From 83e011a7ceeab23b9ae5eab741f41ea15c983459 Mon Sep 17 00:00:00 2001 From: Light Chain Date: Sat, 12 Sep 2026 20:09:29 +0800 Subject: [PATCH] P398 double-pulse: odd sector fully visible to two separated insertions Existing sources/readouts; intervention program only. H=join@0-join@(w-2) (twice archived H_odd). First-order even-to-even response vanishes; the delay kernel S H exp(tau G) H F has exact minimal order 4/16 at widths 4/5, exhausting the odd subspaces. Baseline orders 10/26; full G/H words 14/42; source-minus-uniform contrasts 13/41. Width-4 wrapped-pair/wrap scalar: Khat(z)=2(z^2+11z+27)/[(z^2+10z+23)(z^2+11z+26)], Hankel det=-16, so zero delay can miss a four-mode delay response. Parity does not fix the quadratic sign. P398 calibration, not percolation. Additive. Do not merge. --- notes/p398-crossline-review-20260912-zh.md | 110 + notes/p398-double-pulse-handoff-20260912.md | 46 + ...398-double-pulse-observability-20260912.md | 293 + .../p398-double-pulse-bareiss-check.json | 76 + .../p398-double-pulse-exact.json | 13811 ++++++++++++++++ .../p398-double-pulse-numerical-control.json | 33 + .../p398-double-pulse-summary.json | 72 + scripts/p398_double_pulse_exact.py | 328 + .../p398_double_pulse_numerical_control.py | 76 + .../verify_p398_double_pulse_certificate.py | 118 + tests/test_p398_double_pulse_exact.py | 97 + 11 files changed, 15060 insertions(+) create mode 100644 notes/p398-crossline-review-20260912-zh.md create mode 100644 notes/p398-double-pulse-handoff-20260912.md create mode 100644 notes/p398-double-pulse-observability-20260912.md create mode 100644 results/research-control-20260912/p398-double-pulse-bareiss-check.json create mode 100644 results/research-control-20260912/p398-double-pulse-exact.json create mode 100644 results/research-control-20260912/p398-double-pulse-numerical-control.json create mode 100644 results/research-control-20260912/p398-double-pulse-summary.json create mode 100644 scripts/p398_double_pulse_exact.py create mode 100644 scripts/p398_double_pulse_numerical_control.py create mode 100644 scripts/verify_p398_double_pulse_certificate.py create mode 100644 tests/test_p398_double_pulse_exact.py diff --git a/notes/p398-crossline-review-20260912-zh.md b/notes/p398-crossline-review-20260912-zh.md new file mode 100644 index 00000000..c2624b24 --- /dev/null +++ b/notes/p398-crossline-review-20260912-zh.md @@ -0,0 +1,110 @@ +# Matching One 横向推进：从被动商到主动可观测性 + +日期：2026-09-12。交付为新增代码、证明与精确证书；未修改远端分支，未合并现有 PR，未修改 STATUS/ROADMAP，未进行随机生产。 + +## 一、这轮实际阅读与取舍 + +最新的 #708（head `f782061c1a592ed2f9fd0e9dabaa45f0e54bc4e7`）已提交宽度四的 509 个确定性续接类与固定 p=1/2 的 15 阶标量谱结果。读取其提交说明及 CI 状态：运行 `34691460988` 的结论为 success。本轮不把它当作新交付，不重做其状态枚举，也不把其 CI 当成本补丁的 CI。 + +横向阅读的重点转到 #594 的干预状态问题、#598/#600 的反射商、#601/#610 的二阶响应与先例，以及 #644/#697 的 P5 解释。实际核对了 main@`eb89e942` 的生成元、原有源、原有三个主读出、join/detach 操作和反射说明。另读取 #700 的 P4 文献/规模说明，并重新检查 #275 的 original-U 验收条件。没有声称通读全库或重审投稿 PDF。 + +选择标准是：不再为了延长宽度表而延长宽度表。P398 已有一阶选择律和二阶幅度曲线，却尚未回答“第二次干预能读到多少被第一次商掉的动力学”。这是一个可以在现成小对象上精确回答、并能改变状态解释的问题。 + +## 二、已直接完成的新结果 + +源和读出完全沿用旧模型；改变的是允许的干预时间程序。使用 + + H=join@0-join@(w-2)，G+epsilon H 在 |epsilon|<=1 内保持非负跃迁率。 + +注意 H 是旧 H_odd 的两倍，二阶核相差四倍。报告和代码均显式说明。 + +反射偶源与偶读出之间，一次奇扰动严格不可见。但两次扰动的延迟核 + + K(tau)=S H exp(tau G) H F + +经过奇子空间传播。我们检验其可达、可观测子空间，而不是从“偶×奇×奇×偶”允许非零推断它必然充分。 + +精确结果： + +| 宽度 | 微观状态数 | 无干预原始响应最小线性阶数 | 两次插入延迟核阶数 | 任意 G/H 时间程序的最小齐次双线性阶数 | 源减 uniform 后的对应阶数 | +|---|---:|---:|---:|---:|---:| +| 4 | 14 | 10 | 4 | 14 | 13 | +| 5 | 42 | 26 | 16 | 42 | 41 | + +两次插入在这两个有限实例中读出了全部奇子空间。原始受控响应保留常数模；减去 uniform 的源对比不保留该模，所以同时给出 13/41。这里的“最小”限于共同的齐次线性/双线性输入输出表示，不是任意非线性表示、最小正实现或场数。 + +这也不反驳此前给定容差下的近似 rank-6 输运。精确语言改变后的阶数，与有限噪声、有限时间、给定容差下的有效阶数，是不同问题。 + +### 一个原有标量观测就足够 + +宽度四，用旧 delta_wrapped_pair 源和旧 wrap 读出： + + Khat(z)=2(z^2+11z+27)/[(z^2+10z+23)(z^2+11z+26)]。 + +其 Taylor 矩从 0 开始为 + + 0,2,-20,156,-1122,7822,-53932,371172,... + +4×4 Hankel 行列式为 -16。因此同一个标量延迟曲线需要四阶线性表示，而 K(0)=0。只测零延迟的二阶数值，会把这个探针误读为没有信息。 + +宽度五，旧 all-singletons 源与 wrap 读出单独即可达到16阶，证书中保存了完整整数 Hankel 子式。并不是所有源都可以；single-block 初态没有初始 join 响应，提供了内置零对照。 + +### 二阶响应没有由奇偶性固定的符号 + +在宽度四，all-singletons→blocks 的 K(0)=2。all-singletons→wrap 的 K(0)=0、K'(0)=-2；wrapped-pair→wrap 的 K(0)=0、K'(0)=+2。故对足够小的正延迟，后两者符号相反。选择律只决定哪些阶数被禁止；“未禁止”也不等于“必定非零”。 + +## 三、不是使用一个负速率的虚构脉冲 + +H 自己不是 Markov 生成元。实际协议使用两段非负速率窗口： + + Y_delta(e1,e2;tau) + =S exp(delta(G+e1 H)) exp(tau G) exp(delta(G+e2 H))F。 + +交叉导数等于 S J_delta exp(tau G) J_delta F。只有在除以 delta² 并令 delta→0 后，才得到上面的理想核。证明还说明非零 Hankel 子式会在所有足够小的正脉宽下保持非零；可观测性不是只存在于零时长极限。 + +另一个独立矩阵指数对照已执行。在 tau=1/4、delta=1/32，有限窗口的交叉导数为 0.0001171819768。四符号差分在幅度 1/8、1/16、1/32 下的误差分别约为 6.27e-10、1.57e-10、3.92e-11，显示预期的二次误差下降。 + +但有限脉宽结果除以 delta² 约为0.11999434，而理想短脉冲核为0.14768527。这个差异没有被掩盖。计算为45位精度数值对照，不是严格区间证明，不是新采样。 + +精确满秩也不等于所有模都能在实际噪声下便宜测出。尚未进行样本量、条件数/辨识误差或设备成本承诺。 + +## 四、证书怎样支撑“精确” + +主计算只使用 Python 标准库、整数和 Fraction。模素数选择候选基后，保存了真正的整数 Hankel 子式；模行列式非零给出有理数域的下界。上界分别来自奇偶子空间维数、整个微观维数，以及源对比看不到的常数方向。没有拿模秩稳定代替有理数上界。 + +进一步用独立的分数自由 Bareiss 消元核对全部10个行列式，每一步整除都检查。另一个可执行验证器从所存 G/H 词重建子式，再做整数消元。宽度四标量 -16 还由 Fraction 消元单独确认。 + +7项本地数学测试通过。完整精确结果、Bareiss 验证及有限脉宽对照都可由交付脚本复现。未运行全仓库 CI；#708 的通过记录不是本补丁的通过记录。 + +## 五、两处横向解释需要修正 + +### 1. P5：没有检索到某个名称，不构成新颖性证明 + +本轮有界读取了： + +- Petreczky–Wisniewski–Leth, arXiv:1605.04414v1，§2.1–2.2/§3：双线性词系数、可达性、可观测性与最小实现；第一手正文已读。 +- Lucarini, arXiv:2502.07908v1，§II.2，(13)–(19)：Markov 链时变二阶响应及模对展开；第一手正文已读。出版社 DOI 直接打开403，未以此伪称全出版社版本已审读。 +- Müller–Basu–Sollich–Krüger, Phys. Rev. Research 2,043123 (2020)：读取出版社摘要及部分摘录，存在二阶粗粒化响应的明确先例；未将其平衡假设移植到 P398。 + +因此应以具体 P398 的可观测性证书、原有标量探针和干预语言改变为本次结果，而不是宣传通用选择律或响应公式为新发明。也未进行足以裁决本次模型特定结果新颖性的全面文献检索。 + +### 2. P4/#700：N 是站点数，不是线性尺度 + +PR #700 的规模讨论中，N=145–425 sites 与文献线性尺寸 L 的比较混在一起；其“exact-enumeration branch L<=128”也超出了本身表格的“exact L=3–11，MC 到128”分类。对当前原始高斯方形商： + + N=145 -> ell=sqrt(145)≈12.04； + N=425 -> ell=sqrt(425)≈20.62； + N=725 -> ell=sqrt(725)≈26.93。 + +L=512 的正方形有262144个站点，不是512个。需要修正单位及方法分类，而不是将此转成一条“审稿人必定要求 L=512”的采购定律。小有限尺寸不自动推翻已经冻结的有限尺寸信号，大副本数也不能代替系统误差控制。 + +本轮没有重新裁定期刊定位，没有按这段文字开大尺寸任务。 + +## 六、队列建议：交付结果，不制造新待办 + +1. 在既有 #594/#598/#610 记录本次结果：被动偶商对原任务成立；允许原有奇方向的两次分时扰动后，奇动力学可以成为可见状态。不要再派发同样的 w4/w5 二阶可达性任务，也不默认扩展到 w8/w10。 +2. #636 的 width-four Q(p) 可见谱问题继续保留，但不是当前唯一研究方向。本轮不重写已经提交的 #708，也不拿它的509/15去比较本次14/42；系统、语言和观测量都不同。 +3. #275 仍需要两个实际候选到同一 original-U 的前向映射。本次 P398 控制结果没有补出那两列，不能用新增干预来事后挽救旧合同。若将来物理候选确实只在某个受控响应上不同，先给该响应的候选预测与可观测性，再谈采样。 +4. 无大规模检索或大计算任务已被证明必要，因此没有新开计算单。本次直接提供可提交补丁及一段既有 issue 更新文字。 + +来源：上述 main 固定版本；#708 head `f782061c`；#700 head `925e98bd825eb73588cf0fa3f111cf146695488a` 的 `notes/venue-retrieval-p4-20260909.md`；以及数学说明末尾列出的第一手文献。所有历史数据与既有结果保持不变。 diff --git a/notes/p398-double-pulse-handoff-20260912.md b/notes/p398-double-pulse-handoff-20260912.md new file mode 100644 index 00000000..bd83f5aa --- /dev/null +++ b/notes/p398-double-pulse-handoff-20260912.md @@ -0,0 +1,46 @@ +# Existing-issue update text: completed P398 double-pulse analysis + +Use in #594 (with references to #598/#600/#610), not as a duplicate new task. +This text has not been posted to GitHub by this session. + +--- + +Completed a bounded exact analysis using the ORIGINAL four sources and three +primary readouts of P398, widths 4/5. The change is the allowed intervention +time program, not a new source/readout or a reinterpretation of the old rank-6 +approximation. + +With H=join@0-join@(w-2) (twice archived H_odd), baseline G commutes with R, +H is odd, and G+epsilon H is physical for |epsilon|<=1. First-order even-to-even +response vanishes, as already known. The new result is that the two-insertion +kernel S H exp(tau G) H F has exact minimal delay order 4/16, exhausting the +odd subspaces. Finite positive pulse durations preserve these ranks for all +sufficiently small durations by analyticity of a nonzero Hankel minor. + +Exact raw baseline order is 10/26. The full noncommutative G/H word response +has minimum homogeneous bilinear order 14/42; original source-minus-uniform +contrasts give 13/41 because the constant mode is invisible. Every rank has an +explicit nonzero integer Hankel minor and an independent mathematical upper +bound; all ten determinant witnesses were rechecked with integer Bareiss +elimination, including reconstruction of their G/H words. + +At width 4, the old wrapped-pair source and wrap readout already give + + Khat(z)=2(z^2+11z+27)/[(z^2+10z+23)(z^2+11z+26)], + K(0)=0, K'(0)=2, Hankel determinant=-16. + +Thus zero delay can miss a fully observable four-mode delay response. Exact +opposite-sign examples in the old dictionary also show that parity fixes no +universal sign of the quadratic term. + +Files: `notes/p398-double-pulse-observability-20260912.md`, the exact generator/ +certificate reader and independent verifier, a finite-window numerical control, +and additive JSON results. Seven local mathematical tests pass. No new MC, +full-repository CI, large widths, GPU, physical p_c or continuum claim. Generic +response and bilinear realization formulas have primary prior art; the result +is the finite P398 observability certificate, not a claim that those tools are new. + +This completes the small second-order accessibility question. No default width +extension is requested. A noise-limited identification task, if later justified, +needs an explicit alternative and error budget; exact full rank alone is not a +sample-size plan. #275's original-U candidate-map requirement remains separate. diff --git a/notes/p398-double-pulse-observability-20260912.md b/notes/p398-double-pulse-observability-20260912.md new file mode 100644 index 00000000..e4f54027 --- /dev/null +++ b/notes/p398-double-pulse-observability-20260912.md @@ -0,0 +1,293 @@ +# P398: a baseline-invisible odd sector is fully visible to two separated insertions + +Date: 2026-09-12. Completed bounded exact analysis at widths 4 and 5. +Related existing channels: #594, #598, #600, #601, #610, #644. No new production, +new width campaign or generic literature ticket is requested. This is P398's +calibration process, **not square-site percolation**. No mathematical novelty +claim is made for parity, response expansions or bilinear realization theory. + +## 1. The question changed; the original readout dictionary did not + +The archived `notes/p398-reflection-parity-20260906.md` already establishes the +reflection-even baseline quotient, vanishing first-order response to an odd +perturbation, and an observed quadratic perturbation-size ladder. We do not +claim those again. The remaining question answered here is different: + +> Which dynamics hidden at baseline can be recovered from the *delay dependence* +> of