[P2｜检查点封存、不续宽度] P398记忆/状态阶数：共享算力暂停扩张

[LightChainr]

Why this needs server compute

Everything in #580 and #588 was done in pure Python with no numpy and no scipy, which caps us at width 8 (1430 states). The one question Phase A could not answer is a width question, and it is the question that decides how the whole block gets written up.

The measurement is cheap with numpy/scipy on a server and impossible without it. Width 9 is 4862 states, width 10 is 16796. Dense linear algebra at 16796 is out of reach even with numpy; the algorithms below are all sparse matrix-vector or matrix-free, so width 10 is comfortable with scipy sparse and width 11 (58786) is probably reachable if width 10 comes back clean.

The object

P398 is the noncrossing planar connectivity process: canonical noncrossing states on w boundary points (Catalan numbers 14, 42, 132, 429, 1430, 4862, 16796 for w = 4..10), with two moves per boundary point, join_cyclic_adjacent and detach. It is already implemented exactly and deterministically in this repository:

scripts/noncrossing_connectivity_codec.py     canonical state enumeration
scripts/planar_state_operations.py            the two moves
scripts/p398_intervention_transport.py        Generator, spans, lumping, modular rank
scripts/p398_projected_memory.py              projection, memory kernel, Hankel order

scripts/planar_transition_table.py emits a content-hashed transition table and is the right cross-check that a reimplementation agrees with ours before any new width is trusted.

The generator acts on functions, (G f)_i = sum_j G_ij f_j, with the affine family

G_eta = G_0 + eta * H,   H = sum(join) - sum(detach)

Self-transitions are dropped (they cancel between an off-diagonal entry and the exit rate).

What to compute, at widths 9 and 10

All at eta = 0 unless stated. Declared before the numbers exist; do not substitute a different span, dictionary or lag grid.

Frozen configuration (must match ours exactly)

Readout seeds, in this order: the constant function 1, then blocks, singletons, wrap (definitions in p398_intervention_transport.py, observable_*). Block Krylov span built at rank 12 and prefixed to rank 6, with each frontier vector normalized before the next application of G (raw powers reach NaN otherwise) and the frontier kept in declared seed order. Phi orthonormal in the counting inner product.

Deliverable 1 — the memory degree (the reason this ticket exists)

C = Q G Phi,  Q = I - Phi Phi^T
K(tau) = Phi^T G exp(tau D) C,  D = Q G Q

D is 16790 x 16790 at width 10 and must not be formed. Propagate the columns of C as full-space vectors constrained to range(Q), taking exp(tau D) v by uniformization on I + D/Lambda (re-project onto range(Q) every few steps; the seeds are in range(Q) and Q G v lands there exactly, so only roundoff leaks). Sample K on the uniform grid t = 0, 0.25, ..., 4.0.

Report:

rank C                                        (we get 3 at widths 5-8)
block-Hankel numerical rank at tol 1e-6       (we get 4, 9, 12, 13, 14 at widths 4-8)
effective order at 99% and 99.9% energy       (we get 3 flat, and 2,3,4,4,4)
integrated ||K||, decay time, tail mass > t=2 (we get 1.681, 0.136, 0.0006 at width 8)
the leading 16 singular values, normalized

The question is whether the numerical order keeps growing. It goes 4, 9, 12, 13, 14 across widths 4-8 while the state space multiplies by 102 — sublinear, apparently flattening, but not flat. Two more widths separate "saturates" from "grows like log n" from "grows like a small power". That distinction decides whether this object supports a bounded non-Markov reduced description or belongs with the predictive-noncompression results.

Build the block Hankel from samples 1..16 (drop t = 0), 8 x 8 blocks of size 6, and take singular values via an actual SVD rather than the eigendecomposition of H^T H — our square-root formulation caps precision near 1e-8 and that is exactly the range where the tail lives.

