From f782061c1a592ed2f9fd0e9dabaa45f0e54bc4e7 Mon Sep 17 00:00:00 2001 From: Light Chain Date: Sat, 12 Sep 2026 19:35:42 +0800 Subject: [PATCH] Width-four rank closure: 1448 states, 509 futures, 15-mode scalar at p=1/2 Pinned-boundary connectivity with path gains and retained global cycle span is sufficient for future ambient rank; a common Dehn shear preserves rank but not directional labels. Exhaustive width-four row language: 1448 representatives, 23168 transitions independently reconstructed; deterministic continuation quotient 3 -> 164 -> 509 -> 509, minimal for this language. M_{4,m}(p)=b(p)^T K(p)^{m-1} c for m>=2. At p=1/2, 16^m M has minimal constant-coefficient order 15 (nonzero physical-tail Hankel, 509 annihilator moments, p(A)c not the zero vector). Not a new p_c, not an all-p identity, not a pTL intertwiner. Additive. Do not merge. Full-repo CI not run for this patch. --- ...636-width4-completion-and-next-20260912.md | 49 + notes/width4-lifted-rank-closure-20260912.md | 325 ++ notes/width4-review-control-20260912-zh.md | 55 + .../width4-half-probability-spectrum.json | 124 + .../width4-rank-closure-certificate.json | 3427 +++++++++++++++++ .../width4-rank-closure-summary.json | 660 ++++ scripts/lifted_boundary_rank.py | 278 ++ scripts/verify_width4_rank_closure.py | 222 ++ scripts/width4_half_probability_spectrum.py | 156 + tests/test_lifted_boundary_rank.py | 69 + 10 files changed, 5365 insertions(+) create mode 100644 notes/issue-636-width4-completion-and-next-20260912.md create mode 100644 notes/width4-lifted-rank-closure-20260912.md create mode 100644 notes/width4-review-control-20260912-zh.md create mode 100644 results/research-control-20260912/width4-half-probability-spectrum.json create mode 100644 results/research-control-20260912/width4-rank-closure-certificate.json create mode 100644 results/research-control-20260912/width4-rank-closure-summary.json create mode 100644 scripts/lifted_boundary_rank.py create mode 100644 scripts/verify_width4_rank_closure.py create mode 100644 scripts/width4_half_probability_spectrum.py create mode 100644 tests/test_lifted_boundary_rank.py diff --git a/notes/issue-636-width4-completion-and-next-20260912.md b/notes/issue-636-width4-completion-and-next-20260912.md new file mode 100644 index 00000000..dd8fb025 --- /dev/null +++ b/notes/issue-636-width4-completion-and-next-20260912.md @@ -0,0 +1,49 @@ +## 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. diff --git a/notes/width4-lifted-rank-closure-20260912.md b/notes/width4-lifted-rank-closure-20260912.md new file mode 100644 index 00000000..4d7250bc --- /dev/null +++ b/notes/width4-lifted-rank-closure-20260912.md @@ -0,0 +1,325 @@ +# Width four: a sufficient boundary state, a 509-class continuation quotient, and a 15-mode scalar control + +Date: 2026-09-12. Direct response to #636's latest comment and draft #707. +Status: completed exact finite algebra with a general sufficiency argument and +an exhaustive finite closure certificate. No new Monte Carlo, no GPU, no +claim of a new critical probability, and no novelty claim. + +The main result is NOT another width extrapolation. The previously missing +state is now constructed. On the width-four square-site torus, a bounded +boundary representation closes under EVERY row mask. It has 1,448 reachable +geometric representatives and 509 distinguishable deterministic rank futures. +At p=1/2 the single scalar matching sequence has minimal linear recurrence +order 15. These three dimensions answer three different questions. + +## 1. Physical object and exact scope + +Take the NN occupied graph on the square-cell torus with periods (w,0),(0,m), +w,m>=2. Parallel periodic edges at period two are distinct lifted edges. +The output is the rational rank r of the ambient homology image, and the +repository's digital-Alexander identity identifies D=r-1 and M=E[D]. + +Grow complete rows on an open cylinder, retaining row zero as a pinned seam. +The exposed terminal set consists of occupied vertices of the first row and +of the current row: at most 2w terminals. A first-row alias is a virtual zero- +gain edge, NOT an extra occupied site. The final vertical edges join the current +row back to the pinned row. All tested physical lengths are at least two. + +Use cut gains, not Euclidean displacements: a positive horizontal edge crossing +w-1 -> 0 has gain (1,0); other horizontal edges and interior vertical edges have +gain zero. The final current-to-pinned seam edges have gain (0,1). Cycle sums +are exactly the integer winding coordinates. The equivalent physical-displacement +oracle is kept separate from this cut-gain convention. + +Only rank is promised. The shear quotient below does NOT retain the actual +rank-one slope, x/y/spiral label, primitive character, or component count. +This distinction matters for any reuse outside A_top/E_top or rank probabilities. + +## 2. Boundary elimination