[P1 reserve] Source-visible topology and all-width closure beyond the completed width-four laboratory

[LightChainr]

当前分配：保留高价值表示论方向，暂停新增扩张 — 2026-09-13

按所有者委托的 #738 入口重整，并与 #650 较新的“一个活跃交付”决定对齐：本题为 P1 reserve，保持开放，不是本轮第二个活跃分析任务。本轮唯一交付是 #735 与 #613/#736 的概率论文及独立证明审查。只有主线发现一个确切需要，或下一次明确调整分配，才重启这里的新计算。

已经完成的输入，不再重复委派

• Exact width-two torus/cylinder bridge: full spectrum and finite-length root displacement #705 已合并：宽度二局部 transfer 与有限根谱余项。
• Width-four exact rank closure: 509 futures and 15-mode scalar (do not merge) #708 f782061c1a592ed2f9fd0e9dabaa45f0e54bc4e7：宽度四 lifted rank-future 充分状态，509 个确定性 continuation 类。
• All-p width-four spectrum and diamond 3L−1 onset (stacked on #708; do not merge) #710 d543ba1afe052a36682d2c1723c8a07281c28950：全 p 的宽度四可见谱，P0/P2 共用模式在 M 中抵消；一般标量阶16、特殊参数15/14。
• Width-four site sources and exact rank conditioning (do not merge) #733 525c8e98d99c44d4c76280a1f0f83064df9bfa72：源兼容 lumping、实际 square-site 两行源、最终 rank 条件精确反向采样。
• 新 Multi-advance 20260913: spatial-source Hessian, invisible marks, annealed/quenched roots, rank-source zero-free strip, conditional-odds integration, limit-order audit (do not merge) #737 7b459c4809baee9dfc2b910091e00a3a8d508d1c：完整有限空间源 Hessian、不可见空间标记、条件结构因子公式。实际文件在 delivery/ 下，PR 明确披露三个未提交 delivery artifacts，复用前核对所需路径与输入，不能把原压缩包运行日志等同于 fresh-checkout 完整性。

后四项目前未合并；这是可复用研究资产，不是要求重建同样的 engine、状态枚举、Hessian 或标记例子。

重新激活后，哪个问题仍有价值

对实际 square-site 局部转移及 torus closure，给出全宽度的 closure weights、leading prefactors、可见次领先模式及抵消结构；或者用同一个物理源区分两个明确仍存活的 closure 候选。

必须区分普通 trace、modified/Markov trace、projector 与 boundary overlap。等 leading weights 不消掉全部 subleading modes；全算子最小谱隙不一定是 M 的可见 gap；固定宽度指数收敛不推出固定长宽比的 L^-4 律。

只有一个新的宽度能否定事先明确的结构猜想时，它才值得计算。不是默认从4扫到24，不再声称仓库没有 p-weighted local transfer，也不重复已撤回的单算子不可能定理。

#737 暴露的物理路径及限制

对于均值为零的 logit source h，H=sum_i h_i n_i，有限 logit-root 曲率为

z_*''(h) = -[Var(H | r=2)-Var(H | r=0)]
             / [E(K | r=2)-E(K | r=0)].

这是移动根到 rank-conditioned 空间结构因子的具体联系。Additive-p 与 additive-logit 扰动有不同二阶修正；不能悄悄换坐标。Invisible marks 使用正的 dependent marked measures，它们不使已知 Bernoulli 模型变得不确定，也不提供 #275 的两个 continuum 候选。

本次独立轨道计算复核了 square/L/T 轨道16/128/64与 contrast matrix [[-512,-128],[0,-256]]、det=131072；没有独立重跑全部 Hessian 或全仓库 CI。

边界

确定性类数、强 lumping、scalar Hankel order、positive realization 与有限噪声 effective order 不可互换。P398 已识别为 periodic IC O(1) TL 校准过程，不是 square-site 微观过程。热 jet 的 m*lambda^m 不是物理 Jordan 识别。

原数据、分支、历史结果和评论保留。无新 Monte Carlo/GPU/大枚举，无硬件或凭证变动，无研究 PR 自动合并。下一次有明确科学区别需要解决时，从这些已完成资产出发，不从旧宽度二任务重新开始。

[LightChainr]

Proposed next brief for this existing issue, not a new ticket and not an engine authorization. Full text: notes/issue-636-width4-next-brief-20260912.md on draft PR #707 (do not merge).

