All-p width-four spectrum and diamond 3L−1 onset (stacked on #708; do not merge)

notes/completed-636-673-handoff-20260912.md

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+# Updates for existing #636 / #673; no duplicate task
+
+## #636 — the all-p width-four task is completed
+
+Starting from #708's fixed input certificate, a 94-state common weighted
+quotient and three explicit nonnegative local blocks of sizes 5,15,16 give
+all-length polynomial identities:
+
+```
+P0 numerator = tr(B5^m)-tr(B16^m)+2t^(2m)
+P2 numerator = tr(B15^m)-tr(B16^m)+2t^(2m)
+M numerator  = tr(B15^m)-tr(B5^m)
+```
+
+Normalize by (1+t)^(4m), t=p/(1-p). The result is proved for all p, all m>=2
+using 126 exact polynomial zero moments, a coefficient-bounded Kronecker
+certificate and Cayley-Hamilton, not a parameter scan. Generic minimum scalar
+orders are P0=17, P2=23, M=16; special p=1/2 gives 15, p=2/3 gives 14.
+
+The leading Perron weights near the unique cylinder crossing are one on both
+sides. The crossing has a degree-17 defining polynomial and agrees with the
+already-published n=4 value q4=.59141717085313848.... A slower shared 0.773935
+relative mode cancels identically from M; its visible relative decay is
+0.251749755, with coefficient two. Rational certificates prove every finite
+root lies below q4 and the fixed-width displacement formula in the note.
+
+Do not re-dispatch this same visible-spectrum task or buy another width engine.
+The genuinely larger question is an all-width closure/observable-weight theorem
+and width-uniform bounds, not a list of more roots. No new hardware or scan is
+commissioned by this comment. The thermal-derivative m*lambda^m control can be
+cross-linked to #275 without changing that issue's candidate contract.
+
+## #673 / #692 — rank-two onset and equality classification completed
+
+The old shared-vertex union lower-bound argument is invalid, but its axis
+conclusion is correct. Replace it with the two-cycle-core lemma. A minimal
+ambient-rank-two core is a wedge or theta; a dumbbell is excluded by the
+intersection form. Systolic lengths yield the axis onset 2L-1 and L^2 crosses.
+For the diamond, all three theta cycle lengths are at least2L, yielding
+3L-1 vertices; equality and period arithmetic force a full2L line with one
+straight transverse L-1-site plug. There are exactly4L^2 minimizers for every
+L>=2. Diamond L5 k14,count100 is a theorem, not a task needing2^50 enumeration.
+
+Read `notes/two-cycle-core-onsets-20260912.md`; preserve the original note and
+result history but remove the invalid proof dependency. No additional census
+or new duplicate issue is needed. This does not classify every higher-mass
+wrapping cell.
+
+## Validation and integration boundary
+
+Ten new local mathematical tests passed. The all-p proof, resultant, Hankel
+minors and rational root inequalities were actually executed. A clean patch
+application check is included in VALIDATION.json. Full repository CI has not
+been run for this addition. No remote branch or issue was modified by the
+analysis session. The parametric scripts use #708's existing certificate;
+the onset proof/script is independent of #708 and may be integrated separately.