lemma + +Here is the mathematical reason that old interior vertices can be forgotten. +For a graph G with oriented additive gains in Q^d and exposed terminal set T, +retain: + +* the partition Pi of T by ordinary graph connectivity; +* the GLOBAL rational span H of gains of all cycles, including cycles in + components no longer touching T; +* for every terminal block, a root o and path potentials d_v modulo H, where + d_v is the gain of any path o -> v and d_o=0. + +For two graphs with the same labelled T and the same retained data, gluing +ANY common future graph along T gives the same total cycle-gain span. + +Proof. Replace each old connected component meeting T by a star on its +terminals, with root-to-v edge gain d_v. Retain H independently. A closed +walk in the glued graph can be split into maximal old-graph paths and future- +graph paths. Replace each old path from u to v by the star path of gain +-d_u+d_v. The difference is a gain in H, because the two old routes differ by +a cycle. Conversely each star route has an old-graph realization with the +same gain modulo H. The spans generated by H and the new closed walks are +therefore identical in both directions. Components with no exposed vertex +cannot be used by a new path, but their existing cycles must remain in H. +This proves both elimination and arbitrary-future sufficiency. + +Adding an edge u -> v with gain g merges two terminal blocks, or, if they +are already connected, adds d_u+g-d_v to H. Re-rooting is a gauge choice. +Whenever H grows, reduce all stored path potentials modulo the new H. +Nothing in this lemma presumes planarity or a probabilistic independence law. + +On the open cylinder, H is either zero or the horizontal line. Thus before +longitudinal closure it is enough to retain a Boolean horizontal-cycle flag +and integer horizontal path gains. Once the flag is true, all horizontal +path gains can be dropped: their differences lie in H already. The flag must +survive even if every vertex of its original component has been forgotten. +The test [0,15,0] exercises exactly this case. + +## 3. A second, rank-specific quotient removes accumulated shear + +Path gains can otherwise drift as a strip grows. Mark pinned terminals by +epsilon=0 and current terminals by epsilon=1. For one common integer c, +replace every path gain within each block by + + d'_v-d'_u = d_v-d_u + c*(epsilon_v-epsilon_u). + +This preserves the rank of every future torus closure. To see it, assign +epsilon=1 to all future vertices; future internal edges have zero epsilon +change, while each positive closing seam edge has epsilon change -1 and +vertical gain +1. On a completed cycle, the total epsilon change is zero. +The sum of epsilon changes on the old portions is consequently its vertical +winding h_y. The changed horizontal winding is + + (h_x,h_y) -> (h_x+c*h_y,h_y). + +This is an invertible integer shear. It preserves rank, not the actual +homology line. Old horizontal cycles are fixed by it. Thus the assertion holds +for EVERY suffix, not merely for the suffix that motivated the construction. + +The implemented normalization chooses the lexicographically first current +terminal connected to a pinned terminal, and shifts their relative gain to +zero. If no such connection exists, the shift changes no stored relative +potential. This is a quotient of the observable's state, NOT a claim that the +physical sampling law is invariant under a lattice shear. Different history +weights are still added with their original row occupation factors. + +The code first performs the boundary elimination of section 2, then this +rank-specific normalization. Both transformations preserve arbitrary rank +futures, so they can be applied after every appended row. + +## 4. Complete finite closure at width four + +Starting from all 16 first-row masks, breadth-first exploration of the EXACT +integer update terminates at 1,448 states. Every state has all 16 successors +inside that set: 23,168 transitions, none truncated. In these representatives +all stored gains belong to {-1,0,1}. A cap exists to fail explicitly should +exploration not terminate, but it was NOT reached. + +The complete state/transition/closure table is supplied in +`results/research-control-20260912/width4-rank-closure-certificate.json`. +Every transition was independently reconstructed by graph traversal of the +boundary-star graph, rather than by the weighted union/find implementation +used to generate the table. This verifies the update arithmetic on the WHOLE +finite closure, not only on a few initial states. + +This is a computer-assisted finite closure proof: sections 2-3 prove what the +state preserves, and the finite table proves that its reachable width-four +state set is closed. Merely finding unchanged counts at two successive strip +lengths would NOT establish closure; all successors have actually been checked. +The counts by length