two odd insertions, with the original sources and readouts unchanged? + +The distinction is experimentally material: measuring one small quadratic +coefficient is not measuring the dynamics between two interventions. + +We read the implementation at main commit +`eb89e9422791d9e3c3a78f0e65d56912b815a7bd`: + +- `scripts/p398_intervention_transport.py`, blob + `ac13194751b8ba727a45499900e3f12ce1b90461`: generator, original sources/readouts; +- `scripts/planar_state_operations.py`, blob + `c5f57dcb8606dbb62306cde839edefae680f863d`: cyclic join and point detach; +- `scripts/p398_reflection_parity.py` and the archived parity note: normalization + and reflection convention. + +The new standalone standard-library script independently implements these rules. +It is not a byte-for-byte import of the entire original module, nor a full +repository checkout/test execution. + +## 2. Fixed finite model and physical perturbation + +States are noncrossing partitions of w cyclically labelled points, encoded as +canonical restricted growth strings. For each point j let J_j be the row-oriented +join-move generator (merge j and j+1 modulo w), and D_j the detach generator. +Each is a deterministic move minus the identity; a no-op cancels exactly. + + G = sum_j (J_j + D_j), + R : j -> w-1-j, + H = J_0 - J_(w-2). + +Thus RGR=G and RHR=-H. The intervention is physical: G+epsilon H has join rates +1+epsilon and 1-epsilon at the two named sites, and rate one everywhere else. +It is a Markov generator for |epsilon|<=1, and is irreducible for |epsilon|<1 +at the two widths checked. H by itself is **not** a Markov generator. + +Our H is twice the archived `H_odd=(J_0-J_(w-2))/2`. Every two-insertion kernel +below is therefore four times its value in the archived normalization. Ranks +and pole locations are unchanged by this nonzero rescaling. + +F has the ORIGINAL three readout columns: block count, singleton count, +`wrap=1[state[0]==state[-1]]`. S has the ORIGINAL four probability-source rows: +delta all singletons, delta single block, delta wrapped pair, and the uniform +law. All are R-even. Exact computations store n*S, with the common denominator +n recorded, to avoid introducing a floating-point uniform source. + +## 3. General algebra: forbidden once, allowed twice + +Let P_+=(I+R)/2 and P_-=(I-R)/2. Since G commutes with R, it restricts to G_+ +and G_-. An odd H maps each parity to the other. Hence for all nonnegative +waiting times, + + S exp(t0 G) H exp(t1 G) F = 0. + +Inserting R on both sides proves this identity pointwise. More generally any +word with an odd number of H factors has zero even-to-even matrix element. +"Not forbidden" does NOT imply nonzero: parity neither guarantees that an +allowed response fires nor fixes its sign. + +Now define the matrix-valued delay kernel + + K(tau) = S H exp(tau G) H F + = C_- exp(tau G_-) B_-, + +where B_- is H F in odd coordinates and C_- is S H restricted to them. Its +Taylor coefficients are S H G^k H F. An exact linear realization of this delay +kernel has order equal to its reachable-and-observable odd subspace, NOT +necessarily the entire odd dimension. The equality is tested below rather +than assumed from the selection rule. + +### Relation to projected memory + +In parity coordinates write H_+-: E_- -> E_+ and H_-+: E_+ -> E_-. For a fixed +amplitude epsilon, wherever the resolvents exist, the exact Schur complement is + + P_+ (zI-G-epsilon H)^(-1) P_+ + = [zI-G_+ - epsilon^2 H_+-(zI-G_-)^(-1)H_-+]^(-1) + +on the even subspace. Eliminating the odd component from +x'=(G+u(t)H)x with x_-(0)=0 equivalently gives + + x_+'(t) = G_+ x_+(t) + + u(t) H_+- int_0^t exp((t-s)G_-) u(s) H_-+ x_+(s) ds. + +Thus the baseline quotient is exact when u=0, but the controlled even dynamics +has an odd-propagation memory kernel. No time-local even-only replacement is +asserted. This connects the archived parity and projected-memory questions +without conflating their particular numerical model reductions. + +## 4. Two actual rate pulses, not a signed fictitious propagator + +For duration delta>0 use two separately tunable physical windows: + + Y_delta(e1,e2;tau) + = S exp(delta(G+e1 H)) exp(tau G) exp(delta(G+e2 H)) F. + +All three factors are Markov semigroups for |e1|,|e2|<=1. Define + + J_delta = int_0^delta exp(sG) H exp((delta-s)G) ds. + +Then the mixed derivative at zero is exactly + + d_e1 d_e2 Y_delta(0,0;tau) + = S J_delta exp(tau G) J_delta F, + +and J_delta/delta -> H as delta->0. Only after division by delta^2 and this +short-duration limit is the result K(tau). Finite-duration corrections must +not be ignored or relabelled as sampling error. + +The four-sign contrast + + [Y(e,e)-Y(e,-e)-Y(-e,e)+Y(-e,-e)]/(4e^2) + +converges to the finite-window mixed derivative with O(e^2) error. Reflection +implies Y(e,e)=Y(-e,-e) and Y(e,-e)=Y(-e,e). These equalities do not require +that G be reversible, or that the source be its stationary law. + +**Finite-duration persistence.