notes/crossline-next-results-20260912-zh.md

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+# Matching One：两条研究线的实质推进
+
+日期：2026-09-12。本轮读取了 #708、#636 的已有闭合资产，另读 #692 的起点证明、#275 最新三条实质评论与 Jacobsen 2015 原文。远端分支、历史结果、冻结记录均未修改；本包供提交处理，不是已经合入的 PR。
+
+## 一、全参数的宽度四可见谱已经完成
+
+原 #708 只给出 p=1/2 的 15 阶标量递推。本轮先在原 509 个确定性续接类上构造对所有行占据权重同时有效的强可合并划分，得到 `3→35→94→94`。94 不是任意正实现的最小维数，而是这张状态表及其指定读出上的共同强合并结果。
+
+设 t=p/(1-p)，每行权重 t^k。三个非负局部块 B5、B15、B16 满足，对所有 m≥2、所有 p：
+
+```
+(1+t)^(4m) P0 = tr(B5^m)  - tr(B16^m) + 2 t^(2m)
+(1+t)^(4m) P2 = tr(B15^m) - tr(B16^m) + 2 t^(2m)
+(1+t)^(4m) M  = tr(B15^m) - tr(B5^m)
+```
+
+这不是几个 p 点上的数值吻合。标准库验证器用有严格系数界的整数 Kronecker 编码，证明了前 126 项为多项式恒等式；94+15+16+1 的有限实现上界与 Cayley–Hamilton 将其推广到所有长度。三个小块的特征多项式又由直接矩阵乘法的前 5、15、16 个迹独立核对。
+
+全参数的一般最小递推阶数为：P0=17，P2=23，M=16。正是共同模式抵消让差值更低阶。精确 Hankel 子式给出下界，完整迹分解给出上界。p=1/2 时阶数降为15；p=2/3 时另一对反号模式碰合并抵消，降为14。单点阶数不能代替一般阶数。
+
+完整块矩阵、分解、94 态合并、递推及整数行列式均在 JSON 中；不要求读者信任文字结论。
+
+## 二、精确得到宽度四的圆柱根与有限长度位移
+
+B5、B15 的 Perron 根分别由一个二次因子和一个五次因子承担，正的可合并商对此给出证明。其 resultant 是 t^10(t+1)^2 乘一个17次整系数多项式。后者只有一次 Descartes 符号变化；Perron 分支的端点顺序交换，故正交点唯一。
+
+```
+q4 = 0.591417170853138481798834101735923177964270443...
+```
+
+它与 Jacobsen 2015 Table 2 的已发表 n=4 数值一致，绝不是无限方格 p_c 的新值。
+
+共同 B16 模式在交点的相对衰减率约为0.7739350472，但它在 P0 和 P2 中权重相同，在 M 中精确抵消。M 真正可见的领先修正率为
+
+```
+rho = 0.25174975499192538...
+h'(q4) = 2.4643526094735922...
+```
+
+两侧领先权重在交点邻域均为1；完整余谱没有消失。已用有理数根区间、复根对的 Vieta 界和正 Collatz 向量核对相关不等式。结论是所有有限 m≥2 的根均严格小于 q4，而且
+
+```
+p_(4,m)-q4 = -2 rho^m/[m h'(q4)] × [1+O(0.674^m)+O(rho^m)].
+```
+
+这是固定宽度定理，不给出宽度一致界，也不证明固定长宽比的 L^-4 规律。
+
+一个和 #275 有关的进一步结论：热导数在交点自动产生 m*lambda^m 项，即使相交的两个领先 Perron 分支在块对角表示中完全半单。因此，导数响应中的长度多项式因子本身不能识别 Jordan 机制。原始匹配函数的普通迹分解更说明：每个固定 p 下它都是纯特征值幂之和；底层算子可能有不可见的非对角化部分。这里改变的是读出与求导，不是发现了一个新场。
+
+## 三、横向解决 #673 / #692 的菱形起点猜想
+
+#692 以“两个周期共有至少一个顶点”推出联合顶点数至少2L-1，集合大小不等式方向不对。数值结论可以保留，但证明必须换掉。
+
+替换工具是一个可复用的双周期核心引理：任意环境同调秩为二的嵌入图，均含图圈秩恰为二且环境同调映射单射的核心。去叶、压缩二度顶点后，只可能是两圈共一点的楔形或三条内部不交路径组成的 theta；哑铃形因两个不交周期不能有独立环面同调而排除。
+
+轴向环面中，theta 必含同时具有两个非零绕行坐标的简单周期，需要至少2L个顶点；楔形至少2L-1，等号恰为一整行与一整列。于是起点2L-1、计数L²得到正确证明。
+
+菱形环面周期为(L,L)、(L,-L)，每个非平凡周期至少2L条边。theta 的三个两两路径圈各至少2L，推出三路径总长至少3L、顶点至少3L-1。等号时三路径长度皆L。进一步利用提升位移的 l1 三角等号及模L的坐标同余，证明三条路径必须是三条直的坐标轴方向臂，不能有额外弯曲的等号情况。
+