for m=1,...,6 are 16,256,888,1408,1448,1448, but the proof +uses the transition closure rather than this apparent saturation. + +### Deterministic continuation equivalence + +Give a state output equal to the rank obtained by closing its seam immediately. +Partition refinement by output and all 16 successor classes gives counts + + 3 -> 164 -> 509 -> 509. + +The final partition is an exact transition congruence: output and every +successor agree for ALL members of each class. Conversely, the 509 classes +have 509 different output signatures on the 273 suffixes of lengths 0,1,2. +Thus every pair is separated by at most two appended rows. All class +representatives are reachable at a physical length m>=2, not just at the +convenient first-row initializer. + +Together these facts certify **509 minimal deterministic continuation classes** +for this exact language: fixed width four, fixed column-labelled row-mask +alphabet, arbitrary complete-row suffixes, and final rational rank output. +The suffix bound concerns distinguishing experiments, NOT the length of past +history needed to reconstruct a state. The minimum is for a homogeneous +deterministic machine; strip length is not supplied as free auxiliary state. +It is not a minimal positive stochastic realization, a minimal linear +realization, a pTL module count, or a continuum state count. Permuting the +input alphabet while carrying hidden extra information is a different language. + +### The earlier failure is still present and correctly resolved + +Draft #707's prefixes [13,5] and [7,5], followed by [13,0], return ranks 0 +and 1. The new representation distinguishes them. + +An equivalent version has IDENTICAL pinned occupancy as well: + + prefix A = [0,7,5], prefix B = [0,13,5]. + +Both have empty pinned row, current mask 5, one current connectivity block +{0,2}, five occupied sites, and no completed winding. The current 0-to-2 +cut gain is 0 versus -1. Append the same row [7] and close: + + [0,7,5,7] -> rank 0; + [0,13,5,7] -> rank 1. + +So even after introducing the pinned seam, plain labelled partitions plus +an existing-wrap flag still lose necessary information. The gain is not a +cosmetic field added only because the original first-row masks differed. + +## 5. Exact p-dependent local operator now exists + +Let tau(s,b) be the table successor for row mask b, k(b) its population, and +q_b(p)=p^k(b)*(1-p)^(4-k(b)). On the 509-class quotient define + + K(p)[s,t] = sum_{b: tau(s,b)=t} q_b(p), + b(p)[s] = sum_{b: initial(b)=s} q_b(p), + c[s] = close_rank(s)-1. + +Then for every p in [0,1] and every m>=2, + + M_(4,m)(p) = b(p)^T K(p)^(m-1) c. (A) + +K is row-stochastic and b is a probability vector. The sites in the initial +row are weighted once, each new row once, and no probability weight is +attached to the final virtual closure. Integer occupation-count polynomials +are propagated separately when exact Bernstein coefficients are requested. + +This is an explicit efficient-enough finite reference representation, not an +identification with Jacobsen's particular pTL sectors. In particular ordinary +Perron eigenvalue 1 of this stochastic representation must NOT be mistaken for +the observable survival eigenvalue: the matching observable can cancel a +stationary direction. The input vector, closure functional and visible modes +are part of the spectral question. + +## 6. Exact scalar spectrum at p=1/2 + +At p=1/2 write A=16*K, an integer matrix whose row sums are 16, and take b +to count first-row masks. Put + + s_m = 16^m M_(4,m)(1/2) = b^T A^(m-1) c. + +For a monic polynomial f let t_f(m) be the sum of m-th powers of its roots, +including multiplicities. The complete scalar identity is + + s_m = -2*2^m + (-1)^m - t_f2(m) + + 2*t_f3a(m) + t_f3b(m) + t_f5(m), (B) + +where + + f2(x) = x^2-15x+4, + f3a(x) = x^3-3x^2-4x+2, + f3b(x) = x^3-3x^2+2x-2, + f5(x) = x^5-11x^4-9x^3+3x^2-10x-2. + +Thus the scalar recurrence polynomial is + + P(x)=(x-2)(x+1) f2(x) f3a(x) f3b(x) f5(x), + +of degree 15. Its descending coefficients are + + [1,-33,353,-1301,697,4327,-5785,-551, + 5534,-8814,872,2716,-3992,1480,80,-64]. + +### Why this is an all-length identity, not a recurrence fitted to 30 points + +Rational Berlekamp-Massey exploration suggested a candidate. The deliverable +DOES NOT rely on that exploratory identification. It verifies the proposed +identity by integer finite-dimensional algebra: + +1. Form R=P(A)c exactly. It has 82 nonzero components. We explicitly do NOT + assert P(A)c=0 on the full state space. +2. Check b^T A^k R=0 for every k=0,...,508, using exact integers. Since A is + a 509-dimensional matrix, Cayley-Hamilton makes every subsequent scalar + moment a linear combination of these. The recurrence follows for all m. +3. Verify the trace-power expression (B) at 31 initial indices by Newton + identities. It satisfies the same recurrence, so the same all-m formula + follows. The