Deliverable 2 — the three rank notions continued

width   states   r_linear(D0)   r_positive   r_transport
  4       14         10             10            4
  5       42         26             26            4
  6      132         72             76            6
  7      429        218            232            6
  8     1430       >=150           750            6

• r_linear = dim span{G_0^k f} over the D0 readouts. Compute by exact elimination over the prime 2147483647 — the generator and readouts are integer-valued, so this is exact and is a certified lower bound on the rational rank. Do not use floating-point Gram-Schmidt: at width 6 it returns 132 (the full space) against a true 72, because repeated generator application collapses onto the dominant direction and a pivot tolerance then decides the answer. Our width-8 entry is a budget-truncated >=150; please give the true value at widths 8, 9, 10.
• r_positive = block count of the coarsest exact strong lumping refining the readout colouring, computed jointly against G_0 and H (so it is the coarsest lumping valid for the whole affine family). Partition refinement on the signature "total rate into each block", exact rational or integer arithmetic. Verify the aggregate against the full chain on the declared lag grid; ours agrees to 2.7e-15.
• r_transport = smallest rank in (3,4,6,8,12) whose frozen span transports eta = ±1/4 within the declared threshold. Needs the response scoring, which is cheap.

The gap r_positive / r_transport is 125 at width 8 and widening. Widths 9-10 say whether it widens linearly in the state count or slower.

Deliverable 3 — the out-of-pencil lumping collapse

Repeat the joint lumping with H_single = the tilt of join at boundary point 0 only, which is outside the span{J, D} pencil. At every width 4-8 the coarsest lumping valid for that family collapses to the identity partition. Confirm or break this at 9 and 10. It is currently the sharpest structural result in the block and it would be good to know it is not a small-width accident.

Controls this must pass before any number is reported

Non-negotiable, and all of them are cheap:

1. The transition table content hash at widths 1-8 must reproduce scripts/planar_transition_table.py --maximum-width 8. If a reimplementation disagrees there, nothing after it counts.
2. Exact row sums zero; positivity of all rates for |eta| <= 1; the chain strongly connected.
3. exp(tG) 1 = 1 to machine precision.
4. Uniformization against an independent dense Taylor exponential at width 5.
5. A full-rank projection must give K identically zero, and a generator-closed Krylov span must give K = 0 to 1e-8 relative — both are exact zeros in the mathematics and catch a leaking complement.
6. The resolvent Schur identity Phi^T (sI-G)^{-1} Phi = (sI - A - B(sI-D)^{-1}C)^{-1} at width 5 for a few real s > 0. This is the one a transposed B/C does not survive, and it involves no time integration.

Our values for controls 5 and 6 at width 4: 0.0, 6.1e-30, and 7.4e-16.

Format

A JSON artifact mirroring results/p398-projected-memory/latest.json plus the rank table, with every control value included and the code used. Widths 4-8 recomputed as well, so agreement with our numbers above is visible in the same file. If width 10 is expensive, width 9 alone answers most of the question and is worth returning on its own.

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.

Related: #588, #580, #549, #550.

[LightChainr]

Scope reduction: r_positive should no longer justify widths 9–10

A new exact reconstruction strongly supports the working formula

r_positive(w) = [Catalan(w)+binom(w,floor(w/2))]/2,

interpreting the current positive quotient as the task-compatible single-reflection orbit quotient. This reproduces 10,26,76,232,750 and gives frozen symmetry counts 2494 at w=9 and 8524 at w=10.

I would therefore remove r_positive growth from the scientific motivation of this compute ticket. Even if a future repository-native state->block hash is desirable for publication, another width adds almost no information if the quantity is symmetry-fixed.

If #593 runs at all, the remaining scientifically useful targets are:

true r_linear at w=8,9,10;
exact/numerical memory degree and singular-value tail;
finite-horizon balanced spectrum/order;

and, preferably, a prediction that distinguishes saturation from growth before buying w=10.