+因此，对所有整数 L≥2：
+
+```
+菱形 rank-2 起点 = 3L-1
+最小配置 = 一条完整2L点直线 + 垂直方向长度L的直路径的L-1个内部点
+最小配置总数 = 4L²
+```
+
+菱形 L=5 的 k=14、100个配置现在由证明给出，无须枚举2^50个配置。新证明也修复了 #692 对2L点 spiral 起点的另一处不正确共有顶点论证。每个方向有L条完整线（两族共2L），不是每族2L；高质量端点反例须使用L点NNN链，而不是2L点NN直线。
+
+## 四、实际执行与下一步取舍
+
+全参数标准库证书运行约1.4秒。对旧小尺寸轴向L=2,3,4和菱形L=2,3，在起点及以下核对了133,711个配置；没有新增大尺寸全枚举。另对4×2、4×3的全部配置独立核对概率系数。L=2,…,8只生成并核对定理给出的最小配置，不是额外的全空间 census。所有10项本地数学测试通过；完整项目CI尚未运行。
+
+#636 本次指定的“Q(p) 可见谱、分别处理P0/P2/M”已有完成结果，不应再派同一单。所有宽度的规范闭合与宽度一致谱界仍是更大问题，不能自动变成宽度5、6、…扫描。
+
+#673 的菱形起点及 plug 分类可以按本证明结案，#692 应修正文义后再整合；无需为菱形L5开计算任务。
+
+#275 保持其原始候选前向映射要求。新热导数对照可以作为防止误读的精确控制，但不是给旧模型增加第三候选或改动冻结评分的许可。
+
+本轮没有触及需要大规模CPU/GPU或广泛文献检索的任务，因此没有新增工单或启动外部机器。证明、脚本、整数结果、更新文字均已完成并装入提交包。

notes/two-cycle-core-onsets-20260912.md

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+# Two-cycle cores settle the axis and diamond rank-two onsets
+
+Date: 2026-09-12. Direct completion of the remaining geometric question in
+#673 / draft #692. This is a proof, with small existing-size checks, not a
+proposal to enumerate another torus. No novelty claim is made.
+
+## Result and conventions
+
+On the nearest-neighbour square-site graph with period lattice
+
+* axis: `Lambda = <(L,0),(0,L)>`,
+* diamond: `Lambda = <(L,L),(L,-L)>`,
+
+let `r(B)` be the rank over Q of the occupied graph's image in torus H1.
+For every integer `L >= 2` (parallel lifted edges at period two retained),
+
+| Geometry | Least occupied mass with r=2 | All minimizers | Number |
+|---|---:|---|---:|
+| axis | 2L-1 | a full physical horizontal row union a full vertical column | L^2 |
+| diamond | 3L-1 | a full physical horizontal/vertical line of 2L vertices, plus the L-1 internal vertices of one straight transverse length-L arc | 4L^2 |
+
+The existing digital-Alexander theorem identifies these as the `b x n`
+(exclusive cross) cell: black rank two forces complementary matching rank
+zero. It is NOT the rank-one spiral / `both x both` cell.
+
+In the `(u,v)=(x+y,y-x)` convention of #692, a physical horizontal or vertical
+line is a straight diagonal line. A physical transverse arc is the straight
+plug described empirically there. Thus this theorem addresses exactly the
+existing conjecture, not a different lattice or event.
