diff --git a/notes/issue-636-width4-next-brief-20260912.md b/notes/issue-636-width4-next-brief-20260912.md
new file mode 100644
index 00000000..8689a901
--- /dev/null
+++ b/notes/issue-636-width4-next-brief-20260912.md
@@ -0,0 +1,54 @@
+# Suggested re-scope of existing #636: width-four annular/lifted continuation closure
+
+This is a proposed next brief for the existing issue, not a new duplicate issue,
+production authorization, or claim that a weighted engine has already been built.
+It follows the delivered width-three exact analysis. Ordinary CPU only; no GPU,
+no width scan, no new Monte Carlo, no free-exponent fit.
+
+## One decision
+
+Can a specified local frontier state, carrying appropriate connectivity and
+horizontal topological information, preserve the square-site ambient rank under
+arbitrary continuation and periodic closure at circumference four? Supply a
+sufficient construction with proof, or a counterexample to a precisely stated
+candidate quotient. Mere small-size agreement is not the closure proof.
+
+## Two compulsory witnesses, already solved
+
+Bit j denotes column j. First compare open prefixes [0,5] and [7,5], followed
+by the common suffix [13,0]. Their final 4x4 ranks are 0 and 1. Frontier occupancy
+alone is insufficient.
+
+The stronger pair is [13,5] and [7,5]. Both have five occupied sites, frontier
+mask 5={0,2}, ordinary frontier partition {{0,2}}, and no existing transverse
+winding. Their lifted 0-to-2 connecting paths have displacements -2 and +2.
+Append [13,0] and close through the empty row: ranks 0 and 1, D=-1 and 0.
+A state that tracks only the ordinary partition, occupation count and already
+completed wrapping flag STILL fails. Both prefix weights are p^5(1-p)^3, so
+weighting cannot compensate for identifying the two histories.
+
+## Deliverable
+
+Define the state, local row update, equivalence relation and periodic closure
+functional; explicitly retain enough annular or lift information to distinguish
+these histories. Prove future equivalence for every pair the representation does
+identify. Explain how primitive winding, independent windings and rank saturation
+are preserved without silently truncating necessary integer offsets.
+
+Then implement only a bounded width-four reference calculation. Check the existing
+4x4 Bernstein polynomial and a small non-square torus by an independent lifted
+traversal. Preserve parallel periodic edges. Use integer/rational coefficients;
+no asymptotic exponent or p_c claim. If a finite state bound has not been proved,
+label it and do not start enumerating more widths.
+
+If a pTL formulation from Jacobsen is reused, give the actual map between its
+states/closure and the black-site rank observable; matching a root numerically
+is not an intertwiner. Output a full finite probability closure before discussing
+leading eigenvalues; amplitudes and all subleading terms must remain explicit.
+
+## Stop
+
+Stop at a demonstrated sufficient state and closure, or one precise obstruction.
+Do not claim minimal state count unless a distinguishability argument proves it.
+Do not re-prove the width-three formula or pay for another all-honest-torus
+five-cell census. A broader engine needs a separate information-gain decision.
diff --git a/notes/review-next-frontier-20260912-zh.md b/notes/review-next-frontier-20260912-zh.md
new file mode 100644
index 00000000..96142071
--- /dev/null
+++ b/notes/review-next-frontier-20260912-zh.md
@@ -0,0 +1,101 @@
+# Matching One：新交付复核与下一步实质推进
+
+日期：2026-09-12。读取对象：PR #705、#706、其当前文件、评审评论和实际 Actions 结果；本报告不是只复述 PR 开头正文。
+
+## 一、总体判断
+
+#705 已合并，给出了宽度二的完整有限长度谱公式，而不是只有一个数值根。#706 的有效结论是：N145/290/725 同一方格点渗流观测量的归一化形状仍不对称，且不对称范数沿这三个尺寸减弱；但“新修正场已被识别”或“任意共同非线性坐标都被排除”仍不成立。
+
+本次没有再给已经完成的工作开单。新增工作的主要结果是：完成宽度三的全长度精确公式、有限根位移定理，并在宽度四给出必须保留历史连通性和路径绕行信息的显式反例。代码、整数多项式、精确区间证书及测试随包提供，交由项目方提交。本轮未修改远端仓库，也未运行新生产模拟。
+
+## 二、#706 应如何接收
+
+读取 head：59fff91c6592e8c947ccd1e3df25e15df6a291e5。完整 CI 34685638158 已完成且成功；实际 jobs 中 Tests、smoke、Python 3.9/3.13 编译与 C++17 构建均成功。不能把 CI 成功解释成极端名义显著性的有限样本标定已经完成。
+
+PR 开头正文仍有被后续文件修正的旧数字，合并时应改正文和旧 note 的显眼导航，而不是重新生成或覆盖原始结果。
+
+1. 宽度标准误差的原公式把 delete-one 样本当成独立观测均值，低估了 99 倍。正确标准误差依次是 3.6730e-6、2.7377e-6、2.3226e-6，已经由非线性 jackknife 复核。
+2. 两个相邻区间共享 N290。比较区间位移范数时必须保留负交叉协方差。修正后 Z 的范数变化约 9.04 个名义标准误差，而不是正文的 10.9；A 的对应变化约 0.897 个标准误差，仍未分辨出来。
+3. “完整逆协方差统计量小于对角和”不能证明逆矩阵不受近零方向影响。单坐标的大信号与完整逆矩阵的校准是不同问题。
+
+这些修正已经在 notes/shape-lineage-review-20260912.md 和 results/research-control-20260912/shape-lineage-reviewed-summary.json 中，不应再作为新的待修复发现派发。
+
+## 三、比继续拟合一个指数更重要的解释边界
+
+修订结果给出 ||A||/W = 0.255963、0.256397、0.256847。这是有用的结构迹象，但完整向量的 A/W 恒定模型并不成立。
+
+仅用 N145 中位对称条件确定的二次坐标 phi(p)=p+0.2684067(p-1/2)^2，在 N290/N725 上把不对称残差范数降到原来的约 0.267%/0.068%。这说明大部分范数可以被一个简单坐标形变吸收；但该精确二次模型仍被完整向量诊断拒绝。范数减少不是解释方差、因果归因，也不是已经给出了物理温度坐标。
+
+对任意共同递增坐标，现有两个不变量方向——共同对称化反射的交换关系及完整位置-尺度传输的无交叉条件——均未分辨出反例。不能用未拒绝支持“形状纯粹是坐标”，也不应不断提高多项式阶数寻找事后兼容。
+
+最后一个传输检验已经实际执行，但原 note 仍写“结果须从 job 读取”。本轮读取 run 34685638146 / job 103532033981 的成功日志并归档：在固定 u=.3,.5,.7 处，组合差分别约 -4.2494e-6、-4.4743e-6、-4.5533e-6，标准误差 1.9009e-6、1.8363e-6、1.8494e-6。三点同时区间均包含零；最大边际标准化值 2.462。没有分辨出符号交叉，因此该必要条件未能排除完整共同坐标塌缩。不是重新跑了一次生产，也不是新的独立证据。
+
+本包的 shape-transport-order-finalized-summary.json 是上述已完成日志的字段摘录，明确保留 run、job、head、checkout 和协方差来源。没有把它伪装成本地原始直方图重算。
+
+## 四、本次直接完成的数学推进：宽度三
+
+对 3×m 正常环面，m≥2，令 F 为“出现全占据行”，V 为“每个相邻行接口都有占据重叠”。我证明了配置级恒等式
+
+    r = 1_F + 1_V.
+
+宽度三的每个非空行都连通，非满行则是可缩树；这一特殊几何事实是证明核心。由七个非空行状态的局部转移矩阵 T，得到
+
+    M_(3,m)(p) = tr(T(p)^m) - (1-p^3)^m.
+
+这不是穷举外推，而是对所有 m 的事件分类证明。S3 对称性进一步把完整迹分成一个三维块 B 和出现两次的二维块 C：
+
+    tr(T^m)=tr(B^m)+2 tr(C^m).
+
+圆柱交点由六次方程
+
+    p^6-3p^5-5p^4-4p^3+p+1=0
+
+的唯一 (0,1) 根给出：q3=0.58888069991785299805144269575170493372217…，与 Jacobsen 2015 Table 2 的已发表 n=3 数值一致。这个数不是无限方格点渗流 p_c 的新估计，也不声称发现了新临界常数。
+
+保留全部子领先谱后，又得到：每个有限长度根都严格小于 q3；其领先位移为
+
+    p_(3,m)-q3 ~ -2*rho^m/(m*h'(q3)),
+    rho=0.22476579734522079…,
+    h'(q3)=2.69049781437123949….