Width three now has an all-length local formula (r=1_F+1_V, M_{3,m}=tr(T^m)-(1-p^3)^m). Width four does not inherit it. Compulsory witnesses already solved:

• [0,5] vs [7,5] then common [13,0] → ranks 0 and 1 (frontier occupancy alone fails).
• Stronger: [13,5] vs [7,5]: same occupation 5, frontier {0,2}, ordinary partition {{0,2}}, winding flag 0, same weight p^5(1-p)^3, lifts -2 vs +2; common [13,0] → ranks 0 and 1.

Deliverable: a specified local state that retains enough annular/lifted information to distinguish these histories, with a future-equivalence proof, then a bounded width-four integer check. No GPU, no width scan, no Monte Carlo, no Catalan counting as a substitute. #636 stays uncommissioned as a weighted engine until that closure exists.

[LightChainr]

The latest-comment sufficiency task is delivered as draft PR #708 (do not merge). Do not re-commission another width-four representation or census for that question.

Fixed language: 1448 boundary representatives, 23168 transitions independently reconstructed; deterministic rank futures 509, minimal for this language. M_{4,m}(p)=b^T K^{m-1} c. At p=1/2 the scalar 16^m M has minimal order 15 (Hankel nonzero; P_{15}(A)c is not the zero vector).

Remaining bounded task on this same issue (text in notes/issue-636-width4-completion-and-next-20260912.md on #708): visible spectrum of the supplied operator over Q(p) for P_0, P_2, and P_2-P_0. Not another engine, width scan, GPU job, or histogram. #275 remains separate.

[LightChainr]

The all-p visible-spectrum task from the last #708 comment is delivered as stacked draft #710 (base #708). Do not re-dispatch it or buy another width engine.

M=(tr(B15^m)-tr(B5^m))/(1+t)^{4m} for all p, m≥2. Generic orders P0=17, P2=23, M=16. Shared slow B16 mode cancels in M. q4 is the published Jacobsen n=4 value, not a new pc.

Genuinely larger remaining question: all-width observable weights / uniform bounds, not more roots. Thermal mλ^m at the crossing is semisimple — cross-link #275, do not change that issue’s candidate contract.

[LightChainr]

Do not re-dispatch the width-4 visible-spectrum engine. Draft #716 (on #710) records that E=P0+P2 sees 12 modes that M cancels, and that the cylinder root organises two rare sectors, not a bulk jump. Next new #636 question, if any, is all-width weights with those three objects controlled together — not more q_w digits.

[LightChainr]

Delivered: source compatibility and rare-sector conditioning on PR708's automaton are in PR #733 (base analysis/width4-rank-closure-20260912, do not merge).

• All seven D4 column-partition types classified exactly on the existing PR708 finite automaton. Common strong lumpings: 94/303/179/262/509/303/509 states; each equals the corresponding colour-preserving D4 orbit partition blockwise. Two independently addressed adjacent columns already require all 509 deterministic rank classes within this lumping class — no further width-four state enumeration is needed.
• Genuine square-site two-row dipole source: zero linear response, nonzero mixed response (4x4 coefficients 327/1024 and 633/2048 at p=1/2); the finite root's mixed shifts carry strict rational sign certificates. Does not inherit P398's continuous-time response formula.
• Exact integer backward sampler conditions on any final rank; the complete nonuniform 4x3 conditional law is verified configurationwise; the 4x128 rare case has exact normalizers and fixed-seed algorithm controls.

Boundary: additive on #708, independent of unmerged #710 and #716 and of the root/TL bundle. Requires PR708's certificate blob 50b7297, deliberately not duplicated. 14 local tests pass on that tree; the repository's complete CI suite was not executed for these files. All-width structural closure weights and width-uniform spectral estimates remain uncompleted. This does not authorize a next-width scan or a new transfer engine.