Width-four site sources and exact rank conditioning (do not merge)

notes/finite-critical-polynomial-dictionary-20260912.md

@@ -0,0 +1,136 @@
+# Finite critical-polynomial events are the rank sectors, not a different observer
+
+2026-09-12. Review of #717 at c2f58b6756b0f9103c4b292dca3f045f497abd32.
+This note corrects a literature/object dictionary. The identification is not
+claimed as new. It does not identify full transfer representations or establish
+an all-width leading-weight theorem.
+
+## 1. What the primary texts actually support
+
+Mertens--Ziff, arXiv:1603.07289v2, introduction following Eq. (4), explicitly
+connect the finite cross-wrapping difference to the critical polynomial of
+Scullard--Jacobsen. Section II, Eqs. (12), (20), (21) and the text following
+(21), identifies R^c-R^0 with that matching difference. Therefore #717 §2.3's
+categorical statement that 2D/0D are not the matching P2/P0 is incorrect for
+this matched finite site ensemble and embedding. Different names do not
+establish different events. One must first compare the configurations and
+weights.
+
+Scullard--Jacobsen, arXiv:1209.1451v1, §2.1 defines the periodic-extension
+classification; §3.3 gives an intrinsic torus-loop test. That test puts a
+configuration with a noncontractible surrounding loop in 1D. Thus a spiral
+with one nonzero primitive winding vector is not a rank-two cross simply
+because both coordinate projections are nonzero.
+
+Jacobsen, arXiv:1507.03027v1, §2 Eq. (4) and §6.1 Eq. (32) distinguish the
+finite topological partition sum from the local square-site realization.
+The generic FK edge-subset sum at q != 1 must not be silently substituted for
+a site-subset sum. The correct site local weights are part of the dictionary.
+The 2012 text also explicitly permits site problems through suitable local
+connectivity weights/correlations. The normalized local two-choice operator
+in the 2015 square-site convention is
+
+    R_i(p_i) = (1-p_i) E_(i+2) E_i + p_i E_(i+1).
+
+This gives the right local Bernoulli weights; by itself it does not verify a
+particular implementation's staggering, torus closure, or source/readout map.
+
+A second wording correction: the intermediate value argument in Jacobsen §4
+is not by itself a uniqueness proof. Immediately before Eq. (13), existence
+of a positive unique finite zero is described as observed. Keep actual
+monotonicity proofs separate. No universal cylinder uniqueness theorem is
+inferred from the intermediate value theorem in this note.
+
+## 2. Configuration-level dictionary with the needed topology
+
+Let a finite embedded square-site NN occupied graph G live on an honest
+oriented torus, with periodic lift to the plane. For a connected component C,
+let L_C <= Z^2 be the gain subgroup of its closed walks, expressed in the
+period basis. Covering path lifting gives:
+
+* connected components of the lift of C correspond to cosets of L_C;
+* the deck-translation stabilizer of a chosen lifted component is L_C;
+* its growth direction rank is rank L_C.
+
+These follow by comparing two root-to-root lifts: their endpoints differ by
+precisely the gain of a closed walk. No large repeated-box simulation is needed.
+
+For embedded graph components on a torus, the nonzero image is saturated.
+At rank one, a nonzero closed walk decomposes into simple cycles. A nontrivial
+embedded simple torus cycle has primitive class, so the common rank-one image
+contains its primitive generator. At rank two, a regular neighborhood of C
+cannot have genus zero: a genus-zero neighborhood's H1 is generated by disjoint
+boundary circles, an isotropic family of torus classes of rank at most one.
+A genus-one neighborhood contains two curves meeting once. Their ambient
+intersection is one, so their classes form an integral basis of H1(T^2).
+Thus L_C=Z^2 at rank two, not a proper finite-index subgroup.
+
+Different graph components are disjoint. Their ambient classes have zero
+intersection. Hence independent rank-one classes cannot be carried by separate
+components; if total rank is two, one component itself has rank two and every
+other component has trivial ambient image.