+
+系数 2 来自普通表示重数，不是 Jordan 块。转移矩阵与实对称矩阵相似，因而在 0<p<1 可对角化；负子领先特征值造成交替的有限长度修正，也不需要 Jordan 机制。完整余项公式和精确有理符号证书见英文证明。
+
+## 五、最重要的新边界：宽度四不能只保留普通分区
+
+先有一个简单反例：四行掩码 [1,5,4,5] 的相邻接口全部重叠，却没有非平凡绕行；反向反例 [11,14] 没有满行却有秩二。宽度三的“满行 + 接口重叠”字典不能照搬。
+
+更强的配置级证明是两条开放圆柱历史：
+
+    A: [13,5]，B: [7,5]。
+
+这里 bit j 对应第 j 列。两者均含 5 个占据点，末行都是 {0,2}，末行两点已经连通，已有绕行秩都是零。因此“占据掩码、普通连通分区、占据数、已发生绕行标志”完全相同，甚至前缀概率权重也相同。
+
+但 A 的 0→2 连接路径的横向提升位移为 -2，B 为 +2。给两者都追加 [13,0]，A 的最终环面秩为 0，B 为 1。共同后续提供的路径位移为 -2；B 闭合出 +2-(-2)=4，也就是一个整周，A 则闭合为零。
+
+所以，任何将这两条历史视为同一状态的确定性续接表示，都不能保留该绕行观测量。普通分区加一个当前拓扑标志仍不够；需要保留适当的环形嵌入或提升路径信息。这是明确可复核的表示下界，不是“单算子不可能”的广义 no-go。
+
+它把后续问题从“再多枚举一层”推进为“究竟哪些前沿拓扑记忆可以被合法商掉”。有限状态数、非交叉分区数、概率根和谱根之间的桥，必须经过这个定义层，而不能靠看似很接近的数值根跳过。
+
+## 六、验证范围
+
+独立提升同调遍历实际覆盖 3×2 至 3×6 的 299,584 个配置，配置级字典零失败，全部多项式系数与转移递推一致。3×3 的既有 Bernstein 整数逐项重现。m=2…10 的独立 7×7 整数矩阵迹检查通过。根区间、特征根隔离带、位移符号所需不等式和 h' 正性由 Fraction 区间运算核对。七个本地数学测试通过。
+
+这不是新随机块。完整项目 CI 未在本次新补丁上运行；远端 #706 的成功 CI 不得冒充本补丁 CI。本次也未证明所有宽度的 pTL 交织映射、宽度一致的误差界、固定长宽比 L^-4 收敛率或无限模型的场身份。
+
+## 七、后续取舍
+
+不再派发原 #622 的相同数据重建，不为已有三次未决坐标检验补样，也不把原 #681 的有界检索当作仍未交付。#650 的队列说明应同步已完成状态；不是关闭这些科学问题本身。
+
+本次宽度三推导和宽度四反例已经完成，无需为它们开计算单。若继续 #636，应把第一项验收收紧为：建立能区分上述两种相同普通分区历史的局部状态表示，给出续接与周期闭合为何保留绕行的证明；再用既有小环面整数核对。不启动宽度扫描或 GPU。一个“普通分区类数少、所以够用”的答复不满足验收。
+
+物理解释方向仍应回到 #275 的候选前向映射：需要给出一个可独立固定的坐标/归一化律或不同源响应，才能区分解析坐标效应与真正的形状修正。单凭当前递减范数无法提供这个区分。这里不新增一个标题宏大、输入却未定义的理论工单。
+
+## 来源位置
+
+- #705: https://github.com/LightChainr/Matching-One/pull/705 ，head fc19cc74，已合并为 eb89e942。
+- #706: https://github.com/LightChainr/Matching-One/pull/706 ，本轮读取 head 59fff91c。
+- #706 修订结果：results/research-control-20260912/shape-lineage-reviewed-summary.json。
+- #706 对称化边界：notes/common-chart-commutator-20260912.md。
+- 最后传输运行：https://github.com/LightChainr/Matching-One/actions/runs/34685638146 ，job 103532033981。
+- #706 完整 CI：https://github.com/LightChainr/Matching-One/actions/runs/34685638158。
+- Jacobsen 第一手文本：https://arxiv.org/html/1507.03027v1 ，section 6.1, Table 2；根数值为文献已有，不作新颖性宣称。
diff --git a/notes/width3-row-closure-20260912.md b/notes/width3-row-closure-20260912.md
new file mode 100644
index 00000000..f5620c84
--- /dev/null
+++ b/notes/width3-row-closure-20260912.md
@@ -0,0 +1,282 @@
+# Width three is the last connected-row case: exact torus/cylinder closure and a width-four memory witness
+
+Date: 2026-09-12. Direct continuation of the reviewed width-two result in PR #705.
+This is a completed mathematical analysis with small independent checks, not a
+request to build a large transfer matrix. No novelty claim is made. The printed
+cylinder root was already published by Jacobsen; the deliverable is an explicit
+finite event dictionary, complete observable spectrum, root-displacement formula,
+and a precise reason that the same occupancy-only simplification stops at width four.
+
+## 1. Scope, weights and the local transfer
+
+Use the square-site NN graph on the honest axis torus with periods (3,0),(0,m),
+m >= 2. Preserve parallel lifted edges when m=2. Let r be its ambient rational
+homology rank. The existing digital-Alexander identity gives D=r-1 and M=E[D].
+Write P_j=P(r=j). Occupation probability is p, not an FK bond fugacity.
+
+There are seven nonempty occupied subsets S of a three-site row. Define
+
+    u=p(1-p)^2,  v=p^2(1-p),  w=p^3,
+    t(S)=p^|S| (1-p)^(3-|S|),
+    T[S,S'] = 1{S intersects S'} t(S').
+
+Thus tr(T^m) is the Bernoulli probability that every row is nonempty and every
+successive pair, including the seam, has occupied overlap. T is nonnegative,
+irreducible, and aperiodic for 0<p<1: every state has a loop and communicates
+with the full row.
+
+## 2. Configurationwise lemma: r=F+V
+
+Let F indicate that at least one row is fully occupied. Let V indicate that every
+periodic interface has occupied overlap. Then, at width three,
+
+    r = 1_F + 1_V.                                      (1)
+
+The proof must use the width restriction, not only a picture of overlapping rows.
+
+If F occurs, a full row carries the transverse generator. If V also occurs,
+choose one occupied edge across each interface. Within each nonempty row the
+incoming and outgoing endpoints can be joined: every nonempty subset of a
+three-cycle is connected. Joining around the torus gives a closed walk with
+longitudinal winding one; its class is independent of the transverse generator.
+Hence r=2. If V fails, an interface has no occupied vertical edge. Cutting there
+puts the graph in an annulus, so longitudinal winding is impossible and r=1.
+
+Now suppose F does not occur. Each nonempty occupied row is a point or a single
+edge, hence a contractible tree. Contract these row trees. Two adjacent proper
+rows have two common occupied columns only when they are the same two-site set.
+The two vertical interface edges in that case differ, after contraction, by a
+contractible plaquette boundary, not by an extra ambient cycle. After removing
+these homologically redundant parallel edges, the row quotient is a subgraph
+of the periodic row cycle. Distinct periodic interfaces are NOT identified with
+each other when m=2. If V fails this quotient is a union of paths, giving r=0.
+If V holds it has exactly one independent cycle, whose longitudinal winding is
+one, giving r=1. Its transverse winding may be nonzero, but it remains rank one.
+This proves (1). In particular, the argument does not confuse a spiral with a
+rank-two cross.
+
+Taking expectations immediately gives the all-length identity
+
+    M_(3,m)(p) = tr(T(p)^m) - (1-p^3)^m.                 (2)
+
+An equivalent sector-level description uses T_proper, the six-state restriction
+with the full row omitted:
+
+    P_2 = tr(T^m) - tr(T_proper^m),
+    P_0 = (1-p^3)^m - tr(T_proper^m).
