget-truncated (>= 150).

Artifacts

• notes/p593-phaseA-rlinear-definition-20260913.md — result table and conclusion
• results/p593-phaseA-rlinear/ — REPORT.md, metadata.json, commands.txt, raw/candidates.json
• scripts/p593_phaseA_rlinear_candidates.py — the candidate scan (widths 4..8)

All arithmetic is exact over Z/2147483647; no frozen span, dictionary, lag grid or normalization convention was changed.

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