+
+Consequently:
+
+    0D  <=>  every occupied component has zero ambient image  <=> r=0;
+    1D  <=>  a primitive one-direction family (including spiral) <=> r=1;
+    2D  <=>  a rank-two cross component <=> r=2.
+
+The finite 4x4 word [3,6,12,9] is a useful tripwire: its winding generators are
+(+1,+1) and (-1,-1), so it is 1D, not 2D.
+
+The source theorem is for embedded NN primary graphs. Applying the argument
+blindly to diagonals drawn as crossing abstract edges in the matching graph
+is not justified; its complementary digital neighborhood is a separate map.
+
+## 3. Weights and normalizer, including arbitrary local site sources
+
+For independent site probabilities p_v, let t_v=p_v/(1-p_v) in the interior.
+Define the site event polynomial (q=1, not an unspecified q-extension)
+
+    C_B(t) = sum_omega [1_(2D)(omega)-1_(0D)(omega)] product_(v in omega) t_v.
+
+Then the exact finite identity is
+
+    C_B(t) / product_v(1+t_v)
+      = P_2(p)-P_0(p) = E_p[r-1] = M_B(p).
+
+This is a polynomial identity after denominators are cleared, so endpoint
+values follow by continuity. It holds for arbitrary finite sizes and local
+probabilities in the stated embedded site ensemble, not merely at a root or
+in a cylinder limit. A different lattice cell convention may add a common
+nonzero normalizing factor; that factor must be recorded, not suppressed.
+
+Because equality holds for the full multivariate polynomial, all specified
+local-source and thermal derivatives agree after the same normalization.
+Uniform-p derivatives include the partition-normalizer derivative. The identity
+at q=1 does not determine a generic-Potts-q derivative: q-continuations remain
+additional structure. Likewise it does not fix #275's homology-marked,
+restricted-trace moving-root U from the unmarked scalar M.
+
+## 4. What remains genuinely new work in #708/#710
+
+The finite event identity is literature-backed and should not be sold as an
+unpublished bridge. The small B5/B15/B16 matrices, exact factor cancellations,
+full-p visible-order certificates and explicit finite-length root remainder
+are different deliverables. Equality of one scalar partition sum does not
+make their matrices similar to the augmented pTL matrices. It does not give
+an all-width equality of leading closure coefficients, an all-width spectral
+remainder, or a fixed-aspect correction exponent.
+
+The new source work in this package uses PR708's source-safe deterministic
+rank states rather than assuming its smaller homogeneous strong lumping
+remains valid under local occupation fields.
+
+## Sources read in this continuation
+
+All three are PRIMARY_TEXT_READ at the stated sections in HTML. No PDF or
+figure was needed for the claims here. Equation labels refer to the rendered
+HTML and should be checked when citing a publisher/PDF version.
+
+* https://arxiv.org/html/1603.07289v2 — introduction, §II, Eqs. (12),(20),(21).
+* https://arxiv.org/html/1209.1451v1 — §2.1 and §3.3, local-site discussion §3.2.
+* https://arxiv.org/html/1507.03027v1 — §§2--4 and §6.1.
+
+No broad novelty search was performed. A missing formula in the texts read is
+not a statement that no prior formula exists.

notes/site-source-handoff-20260912.md

@@ -0,0 +1,66 @@
+# Completed work and corrections for the existing issue channels
+
+2026-09-12. Prepared handoff; not yet posted to GitHub. No duplicate issue,
+new production, hardware allocation, merge, or STATUS promotion is requested.
+
+## #711 / #717: dictionary correction, not another bibliography search
+
+Withdraw #717 §2.3's categorical distinction between critical-polynomial 2D/0D
+and the same finite site ensemble's rank2/rank0. Read Mertens--Ziff 2016,
+arXiv:1603.07289v2 introduction and §II Eqs. (12),(20),(21) with the following
+paragraph. The finite event identity is already explicitly connected there.
+Use `notes/finite-critical-polynomial-dictionary-20260912.md` for the exact
+periodic-lift and site-weight dictionary, including the rank-one spiral trap.
+Preserve the distinct correct boundary: no all-width B5/B15/B16 matrix
+intertwiner or closure-weight theorem follows. Also do not attribute uniqueness
+to the intermediate value theorem alone.