+
+The proper-row contribution cancels exactly in P_2-P_0. This cancellation is not
+permission to discard every subleading eigenvalue of T. Also
+F_(3,m)(p)=E[r]/2=(1+M_(3,m)(p))/2 supplies an exact finite quantile laboratory,
+not an additional large-N percolation experiment.
+
+## 3. Complete spectrum, including the repeated sector
+
+The row permutation group is S_3 (the automorphism group of the three-cycle).
+On functions constant on subsets of each size, T reduces to
+
+           [u   2v   w]
+    B =    [2u  3v   w].
+           [3u  3v   w]
+
+The singleton and doubleton permutation representations each contain a
+2-dimensional standard representation. Label a doubleton by its omitted vertex.
+On their standard components the reduction is
+
+    C = [u  -v; -u  0],
+
+and each eigenvalue of C occurs twice in the full seven-state spectrum. Thus
+
+    tr(T^m) = tr(B^m) + 2 tr(C^m),
+    f_B(L)=L^3-(u+3v+w)L^2-(uv+2uw)L+uvw,
+    mu_±=(u ± sqrt(u^2+4uv))/2.                          (3)
+
+Let beta_1 be the Perron eigenvalue of B and beta_2,beta_3 its other roots. The
+exact observable spectral formula is
+
+    M_(3,m)=beta_1^m+beta_2^m+beta_3^m
+            +2 mu_+^m+2 mu_-^m-(1-p^3)^m.              (4)
+
+The leading open and closed coefficients are both one. The leading SUBLEADING
+coefficient will be two, not one. This is ordinary semisimple multiplicity:
+T=K diag(t) is similar to the real symmetric matrix sqrt(diag(t)) K sqrt(diag(t)).
+There are no nontrivial Jordan blocks for 0<p<1. Neither a multiplicity of two nor
+an alternating finite-length correction is evidence of a continuum Jordan pair.
+
+## 4. Cylinder crossing, exact algebra and uniqueness
+
+Set lambda_c=1-p^3. Substituting into f_B gives
+
+    f_B(lambda_c)=(p-1)^2 P(p),
+    P(p)=p^6-3p^5-5p^4-4p^3+p+1.                       (5)
+
+There is exactly one root q in (0,1). One convenient proof uses s=p/(1-p):
+
+    (1+s)^6 P(s/(1+s))
+      =1+7s+20s^2+26s^3+8s^4-14s^5-9s^6.
+
+In descending coefficient order this has exactly one sign change, so Descartes'
+rule bounds the number of positive roots by one. Existence follows from P(0)>0
+and P(1)<0. The rational certificate in the script isolates q and the three B
+roots, showing that lambda_c(q) is the Perron root, not a subleading crossing.
+
+    q=0.58888069991785299805144269575170493372217050345701...
+
+Jacobsen (2015), section 6.1 Table 2, already prints
+0.5888806999178529980514426957517049337221 for n=3. The agreement is with a
+published FINITE-WIDTH estimator. It neither finds a new value of p_c nor proves
+an all-width pTL intertwiner. Equation (5) is called a defining sextic here;
+no unproved assertion of polynomial minimality is needed.
+
+## 5. Every finite root lies below q; exponential displacement with alternating corrections
+
+At q the five distinct eigenvalues are approximately
+
+    beta_1=lambda_c=0.7957876689642490361,
+    beta_2= 0.0362139729093206924,
+    beta_3=-0.1005527942557272624,
+    mu_+  = 0.1788658499322440484,  multiplicity 2,
+    mu_-  =-0.0793337764501209504,  multiplicity 2.
+
+The accompanying Fraction interval calculation proves, rather than infers from
+these decimals,
+
+    beta_2>0, beta_3<0, mu_+>0, mu_-<0,
+    |beta_3|/mu_+ < 3/5,  |mu_-|/mu_+ < 1/2,
+    beta_2 < mu_+ < lambda_c.
+
+Consequently M_(3,m)(q)>0 for every m>=2. For even m every remaining term in
+(4) is positive. For odd m, divide by mu_+^m and bound the negative terms by
+(3/5)^m+2(1/2)^m < 2. The finite M is strictly increasing by the monotone-rank
+argument and has endpoint values -1,+1. Its unique root p_(3,m) therefore
+satisfies p_(3,m)<q for every finite length.
+
+Define
+
+    h(p)=log(beta_1(p)/lambda_c(p)),
+    rho=mu_+(q)/lambda_c(q)=0.22476579734522079...,
+    h'(q)=2.69049781437123949... >0.
+
+The derivative sign is also certified exactly: at q,
+
+    h'(q)=-(q-1)^2 P'(q)/(f_B,L(lambda_c(q),q)*lambda_c(q)).
+
+The numerator derivative is negative and the denominator positive. Analyticity
+of the separated eigenvalues near q and a Taylor expansion of (4)/lambda_c^m
+give
+
+    p_(3,m)-q = -2 rho^m/(m h'(q)) * [
+         1 + (mu_-/mu_+)^m
+           + (beta_2/mu_+)^m/2 + (beta_3/mu_+)^m/2
+           + O(rho^m)].                               (6)
+
+All ratios on the right are evaluated at q. The O(rho^m) relative error accounts
+for evaluation at the displaced root and the nonlinear leading exponential.
+This is a fixed-width asymptotic, with a locally uniform spectral gap; it says
+nothing about a width-uniform bound or the fixed-aspect L^-4 conjecture.
+
+The beta_3 term supplies the largest relative spectral correction and alternates
+with parity. Examples are generated, not fitted:
+
+| m | p_(3,m), diagnostic | actual shift / leading shift |
+|---|---|---|
+| 3 | 0.5865114551126756357 | 0.84206024 |
+| 4 | 0.5883619852843523124 | 1.09362955 |
+| 8 | 0.5888800907157023131 | 1.00649648 |
+| 12 | 0.5888806988874187696 | 1.00055615 |
+| 20 | 0.5888806999178489730 | 1.00000506 |
+
+Exact rational root brackets are stored alongside each diagnostic decimal.
+
+## 6. Why the occupancy-only simplification ends at width four
+
+At width four a proper occupied row may be disconnected. Two short counterexamples
+show that merely counting full rows and successive overlaps can err in either direction.
+Here integer masks use bit j for column j (columns 0 through 3).
+
+* Rows [1,5,4,5] on the 4x4 torus: all interfaces overlap, no row is full,
+  but the ambient rank is zero, not one. The middle masks 5={0,2} do not join
+  the incoming and outgoing paths.
+* Rows [11,14] on the honest 4x2 torus: again no full row, all interfaces
+  overlap, but rank is two, not one. The two complementary paths within the
+  rows create transverse winding even though neither row is full. Parallel
+  length-two edges are retained.
+
+There is a stronger continuation witness, not just failure of one guessed formula.
+Take two open prefixes, ending at the identical frontier 5={0,2}:
+
+    history A: [0,5],      history B: [7,5].
+
+Neither prefix has ambient winding. History B connects the two frontier vertices
+via row 7={0,1,2}; history A does not. Append the SAME suffix [13,0], where
+13={0,2,3}, and close longitudinally through the empty row:
+
+    [0,5,13,0] has r=0 and D=-1;
+    [7,5,13,0] has r=1 and D=0.
+
+In the second history the new route via column 3 closes a transverse cycle
+against the old route via column 1. Thus ANY deterministic continuation summary
+that identifies these two prefixes cannot preserve this readout. Frontier
+occupancy and a current wrapping flag alone are insufficient; connectivity of
+frontier vertices is genuine needed memory. This is a scoped lower bound, NOT
+an impossibility theorem for local transfer operators or a proof that all full
+boundary partitions are necessary/minimal.
+
+The ordinary labelled frontier partition is not the end of the story. An even
+stronger pair has the same occupation count, same frontier, same ordinary
+partition, and same current winding flag:
+
+    history A': [13,5],     history B': [7,5].
+
+Both contain five of eight sites (identical weight p^5(1-p)^3), both connect
+frontier vertices {0,2}, and neither has a transverse cycle. In A' their lifted
+connecting path has horizontal displacement -2; in B' it has displacement +2.
+Attach the same suffix [13,0]. The new 0-to-2 route has displacement -2, hence
+
+    [13,5,13,0] has r=0, D=-1;
+    [ 7,5,13,0] has r=1, D= 0.
+
+The second closed route has net displacement +2-(-2)=4, one circumference;
+the first has displacement zero. The script independently traverses the OPEN
+prefixes and records all equal ordinary fields and the differing integer lifts,
+then verifies the completed torus ranks. This is genuine topological memory
+that a bare set partition and a flag recording already completed wrapping lose.