** J_delta is odd and analytic in delta. If a +d-by-d Hankel minor of K is nonzero, the same minor for the finite-window +kernel equals delta^(2d) times that nonzero minor plus O(delta^(2d+1)). Hence +the full d-mode order survives for every sufficiently small positive duration; +it is not confined to a physically unattainable zero-duration pulse. This is +an existence statement, not a noise budget or a numerical lower bound on the +allowed pulse duration. + +## 5. Executed exact results + +| width | microscopic n | unforced raw I/O order | two-insertion delay order | full G/H controlled word order | source-minus-uniform controlled order | +|---|---:|---:|---:|---:|---:| +| 4 | 14 | 10 | 4 | 14 | 13 | +| 5 | 42 | 26 | 16 | 42 | 41 | + +The unforced raw orders exhaust the even dimensions. The two-insertion orders +exhaust the odd dimensions. The fourth column concerns only K(tau); it is NOT +added to the baseline rank by assumption. The fifth column has its own complete +noncommutative Hankel certificate for all words in G and H. + +The order notion in the last two columns is the minimum dimension of a +**homogeneous linear/bilinear input-output realization**, allowing the declared +initial preparations. The original model supplies the upper bound n. For source +contrasts the constant function is invariant under G, killed by H, and invisible +to the contrasts, giving upper bound n-1. Explicit nonzero minors attain both +bounds. A model allowing an explicit affine/output offset need not count that +constant in the same way; hence both raw and contrasted results are reported. +No assertion about minimum nonlinear, positive, or microscopic state dimension +follows. All control derivatives are around zero inside the physical rate +interval; a formal word in H is not an independently executable negative-rate +transition. + +### A single ORIGINAL scalar observer sees all four odd modes at w=4 + +Take the original delta_wrapped_pair source and wrap readout. Its delay kernel +has the exact Laplace transform + + Khat(z) = 2(z^2+11z+27) / + [(z^2+10z+23)(z^2+11z+26)]. + +The first coefficients K^(k)(0) are + + 0, 2, -20, 156, -1122, 7822, -53932, 371172, -2561154. + +The 4x4 Hankel determinant from coefficient zero is exactly -16. The denominator +is the odd generator's characteristic polynomial, checked by integer +Faddeev-LeVerrier and exact annihilation. The numerator follows from these +initial moments. The nonzero Hankel determinant proves that no lower-order +constant linear realization reproduces the whole delay curve. + +In particular K(0)=0 while K'(0)=2: a coincident-pulse measurement can be zero +although the separated-pulse experiment contains four dynamic modes. This is +why delay is an information-bearing coordinate, not a plot embellishment. +At w=5 the original all-singletons source and wrap readout alone attain order +16, certified by a nonzero 16x16 integer Hankel determinant modulo the stated +prime. This does not claim that every one of the source/readout pairs does so. +The single-block source has no initial join response and is a built-in zero. + +### No universal sign of the quadratic effect + +Already at w=4, all-singletons -> blocks has K(0)=2. All-singletons -> wrap has +K(0)=0 and K'(0)=-2, whereas wrapped-pair -> wrap has K(0)=0 and K'(0)=+2. +Analyticity gives opposite signs at sufficiently small positive delay for the +last two. Thus reflection parity fixes allowed orders, not the sign of a +quadratic response. These are exact examples in the existing dictionary. + +## 6. Why the rank certificate is exact + +The output contains integer G,H, the reflected state permutation, the original +scaled sources/readouts, exact odd/even coordinates, and explicit Hankel minors. +For efficiency, candidate independent rows and columns are selected modulo +1000000007. Every printed minor is also stored as an INTEGER matrix and its +determinant modulo this prime is nonzero. Consequently it is nonzero over Q. + +Modular stabilization alone would NOT prove a rational upper bound. Here the +independent mathematical bounds are parity dimension, the microscopic n, and +the invisible constant. The script refuses a conclusion when a lower bound +does not attain the stated bound. No floating rank tolerance is used. + +All ten stored rank-witness minors were also independently recomputed by +fraction-free Bareiss elimination over the integers, with exact division at +every step. Every determinant is nonzero and its modular residue agrees. The +separate verifier reconstructs the selected word coefficients before checking +the minors. The w=4 scalar determinant is additionally checked by Fraction +elimination (-16). For each odd generator, integer characteristic +polynomial coefficients divide exactly at every Faddeev-LeVerrier step, the +terminal matrix is zero, and the scalar moment recurrence is checked. Seven +small mathematical tests execute these facts, including the quarter-strength +normalization under replacement H -> H/2 and the actual positive/negative +examples. The tests are not a claim of full repository CI. + +## 7. Executed physical-window numerical control + +`p398_double_pulse_numerical_control.py` uses mpmath at 45 decimal digits on the +w=4 scalar observer above. At tau=1/4 and delta=1/32 it compares actual matrix +exponentials against an independent 35-term Frechet-derivative power series. +The latter is a numerical convergence control, not a rigorous remainder bound. + +Finite-window mixed derivative: + + 0.00011718197680195303035423582807185137. + +Centered