physical statement is restricted to m>=2. +4. Compute the 15x15 Hankel determinant beginning at s_2, namely + det[s_(2+i+j)]_(0<=i,j<15). It is nonzero. No recurrence of order <15 can + represent the physical scalar tail. The exact integer is in the JSON. +5. Euclidean polynomial gcd gives gcd(P,P')=1 exactly. Therefore this scalar + observable at THIS p has no necessary m*lambda^m Jordan terms. Invisible + modes of the full matrix and other p values are not characterized by this. + +The certificate is in `width4-half-probability-spectrum.json`; its generator +is `scripts/width4_half_probability_spectrum.py`. The weights in (B) are +(-2,1,-1,2,1,1), not a two-mode truncation. No result here asserts all-p +factorization, equal leading weights at every p, or the width-uniform outer +limit. Repeated coefficients, negative/complex modes and closure cancellations +must be understood before any continuum naming. + +## 7. Executed independent checks + +All 74,640 configurations in 2x2,2x3,3x2,3x3,3x4,4x2,4x3,4x4 were independently +traversed as physical lifted graphs. At every configuration their rank agreed +with both the un-sheared boundary implementation and the shear-normalized one. +All per-occupation rank counts agreed with exact polynomial propagation. + +For 4x4 the rank totals are + + rank 0: 36559; rank 1: 19932; rank 2: 9045. + +The matching Bernstein integers are + + [-1,-16,-120,-560,-1812,-4272,-7448,-9424,-7874, + -2896,1720,2832,1660,560,120,16,1]. + +Additional DP-only 4x5 and 4x6 polynomials are stored; their rank totals are +(591681,332454,124441) and (9208783,5928144,1640289). Those are integer +sums over the certified states, NOT an independent 2^20/2^24 graph census +and NOT stochastic evidence. Count conservation holds coefficientwise. + +Seven local tests passed. All 23,168 certified state transitions were checked +by the second graph traversal; quotient congruence and pair-separating signatures +were checked exactly; the scalar certificate uses 509 exact annihilator +moments and a nonzero physical-tail Hankel determinant. The full repository +suite has NOT been run for these new files. No external machine was used. + +## 8. Review of received material and provenance + +Reviewed #707 head 0d249ff6700e0c8f9bc39bb0677f64a279b3c90e. Its width-three +script Git blob is 09b766a766205c17352bdda431d1640f19a78188; its note blob is +f5620c84e35ff149d1b500560e21fa2c3f112b0d. The supplied archive matches those +actual fetched bytes, and the seven received tests were rerun successfully. +The original all-length row proof remains valid; no defect was found in its +rank dictionary, trace-block decomposition or root-shift argument. +The actual CI API now reports run 34689086221 successful on this #707 head +(completed 2026-09-12T10:56:18Z). That is the received PR's CI, NOT CI for +this new patch, which has only local validation. +The reported independent 299,584-configuration Grok check is read from the +PR, not misreported as a full rerun in this session. + +#706 still has stale opening-body errors which its later reviewed files +already correct. This delivery does not overwrite those artifacts, reinterpret +the unresolved common-chart tests, or claim a new source/field identification. + +Primary background read: Newman-Ziff, PRE 64 (2001) 016706, +arXiv:cond-mat/0101295v2, section II.4, cluster-wrapping discussion: +https://arxiv.org/html/cond-mat/0101295 . Displacement-aware union/find for +wrapping is established technology; it is not claimed as newly invented here. +Jacobsen 2015 arXiv:1507.03027v1, sections 4 and 6.1: +https://arxiv.org/html/1507.03027v1 . Its particular spectral representation is +not equated with (A) without an explicit map. No broad novelty search was made. + +## 9. Consequence for #636 + +The current width-four state-sufficiency and bounded-integer-check task is +completed by this delivery. Do not commission a duplicate engine or a fresh +width-four census. The exact transition certificate is an input for the +remaining question, not another reason to broaden a numerical size ladder. + +One useful next deliverable is a parameter-dependent observable reduction of +(A) over Q(p): identify the minimal visible denominator and closure weights, +separately for P0, P2 and P2-P0, with exact specialization back to (B). +A conjectured rational function must have an operator/Krylov identity or an +explicit sufficient polynomial-degree bound, not only interpolation at sample p. +This would decide whether the equal-leading-weight pattern extends beyond the +width-two/three cases, or where a genuine closure/prefactor exception appears. +Use existing #636 for that bounded proof/algebra question; no new ticket, +GPU, width scan, or comparison with p_c is needed. Width-uniform bounds and +#275's candidate-specific original-U forward maps remain separate problems. diff --git a/notes/width4-review-control-20260912-zh.md b/notes/width4-review-control-20260912-zh.md new file mode 100644 index 00000000..14a4d3a1 --- /dev/null +++ b/notes/width4-review-control-20260912-zh.md @@ -0,0 +1,55 @@ +# 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。 