+Embedded annular or suitably lifted connectivity states can distinguish it;
+this is NOT a no-go for those representations.
+
+The next useful transfer question, if pursued, is therefore a precise width-four
+connectivity-and-topology closure, tested against both continuation witnesses.
+Plain boundary-partition counts alone do not settle the needed state space.
+It is not another occupancy-mask eigenvalue calculation or a large width scan.
+
+## 7. Executed validation and sources
+
+The script derived unnormalized Bernstein integer coefficients by Newton trace
+recurrences in site fugacity. An independent lifted-edge traversal checked all
+299,584 configurations on 3x2,...,3x6, both the pointwise identity (1) and EVERY
+polynomial coefficient. This generalizes the geometric checking technique in
+#705 and is independent of the row-transfer formula, not claimed as an unrelated
+second research program. The existing 3x3 coefficient vector
+[-1,-9,-36,-78,-90,-36,36,36,9,1] is reproduced exactly.
+
+An independent integer 7x7 matrix-power trace at p=1/2 matches the block recurrence
+for m=2,...,10. Fraction interval signs certify the q bracket, the five eigenvalue
+bands, positivity of h'(q), and the inequalities used in the all-m proof. Seven
+small mathematical tests passed locally. No full-repository CI was run for this
+new patch; no new production, GPU, or fitted exponent was used.
+
+Files:
+- scripts/width3_cylinder_exact.py
+- tests/test_width3_cylinder_exact.py
+- results/research-control-20260912/width3-cylinder-exact.json
+
+Repository inputs read: PR #705 at fc19cc748527249f0ce1c69d7cd2a88a4879427b,
+merged as eb89e9422791d9e3c3a78f0e65d56912b815a7bd; original digital-Alexander
+proof; the 3x3 Bernstein rung via #668/#684.
+
+External primary source, PRIMARY_TEXT_READ on 2026-09-12:
+Jacobsen, Critical points of Potts and O(N) models from eigenvalue identities in
+periodic Temperley-Lieb algebras, arXiv:1507.03027v1, section 6.1 Table 2 and
+sections 4,7.1. https://arxiv.org/html/1507.03027v1
+The table value and limit order are literature facts; the all-width or continuum
+claims not established above remain unclaimed. No broad novelty search was done.
diff --git a/results/research-control-20260912/shape-transport-order-finalized-summary.json b/results/research-control-20260912/shape-transport-order-finalized-summary.json
new file mode 100644
index 00000000..cecaaf5c
--- /dev/null
+++ b/results/research-control-20260912/shape-transport-order-finalized-summary.json
@@ -0,0 +1,82 @@
+{
+  "schema": "matching-one.shape-transport-order-finalized-summary.v1",
+  "provenance": {
+    "head": "59fff91c6592e8c947ccd1e3df25e15df6a291e5",
+    "checkout": "93f44c5dfb52d178046f546e9d644fd84e50dfcd",
+    "run_id": 34685638146,
+    "job_id": 103532033981,
+    "command": "python scripts/shape_transport_order.py",
+    "extraction": "Fields read from completed successful workflow stdout. Not a new local raw-histogram run.",
+    "elapsed_seconds": 17.045987239,
+    "mathematical_tests_in_job": 19
+  },
+  "finalization_date": "2026-09-12",
+  "reference_levels": [
+    0.3,
+    0.5,
+    0.7
+  ],
+  "reference_points": [
+    0.5638915164071909,
+    0.5927517136919742,
+    0.6211934034174541
+  ],
+  "composition_difference_high_precision": [
+    -4.249439184159747e-06,
+    -4.474271110694973e-06,
+    -4.553323687558643e-06
+  ],
+  "standard_errors": [
+    1.900866250374583e-06,
+    1.836344494162728e-06,
+    1.8494040220848198e-06
+  ],
+  "covariance": [
+    [
+      3.613292501813127e-12,
+      3.4593671499807256e-12,
+      3.46768682759206e-12
+    ],
+    [
+      3.4593671499807256e-12,
+      3.372161101241766e-12,
+      3.3685088503157967e-12
+    ],
+    [
+      3.46768682759206e-12,
+      3.3685088503157967e-12,
+      3.4202952369035086e-12
+    ]
+  ],
+  "jackknife_bias_estimate": [
+    -1.0533094393337672e-11,
+    -1.2296963397197432e-11,
+    -2.1912731669465483e-11
+  ],
+  "simultaneous_nominal_alpha": 0.0027,
+  "bonferroni_critical_z": 3.3200541166994784,
+  "lower": [
+    -1.0560418004010983e-05,
+    -1.057103420821836e-05,
+    -1.0693445124521922e-05
+  ],
+  "upper": [
+    2.0615396356914884e-06,
+    1.622491986828414e-06,
+    1.586797749404636e-06
+  ],
+  "max_abs_marginal_z": 2.462049197030357,
+  "resolved_crossing": false,
+  "verdict": "UNRESOLVED_FULL_COMMON_COORDINATE_COLLAPSE",
+  "pooled_float_vs_55dps_max": 1.48427446819855e-15,
+  "composition_quantiles_range_all_deletions": [
+    0.24880019486659927,
+    0.7510794674016856
+  ],
+  "limits": [
+    "No resolved crossing is not evidence establishing a common coordinate.",
+    "A nonzero constant-sign commutator would not reject affine conjugacy: Aff(1) is nonabelian.",
+    "Estimated nonlinear jackknife covariance and nominal marginal intervals; no exact finite-sample coverage claim.",
+    "Same effective spin0 laws and fixed central points, not a new independent experiment."
+  ]
+}
diff --git a/results/research-control-20260912/width3-cylinder-exact.json b/results/research-control-20260912/width3-cylinder-exact.json
new file mode 100644
index 00000000..45a5d91b
--- /dev/null
+++ b/results/research-control-20260912/width3-cylinder-exact.json
@@ -0,0 +1,486 @@
+{
+  "schema": "matching-one.width3-cylinder-exact.v1",
+  "scope": "square-site width 3, length m>=2; NOT infinite-lattice pc or an all-width pTL intertwiner",
+  "cylinder_sextic_low_first": [
+    1,
+    1,
+    0,
+    -4,
+    -5,
+    -3,
+    1
+  ],
+  "cylinder_exact_rational_bracket": [
+    "6566239220600206483872888902707641265663279/11150372599265311570767859136324180752990208",
+    "840478620236826429935729779546578082004899713/1427247692705959881058285969449495136382746624"
+  ],
+  "cylinder_q_diagnostic": "0.58888069991785299805144269575170493372217050345701",
+  "published_n3_value": "0.5888806999178529980514426957517049337221",
+  "source": "Jacobsen 2015 arXiv:1507.03027v1 section 6.1 Table 2, primary HTML read 2026-09-12",
+  "published_match_below_1e_minus_40": true,
+  "block_B_eigenvalues_diagnostic": [
+    "0.79578766896424903605620104266078116604873503565708",
+    "0.036213972909320692438479008027583384373767333677894",
+    "-0.10055279425572726240102376097844011798302112947222"
+  ],
+  "block_C_eigenvalues_diagnostic": [
+    "0.17886584993224404842481657728063602638614720762375",
+    "-0.079333776450120950401600026784588924049863141321157"
+  ],
+  "block_C_eigenvalue_multiplicity": 2,
+  "closed_rate_diagnostic": "0.79578766896424903605620104266078116604873503565708",
+  "rho_diagnostic": "0.22476579734522079198176767579637432574092549867206",
+  "hprime_diagnostic": "2.6904978143712394890696018902702841665091753340637",
+  "all_leading_coefficients": 1,
+  "dominant_subleading_coefficient": 2,
+  "exact_enumeration_checks": {
+    "2": {
+      "configurations": 64,
+      "classification_failures": 0,
+      "bernstein_counts": [
+        -1,
+        -6,
+        -12,
+        -6,
+        6,
+        6,
+        1
+      ],
+      "power_coefficients": [
+        -1,
+        0,
+        3,
+        2,
+        -3
+      ]
+    },
+    "3": {
+      "configurations": 512,
+      "classification_failures": 0,
+      "bernstein_counts": [
+        -1,
+        -9,
+        -36,
+        -78,
+        -90,
+        -36,
+        36,
+        36,
+        9,
+        1
+      ],
+      "power_coefficients": [
+        -1,
+        0,
+        0,
+        6,
+        0,
+        0,
+        0,
+        -18,
+        18,
+        -4
+      ]
+    },
+    "4": {
+      "configurations": 4096,
+      "classification_failures": 0,
+      "bernstein_counts": [
+        -1,
+        -12,
+        -66,
+        -216,
+        -456,
+        -624,
+        -498,
+        -84,
+        222,
+        192,
+        66,
+        12,
+        1
+      ],
+      "power_coefficients": [
+        -1,
+        0,
+        0,
+        4,
+        3,
+        0,
+        6,
+        0,
+        -87,
+        136,
+        -72,
+        12
+      ]
+    },
+    "5": {
+      "configurations": 32768,
+      "classification_failures": 0,
+      "bernstein_counts": [
+        -1,
+        -15,
+        -105,
+        -450,
+        -1305,
+        -2670,
+        -3885,
+        -3885,
+        -2235,
+        45,
+        1245,