+
+The remaining literature task should compare actual local site weights,
+embedding, topological closure and scalar spectral decompositions, not just
+whether the papers use identical names. Missing small-block formulas in a
+bounded reading are not proof of novelty.
+
+## #636: source compatibility and rare-sector conditioning delivered
+
+On the existing PR708 finite automaton, all seven D4 types of column-probability
+groupings are now classified exactly. Common strong lumpings have 94/303/179/
+262/509/303/509 states, respectively; every one 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.
+
+A genuine square-site two-row dipole source has zero linear response but
+nonzero mixed response. The 4x4 coefficients are 327/1024 and 633/2048 at
+p=1/2; the finite root's mixed shifts have strict rational sign certificates.
+Four-sign extraction of M is exact at finite amplitudes, by separate degree<=2
+multiaffinity. This does not inherit P398's continuous-time response formula.
+
+An 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. Do not commission
+billions of direct snapshots merely to see rare sectors that this oracle can
+condition on. Conversely do not claim the fixed-width oracle scales cheaply
+to large circumference.
+
+Files are additive on PR708, no dependence on unmerged #710 or #716.
+All-width structural closure weights and width-uniform spectral estimates
+remain different, uncompleted questions. This deliverable does not authorize
+a next-width scan or a new transfer engine.
+
+## #275 / #337: what this does and does not supply
+
+The full multivariate rank event polynomial fixes occupation-source derivatives
+with the correct physical normalizer. It does not fix a generic-q continuation,
+a homology-line-marked source, or the candidate-specific restricted-trace and
+moving-root original-U map. Do not use the extra site sources as a post-hoc
+rescue of a frozen candidate or claim continuum identification from a nonzero
+mixed finite response. The source-safe exact oracle can verify a specified
+future forward map once that map is actually supplied.
+
+## Verification boundary
+
+The completed archive includes proofs, executable standard-library-only code,
+full labels/actions, rational source/root certificates, exact conditioned path
+probabilities, and a minimal-workspace patch application/reproduction check.
+The repository's complete CI suite was NOT executed for these new files.
+Past successful CI on #708/#710 is not a substitute.

notes/site-source-primary-audit-20260912.md

@@ -0,0 +1,42 @@
+# Primary-source sections actually read
+
+2026-09-12. This is a bounded verification log, not a broad novelty review.
+No new paper is labelled read in full when only the listed sections were used.
+
+1. Mertens--Ziff, *Percolation in Finite Matching Lattices*, PRE 94 (2016)
+   062152, arXiv:1603.07289v2. PRIMARY SECTIONS READ: introduction and §II,
+   especially introduction following (4), (12), (20), (21) and its following
+   paragraph. Supplies the explicit finite matching/critical-polynomial
+   relationship. https://arxiv.org/html/1603.07289v2
+2. Scullard--Jacobsen, *Transfer matrix computation of generalised critical
+   polynomials in percolation*, J. Phys. A 45 (2012) 494004,
+   arXiv:1209.1451v1. PRIMARY SECTIONS READ: §2.1, site local-weight examples,
+   §3.3. Defines 0D/1D/2D and the noncontractible surrounding-loop criterion.
+   https://arxiv.org/html/1209.1451v1
+3. Jacobsen, *Critical points of Potts and O(N) models from eigenvalue
+   identities in periodic Temperley-Lieb algebras*, J. Phys. A 48 (2015)
+   454003, arXiv:1507.03027v1. PRIMARY SECTIONS READ: §§2--4 and §6.1.
+   Separates the finite partition object, augmented annular state information,
+   cylinder asymptotics and the two-term square-site R operator.
+   https://arxiv.org/html/1507.03027v1
+4. Corstanje--van der Meulen--Schauer, *Conditioning continuous-time Markov
+   processes by guiding*, arXiv:2111.11377v2. PRIMARY SECTIONS READ: introduction
+   and §2, especially Definition 2.3 and Example 2.4. Background attribution
+   for h-transform conditioning; the discrete integer recursion here is proved
+   directly rather than imported from a continuous-time generator formula.