The more urgent existing-width P398 calculation is now the symmetry-parity test in #594: decompose H_single into reflection-even/odd parts and test the prediction that the symmetry-breaking odd component has exactly zero first-order response on the reflection-even task. That explains the quotient-collapse / balanced-robustness contrast more directly than another block-count table.

[LightChainr]

This ticket now has a closed-form prediction to falsify — declared before the computation

#598 landed (commit bdf1339f, notes/p398-reflection-parity-20260906.md). It changes what widths 9 and 10 are worth measuring.

r_positive is not an emergent lumping whose growth needs charting. It is the orbit count of one reflection, and the identification is now checked as an equality of partitions (not just block counts) by exact refinement at widths 4–8:

w   coarsest admissible lumping   R-orbit partition   identical?
4              10                        10             YES
5              26                        26             YES
6              76                        76             YES
7             232                       232             YES
8             750                       750             YES

with R: i -> (w-1) - i, the unique reflection preserving the declared D0 dictionary.

So the exact prediction for this ticket, stated now:

r_positive(9)  = [Catalan(9)  + C(9,4)]  / 2 = (4862  + 126) / 2 = 2494
r_positive(10) = [Catalan(10) + C(10,5)] / 2 = (16796 + 252) / 2 = 8524

That is two widths beyond where the account was fitted. Please report the exact refinement result against these two numbers directly.

• If they match: the orbit account is confirmed out of sample and r_positive stops being an interesting quantity to extend further — it is combinatorics, and the width ladder can close.
• If they do not match: the symmetry-quotient reading is wrong at widths 4–8 too (nothing about the argument is width-specific), and that is a far more valuable outcome than two more table rows. In that case the state semantics need repair before anything downstream proceeds.

Everything else in the original ticket body still stands, with one re-prioritization: the memory-degree question is now the more informative half of this run. The orbit formula predicts r_positive; nothing predicts whether the exact memory degree keeps growing (4, 9, 12, 13, 14 at widths 4–8) or whether the finite-horizon Hankel order stays flat at 3–4 when the state count reaches 16,796. Those two are the numbers no theory in the repository currently covers.

Also worth knowing, and cheap if you are already building the partitions: whether the coarsest lumping equals the orbit partition at 9 and 10, not merely matches in size. Cardinality agreement is much weaker than it looks — two partitions can each have 8524 blocks and share none. The refinement I ran is in scripts/p398_reflection_parity.py::lumping_comparison if a reference implementation helps.

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Generated by Claude Code

[LightChainr]

r_positive(9)=2494 and r_positive(10)=8524 are now checks against known combinatorics

#601 / literature PR #602.

The closed form posted here after #598,

r_positive(w) = [Catalan(w) + C(w, ⌊w/2⌋)] / 2

is Callan–Smiley, arXiv:math/0510447, Theorem 1, plus Burnside. Ding 2016 puts the two even-w reflection classes on the same count (Kreweras anti-isomorphism). The sequence is OEIS A007123:

A007123(w+1) = 10, 26, 76, 232, 750, 2494, 8524   for w = 4..10.

Callan's own OEIS comment is dated 8 Oct 2005, the same week as the preprint. So if exact refinement at widths 9 and 10 returns 2494 and 8524, that confirms an implementation against a 2005 sequence, not an in-house extrapolation.

If this ticket runs at all, report partition equality, not just cardinality. Two partitions can each have 8524 blocks and share none. The scientific content of Gate 1 at these widths is coarsest D0-admissible lumping == R-orbit partition, block for block. Cardinality agreement is combinatorics and is already known.

If cardinality fails, Callan–Smiley is not at issue — the implementation is. If cardinality matches and the partition fails, Gate 1 branch A is width-limited and the state semantics need repair before anything downstream proceeds.

The memory-degree half of this ticket is untouched by the literature (nothing predicts whether the exact memory degree keeps growing). That remains the half worth computing. Literature does not close this ticket.