+        1020,
+        420,
+        105,
+        15,
+        1
+      ],
+      "power_coefficients": [
+        -1,
+        0,
+        0,
+        5,
+        0,
+        3,
+        -10,
+        30,
+        0,
+        -200,
+        312,
+        -120,
+        -65,
+        60,
+        -15,
+        2
+      ]
+    },
+    "6": {
+      "configurations": 262144,
+      "classification_failures": 0,
+      "bernstein_counts": [
+        -1,
+        -18,
+        -153,
+        -810,
+        -2970,
+        -7938,
+        -15846,
+        -23778,
+        -26478,
+        -20658,
+        -8766,
+        2232,
+        6750,
+        5382,
+        2520,
+        774,
+        153,
+        18,
+        1
+      ],
+      "power_coefficients": [
+        -1,
+        0,
+        0,
+        6,
+        0,
+        0,
+        -12,
+        0,
+        54,
+        32,
+        -396,
+        486,
+        30,
+        -414,
+        288,
+        -114,
+        72,
+        -36,
+        6
+      ]
+    }
+  },
+  "integer_matrix_trace_checks_m_2_to_10": true,
+  "finite_roots": {
+    "2": {
+      "exact_rational_root_bracket": [
+        "400312927426073777010206031846360697872060411/713623846352979940529142984724747568191373312",
+        "800625854852147554020412063692721395744120823/1427247692705959881058285969449495136382746624"
+      ],
+      "root_diagnostic": "0.5609578904515291298145210442389823818645",
+      "root_minus_cylinder_diagnostic": "-0.02792280946632386823692165",
+      "leading_shift_diagnostic": "-0.01877706920495645957168997",
+      "shift_ratio_diagnostic": "1.487069635923441540396485"
+    },
+    "3": {
+      "exact_rational_root_bracket": [
+        "837097121055181458023268930732850654644401453/1427247692705959881058285969449495136382746624",
+        "418548560527590729011634465366425327322200727/713623846352979940529142984724747568191373312"
+      ],
+      "root_diagnostic": "0.5865114551126756356545589766069017348243",
+      "root_minus_cylinder_diagnostic": "-0.002369244805177362396883719",
+      "leading_shift_diagnostic": "-0.002813628621105619791973803",
+      "shift_ratio_diagnostic": "0.8420602446979530384188936"
+    },
+    "4": {
+      "exact_rational_root_bracket": [
+        "839738285972989758802080916395032008952045671/1427247692705959881058285969449495136382746624",
+        "104967285746623719850260114549379001119005709/178405961588244985132285746181186892047843328"
+      ],
+      "root_diagnostic": "0.5883619852843523124446730988582938294302",
+      "root_minus_cylinder_diagnostic": "-0.0005187146335006856067695969",
+      "leading_shift_diagnostic": "-0.0004743056103421040659819628",
+      "shift_ratio_diagnostic": "1.09362955484871977242227"
+    },
+    "5": {
+      "exact_rational_root_bracket": [
+        "420181142003630538273314870547619800015168973/713623846352979940529142984724747568191373312",
+        "840362284007261076546629741095239600030337947/1427247692705959881058285969449495136382746624"
+      ],
+      "root_diagnostic": "0.5887991890279353635088962923004941727878",
+      "root_minus_cylinder_diagnostic": "-0.00008151088991763454254640345",
+      "leading_shift_diagnostic": "-0.00008528614295508369719285866",
+      "shift_ratio_diagnostic": "0.9557342739788642171284672"
+    },
+    "6": {
+      "exact_rational_root_bracket": [
+        "840455282049249677895950177429479812000281615/1427247692705959881058285969449495136382746624",
+        "52528455128078104868496886089342488250017601/89202980794122492566142873090593446023921664"
+      ],
+      "root_diagnostic": "0.5888643480346472844728520094931130823426",
+      "root_minus_cylinder_diagnostic": "-0.00001635188320571357859068626",
+      "leading_shift_diagnostic": "-0.00001597450660316489351440316",
+      "shift_ratio_diagnostic": "1.023623678146899300822827"
+    },
+    "8": {
+      "exact_rational_root_bracket": [
+        "420238875377231236692311921907541377775002945/713623846352979940529142984724747568191373312",
+        "840477750754462473384623843815082755550005891/1427247692705959881058285969449495136382746624"
+      ],
+      "root_diagnostic": "0.5888800907157023131380716130272317396199",
+      "root_minus_cylinder_diagnostic": "-0.0000006092021506849133710827245",
+      "leading_shift_diagnostic": "-0.0000006052700254996210060063077",
+      "shift_ratio_diagnostic": "1.006496480941785588560711"
+    },
+    "12": {
+      "exact_rational_root_bracket": [
+        "840478618766141554937576837740610249769443589/1427247692705959881058285969449495136382746624",
+        "420239309383070777468788418870305124884721795/713623846352979940529142984724747568191373312"
+      ],
+      "root_diagnostic": "0.5888806988874187696311190589824471686986",
+      "root_minus_cylinder_diagnostic": "-1.030434228420323636769258e-9",
+      "leading_shift_diagnostic": "-1.029861467037597738970488e-9",
+      "shift_ratio_diagnostic": "1.000556153813943004503969"
+    },
+    "20": {
+      "exact_rational_root_bracket": [
+        "13132478441200323205557217575903864723271695/22300745198530623141535718272648361505980416",
+        "840478620236820685155661924857847342289388481/1427247692705959881058285969449495136382746624"
+      ],
+      "root_diagnostic": "0.5888806999178489729756992114548407710935",
+      "root_minus_cylinder_diagnostic": "-4.025075743484296864162629e-15",
+      "leading_shift_diagnostic": "-4.025055391908418730765929e-15",
+      "shift_ratio_diagnostic": "1.000005056222560078609715"
+    }
+  },
+  "width4_obstructions": [
+    {
+      "width": 4,
+      "length": 4,
+      "rows_as_bitmasks": [
+        1,
+        5,
+        4,
+        5
+      ],
+      "occupied_vertices": [
+        0,
+        4,
+        6,
+        10,
+        12,
+        14
+      ],
+      "full_row": false,
+      "every_interface_overlaps": true,
+      "actual_ambient_rank": 0,
+      "naive_rank": 1
+    },
+    {
+      "width": 4,
+      "length": 2,
+      "rows_as_bitmasks": [
+        11,
+        14
+      ],
+      "occupied_vertices": [
+        0,
+        1,
+        3,
+        5,
+        6,
+        7
+      ],
+      "full_row": false,
+      "every_interface_overlaps": true,
+      "actual_ambient_rank": 2,
+      "naive_rank": 1
+    }
+  ],
+  "exact_interval_certificate": {
+    "q_rational_interval": [
+      "11777613998357/20000000000000",
+      "29444034995893/50000000000000"
+    ],
+    "q_polynomial_endpoint_signs": [
+      1,
+      -1
+    ],
+    "eigenvalue_intervals": {
+      "B_negative": {
+        "rational_interval": [
+          "-51/500",
+          "-99/1000"
+        ],
+        "endpoint_signs": [
+          -1,
+          1
+        ]
+      },
+      "B_small_positive": {
+        "rational_interval": [
+          "7/200",
+          "19/500"
+        ],
+        "endpoint_signs": [
+          1,
+          -1
+        ]
+      },
+      "B_perron": {
+        "rational_interval": [
+          "397/500",
+          "797/1000"
+        ],
+        "endpoint_signs": [
+          -1,
+          1
+        ]
+      },
+      "C_negative": {
+        "rational_interval": [
+          "-81/1000",
+          "-39/500"
+        ],
+        "endpoint_signs": [
+          1,
+          -1
+        ]
+      },
+      "C_positive": {
+        "rational_interval": [
+          "89/500",
+          "9/50"
+        ],
+        "endpoint_signs": [
+          -1,
+          1
+        ]
+      }
+    },
+    "negative_beta_over_positive_mu_upper": "3/5",
+    "negative_mu_over_positive_mu_upper": "1/2",
+    "P_prime_at_q_sign": -1,
+    "F_lambda_at_perron_sign": 1,
+    "hprime_positive": true,
+    "all_m_at_least_2_defect_positive": true,
+    "arithmetic": "Fraction interval arithmetic; every endpoint sign certified"
+  },
+  "width4_continuation_memory_witness": {
+    "width": 4,
+    "prefixes": [
+      [
+        0,
+        5
+      ],
+      [
+        7,
+        5
+      ]
+    ],
+    "common_frontier_mask": 5,
+    "common_suffix": [
+      13,
+      0
+    ],
+    "completed_torus_ranks": [
+      0,
+      1
+    ],
+    "completed_matching_D": [
+      -1,
+      0
+    ],
+    "conclusion": "Any deterministic continuation state identifying these prefixes cannot preserve the rank readout."