+
+## 1. The reusable two-cycle-core lemma
+
+If a graph embedded in a torus has ambient rank two, it contains a connected
+subgraph of cycle rank two whose two graph-homology generators have independent
+ambient images. Indeed, different connected components cannot carry independent
+ambient classes: their disjoint embedded cycles have intersection number zero,
+whereas independent classes in H1(T^2;Q) have nonzero intersection. In a component
+carrying rank two choose a spanning tree and two off-tree edges whose fundamental
+cycles have independent ambient images. Retain the tree and those two edges,
+then prune leaves. The graph cycle rank is two and the ambient map is injective.
+
+After suppressing degree-two vertices, the connected core is either a wedge
+of two circles, a theta (three internally disjoint paths between two vertices),
+or a dumbbell (two circles connected by a path). This classification follows
+from `sum(deg(v)-2)=2` after leaves are removed. A dumbbell is impossible: its
+two disjoint circles would have independent ambient images. Thus only wedge
+and theta remain. All their simple cycles have nonzero ambient class; for a
+theta the three classes are, up to signs, `a,b,a-b`, with a,b independent.
+
+This lemma concerns a SUBGRAPH of the occupied induced graph. Extra occupied
+edges cause no problem: a lower bound on this core is already a lower bound
+on occupied vertices. At equality there can be no additional occupied vertex.
+
+## 2. Axis proof, including the missing inequality in #692
+
+A simple cycle of class `(a,b)` has at least `L(|a|+|b|)` edges, by its physical
+integer displacement. For a simple cycle the edge count equals its number of
+vertices, including a two-edge periodic circle.
+
+A wedge has two essential circles, each of length at least L, meeting in
+one vertex; hence it has at least `2L-1` vertices. Equality forces both circles
+to have length L. Independence forces one horizontal and one vertical, and
+geodesic equality forces each to be straight. This gives precisely a row-column
+cross.
+
+For a theta, among `a,b,a-b` at least one class has both coordinates nonzero.
+Otherwise the independent a,b would have to lie on the two different axes,
+and a-b would be mixed. That simple circle has at least 2L vertices. A theta
+therefore cannot occur at occupied mass `2L-1` or below.
+
+The L horizontal rows and L vertical columns give L^2 different crosses,
+each with rank two. This proves the onset, equality classification and count.
+
+The original #692 inference, “the two cycles share at least one vertex, so
+union size >= 2L-1”, has its inequality in the wrong direction: a lower bound
+on intersection size gives an UPPER bound on a union when sizes are fixed.
+The core lemma repairs the proof without invalidating the observed onset.
+It also avoids assuming that every rank-two graph contains separately chosen
+simple cycles in the two prescribed coordinate classes.
+
+## 3. Diamond lower bound
+
+For `Lambda=<(L,L),(L,-L)>`, a period vector is
+
+    L(a+b, a-b),
+
+whose l1 length is `2L max(|a|,|b|)`. Every nontrivial simple cycle therefore
+has at least 2L edges.
+
+A wedge consequently needs at least `4L-1` vertices. In a theta let the three
+path lengths be `l1,l2,l3`. Each pair is an essential simple circle, so
+
+    l1+l2 >= 2L,   l1+l3 >= 2L,   l2+l3 >= 2L.
+
+Adding gives `E=l1+l2+l3 >= 3L`. A connected theta has `V=E-1`, hence
+`V >= 3L-1`. This proves the lower bound on every occupied configuration.
+
+At equality the core must be a theta and all three pair bounds are equalities,
+so `l1=l2=l3=L`. The remaining issue is whether bent geodesic paths create
+additional equality cases. The period arithmetic rules them out.
+
+## 4. Diamond equality: three straight arms, not arbitrary bent plugs
+
+Orient the three length-L paths from the same branch vertex u to v. Lift them
+from the same lattice representative of u and write their physical displacement
+vectors as d1,d2,d3. Their pairwise differences are nonzero period vectors,
+so
+
+    2L <= ||di-dj||_1 <= ||di||_1+||dj||_1 <= 2L.