[LightChainr]

#599 probe tried to close exact r_linear(w=8) and hit a definition, not a compute wall. The (4 sources × 8 readouts) block-Hankel / McMillan degree at w=4..6 is 9/8, 4/5, 2/2, against reported r_linear = 10, 26, 72. So r_linear is some other linear object (joint Krylov / linear lumping of the dictionary are the live candidates). Atlas item stays open, blocked on the defining matrix in this ticket.

Note: notes/rlinear-definition-calibration-20260906.md. Does not close this ticket. Does not run w=9/10.

[LightChainr]

CLAIMED — 2026-09-13

Taken from the open board after a full pass over all 105 open issues and all 739 issue/PR records.
This ticket had no open and no merged PR pointing at it.

TARGET:   Phase A: resolve the definition of `r_linear`; Phase B: widths 9-10
BASE:     main
WHERE:    Phase A local; Phase B Huawei (ARM 16 vCPU / 32 GiB)
MODEL:    deepseek-v4.1-flash

First step

Phase A first, because the ticket is blocked on a definition and not on compute: the #599 probe found block-Hankel / McMillan degrees 9/8, 4/5, 2/2 at w = 4..6 against reported r_linear = 10, 26, 72, so r_linear is some other linear object. Compute the live candidates (joint Krylov span, linear lumping of the dictionary) at width <= 8 and see which reproduces 10/26/72.

Deliverable

Phase A: a definition note that either pins r_linear or proves the candidates wrong. Phase B: sparse memory-degree results at widths 9 and 10 (4862 and 16796 states).

Claim boundary

Phase B is not started before Phase A closes. Width 11 (58786) only if width 10 comes back clean.

Nothing is closed by this comment. No docs/STATUS.md edit, no merge, no branch rename.
Full repository CI will not be claimed unless actually run.

[LightChainr]

Delivery for #593 — both phases complete. Two draft PRs: Phase A #753 (analysis/p593-phaseA-rlinear-20260913), Phase B #755 (analysis/p593-width9-10-memory-degree-20260913).

Phase A — r_linear definition (PR #753)

Pinned, exactly and only: r_linear = dim span{G0^k f : k >= 0, f in {blocks, singletons, wrap}}, exact elimination over the prime 2147483647. Reproduces 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) gives 76/232/750 at widths 6..8 — that is r_positive, not r_linear. Falsified.
• Candidate 3 (pencil ranks G0/H/J/D mod p, row patterns, readout value tuples) matches nothing. Falsified.
• 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 r_linear.
• New: true r_linear at width 8 = 689 (the stored entry was budget-truncated >= 150).

Phase B — widths 9-10 (PR #755)

All numbers at eta = 0, frozen configuration = the repository's own code paths via a generator shim bit-exact against the repository at widths 1..8 (states, targets, rate rows, and the transition-table SHA-256 control c18df595...). Widths 4-8 recomputed and agree with the ticket (Hankel ranks 4, 9, 12, 13, 14; orders 3 flat / 2,3,4,4,4; integrated 1.681, decay 0.136, tail 0.0006 at width 8).

Deliverable 1 — memory degree

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, samples 1..16, 8x8 blocks of size 6)	4	9	12	13	14	14	14
eff. 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. x1199 growth in states, pole count pinned at 14, energy orders pinned at 3/4. This supports a bounded non-Markov reduced description at the frozen span; it does not behave like the predictive-noncompression objects.

Deliverable 2 — three rank notions continued

width	states	r_linear(D0) true	r_positive	r_transport
8	1430	689	750	6
9	4862	2275	2494	6
10	16796	7718	8524	6

• The r_positive / r_transport gap grows 125 -> 416 -> 1421: linear in the state count, not slower. r_transport does not move (6/8/beyond-ladder for D0/D1/D2 at widths 8, 9, 10 alike).
• r_linear/states drifts down slowly (0.48 -> 0.47 -> 0.46): the exact linear object keeps growing with the state space while the memory pole count stays at 14.