+  },
+  "width4_topology_memory_witness": {
+    "width": 4,
+    "prefixes": [
+      [
+        13,
+        5
+      ],
+      [
+        7,
+        5
+      ]
+    ],
+    "common_suffix": [
+      13,
+      0
+    ],
+    "prefix_signatures": [
+      {
+        "occupied_count": 5,
+        "frontier_mask": 5,
+        "ordinary_frontier_partition": [
+          [
+            0,
+            2
+          ]
+        ],
+        "existing_transverse_winding": false,
+        "lifted_frontier_displacements": {
+          "0->2": -2
+        }
+      },
+      {
+        "occupied_count": 5,
+        "frontier_mask": 5,
+        "ordinary_frontier_partition": [
+          [
+            0,
+            2
+          ]
+        ],
+        "existing_transverse_winding": false,
+        "lifted_frontier_displacements": {
+          "0->2": 2
+        }
+      }
+    ],
+    "completed_torus_ranks": [
+      0,
+      1
+    ],
+    "completed_matching_D": [
+      -1,
+      0
+    ],
+    "exact_common_prefix_probability": "p^5*(1-p)^3",
+    "conclusion": "Even frontier mask, ordinary set partition, occupation count, and current winding flag do not preserve continuation; lifted/annular path information is needed."
+  },
+  "boundary": "Rows remember occupancy only; width 4 requires within-row/accumulated connectivity. Width-three proof does not justify mask-only transfer at width 4."
+}
diff --git a/scripts/width3_cylinder_exact.py b/scripts/width3_cylinder_exact.py
new file mode 100644
index 00000000..09b766a7
--- /dev/null
+++ b/scripts/width3_cylinder_exact.py
@@ -0,0 +1,393 @@
+#!/usr/bin/env python3
+"""Width-three square-site torus: exact row-transfer spectrum and its scope.
+
+The variable t used by coefficient recurrences is site fugacity. Coefficients
+are the unnormalized Bernstein counts in M(p)=sum c[k] p**k (1-p)**(3*m-k).
+No Monte Carlo, fitted exponent or infinite-lattice threshold estimate is used.
+"""
+from __future__ import annotations
+import argparse
+from fractions import Fraction as F
+from itertools import product
+import json
+from math import comb
+from pathlib import Path
+
+
+def add(*arrays):
+    result = [0] * max(map(len, arrays))
+    for a in arrays:
+        for i, v in enumerate(a):
+            result[i] += v
+    while len(result) > 1 and result[-1] == 0:
+        result.pop()
+    return result
+
+
+def mul(a, b):
+    result = [0] * (len(a) + len(b) - 1)
+    for i, x in enumerate(a):
+        for j, y in enumerate(b):
+            result[i+j] += x*y
+    return add(result)
+
+
+def scale(a, c):
+    return [c*x for x in a]
+
+
+def power(a, n):
+    result = [1]
+    for _ in range(n):
+        result = mul(result, a)
+    return result
+
+
+def evaluate(a, x):
+    result = 0
+    for c in reversed(a):
+        result = result*x + c
+    return result
+
+
+def block_traces(m):
+    """Integer polynomials for tr(B**m), tr(C**m), from Newton identities."""
+    if not isinstance(m, int) or m < 0:
+        raise ValueError('m must be a nonnegative integer')
+    # B eigen-equation L^3 - h L^2 - j L + t^6 = 0.
+    h, j, detneg = [0,1,3,1], [0,0,0,1,2], [0,0,0,0,0,0,1]
+    b = [[3], h, add(mul(h,h),scale(j,2))]
+    for k in range(3, m+1):
+        b.append(add(mul(h,b[k-1]),mul(j,b[k-2]),scale(mul(detneg,b[k-3]),-1)))
+    # C eigen-equation L^2 - t L - t^3 = 0; multiplicity two in the full transfer.
+    c = [[2], [0,1]]
+    for k in range(2,m+1):
+        c.append(add(mul([0,1],c[k-1]),mul([0,0,0,1],c[k-2])))
+    return b[m], c[m]
+
+
+def bernstein_counts(m):
+    if not isinstance(m,int) or m < 2:
+        raise ValueError('the torus requires integer m >= 2')
+    b,c = block_traces(m)
+    out = add(b,scale(c,2),scale(power([1,3,3],m),-1))
+    return out + [0]*(3*m+1-len(out))
+
+
+def power_coefficients(m):
+    n = 3*m
+    out = [0]*(n+1)
+    for k,c in enumerate(bernstein_counts(m)):
+        for j in range(n-k+1):
+            out[k+j] += c*comb(n-k,j)*(-1)**j
+    return add(out)
+
+
+def ambient_rank(mask, width, length):
+    """Independent lift traversal; retains distinct +/- periodic edges.
+
+    Positions are integer lifts of vertices. Every closing edge supplies an
+    ambient period. Rank is computed by integer determinants, not row masks.
+    """
+    if min(width,length)<2:
+        raise ValueError('honest axis periods must both be at least two')
+    positions = {}
+    first = None
+    for root in range(width*length):
+        if not (mask>>root)&1 or root in positions:
+            continue
+        positions[root] = (0,0)
+        stack = [root]
+        while stack:
+            current = stack.pop()
+            x,y = current%width,current//width
+            px,py = positions[current]
+            for dx,dy in ((1,0),(-1,0),(0,1),(0,-1)):
+                vertex = ((y+dy)%length)*width+(x+dx)%width
+                if not (mask>>vertex)&1:
+                    continue
+                proposed = (px+dx,py+dy)
+                if vertex not in positions:
+                    positions[vertex] = proposed
+                    stack.append(vertex)
+                else:
+                    wx,wy = proposed[0]-positions[vertex][0],proposed[1]-positions[vertex][1]
+                    if wx%width or wy%length:
+                        raise AssertionError('cycle displacement is not a torus period')
+                    wx,wy=wx//width,wy//length
+                    if wx or wy:
+                        if first is None:
+                            first=(wx,wy)
+                        elif first[0]*wy-first[1]*wx:
+                            return 2
+    return int(first is not None)
+
+
+def row_events(mask, width, length):
+    rows=[(mask>>(width*j))&((1<<width)-1) for j in range(length)]
+    full=any(row==(1<<width)-1 for row in rows)
+    overlaps=all(rows[j]&rows[(j+1)%length] for j in range(length))
+    return full,overlaps
+
+
+def check_census(m):
+    counts=[0]*(3*m+1)
+    failures=0
+    for mask in range(1<<(3*m)):
+        rank=ambient_rank(mask,3,m)
+        full,overlap=row_events(mask,3,m)
+        failures+=rank != int(full)+int(overlap)
+        counts[mask.bit_count()]+=rank-1
+    if failures or counts != bernstein_counts(m):
+        raise AssertionError(f'width 3, length {m}: row classification or coefficient failure')
+    return {'configurations':1<<(3*m),'classification_failures':failures,
+            'bernstein_counts':counts,'power_coefficients':power_coefficients(m)}
+
+
+def overlap_trace_half(width,length):
+    """Independent integer-matrix trace at p=1/2, not the block recurrence."""