four-sign errors at e=1/8,1/16,1/32 are respectively + + 6.26516675e-10, 1.56628921e-10, 3.91572148e-11, + +approximately quartering when the amplitude halves, as the expansion predicts. +Reflection-related matrix-element pairs agree at the recorded precision. + +At this finite duration the mixed derivative divided by delta^2 is +0.1199943442, while the ideal short-pulse K(1/4) is 0.1476852668. This visible +difference is intentional: the executable protocol and its limiting formula +are distinct objects. Exact observability likewise is not a claim that noisy +finite-amplitude data can estimate all modes cheaply. No sample top-up is priced. + +## 8. Prior art and reading limits + +This is a finite model-specific certificate and experiment-language comparison, +not a discovery of nonlinear response or bilinear Hankel theory. + +1. Petreczky, Wisniewski and Leth, *Moment matching for bilinear systems with nice + selections*, arXiv:1605.04414v1; IFAC-PapersOnLine 49(18), 838-843 (2016), + DOI 10.1016/j.ifacol.2016.10.270. PRIMARY_TEXT_READ for sections 2.1-2.2 and + 3: homogeneous bilinear form, word coefficients, reachability/observability + and minimum realizations. https://arxiv.org/html/1605.04414v1 +2. Lucarini, *Interpretable and Equation-Free Response Theory for Complex + Systems*, arXiv:2502.07908. PRIMARY_TEXT_READ for the retrieved v1 HTML, + section II.2, equations (13)-(19): Markov-chain second-order response, + time-ordering and two-mode spectral weights. Publisher indexed text was also + retrieved (DOI 10.1098/rsta.2025.0081), but direct DOI opening returned 403; + journal equation numbering is not substituted for the arXiv numbering. + https://arxiv.org/html/2502.07908v1 +3. Mueller, Basu, Sollich and Krueger, *Coarse-grained second-order response + theory*, Physical Review Research 2, 043123 (2020), + DOI 10.1103/PhysRevResearch.2.043123. ABSTRACT_ONLY plus selected publisher + excerpts: an equilibrium second-order coarse-grained framework and its + non-Markovian issues. Its equilibrium hypotheses are not silently imported + to P398. https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.2.043123 + +These are bounded primary readings, not a literature-completeness/novelty +search. The older inference "no named Markov selection rule was found, hence +our statement is new" is unsupported: absence of a retrieved name is not an +originality certificate. The concrete P398 ranks and scalar observer above +remain the result to evaluate on their own merits. + +## 9. Decision and reproduction + + python scripts/p398_double_pulse_exact.py --out /tmp/p398-pulses.json + python scripts/verify_p398_double_pulse_certificate.py + python -m unittest discover -s tests -p 'test_p398_double_pulse_exact.py' + python scripts/p398_double_pulse_numerical_control.py + +The first three commands need only the standard library. The last needs mpmath. +Output paths use exclusive creation; existing artifacts are not overwritten. + +This completes a bounded second-order *accessibility* question in #594/#598/ +#610, rather than commissioning more widths or repeating the old quadratic +amplitude ladder. It does not settle P398's approximate noisy mode-resolution +cost, an all-width growth law, or any square-site original-U map in #275. +A further large run requires an explicit alternative and signal/noise objective; +there is no such justified purchase in this delivery. diff --git a/results/research-control-20260912/p398-double-pulse-bareiss-check.json b/results/research-control-20260912/p398-double-pulse-bareiss-check.json new file mode 100644 index 00000000..ed916138 --- /dev/null +++ b/results/research-control-20260912/p398-double-pulse-bareiss-check.json @@ -0,0 +1,76 @@ +{ + "schema": "matching-one.p398-double-pulse-bareiss-check.v1", + "method": "Independent fraction-free Bareiss elimination of stored integer Hankel matrices, with exact division at every step.", + "checks": [ + { + "width": 4, + "object": "baseline", + "dimension": 10, + "exact_integer_determinant": "190412207161344", + "modular_residue_matches": true + }, + { + "width": 4, + "object": "double_pulse", + "dimension": 4, + "exact_integer_determinant": "-131712", + "modular_residue_matches": true + }, + { + "width": 4, + "object": "controlled", + "dimension": 14, + "exact_integer_determinant": "25079572629634940928", + "modular_residue_matches": true + }, + { + "width": 4, + "object": "contrasts", + "dimension": 13, + "exact_integer_determinant": "9139785943744512", + "modular_residue_matches": true + }, + { + "width": 4, + "object": "scalar_pulse", + "dimension": 4, + "exact_integer_determinant": "-16", + "modular_residue_matches": true + }, + { + "width": 5, + "object": "baseline", + "dimension": 26, + "exact_integer_determinant": "181461345108945382405059034718742430957841106782936682773049890032294986062007730129156072676524032000", + "modular_residue_matches": true + }, + { + "width": 5, + "object": "double_pulse", + "dimension": 16, + "exact_integer_determinant": "-55257277593008057571282631920143302656", + "modular_residue_matches": true + }, + { + "width": 5, + "object": 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"0.1199943442451999030827374879455758", + "ladder": [ + { + "epsilon": "1/8", + "centered_mixed_difference": "0.00011718260331862801207829848206207703", + "finite_window_derivative_error": "6.265166749817240626539902e-10", + "parity_pair_max_error": "0.0" + }, + { + "epsilon": "1/16", + "centered_mixed_difference": "0.00011718213343087405684069949919877212", + "finite_window_derivative_error": "1.566289210264864636711269e-10", + "parity_pair_max_error": "0.0" + }, + { + "epsilon": "1/32", + "centered_mixed_difference": "0.00011718201595916780455799284674114656", + "finite_window_derivative_error": "3.91572147742037570186693e-11", + "parity_pair_max_error": "0.0" + } + ] +} diff --git a/results/research-control-20260912/p398-double-pulse-summary.json b/results/research-control-20260912/p398-double-pulse-summary.json new file mode 100644 index 00000000..dd0f1ab1 --- /dev/null +++ b/results/research-control-20260912/p398-double-pulse-summary.json @@ -0,0 +1,72 @@ +{ + "schema": "matching-one.p398-double-pulse-summary.v1", + "full_certificate": "results/research-control-20260912/p398-double-pulse-exact.json", + "exact_integer_determinants": "results/research-control-20260912/p398-double-pulse-bareiss-check.json", + "widths": [ + { + "width": 4, + "microscopic_states": 14, + "baseline_order": 10, + "double_pulse_order": 4, + "controlled_word_order": 14, + "controlled_contrast_order": 13, + "scalar_laplace_numerator_high_first": [ + 2, + 22, + 54 + ], + "scalar_laplace_denominator_high_first": [ + 1, + 21, + 159, + 513, + 598 + ] + }, + { + "width": 5, + "microscopic_states": 42, + "baseline_order": 26, + "double_pulse_order": 16, + "controlled_word_order": 42, + "controlled_contrast_order": 41, + "scalar_laplace_numerator_high_first": [ + -2, + -176, + -7062, + -170922, + -2781008, + -32079704, + -269465528, + -1665384800, + -7552656564, + -24735264292, + -56585225488, + -85021400912, + -74393252992, + -28111385088 + ], + "scalar_laplace_denominator_high_first": [ + 1, + 108, + 5424, + 168112, + 3598246, + 56381652, + 668841193, + 6125591806, + 43758627120, + 244548035884, + 1065231609598, + 3577186030660, + 9074962639953, + 16805918185608, + 21416462410576, + 16771594235392, + 6077795182336 + ] + } + ], + "verified_nonzero_integer_minors": 10, + "scope": "Existing P398 calibration model. Exact homogeneous linear/bilinear order, not effective noisy dimension, positive realization minimum, nonlinear state dimension, or percolation." +} diff --git a/scripts/p398_double_pulse_exact.py b/scripts/p398_double_pulse_exact.py new file mode 100644 index 00000000..e211c5cf --- /dev/null +++ b/scripts/p398_double_pulse_exact.py @@ -0,0 +1,328 @@ +#!/usr/bin/env python3 +"""Exact second-order observability on the existing P398 process, widths 4/5. + +Uses only the Python standard library. This is an independent implementation of +P398's declared noncrossing join/detach model and original three readouts/four +sources, not a new percolation model. All claim-bearing arithmetic is integer +or Fraction. A nonzero finite-field minor certifies a rational rank lower bound; +parity/ambient dimension supplies the matching upper bound. + +H = join@0 - join@(w-2) = 2 * the repository's H_odd. G+epsilon*H is a +Markov generator for |epsilon|<=1 (irreducible for |epsilon|<1 here). +""" +from __future__ import annotations + +import argparse +from fractions import Fraction +from itertools import combinations +import json +from pathlib import Path +from typing import Sequence + +PRIME = 1_000_000_007 +Matrix = list[list[int]] + + +def canonical(values: Sequence[int]) -> tuple[int, ...]: + names: dict[int, int] = {} + return tuple(names.setdefault(x, len(names)) for x in values) + + +def noncrossing_states(width: int) -> list[tuple[int, ...]]: + if width not in (4, 5): + raise ValueError('This bounded analysis is declared only at widths 4 and 5.') + quadruples = list(combinations(range(width), 4)) + def grow(prefix): + if len(prefix) == width: + if not any(prefix[a] == prefix[c] and prefix[b] == prefix[d] + and prefix[a] != prefix[b] for a, b, c, d in quadruples): + yield prefix + return + for label in range(max(prefix, default=-1) + 2): + yield from grow(prefix + (label,)) + return list(grow(())) + + +def zeros(n: int, m: int) -> Matrix: + return [[0] * m for _ in range(n)] + + +def transpose(a: Matrix) -> Matrix: + return [list(c) for c in zip(*a)] + + +def matvec(a: Matrix, x: Sequence[int]) -> list[int]: + return [sum(t * z for t, z in zip(row, x)) for row in a] + + +def multiply(a: Matrix, b: Matrix) -> Matrix: + bt = transpose(b) + return [[sum(x * y for x, y in zip(row, col)) for col in bt] for row in a] + + +def build_model(width: int) -> dict: + states = noncrossing_states(width) + index = {state: i for i, state in enumerate(states)} + n = len(states) + joins, detaches = [], [] + for point in range(width): + j, d = zeros(n, n), zeros(n, n) + for i, state in enumerate(states): + image = canonical([state[point] if x == state[(point + 1) % width] else x + for x in state]) + j[i][index[image]] += 1 + j[i][i] -= 1 + q = list(state) + q[point] = width + d[i][index[canonical(q)]] += 1 + d[i][i] -= 1 + joins.append(j) + detaches.append(d) + g = [[sum(a[i][j] for a in joins + detaches) for j in range(n)] for i in range(n)] + h = [[joins[0][i][j] - joins[width - 2][i][j] for j in range(n)] for i in