diff --git a/results/research-control-20260912/width4-half-probability-spectrum.json b/results/research-control-20260912/width4-half-probability-spectrum.json new file mode 100644 index 00000000..153bd1c2 --- /dev/null +++ b/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" + ] +} diff --git a/results/research-control-20260912/width4-rank-closure-certificate.json b/results/research-control-20260912/width4-rank-closure-certificate.json new file mode 100644 index 00000000..50b7297d --- /dev/null +++ b/results/research-control-20260912/width4-rank-closure-certificate.json @@ -0,0 +1,3427 @@ +{ + "schema": "matching-one.width4-rank-closure.v1", + "scope": "square-site NN, periods (4,0),(0,m), m>=2; rank only", + "state_order": "BFS: initial masks 0..15, successors 0..15", + "gain_convention": "oriented horizontal cut crossing; internal vertical gain zero; closure seam gain (0,1)", + "states": [ + [0,[[-1,0],[-1,0],[-1,0],[-1,0],[-1,0],[-1,0],[-1,0],[-1,0]]], + [0,[[0,0],[-1,0],[-1,0],[-1,0],[0,0],[-1,0],[-1,0],[-1,0]]], + [0,[[-1,0],[1,0],[-1,0],[-1,0],[-1,0],[1,0],[-1,0],[-1,0]]], + [0,[[0,0],[0,0],[-1,0],[-1,0],[0,0],[0,0],[-1,0],[-1,0]]], + [0,[[-1,0],[-1,0],[2,0],[-1,0],[-1,0],[-1,0],[2,0],[-1,0]]], + 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0, + 0, + 0, + 0, + 0 + ], + [ + 0, + 0, + 0, + 0, + 6, + 120, + 1168, + 7296, + 32643, + 110712, + 294264, + 622680, + 1042250, + 1335456, + 1243260, + 793632, + 334035, + 91776, + 16764, + 1968, + 114, + 0, + 0, + 0, + 0 + ], + [ + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 24, + 552, + 5592, + 32544, + 117600, + 270348, + 400712, + 387864, + 253848, + 117832, + 40536, + 10512, + 2024, + 276, + 24, + 1 + ] + ], + "rank_totals": [ + 9208783, + 5928144, + 1640289 + ], + "prefix_state_counts": [ + 16, + 256, + 888, + 1408, + 1448, + 1448 + ], + "new_enumeration": false + } + }, + "compulsory_witnesses": [ + { + "rows": [ + 13, + 5, + 13, + 0 + ], + "physical_rank": 0, + "compressed_rank": 0 + }, + { + "rows": [ + 7, + 5, + 13, + 0 + ], + "physical_rank": 1, + "compressed_rank": 1 + }, + { + "rows": [ + 1, + 5, + 4, + 5 + ], + "physical_rank": 0, + "compressed_rank": 0 + }, + { + "rows": [ + 11, + 14 + ], + "physical_rank": 2, + "compressed_rank": 2 + }, + { + "rows": [ + 0, + 7, + 5, + 7 + ], + "physical_rank": 0, + "compressed_rank": 0 + }, + { + "rows": [ + 0, + 13, + 5, + 7 + ], + "physical_rank": 1, + "compressed_rank": 1 + } + ], + "not_claimed": [ + "minimal linear realization", + "pTL intertwiner", + "continuum state count", + "width-independent bounds", + "new infinite-lattice critical probability" + ] +} diff --git a/scripts/lifted_boundary_rank.py b/scripts/lifted_boundary_rank.py new file mode 100644 index 00000000..c546ab4a --- /dev/null +++ b/scripts/lifted_boundary_rank.py @@ -0,0 +1,278 @@ +#!/usr/bin/env python3 +"""Exact rank-preserving row composition with a pinned torus seam. + +Boundary labels 0..w-1 pin row zero; w..2w-1 are the live row. Gains +count crossings of the horizontal cut, NOT Euclidean displacements. All +arithmetic used by transitions and rank tests is integer. This is a small +reference algebra, not an optimized pTL implementation. +""" +from __future__ import annotations +from functools import lru_cache +from math import comb +from typing import NamedTuple + + +class State(NamedTuple): + horizontal: bool + entries: tuple[tuple[int, int], ...] # canonical component root, relative gain + + +class GainDSU: + def __init__(self, size: int, horizontal: bool = False): + self.parent = list(range(size)) + self.gain = [0] * size + self.horizontal = horizontal + + def find(self, vertex: int) -> tuple[int, int]: + p = self.parent[vertex] + if p == vertex: + return vertex, 0 + root, gain = self.find(p) + self.gain[vertex] += gain + self.parent[vertex] = root + return root, self.gain[vertex] + + def edge(self, u: int, v: int, gain: int) -> None: + ru, du = self.find(u) + rv, dv = self.find(v) + if ru == rv: + self.horizontal |= du + gain != dv + else: + self.parent[rv] = ru + self.gain[rv] = du + gain - dv + + +def _validate(width: int, mask: int) -> None: + if not isinstance(width, int) or width < 2: + raise ValueError('integer width >= 2 required') + if not isinstance(mask, int) or not 0 <= mask < (1 << width): + raise ValueError('row mask outside width') + + +def _row(dsu: GainDSU, offset: int, width: int, mask: int) -> None: + # Distinct periodic edges survive w=2: 0->1 with gain 0; 1->0 with gain 1. + for x in range(width): + y = (x + 1) % width + if mask >> x & 1 and mask >> y & 1: + dsu.edge(offset+x, offset+y, int(x == width-1)) + + +def _canonical(dsu: GainDSU, keep: list[tuple[int, int]], width: int, + shear: bool) -> State: + entries = [(-1, 0)] * (2*width) + groups: dict[int, list[tuple[int, int]]] = {} + for old, new in keep: + r, d = dsu.find(old) + groups.setdefault(r, []).append((new, d)) + for group in groups.values(): + root, d0 = min(group) + for v, d in group: + entries[v] = (root, 0 if dsu.horizontal else d-d0) + if shear and not dsu.horizontal: + # A global Dehn shear: add one common integer to ALL live endpoints. + # This changes only the horizontal part of eventual vertical cycles, + # via (hx,hy)->(hx+c*hy,hy), so total rational rank is preserved. + mixed = [(j, r, d) for j, (r, d) in enumerate(entries) + if j >= width and 0 <= r < width] + if mixed: + _, _, d = min(mixed) + c = -d + entries = [(r, g + c*(int(j >= width)-int(r >= width))) + if r >= 0 else (r, g) + for j, (r, g) in enumerate(entries)] + return State(dsu.horizontal, tuple(entries)) + + +def initial_state(width: int, mask: int, *, shear: bool = True) -> State: + _validate(width, mask) + dsu = GainDSU(2*width) + keep = [] + for x in range(width): + if mask >> x & 1: + dsu.edge(x, width+x, 0) # virtual alias; no site is counted twice + keep.extend(((x, x), (width+x, width+x))) + _row(dsu, width, width, mask) + return _canonical(dsu, keep, width, shear) + + +@lru_cache(maxsize=None) +def advance(state: State, mask: int, shear: bool = True) -> State: + width = len(state.entries)//2 + _validate(width, mask) + dsu = GainDSU(3*width, state.horizontal) + for v, (root, gain) in enumerate(state.entries): + if root >= 0 and v != root: + dsu.edge(root, v, gain) + keep = [(x, x) for x in range(width) if state.entries[x][0] >= 0] + for x in range(width): + if mask >> x & 1: + keep.append((2*width+x, width+x)) + if state.entries[width+x][0] >= 0: + dsu.edge(width+x, 2*width+x, 0) + _row(dsu, 2*width, width, mask) + return _canonical(dsu, keep, width, shear) + + +def close_rank(state: State) -> int: + """Close live row onto pinned first row by edges with vertical cut gain 1.""" + width = len(state.entries)//2 + adjacency: dict[int, list[tuple[int, int, int]]] = { + i: [] for i, (r, _) in enumerate(state.entries) if r >= 0} + def edge(u, v, x, y): + adjacency[u].append((v, x, y)) + adjacency[v].append((u, -x, -y)) + for v, (root, gain) in enumerate(state.entries): + if root >= 0 and v != root: + edge(root, v, gain, 0) + for x in range(width): + if x in adjacency and width+x in adjacency: + edge(width+x, x, 0, 1) + first = (1, 0) if state.horizontal else None + positions = {} + for root in adjacency: + if root in positions: + continue + positions[root] = (0, 0) + stack = [root] + while stack: + u = stack.pop() + xu, yu = positions[u] + for v, x, y in adjacency[u]: + proposed = (xu+x, yu+y) + if v not in positions: + positions[v] = proposed + stack.append(v) + else: + dx, dy = proposed[0]-positions[v][0], proposed[1]-positions[v][1] + if dx or dy: + if first is None: + first = (dx, dy) + elif first[0]*dy-first[1]*dx: + return 2 + return int(first is not None) + + +def state_for_rows(width: int, rows: tuple[int, ...] | list[int], *, + shear: bool = True) -> State: + if not rows: + raise ValueError('at least one row required') + state = initial_state(width, rows[0], shear=shear) + for mask in rows[1:]: + state = advance(state, mask, shear) + return state + + +def torus_rank(width: int, rows, *, shear: bool = True) -> int: + if len(rows) < 2: + raise ValueError('length >= 2 required to retain both interfaces') + return close_rank(state_for_rows(width, rows, shear=shear)) + + +def reachable_states(width: int, *, max_states: int = 100000, shear: bool = True): + """Exhaust all row transitions; raises rather than calling a cutoff complete.""" + initial = [initial_state(width, mask, shear=shear) for mask in range(1<= max_states: + raise RuntimeError('state cap reached; closure NOT established') + index[new] = len(states) + states.append(new) + row.append(index[new]) + transitions.append(row) + cursor += 1 + return states, transitions, [index[s] for s in initial] + + +def rank_count_polynomials(width: int, length: int, *, shear: bool = True): + """Exact occupation-count distributions, no p-grid and no random sampling.""" + if length < 2: + raise ValueError('length >= 2 required') + current = {} + for mask in range(1<> x & 1) or v in positions: + continue + positions[v] = (0, 0) + stack = [v] + while stack: + a = stack.pop() + ax, ay = a%width, a//width + px, py = positions[a] + for dx, dy in ((-1,0), (1,0), (0,-1), (0,1)): + bx, by = (ax+dx)%width, (ay+dy)%height + if not (rows[by] >> bx & 1): + continue + b = by*width+bx + q = (px+dx, py+dy) + if b not in positions: + positions[b] = q + stack.append(b) + else: + gx, gy = q[0]-positions[b][0], q[1]-positions[b][1] + if gx%width or gy%height: + raise AssertionError('cycle displacement not a period') + if gx or gy: + if first is None: + first = (gx, gy) + elif first[0]*gy-first[1]*gx: + return 2 + return int(first is not None) + + +def independent_advance(state, mask): + """Graph traversal of a boundary star, independent of weighted union/find.""" + w = len(state.entries)//2 + active = {i for i,(r,_) in enumerate(state.entries) if r>=0} + active.update(2*w+x for x in range(w) if mask>>x & 1) + adj = {i:[] for i in active} + def edge(u,v,g): + adj[u].append((v,g));adj[v].append((u,-g)) + for v,(r,g) in enumerate(state.entries): + if r>=0 and v!