+    states=list(range(1,1<<width)); d=len(states)
+    K=[[int(bool(s&t)) for t in states] for s in states]
+    P=[[int(i==j) for j in range(d)] for i in range(d)]
+    for _ in range(length):
+        P=[[sum(P[i][k]*K[k][j] for k in range(d)) for j in range(d)] for i in range(d)]
+    return F(sum(P[i][i] for i in range(d))-((1<<width)-1)**length, 1<<(width*length))
+
+
+def bisect_sign(poly,lo=F(0),hi=F(1),steps=150):
+    """An exact sign bracket, allowing either crossing orientation."""
+    a,b=evaluate(poly,lo),evaluate(poly,hi)
+    if not a*b < 0:
+        raise ValueError('need opposite endpoint signs')
+    for _ in range(steps):
+        mid=(lo+hi)/2; value=evaluate(poly,mid)
+        if value==0:
+            return [str(mid),str(mid)]
+        if a*value>0:
+            lo,a=mid,value
+        else:
+            hi,b=mid,value
+    return [str(lo),str(hi)]
+
+
+def width4_witnesses():
+    """Shows both missing within-row connectivity and unrecorded transverse loops."""
+    answer=[]
+    # All overlaps, no full row; the two sites of 0101 cannot connect.
+    # Alternating pair of equal 1101/0111 rows makes a transverse winding loop
+    # without any full row: it is a second, distinct width-four failure.
+    for rows in ([1,5,4,5],[11,14]):
+        width=4;m=len(rows)
+        mask=sum(row<<(width*i) for i,row in enumerate(rows))
+        full,overlap=row_events(mask,width,m)
+        r=ambient_rank(mask,width,m)
+        answer.append({'width':width,'length':m,'rows_as_bitmasks':rows,
+                       'occupied_vertices':[i for i in range(width*m) if mask>>i&1],
+                       'full_row':full,'every_interface_overlaps':overlap,
+                       'actual_ambient_rank':r,'naive_rank':int(full)+int(overlap)})
+    return answer
+
+
+
+def interval_certificate():
+    """Exact rational sign enclosures for q and all five distinct eigenvalues.
+
+    Three disjoint sign brackets for a cubic and two for a quadratic exhaust
+    their roots. Coarse fixed bands suffice for the all-m positive-defect proof.
+    """
+    sextic=[1,1,0,-4,-5,-3,1]
+    # A wider fixed rational interval makes this certificate short and readable.
+    lo,hi=F(58888069991785,10**14),F(58888069991786,10**14)
+    assert evaluate(sextic,lo)>0>evaluate(sextic,hi)
+    def point(x): return (F(x),F(x))
+    def plus(a,b):return (a[0]+b[0],a[1]+b[1])
+    def minus(a,b):return (a[0]-b[1],a[1]-b[0])
+    def times(a,b):
+        v=[x*y for x in a for y in b]
+        return (min(v),max(v))
+    def val(poly,x):
+        result=point(0)
+        for c in reversed(poly):result=plus(times(result,x),point(c))
+        return result
+    def sign(a):
+        if a[0]>0:return 1
+        if a[1]<0:return -1
+        raise AssertionError('interval sign unresolved')
+    q=(lo,hi);q2=times(q,q);q3=times(q,q2);qm=minus(point(1),q)
+    U=times(q,times(qm,qm));V=times(q2,qm);W=q3
+    H=plus(plus(U,times(point(3),V)),W)
+    J=plus(times(U,V),times(point(2),times(U,W)))
+    det=times(times(U,V),W)
+    def charb(x):return plus(minus(minus(times(x,times(x,x)),times(H,times(x,x))),times(J,x)),det)
+    def charc(x):return minus(minus(times(x,x),times(U,x)),times(U,V))
+    bands={'B_negative':(charb,F(-102,1000),F(-99,1000)),
+           'B_small_positive':(charb,F(35,1000),F(38,1000)),
+           'B_perron':(charb,F(794,1000),F(797,1000)),
+           'C_negative':(charc,F(-81,1000),F(-78,1000)),
+           'C_positive':(charc,F(178,1000),F(180,1000))}
+    roots={}
+    for name,(f,a,b) in bands.items():
+        signs=[sign(f(point(a))),sign(f(point(b)))]
+        assert signs[0]*signs[1]<0
+        roots[name]={'rational_interval':[str(a),str(b)],'endpoint_signs':signs}
+    # M(q)>0: |beta_-|/mu+ < 3/5, |mu-|/mu+ < 1/2.
+    assert F(102,178)<F(3,5) and F(81,178)<F(1,2)
+    # h'(q)=-(q-1)^2 P'(q)/(F_lambda(q,lambda_c)*lambda_c)>0.
+    lc=minus(point(1),q3)
+    derivative=val([i*sextic[i] for i in range(1,len(sextic))],q)
+    fl=minus(minus(times(point(3),times(lc,lc)),times(point(2),times(H,lc))),J)
+    assert sign(derivative)==-1 and sign(fl)==1 and sign(lc)==1
+    return {'q_rational_interval':[str(lo),str(hi)],
+            'q_polynomial_endpoint_signs':[1,-1], 'eigenvalue_intervals':roots,
+            'negative_beta_over_positive_mu_upper':'3/5',
+            'negative_mu_over_positive_mu_upper':'1/2',
+            'P_prime_at_q_sign':-1,'F_lambda_at_perron_sign':1,
+            'hprime_positive':True,'all_m_at_least_2_defect_positive':True,
+            'arithmetic':'Fraction interval arithmetic; every endpoint sign certified'}
+
+
+def continuation_memory_witness():
+    # Both open prefixes have frontier 0101 and no winding. Only the second
+    # prefix connects its two frontier vertices. A common suffix exposes this.
+    prefixes=[[0,5],[7,5]];suffix=[13,0];ranks=[]
+    for prefix in prefixes:
+        rows=prefix+suffix
+        mask=sum(row<<(4*j) for j,row in enumerate(rows))
+        ranks.append(ambient_rank(mask,4,len(rows)))
+    assert ranks==[0,1]
+    return {'width':4,'prefixes':prefixes,'common_frontier_mask':5,
+            'common_suffix':suffix,'completed_torus_ranks':ranks,
+            'completed_matching_D':[-1,0],
+            'conclusion':'Any deterministic continuation state identifying these prefixes cannot preserve the rank readout.'}
+
+
+
+def open_prefix_signature(rows, width=4):
+    """Exact open-cylinder connectivity with horizontal lifted displacements."""
+    height=len(rows)
+    occupied={width*y+x for y,row in enumerate(rows) for x in range(width) if row>>x&1}
+    positions={};components={};winding=False;component=0
+    for root in sorted(occupied):
+        if root in positions:continue
+        positions[root]=(0,0);components[root]=component;stack=[root]
+        while stack:
+            vertex=stack.pop();x,y=vertex%width,vertex//width
+            for dx,dy in ((1,0),(-1,0),(0,1),(0,-1)):
+                if not 0<=y+dy<height:continue
+                other=width*(y+dy)+(x+dx)%width
+                if other not in occupied:continue
+                proposed=(positions[vertex][0]+dx,positions[vertex][1]+dy)
+                if other not in positions:
+                    positions[other]=proposed;components[other]=component;stack.append(other)
+                elif proposed!=positions[other]:
+                    assert (proposed[0]-positions[other][0])%width==0
+                    assert proposed[1]==positions[other][1]
+                    winding=True
+        component+=1
+    last=width*(height-1)
+    partition={}
+    for x in range(width):
+        if last+x in occupied:partition.setdefault(components[last+x],[]).append(x)
+    groups=sorted(partition.values())
+    marked={}
+    for group in groups:
+        for x in group[1:]:
+            marked[f'{group[0]}->{x}']=positions[last+x][0]-positions[last+group[0]][0]
+    return {'occupied_count':len(occupied),'frontier_mask':rows[-1],
+            'ordinary_frontier_partition':groups,'existing_transverse_winding':winding,
+            'lifted_frontier_displacements':marked}
+
+
+def topology_memory_witness():
+    prefixes=[[13,5],[7,5]];suffix=[13,0]
+    signatures=[open_prefix_signature(rows) for rows in prefixes]
+    ordinary_keys=('occupied_count','frontier_mask','ordinary_frontier_partition',
+                   'existing_transverse_winding')
+    assert all(signatures[0][k]==signatures[1][k] for k in ordinary_keys)
+    assert signatures[0]['lifted_frontier_displacements']=={'0->2':-2}
+    assert signatures[1]['lifted_frontier_displacements']=={'0->2':2}
+    ranks=[]
+    for prefix in prefixes:
+        rows=prefix+suffix;mask=sum(row<<(4*i) for i,row in enumerate(rows))
+        ranks.append(ambient_rank(mask,4,len(rows)))
+    assert ranks==[0,1]
+    return {'width':4,'prefixes':prefixes,'common_suffix':suffix,
+            'prefix_signatures':signatures,'completed_torus_ranks':ranks,
+            'completed_matching_D':[-1,0],
+            'exact_common_prefix_probability':'p^5*(1-p)^3',
+            'conclusion':'Even frontier mask, ordinary set partition, occupation count, and current winding flag do not preserve continuation; lifted/annular path information is needed.'}
+
+
+def report(max_check=5):
+    import mpmath as mp
+    if not 2<=max_check<=6:
+        raise ValueError('local exact check restricted to 2 <= max_check <= 6')
+    checks={str(m):check_census(m) for m in range(2,max_check+1)}
+    for m in range(2,11):
+        assert overlap_trace_half(3,m)==evaluate(power_coefficients(m),F(1,2))
+    # Independent 3x3 table from #668/#684 (the existing square-torus rung).
