Width-four exact rank closure: 509 futures and 15-mode scalar (do not merge)

notes/issue-636-width4-completion-and-next-20260912.md

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+## Width-four sufficiency task completed — proposed update to this existing issue
+
+The latest #636 comment's requested state is now constructed and checked.
+Do not commission another width-four representation or census for that question.
+The additive submission bundle contains the proofs, code and complete finite table;
+it is not yet committed by this message.
+
+For the fixed width-four NN square-site row language, retain pinned seam and
+current boundary, component partition, path gains modulo the accumulated GLOBAL
+cycle span, and the latter span even after its component leaves the boundary.
+Replacing old interiors by gained stars preserves all future cycle spans.
+A common integer shear of current-to-pinned path gains changes final windings
+by (hx,hy)->(hx+c*hy,hy) and preserves rank, but NOT directional/primitive labels.
+
+Complete closure: 1448 representatives, all 23168 row transitions independently
+reconstructed. Deterministic continuation refinement: 3 -> 164 -> 509 -> 509.
+Every class is reachable at a physical length >=2; every pair is distinguished
+by some suffix of at most two rows. This is a rank-language minimum, not a
+linear state count or continuum claim.
+
+The explicit normalized operator is M_(4,m)(p)=b(p)^T K(p)^(m-1)c for all m>=2,
+where c=r-1 and every row mask receives its original Bernoulli weight.
+At p=1/2 the scalar integer sequence 16^m M has minimal recurrence order 15,
+with six trace factors and weights (-2,1,-1,2,1,1). The identity has a 509-
+moment exact operator certificate and a nonzero physical-tail 15x15 Hankel
+determinant; it is not a finite-sequence fit promoted to a theorem.
+
+Executed controls: 74640 full physical-graph configurations, every occupation
+coefficient, all finite transitions, 7 local tests. No Monte Carlo/GPU. Full
+repository CI for the NEW patch has not run. #707's own CI 34689086221 is now
+successful and is a different check.
+
+### One remaining bounded task worth doing here
+
+From the supplied K(p), compute/prove the parameter-dependent observable
+reduction over Q(p), separately for P0, P2 and P2-P0. Give the actual visible
+spectral denominator and closure weights; specialize exactly to the p=1/2
+certificate. A fraction-free Krylov relation or a rigorously degree-bounded
+interpolation followed by exact identity verification is acceptable. A handful
+of numerical p specializations alone is not.
+
+Question to decide: do the two leading observable weights stay equal, and
+which other visible modes survive the closure? This is not supplied by the
+stochastic matrix's Perron eigenvalue 1 or by the mere existence of K(p).
+Keep pTL intertwining and width-uniform/fixed-aspect asymptotics distinct.
+
+Use ordinary CPU and the existing bounded resource envelope. Do not extend
+widths, launch production, replace proofs by another histogram, or open a
+duplicate issue. #275's original-U candidate map remains a separate problem.

notes/width4-review-control-20260912-zh.md

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+# Matching One：#707 复核与宽度四续接闭合
+
+日期：2026-09-12。
+
+## 复核结论
+
+已读取 #707 的实际 head `0d249ff6`、数学说明、源文件和评审评论。收到的脚本与 GitHub blob `09b766a7` 完全一致，七项已有测试已重新执行并通过。宽度三配置级字典、七态转移的分块谱与有限根位移论证未发现新错误。没有把 PR 评论中报告的 299,584 配置复核冒充为本轮重新执行。
+
+#706 正文仍保留已经被其后续文件纠正的旧不确定度说法。应同步正文，不能再用旧宽度标准误差、忽略共享中间尺寸的区间误差，或用“全卡方小于对角和”替代逆协方差稳健性论证。本轮不对同一形状数据追加高次坐标拟合，也不把未决的共同坐标检验升级为正证据。
+
+GitHub 实际运行 `34689086221` 现已成功完成，head 对应 #707 的 `0d249ff6`。这是收到的 PR 的 CI，不是本轮新补丁的 CI。
+
+本轮没有改动或合并 #706/#707；交付是可直接应用的新增文件补丁。
+
+## 已直接完成的三个结果
+
+### 1. 足够的边界状态，不再只是一个反例
+
+状态保留首行接缝与当前前沿的占据端点、它们的连通分区、连接路径的整数提升位移，以及已经产生的全局周期空间。用路径替换证明了：删去内部顶点、保留带位移的边界星形图，不改变任何共同后续产生的同调空间。
+
+再利用本任务只观察 rank 的特点，将所有最终绕行向量同时作整数剪切 `(hx,hy) -> (hx+c*hy,hy)`。它保留 rank，但不保留 rank-one 直线、方向标签或 primitive character；不能把这项压缩搬到不同观测量而不重新证明。
+
+### 2. 宽度四全长度闭合：1448 个代表，509 个最小确定性续接类
+
+完整探索得到 1,448 个边界代表与 23,168 个行转移。每个状态的 16 个后继均在集合内，且每条转移都用独立图遍历重新计算；这是完整闭合证书，不是观察几个尺寸的计数不再变化。
+
+按所有后续的最终 rank 归并，分区细化为 `3 -> 164 -> 509 -> 509`。509 个类的长度 0、1、2 后缀响应全部不同，而且每个类都能在长度至少二的真实环面前缀上达到。因此，509 是**固定宽度四、固定列标号、逐行追加、最终只读 rank**这个语言下的最小确定性续接类数。它不是线性实现阶数或连续场数。
+
+局部概率转移矩阵 `K(p)` 和初始权重 `b(p)` 已明确，最终输出为 `c=r-1`：
+
+`M_(4,m)(p) = b(p)^T K(p)^(m-1) c`，对全部 `m>=2` 与 `p∈[0,1]` 成立。
+
+### 3. p=1/2 的完整标量谱只需 15 阶递推
+
+令 `s_m = 16^m M_(4,m)(1/2)`。已给出六组谱因子的幂和表达，权重为 `(-2,1,-1,2,1,1)`。对应递推阶数恰为 15。
+
+不是拿有限数据拟合递推后外推：实际构造整数矩阵 A，核验 `b^T A^k P(A)c=0` 的 509 个初始值，由 Cayley–Hamilton 推出所有长度；同时计算从真实长度 m=2 开始的 15×15 Hankel 行列式，严格非零。递推多项式与其导数的有理 gcd 为一。
+
+这也提供了一个清楚的尺度区分：1448 是几何代表数，509 是确定性 rank 续接类数，15 是固定 p 下一个标量期望序列的最小线性递推阶数。任何一个都不能无条件读成 continuum field count。
+
+## 验证与边界
+
+独立的物理提升图遍历逐配置核对了 74,640 个配置，覆盖 2×2、2×3、3×2、3×3、3×4、4×2、4×3、4×4；与未剪切、已剪切两种状态实现全部相同。每个占据数上的 rank 计数也全部一致。
+
+4×4 的 rank 0/1/2 总数为 36,559 / 19,932 / 9,045。另附 4×5、4×6 的动态规划精确系数；没有把它们说成新的独立枚举或随机证据。七项新增本地测试全部通过。全部新结果的精确 JSON 不含浮点数。
+
+没有运行新补丁的完整仓库 CI，没有新 Monte Carlo 或 GPU 作业，没有 p_c 新值，没有所有 p 下的 15 阶结论，没有 all-width pTL 映射，也没有固定长宽比收敛率定理。整数提升的基本技术是已有方法；本次不作优先权／新颖性宣称。
+
+## 后续决策
+
+#636 最新评论中的“构造宽度四足够状态并作有界整数核对”已经做完，不应再次派发。已有 509 态概率转移应作为下一问题的输入。
+
+同一 issue 的下一项有价值交付是：在 Q(p) 上约化可见谱，分别给出 P0、P2、P2−P0 的可见分母和闭合权重，并用本次 p=1/2 的完整谱作精确特化控制。目标是判断领先权重相等到底有无结构性原因，不是再做一个实现、再扫几个宽度、或把数值插值冒充恒等式。
+
+#275 仍需要候选机制的 original-U 前向预测；本次有限算法和谱控制没有替它补出物理预测。形状线保留现有有效结果，停止没有具体判别目标的额外坐标试探。