=r:edge(r,v,g) + for x in range(w): + if mask>>x & 1: + if state.entries[w+x][0]>=0:edge(w+x,2*w+x,0) + y=(x+1)%w + if mask>>y & 1:edge(2*w+x,2*w+y,int(x==w-1)) + seen={};groups=[];horizontal=state.horizontal + for root in sorted(active): + if root in seen:continue + seen[root]=0;stack=[root];group=[] + while stack: + u=stack.pop() + if u=2*w:group.append((u-w,seen[u])) + for v,g in adj[u]: + proposed=seen[u]+g + if v not in seen:seen[v]=proposed;stack.append(v) + elif proposed!=seen[v]:horizontal=True + if group:groups.append(group) + entries=[(-1,0)]*(2*w) + for group in groups: + r,d0=min(group) + for v,d in group:entries[v]=(r,0 if horizontal else d-d0) + mixed=[(j,d) for j,(r,d) in enumerate(entries) if j>=w and 0<=r=w)-int(r>=w))) if r>=0 else (r,d) + for j,(r,d) in enumerate(entries)] + return State(horizontal,tuple(entries)) + + +def minimality_signature(q, rank, state): + """All suffixes of lengths 0,1,2 in fixed lexicographic order.""" + return (rank[state], *[rank[x] for x in q[state]], + *[rank[q[x][b]] for x in q[state] for b in range(len(q[0]))]) + + +def certificate(): + states, transitions, initial = reachable_states(4) + labels, reps, quotient, outputs, refinements = minimize_rank_machine(states, transitions) + for i,row in enumerate(transitions): + for mask,target in enumerate(row): + if independent_advance(states[i],mask) != states[target]: + raise AssertionError('DFS and DSU disagree on a certified transition') + # Independently express separation without reusing the refinement step. + signatures = [minimality_signature(quotient, outputs, s) for s in range(len(quotient))] + if len(set(signatures)) != len(quotient): + raise AssertionError('the announced two-row distinguishing depth fails') + # Every state must occur at a physically valid length >=2, not only in + # the convenient aliased first-row initializer. + reached = {transitions[s][mask] for s in initial for mask in range(16)} + todo = deque(reached) + while todo: + s = todo.popleft() + for t in transitions[s]: + if t not in reached: + reached.add(t); todo.append(t) + if reached != set(range(len(states))): + raise AssertionError('one state is not reachable at length >=2') + return { + 'schema': 'matching-one.width4-rank-closure.v1', + 'scope': 'square-site NN, periods (4,0),(0,m), m>=2; rank only', + 'state_order': 'BFS: initial masks 0..15, successors 0..15', + 'gain_convention': 'oriented horizontal cut crossing; internal vertical gain zero; closure seam gain (0,1)', + 'states': [[int(s.horizontal), [list(x) for x in s.entries]] for s in states], + 'transitions': transitions, + 'initial': initial, + 'rank_output': [close_rank(s) for s in states], + 'quotient_map': labels, + 'quotient_representatives': reps, + 'quotient_transitions': quotient, + 'quotient_rank_output': outputs, + 'quotient_initial': [labels[s] for s in initial], + 'refinement_counts': refinements, + 'distinguishing_suffix_length_bound': 2, + 'all_states_reachable_after_at_least_two_rows': True, + 'all_transitions_checked_by_independent_graph_traversal': True, + } + + +def report(): + start = time.perf_counter() + cert = certificate() + print('complete closure',len(cert['states']),'quotient',len(cert['quotient_transitions']), flush=True) + checks = {} + for width, height in ((2,2),(2,3),(3,2),(3,3),(3,4),(4,2),(4,3),(4,4)): + count = 0 + tally = [[0]*(width*height+1) for _ in range(3)] + for rows in product(range(1<=len(b): + c=a[0]/b[0] + for i,x in enumerate(b):a[i]-=c*x + trim(a) + return a + a=list(map(Fraction,coefficients));n=len(a)-1 + b=[(n-i)*x for i,x in enumerate(a[:-1])] + while b:a,b=b,remainder(a,b) + return len(a)==1 + + +def report(): + states, rows, initials = reachable_states(4) + labels, reps, A, output, refinements = minimize_rank_machine(states,rows) + b = Counter(labels[s] for s in initials) + c = [r-1 for r in output] + size = len(A) + def times(v): + return [sum(v[j] for j in row) for row in A] + def read(v): + return sum(weight*v[i] for i,weight in b.items()) + sequence = [] + v = c[:] + for _ in range(2*15+1): + sequence.append(read(v)); v = times(v) + poly = [1] + for _, factor in FACTORS: + poly = polynomial_product(poly,factor) + if not squarefree_monic(poly): + raise AssertionError('observable annihilator is not squarefree') + if poly != RECURRENCE: + raise AssertionError('factor product != claimed recurrence') + powers = [c] + for _ in range(15): powers.append(times(powers[-1])) + remainder = [sum(RECURRENCE[i]*powers[15-i][j] for i in range(16)) + for j in range(size)] + nonzeros = sum(x!