+    assert bernstein_counts(3)==[-1,-9,-36,-78,-90,-36,36,36,9,1]
+    witness=width4_witnesses()
+    assert witness[0]['actual_ambient_rank']==0 and witness[0]['naive_rank']==1
+    sextic=[1,1,0,-4,-5,-3,1]
+    with mp.workdps(80):
+        q=mp.findroot(lambda p:evaluate(sextic,p),(.58,.60))
+        def blocks(p):
+            u=p*(1-p)**2;v=p*p*(1-p);w=p**3
+            return mp.matrix([[u,2*v,w],[2*u,3*v,w],[3*u,3*v,w]]),u,v
+        def leading(p):
+            b,_,_=blocks(p)
+            vals=mp.eig(b,left=False,right=False)
+            return max(mp.re(v) for v in vals)
+        b,u,v=blocks(q)
+        betas=sorted([mp.re(x) for x in mp.eig(b,left=False,right=False)],reverse=True)
+        mu=(u+mp.sqrt(u*u+4*u*v))/2
+        nu=(u-mp.sqrt(u*u+4*u*v))/2
+        lc=1-q**3
+        hprime=mp.diff(lambda p:mp.log(leading(p)/(1-p**3)),q)
+        r=mu/lc
+        roots={}
+        for m in (2,3,4,5,6,8,12,20):
+            bracket=bisect_sign(power_coefficients(m))
+            lf,hf=map(F,bracket)
+            root=(mp.mpf(lf.numerator)/lf.denominator+mp.mpf(hf.numerator)/hf.denominator)/2
+            shift=-2*r**m/(m*hprime)
+            roots[str(m)]={'exact_rational_root_bracket':bracket,
+                          'root_diagnostic':mp.nstr(root,40),
+                          'root_minus_cylinder_diagnostic':mp.nstr(root-q,25),
+                          'leading_shift_diagnostic':mp.nstr(shift,25),
+                          'shift_ratio_diagnostic':mp.nstr((root-q)/shift,25)}
+        s=lambda x:mp.nstr(x,50)
+        return {'schema':'matching-one.width3-cylinder-exact.v1',
+            'scope':'square-site width 3, length m>=2; NOT infinite-lattice pc or an all-width pTL intertwiner',
+            'cylinder_sextic_low_first':sextic,
+            'cylinder_exact_rational_bracket':bisect_sign(sextic),
+            'cylinder_q_diagnostic':s(q),
+            'published_n3_value':'0.5888806999178529980514426957517049337221',
+            'source':'Jacobsen 2015 arXiv:1507.03027v1 section 6.1 Table 2, primary HTML read 2026-09-12',
+            'published_match_below_1e_minus_40':abs(q-mp.mpf('0.5888806999178529980514426957517049337221'))<mp.mpf('1e-40'),
+            'block_B_eigenvalues_diagnostic':[s(x) for x in betas],
+            'block_C_eigenvalues_diagnostic':[s(mu),s(nu)],
+            'block_C_eigenvalue_multiplicity':2,
+            'closed_rate_diagnostic':s(lc),
+            'rho_diagnostic':s(r),'hprime_diagnostic':s(hprime),
+            'all_leading_coefficients':1,'dominant_subleading_coefficient':2,
+            'exact_enumeration_checks':checks,'integer_matrix_trace_checks_m_2_to_10':True,
+            'finite_roots':roots,'width4_obstructions':witness,
+            'exact_interval_certificate':interval_certificate(),
+            'width4_continuation_memory_witness':continuation_memory_witness(),
+            'width4_topology_memory_witness':topology_memory_witness(),
+            'boundary':'Rows remember occupancy only; width 4 requires within-row/accumulated connectivity. Width-three proof does not justify mask-only transfer at width 4.'}
+
+
+if __name__=='__main__':
+    ap=argparse.ArgumentParser(description=__doc__)
+    ap.add_argument('--out',type=Path)
+    ap.add_argument('--max-check',type=int,default=5)
+    args=ap.parse_args()
+    text=json.dumps(report(args.max_check),indent=2,allow_nan=False)+'\n'
+    if args.out:
+        args.out.parent.mkdir(parents=True,exist_ok=True)
+        with args.out.open('x',encoding='utf-8') as f:f.write(text)
+    else:
+        print(text,end='')
diff --git a/tests/test_width3_cylinder_exact.py b/tests/test_width3_cylinder_exact.py
new file mode 100644
index 00000000..6ca6ea8e
--- /dev/null
+++ b/tests/test_width3_cylinder_exact.py
@@ -0,0 +1,47 @@
+"""Small checks preventing wrong finite polynomials and missing frontier memory."""
+from fractions import Fraction
+from pathlib import Path
+import sys
+import unittest
+sys.path.insert(0,str(Path(__file__).resolve().parents[1]/'scripts'))
+import width3_cylinder_exact as w
+
+
+class WidthThreeExactTests(unittest.TestCase):
+    def test_existing_three_by_three_bernstein_rung(self):
+        self.assertEqual(w.bernstein_counts(3),[-1,-9,-36,-78,-90,-36,36,36,9,1])
+
+    def test_small_configurationwise_rank_dictionary(self):
+        for m in (2,3):
+            self.assertEqual(w.check_census(m)['classification_failures'],0)
+
+    def test_full_seven_state_trace_matches_block_recurrence(self):
+        for m in range(2,9):
+            self.assertEqual(w.overlap_trace_half(3,m),w.evaluate(w.power_coefficients(m),Fraction(1,2)))
+
+    def test_all_length_positive_defect_certificate(self):
+        c=w.interval_certificate()
+        self.assertTrue(c['all_m_at_least_2_defect_positive'])
+        self.assertTrue(c['hprime_positive'])
+        self.assertEqual(len(c['eigenvalue_intervals']),5)
+
+    def test_width_four_overlap_can_both_overcount_and_undercount_rank(self):
+        a,b=w.width4_witnesses()
+        self.assertEqual((a['actual_ambient_rank'],a['naive_rank']),(0,1))
+        self.assertEqual((b['actual_ambient_rank'],b['naive_rank']),(2,1))
+
+    def test_equal_frontier_masks_do_not_preserve_continuation(self):
+        witness=w.continuation_memory_witness()
+        self.assertEqual(witness['completed_torus_ranks'],[0,1])
+        self.assertEqual(witness['completed_matching_D'],[-1,0])
+
+    def test_ordinary_partition_and_current_rank_still_lose_topological_memory(self):
+        witness=w.topology_memory_witness()
+        self.assertEqual(witness['completed_torus_ranks'],[0,1])
+        signatures=witness['prefix_signatures']
+        self.assertEqual(signatures[0]['ordinary_frontier_partition'],signatures[1]['ordinary_frontier_partition'])
+        self.assertEqual(signatures[0]['occupied_count'],5)
+        self.assertEqual(signatures[1]['occupied_count'],5)
+
+
+if __name__=='__main__':unittest.main()