+   https://arxiv.org/html/2111.11377v2
+
+The latest #717 note was read through the authenticated GitHub connector at
+head c2f58b6756b0f9103c4b292dca3f045f497abd32. Its broad non-equivalence statement
+is contradicted by source 1; its caution about nonidentical matrix presentations
+remains appropriate.
+
+PR708 certificate metadata was fetched at
+f782061c1a592ed2f9fd0e9dabaa45f0e54bc4e7. Local mounted bytes match the fetched
+Git blob 50b7297deefe7c50215aea2ed534ca5810461af3. The code refuses other bytes.
+Only the certificate is reused; no uncommitted new-analysis script is imported.
+
+No PDF was analyzed, no quotation was invented, and no table/figure not read
+was used as evidence. A failed optional publisher h-transform fetch was not
+relied on; the arXiv primary §2 was available. The new finite proofs and
+computed certificates have not undergone an independent external review.

notes/site-source-review-20260912-zh.md

@@ -0,0 +1,83 @@
+# 本轮推进：有限事件字典、真实点渗流的局部源，以及稀有扇区条件采样
+
+2026-09-12。复核最新 #717，直接消费 #708 的不可变状态证书。本轮不修改远端，不合并 PR，不覆盖旧生产，不增加大规模检索或算力任务。
+
+## 1. 先纠正一个影响研究方向的文献判断
+
+#717 §2.3 将 2D/0D 与 Matching One 的 rank 2/0 判断为不同事件，这不成立。Mertens–Ziff 2016 的引言及 Eq. (21) 前后已经将有限 cross/no-wrap 差值联系到 Scullard–Jacobsen 的 critical polynomial。我们补写了配置级周期提升字典，并明确了点占据的局部权重和物理配分函数。
+
+对同一嵌入、同一 site ensemble，有限恒等式为
+
+    C_B({t_v}) / ∏_v(1+t_v) = P_2({p_v})-P_0({p_v}) = M_B({p_v}).
+
+它不只在根处成立，也不需要先取圆柱极限。rank-one spiral 属于 1D，不能因为两个绕行投影均非零而误判成 2D。4×4 的 [3,6,12,9] 是具体校验。
+
+这一事件恒等式不是新颖性主张；它也不使两个转移矩阵自动相似，不给出全部宽度的领先闭合权重，更不确定 q≠1 的 Potts 延拓或 original-U 的 thermal jet。另须撤回“中值定理本身证明唯一性”的过强文献归纳。
+
+## 2. 新完成：局部占据源要求什么状态
+
+在 #708 的 509 个确定性 rank 续接类上，对每种列概率分组，要求同一压缩对各组概率的整个开集有效，且每行可独立改变这些概率。逐多项式系数的强合并给出：
+
+| 独立概率分组 | 最终类数 |
+|---|---:|
+| 四列同概率 | 94 |
+| 一列单独、其余三列同概率 | 303 |
+| 两组相对列 | 179 |
+| 两组相邻列 | 262 |
+| 两相邻列分别标记、其余两列同概率 | 509 |
+| 两相对列分别标记、其余两列同概率 | 303 |
+| 四列独立 | 509 |
+
+四个列标签的 15 种集合划分在 D4 下恰为这七型，没有按有利结果挑分组。七个最终分区均与各自概率分组稳定子群的轨道分区逐类相等。这里是给定有限状态表示的共同强合并最小值，不是所有正实现或任意非线性状态的最小维数。
+