=0 for x in remainder) + # The polynomial need NOT annihilate c. Only its scalar observable image. + # For a size-d matrix, d consecutive zero moments imply all via C-H. + for k in range(size): + if read(remainder): + raise AssertionError(f'nonzero annihilator moment at {k}') + remainder = times(remainder) + moments = [(weight,power_sums(factor,31)) for weight,factor in FACTORS] + for m in range(1,32): + if sum(w*values[m] for w,values in moments) != sequence[m-1]: + raise AssertionError('trace-power identity initial values mismatch') + # Start at physical length m=2, not the convenient aliased m=1 term. + hankel = [[sequence[i+j+1] for j in range(15)] for i in range(15)] + det = integer_determinant(hankel) + if not det: raise AssertionError('order-15 minimality not established') + return { + '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':FACTORS, + 'minimal_scalar_recurrence_order':15, + 'recurrence_descending':RECURRENCE, + 'recurrence_is_squarefree_over_Q':True, + 'hankel_determinant_15_start_m2':det, + 'finite_dimensional_certificate_dimension':size, + 'annihilator_zero_scalar_moments_verified':size, + 'pA_c_nonzero_entries':nonzeros, + 'initial_trace_power_identities_verified':31, + 'first_31_s_m':sequence, + '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'], + } + + +if __name__=='__main__': + ap=argparse.ArgumentParser();ap.add_argument('--out',type=Path);args=ap.parse_args() + data=report();text=json.dumps(data,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_lifted_boundary_rank.py b/tests/test_lifted_boundary_rank.py new file mode 100644 index 00000000..929fdec1 --- /dev/null +++ b/tests/test_lifted_boundary_rank.py @@ -0,0 +1,69 @@ +"""Concrete topology/count errors these controls prevent; no stochastic tests.""" +import sys +from pathlib import Path +import unittest +sys.path.insert(0,str(Path(__file__).resolve().parents[1]/'scripts')) +from lifted_boundary_rank import (State, initial_state, advance, close_rank, torus_rank, + state_for_rows, reachable_states, rank_count_polynomials, minimize_rank_machine) +from verify_width4_rank_closure import graph_rank, minimality_signature +from width4_half_probability_spectrum import report as spectral_report + + +class BoundaryRankTests(unittest.TestCase): + def test_parallel_periodic_edges_are_not_collapsed(self): + self.assertEqual(torus_rank(2,[3,0]),1) + self.assertEqual(torus_rank(2,[1,1]),1) + self.assertEqual(torus_rank(2,[3,3]),2) + + def test_old_partition_only_memory_collision(self): + a,b=state_for_rows(4,[13,5]),state_for_rows(4,[7,5]) + self.assertNotEqual(a,b) + self.assertEqual((torus_rank(4,[13,5,13,0]),torus_rank(4,[7,5,13,0])),(0,1)) + for rows in ([13,5,13,0],[7,5,13,0],[1,5,4,5],[11,14]): + self.assertEqual(torus_rank(4,rows),graph_rank(4,rows)) + + def test_forgotten_horizontal_cycle_remains_in_global_span(self): + state=state_for_rows(4,[0,15,0]) + self.assertTrue(state.horizontal) + self.assertTrue(all(r<0 for r,_ in state.entries)) + self.assertEqual(close_rank(state),1) + + def test_common_shear_preserves_all_short_future_ranks(self): + # Switch one representative by a deliberately large shear. The four + # occupancy labels are fixed; the lift integer is not a new site. + state=state_for_rows(4,[3,6,12],shear=False) + entries=tuple((r,g+17*(int(j>=4)-int(r>=4))) if r>=0 else (r,g) + for j,(r,g) in enumerate(state.entries)) + shifted=State(state.horizontal,entries) + self.assertNotEqual(state,shifted) + for a in range(16): + for b in range(16): + old=advance(advance(state,a,False),b,False) + new=advance(advance(shifted,a,False),b,False) + self.assertEqual(close_rank(old),close_rank(new)) + + def test_complete_row_closure_and_continuation_separation(self): + states,transition,initial=reachable_states(4) + labels,reps,q,rank,steps=minimize_rank_machine(states,transition) + self.assertEqual((len(states),sum(map(len,transition))),(1448,23168)) + self.assertEqual(steps,[3,164,509,509]) + self.assertEqual(len(set(minimality_signature(q,rank,i) for i in range(len(q)))),509) + + def test_exact_four_square_polynomial(self): + counts,_=rank_count_polynomials(4,4) + self.assertEqual(list(map(sum,counts)),[36559,19932,9045]) + self.assertEqual([b-a for a,b in zip(counts[0],counts[2])], + [-1,-16,-120,-560,-1812,-4272,-7448,-9424,-7874,-2896,1720,2832,1660,560,120,16,1]) + + def test_spectral_identity_has_a_physical_tail_certificate(self): + data=spectral_report() + self.assertEqual(data['minimal_scalar_recurrence_order'],15) + self.assertNotEqual(data['hankel_determinant_15_start_m2'],0) + self.assertEqual(data['annihilator_zero_scalar_moments_verified'],509) + # The scalar recurrence is NOT a claim p(A)c=0 as a whole vector. + self.assertEqual(data['pA_c_nonzero_entries'],82) + counts,_=rank_count_polynomials(4,4) + self.assertEqual(data['first_31_s_m'][3],sum(counts[2])-sum(counts[0])) + + +if __name__=='__main__':unittest.main()