+
+Every inequality is an equality. Thus `||di||_1=L`, and no two displacement
+vectors use the same coordinate with the same nonzero sign: otherwise the
+triangle inequality for their difference would be strict.
+
+There are only four signed coordinate slots: x+, x-, y+, y-. Three nonzero
+vectors cannot each use two slots without overlap. At least one vector uses
+only one coordinate and hence equals `(+-L,0)` or `(0,+-L)`. All di are congruent
+coordinatewise modulo L, because their differences lie in Lambda. Therefore
+all their coordinates are multiples of L. With l1 norm L, EACH di must be
+one of these four axial vectors. They are distinct, so exactly two are an
+opposite pair and the third is perpendicular.
+
+A nearest-neighbour path of length L with displacement `(L,0)`, for example,
+uses L positive horizontal steps; there is no room for a detour. All three
+paths are straight. The opposite pair forms a full line of 2L vertices; the
+remaining arm contributes exactly L-1 internal vertices of a transverse plug.
+
+Conversely, every such line-plus-plug is a theta with independent essential
+cycles, so it has rank two. This proves the full equality classification.
+
+## 5. Exact count 4L^2
+
+There are two physical line directions and L disjoint full lines of each
+direction. For a given full line, choose one endpoint on its 2L vertices and
+one of two transverse directions. Reversing the same length-L arc counts it
+twice, so there are `(2L*2)/2=2L` distinct plugs per full line.
+
+No occupied minimizer is counted under two full lines. Parallel full lines
+are disjoint. A horizontal and a vertical full line on the diamond quotient
+meet in two vertices; their union has `4L-2 > 3L-1` vertices for L>=2. The full
+line in a minimizer is therefore unique. The count is
+
+    2 * L * 2L = 4L^2.
+
+In particular the previously proposed diamond L=5 onset is now a theorem:
+`k=14`, count 100. No `2^50` enumeration, or even a new low-k census, is needed.
+
+## 6. Other #692 statements that the proof repairs or narrows
+
+At diamond mass 2L, rank two is impossible by the new bound. A rank-one cycle
+with both generator coordinates nonzero must attain the systolic bound with
+both `(u,v)` displacements of magnitude 2L. Every step then has the same
+sign pattern, forcing one of the 2L full straight diagonal lines. The
+`both x both` spiral count 2L at this mass follows, using the existing exact
+rank-one label map for the complementary graph. This replaces #692's other
+use of the incorrect shared-vertex union argument.
+
+Two local wording errors in its high-k proof should also be corrected:
+there are L lines PER diagonal family (2L total), not 2L per family. For the
+sharpness example with exactly L white sites on the diamond, use an NNN
+chain with physical steps `(1,1)` (or `(1,-1)`) closing after L steps, not a
+full NN straight line of 2L sites. The high-k conclusion itself survives.
+
+This note does not assert a general classification of all higher-mass cells,
+Galois properties of matching polynomials, or any critical scaling exponent.
+
+## 7. Executed checks and source trail
+
+`python scripts/rank_two_onset.py` exhaustively checks all occupied subsets
+AT OR BELOW the predicted onset on axis L=2,3,4 and diamond L=2,3: 133,711
+configurations total. A direct physical integer-lift graph traversal finds
+zero rank-two configurations below the bound and exactly the predicted sets
+at equality. Counts are 4/9/16 on the axis and 16/36 on the diamond.
+
+Every predicted minimizer at L=2,...,8 was also constructed and its rank
+checked, without enumerating the surrounding configuration space. These are
+checks of the proof and constructor, not new stochastic evidence.
+
+Source read through the connector: draft #692,
+`notes/wrapping-five-cell-onset-proofs-20260908.md`, blob
+`6a3ac9dcf6e1c7f8bbef853379a0e02ea8ef4cad`, especially sections 3b and 5c.
+The geometry there agrees with the physical periods used here. The cell
+translation uses merged #702, not the superseded #690 directional conjecture.
+Full repository CI for this new file has not been run.