results/research-control-20260912/width4-half-probability-spectrum.json

@@ -0,0 +1,124 @@
+{
+  "schema": "matching-one.width4-half-probability-spectrum.v1",
+  "scope": "M_(4,m)(1/2), all integer m>=2; no other p or width asserted",
+  "integer_sequence_definition": "s_m=16^m*M_(4,m)(1/2)=b*A^(m-1)*c",
+  "trace_formula": "s_m = sum_j weight_j * sum_{f_j(lambda)=0} lambda^m (multiplicity included)",
+  "factors_descending_and_weights": [
+    [
+      -2,
+      [
+        1,
+        -2
+      ]
+    ],
+    [
+      1,
+      [
+        1,
+        1
+      ]
+    ],
+    [
+      -1,
+      [
+        1,
+        -15,
+        4
+      ]
+    ],
+    [
+      2,
+      [
+        1,
+        -3,
+        -4,
+        2
+      ]
+    ],
+    [
+      1,
+      [
+        1,
+        -3,
+        2,
+        -2
+      ]
+    ],
+    [
+      1,
+      [
+        1,
+        -11,
+        -9,
+        3,
+        -10,
+        -2
+      ]
+    ]
+  ],
+  "minimal_scalar_recurrence_order": 15,
+  "recurrence_descending": [
+    1,
+    -33,
+    353,
+    -1301,
+    697,
+    4327,
+    -5785,
+    -551,
+    5534,
+    -8814,
+    872,
+    2716,
+    -3992,
+    1480,
+    80,
+    -64
+  ],
+  "recurrence_is_squarefree_over_Q": true,
+  "hankel_determinant_15_start_m2": -1480741158238721640945362024307471335089774150588805816471254212544678061868790579200000,
+  "finite_dimensional_certificate_dimension": 509,
+  "annihilator_zero_scalar_moments_verified": 509,
+  "pA_c_nonzero_entries": 82,
+  "initial_trace_power_identities_verified": 31,
+  "first_31_s_m": [
+    0,
+    -46,
+    -1464,
+    -27514,
+    -467240,
+    -7568494,
+    -119388136,
+    -1850806458,
+    -28342936392,
+    -430168917166,
+    -6485164995144,
+    -97272274510042,
+    -1453300596591272,
+    -21647273727763086,
+    -321679039870596584,
+    -4771299628033839738,
+    -70667055165917120648,
+    -1045436061966429481198,
+    -15451931157489435989384,
+    -228220684705345695257114,
+    -3368828511332610617805928,
+    -49705628452107159678962510,
+    -733120634385154825294067496,
+    -10809872854629806802861614010,
+    -159355274612023382199893728840,
+    -2348730748989230971326312464302,
+    -34612824078688255447045260637896,
+    -510024033529269360261814692900698,
+    -7514571807950217946301809929033256,
+    -110709767796539584122033242382917134,
+    -1630955864389294282204931570018833512
+  ],
+  "normalization": "s_m counts configurations; divide by 16^m, not 16^(m-1)",
+  "limitations": [
+    "one-parameter point only",
+    "not a global generator spectrum",
+    "not a continuum Jordan test",
+    "no all-width prefactor theorem"
+  ]
+}