+一个简单反例足以说明 94 类的限制：历史 [1,1] 和 [2,2] 具有相同占据数、即时 rank，也在同一个均匀概率类内。下一行概率设为 (q,p,p,p)，两者最终 rank 期望之差恰为 q-p。因此均匀概率下正确的压缩，不能不经检查就运输到局部 source。
+
+## 3. 新完成：方格点渗流中的分离双源响应
+
+在空间行 y，仅将列 0、2 的占据概率改成 p+ε_y、p−ε_y，另外两列仍为 p。反射使 ε_y 改号，故原 rank 读出 M 的单源一阶响应严格为零。
+
+两个不同空间行 a,b 的混合响应 χ_ab 却非零。4×4、p=1/2：
+
+    χ_12 = 327/1024，χ_13 = 633/2048。
+
+这已经是真实 site-percolation 的源响应，不是 P398 校准过程。把奇源先压到 94 维不变子空间会使该源算子为零，错误地丢掉两个插入之间的非平凡分量。该投影零式也已按整数矩阵块和直接验证。
+
+而且没有必要将 ε 外推到零：每个源只改动同一行的两个独立 Bernoulli 站点，M 对各 ε 的次数至多二。因此
+
+    [M(h,k)−M(h,−k)−M(−h,k)+M(−h,−k)]/(4hk)=χ_ab
+
+在任意非零、合法的有限幅度下精确成立。该恒等式针对 M，不针对非线性的根，也不等于没有抽样方差。
+
+4×4 的未扰动有限根约为 0.5906721123310283。在有理数根区间上严格证明 χ>0、M_p>0，故根的混合源导数为 −χ/M_p。相邻行约 −0.0577415189712485，间隔一行约 −0.0549132944707030。它们是有限系统根的 mixed coefficients，不是无限体临界点位移。
+
+## 4. 新完成：不用等待稀有事件自然出现的条件采样器
+
+利用确定性 rank 状态，完成了整数后向递推 h_y(s)，可直接从最终 rank 固定为 0、1 或 2 的原始站点乘积测度中抽样。每一步用合法的非负整数权重 W_y(b)h_(y+1)(τ(s,b)) 选下一行；路径概率中的 h 因子望远镜消去，恰得原始配置权重除以指定 rank 的正规化常数。
+
+这是一种标准有限时域 Doob 型条件化的具体实现，不是新的一般采样理论。不需要整张配置的拒绝重试、不需要 burn-in、不用浮点采样权重。代价是必须先计算精确后向常数，且复杂度仍依赖状态数和大整数位长，不能宣称大宽度问题已解决。
+
+在 p=591417/1000000、4×128 上：
+
+| rank | 概率 | 条件占据点数均值 |
+|---|---:|---:|
+| 0 | 4.2444165285438005×10^-10 | 265.21829715618895 |
+| 2 | 4.2441877885758733×10^-10 | 341.4414449341080 |
+
+这些概率和条件矩由整数前后向计算得出，不是从八条验证轨迹估出来的。每个 rank 的四条固定种子轨迹均经独立物理提升图重判；完整配置已保存。它们仅是算法控制，不构成新的渗流证据。
+
+同时实现了正规化与热导数之间的精确关系：
+
+    ∂_p log(P_2/P_0)=[E(K|r=2)−E(K|r=0)]/[p(1−p)]。
+
+本控制值约为 315.4371084311523。K 是总占据数，不是 birth time。非均匀源必须使用对应的逐点条件指标，不能照搬均匀 K 公式。
+
+## 5. 验证范围
+
+- 69,888 个物理 NN 配置逐项核对状态表、分量同调饱和性、全部占据数系数和两源多项式。
+- 65,152 条 D4/转移共轭关系；各概率分组的共同强合并均闭合。
+- 不调用系数细化器的独立有理数网格验证，共 71 个多维参数点、36,139 条完整状态概率行。网格次数充分，因此验证的是整个多项式族，不是随机参数吻合。
+- 非均匀 4×3 的全部 4,096 配置：12,288 个条件路径概率等式及条件占据一、二阶矩全通过。
+- 4×128 稀有控制的前后向正规化常数一致；八条条件轨迹通过独立物理 rank 检查。
+- 本地测试和补丁应用／重生成记录在归档根目录 VALIDATION.json；未运行完整仓库 CI。
+
+本包仅依赖 #708 的状态 JSON；不依赖 #710、#716 或上一轮尚未确认提交的 root/TL 包。没有重新实现更大宽度引擎，没有新生产或 GPU 任务。可将已完成事项记到既有 #636/#711；#275 的候选专用 original-U 前向映射仍未由这些不带标记的 rank 响应自动提供。