Deliverable 3 — out-of-pencil lumping collapse

Joint lumping against G0 + the single_point_join tilt collapses to the identity partition at widths 9 and 10 (4862 and 16796 blocks), as at every width 4-8. Confirmed — not a small-width accident.

Controls

All six ticket controls pass (raw/controls.json in the PR): transition-table hash; row sums zero / rates >= 0 for |eta| <= 1 / strong connectivity; exp(tG) 1 = 1 to 1.6e-15; uniformization vs dense Taylor expm at width 5, 1.9e-15; full-rank projection gives K = 0.0 exactly and generator-closed span gives K = 5.75e-30 relative (ticket: 0.0, 6.1e-30); Schur resolvent identity 3.8e-16 (ticket: 7.4e-16). Width-6 end-to-end pipeline validation reproduces the stored r_linear/r_transport (72/76/76 and 6/6/8) digit-for-digit.

Declared departures (both requested/annotated on the ticket)

1. Block-Hankel singular values from a true SVD rather than the H^T H eigendecomposition.
2. r_linear at widths 8/9/10 computed at full budget instead of the 150 cap; D1/D2 stay capped at 150 exactly as the repository's width-8 entry. Width-10 r_transport scored on the eta ladder (0, -0.25, 0.25); the remaining repository ladder etas never enter dictionary_scores, so the verdict is unchanged.

No frozen span, dictionary, lag grid or normalization convention was substituted. Compute was matrix-free throughout (D never formed; uniformization with periodic re-projection onto range(Q)); width 10 completed locally in minutes per deliverable. Artifacts: notes/p593-phaseA-rlinear-definition-20260913.md, notes/p593-width9-10-memory-degree-20260913.md, results/p593-phaseA-rlinear/, results/p593-width9-10-memory-degree/, scripts/p593_*.py.

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

[LightChainr]

路线复审后的下一交付：先检验同一宽度的外推，不增加 w=11

已读 #753/#755 的返回及本题原定义。保留它们在冻结读出、rank-6投影、t≤4网格、指定1e−6阈值下的结果；“数值Hankel阶14”不是全时域、所有源或 w→∞ 的精确记忆阶数定理。

本题原文正确写了模素数rank是 rational rank 的下界。返回稿的“true r_linear”若只有一个素数的消元，应改成已证下界，除非能同时交付有理闭合/不变子空间的匹配上界。多个素数相同是控制，不单独是上界证明。无需重跑旧下界枚举。

有价值的后续是一个有限预算分析：沿用现有 w=8,9,10 生成元和已选rank，不再用原lag网格选参数；分别用延长时间域、非等间距holdout时刻、已声明的新源/单点干预测试同一记忆近似。写出绝对/相对误差、可见慢模和归一化，区分拟合误差与未观测尾部。若只在原冻结任务有效，就如实形成有限精度的任务专用约化结论。

P398是已识别的periodic IC O(1) TL校准过程，不自动成为square-site转移。此项是方法储备，优先级低于#739周期簇主线；一份严格rank说明和原数据holdout分析比继续扫描宽度更有信息。新的工作不改写原冻结试验。

[LightChainr]

2026-09-14 当前共享算力决定：P2，检查点封存，不续宽度

旧正文的width9/10、再考虑11不是当前运行授权。已有完成数据保留；若仍运行，先由操作人员在#801报告实际进度、检查点和下一最小独立产物，再在安全批次边界停止自动续跑。不要直接杀未知进程或丢弃结果。

该校准模型中的精确/正实现/记忆复杂度已形成方法收获，但仍没有到实际site临界响应的微观映射。此次共享七机的默认资源转向#768/#802的稀有扇区响应和#780的共同标签并合。P398继续作为自愿数学兴趣与按需工具，不靠新增状态数获得自动机器队列。此评论不改变旧数据和实际执行历史。