range(n)] + reflect = [index[canonical(state[::-1])] for state in states] + f = [[max(state) + 1, sum(state.count(x) == 1 for x in set(state)), + int(state[0] == state[-1])] for state in states] + # n times each ORIGINAL probability source: no floating-point uniform law. + src = zeros(4, n) + initial = [tuple(range(width)), (0,) * width, + tuple([0] + list(range(1, width - 1)) + [0])] + for k, state in enumerate(initial): + src[k][index[state]] = n + src[3] = [1] * n + for i in range(n): + assert sum(g[i]) == sum(h[i]) == 0 + assert f[i] == f[reflect[i]] + for j in range(n): + assert g[reflect[i]][reflect[j]] == g[i][j] + assert h[reflect[i]][reflect[j]] == -h[i][j] + if i != j: + assert g[i][j] >= abs(h[i][j]) + assert all(row[i] == row[reflect[i]] for row in src for i in range(n)) + for matrix in (g, transpose(g)): + reached, queue = {0}, [0] + for i in queue: + for j, rate in enumerate(matrix[i]): + if j != i and rate > 0 and j not in reached: + reached.add(j) + queue.append(j) + assert len(reached) == n + return dict(width=width, states=states, G=g, H=h, reflection=reflect, + F=f, sources_scaled=src, source_denominator=n) + + +def parity_basis(model: dict, parity: int) -> tuple[Matrix, list[int]]: + pi = model['reflection'] + representatives = [i for i, j in enumerate(pi) if i < j or (parity == 1 and i == j)] + u = zeros(len(pi), len(representatives)) + for k, i in enumerate(representatives): + u[i][k] = 1 + if pi[i] != i: + u[pi[i]][k] = parity + return u, representatives + + +class ModularBasis: + def __init__(self, prime: int = PRIME): + self.rows: list[tuple[int, list[int]]] = [] + self.prime = prime + + def add(self, vector: Sequence[int]) -> bool: + p = self.prime + v = [x % p for x in vector] + for pivot, row in self.rows: + factor = v[pivot] + if factor: + v = [(x - factor * y) % p for x, y in zip(v, row)] + if not any(v): + return False + pivot = next(i for i, x in enumerate(v) if x) + inverse = pow(v[pivot], p - 2, p) + self.rows.append((pivot, [x * inverse % p for x in v])) + return True + + +def determinant_mod(a: Matrix, prime: int = PRIME) -> int: + n = len(a) + if any(len(row) != n for row in a): + raise ValueError('determinant expects a square matrix') + b, result = [[x % prime for x in row] for row in a], 1 + for k in range(n): + pivot = next((i for i in range(k, n) if b[i][k]), None) + if pivot is None: + return 0 + if pivot != k: + b[k], b[pivot] = b[pivot], b[k] + result = -result + value = b[k][k] + result = result * value % prime + inverse = pow(value, prime - 2, prime) + for i in range(k + 1, n): + factor = b[i][k] * inverse % prime + for j in range(k + 1, n): + b[i][j] = (b[i][j] - factor * b[k][j]) % prime + return result % prime + + +def word_span(operators: list[tuple[str, Matrix]], seeds: Matrix, side: str) -> dict: + """Exact integer vectors selected by independent modular pivots. + + At a certified full ambient bound these span over Q as well. When the + observed rank is below a proven bound, this routine reports only a lower + bound; a modular stabilization alone is NOT a rational upper-bound proof. + """ + basis, selected, frontier = ModularBasis(), [], [] + for j, vector in enumerate(transpose(seeds)): + if basis.add(vector): + item = dict(seed=j, word='', vector=vector) + selected.append(item) + frontier.append(item) + levels = [len(selected)] + while frontier: + new = [] + for name, a in operators: + for item in frontier: + vector = matvec(a, item['vector']) + if basis.add(vector): + word = name + item['word'] if side == 'column' else item['word'] + name + nxt = dict(seed=item['seed'], word=word, vector=vector) + selected.append(nxt) + new.append(nxt) + levels.append(len(selected)) + frontier = new + return dict(selected=selected, rank_lower_bound=len(selected), levels=levels) + + +def realization_certificate(operators: list[tuple[str, Matrix]], f: Matrix, + src: Matrix, upper_bound: int) -> dict: + r = word_span(operators, f, 'column') + o = word_span([(name, transpose(a)) for name, a in operators], transpose(src), 'row') + gram = [[sum(x * y for x, y in zip(row['vector'], col['vector'])) + for col in r['selected']] for row in o['selected']] + basis, ri = ModularBasis(), [] + for i, row in enumerate(gram): + if basis.add(row): + ri.append(i) + basis, ci = ModularBasis(), [] + for j, col in enumerate(transpose([gram[i] for i in ri])): + if basis.add(col): + ci.append(j) + minor = [[gram[i][j] for j in ci] for i in ri] + det = determinant_mod(minor) + rank = len(ri) + if rank != upper_bound or not det: + raise ValueError(f'Lower bound {rank} does not attain asserted upper bound {upper_bound}') + return dict(minimal_order=rank, ambient_upper_bound=upper_bound, + reach_levels=r['levels'], observe_levels=o['levels'], prime=PRIME, + hankel_minor_determinant_mod_prime=det, + row_words=[{k: v for k, v in o['selected'][i].items() if k != 'vector'} for i in ri], + column_words=[{k: v for k, v in r['selected'][j].items() if k != 'vector'} for j in ci], + integer_hankel_minor=minor) + + +def charpoly_integer(a: Matrix) -> list[int]: + """Faddeev-LeVerrier, integer division checked; coefficients high first.""" + n = len(a) + b = [[int(i == j) for j in range(n)] for i in range(n)] + coefficients = [1] + for k in range(1, n + 1): + ab = multiply(a, b) + trace = sum(ab[i][i] for i in range(n)) + assert trace % k == 0 + c = -trace // k + coefficients.append(c) + b = [[ab[i][j] + (c if i == j else 0) for j in range(n)] for i in range(n)] + assert not any(x for row in b for x in row) + return coefficients + + +def pulse_data(model: dict) -> dict: + g, h, f, src = (model[k] for k in ('G', 'H', 'F', 'sources_scaled')) + u, reps = parity_basis(model, -1) + a = [row for i, row in enumerate(multiply(g, u)) if i in reps] + b = [row for i, row in enumerate(multiply(h, f)) if i in reps] + c = multiply(multiply(src, h), u) + assert multiply(g, u) == multiply(u, a) + assert multiply(h, f) == multiply(u, b) + d = len(reps) + certificate = realization_certificate([('G', a)], b, c, d) + current, moments = b, [] + for _ in range(2 * d + 1): + moments.append(multiply(c, current)) + current = multiply(a, current) + # Scalar witnesses chosen from the ORIGINAL dictionary. No new readout. + source, output = (2, 2) if model['width'] == 4 else (0, 2) + scalar = [Fraction(m[source][output], model['source_denominator']) for m in moments] + assert all(x.denominator == 1 for x in scalar) + scalar = [int(x) for x in scalar] + hankel = [[scalar[i + j] for j in range(d)] for i in range(d)] + determinant = determinant_mod(hankel) + assert determinant != 0 + polynomial = charpoly_integer(a) + for k in range(len(scalar) - d): + assert sum(polynomial[j] * scalar[k + d - j] for j in range(d + 1)) == 0 + # Numerator of C(zI-A)^(-1)B from the first d moments, descending order. + numerator = [sum(polynomial[j] * scalar[k-j] for j in range(k + 1)) for k in range(d)] + while len(numerator) > 1 and numerator[0] == 0: + numerator.pop(0) + return dict(odd_dimension=d, G_odd=a, B_odd=b, C_odd_scaled=c, + source_denominator=model['source_denominator'], + odd_characteristic_polynomial_high_first=polynomial, + rank_certificate=certificate, markov_moments_scaled=moments, + scalar_witness=dict(source_index=source, readout_index=output, + moments=scalar, hankel_minor=hankel, + hankel_determinant_mod_prime=determinant, + laplace_numerator_high_first=numerator, + laplace_denominator_high_first=polynomial)) + + +def analyse(width: int) -> dict: + model = build_model(width) + n = len(model['states']) + g, h, f, src = (model[k] for k in ('G', 'H', 'F', 'sources_scaled')) + up, reps = parity_basis(model, 1) + gp = [row for i, row in enumerate(multiply(g, up)) if i in reps] + fp = [f[i] for i in reps] + sp = multiply(src, up) + assert multiply(up, fp) == f + assert multiply(g, up) == multiply(up, gp) + baseline = realization_certificate([('G', gp)], fp, sp, len(reps)) + controlled = realization_certificate([('G', g), ('H', h)], f, src, n) + contrast = [[src[i][j] - src[3][j] for j in range(n)] for i in range(3)] + # Constants are invariant under G and killed by H and the source contrasts. + # Therefore n-1 is a mathematical upper bound, not a numerical rank guess. + contrasted = realization_certificate([('G', g), ('H', h)], f, contrast, n - 1) + return dict(width=width, states=model['states'], reflection=model['reflection'], + G=g, H=h, readouts=f, sources_scaled=src, source_denominator=n, + baseline=baseline, double_pulse=pulse_data(model), + arbitrary_controlled_word=controlled, controlled_source_contrasts=contrasted) + + +def report() -> dict: + return dict(schema='matching-one.p398-double-pulse-exact.v1', + model='P398 noncrossing join/detach calibration process; NOT percolation', + source_ref='eb89e9422791d9e3c3a78f0e65d56912b815a7bd', + source_files={'scripts/p398_intervention_transport.py':'ac13194751b8ba727a45499900e3f12ce1b90461', + 'scripts/planar_state_operations.py':'c5f57dcb8606dbb62306cde839edefae680f863d'}, + H_convention='join@0 - join@(w-2); twice the archived H_odd', + source_order=['delta_all_singletons','delta_single_block','delta_wrapped_pair','uniform'], + readout_order=['blocks','singletons','wrap'], + pulse_kernel='K(tau)=S H exp(tau G) H F; S is the probability-source matrix', + claim_boundary=['Exact homogeneous linear/bilinear realization orders include the constant mode.', + 'A finite-field nonzero minor is a rational lower bound; parity and dimension give the upper bound.', + 'No new sampling, asymptotic width law, positivity-minimality, CFT or threshold claim.', + 'The short-pulse kernel is a limit of derivatives of physical nonnegative-rate windows, not exp(epsilon H).'], + widths=[analyse(4), analyse(5)]) + + +if __name__ == '__main__': + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--out', type=Path) + args = parser.parse_args() + text = json.dumps(report(), 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 handle: + handle.write(text) + else: + print(text, end='') diff --git a/scripts/p398_double_pulse_numerical_control.py b/scripts/p398_double_pulse_numerical_control.py new file mode 100644 index 00000000..a7b9f019 --- /dev/null +++ b/scripts/p398_double_pulse_numerical_control.py @@ -0,0 +1,76 @@ +#!/usr/bin/env python3 +"""Finite-duration Markov-pulse control for the exact P398 double-pulse result. + +Requires mpmath. No sampling. Matrix exponentials use physical G+epsilon*H, +never the signed operator H as a standalone Markov generator. Numerical values +are diagnostics, not rigorous interval certificates. Historical results are +never overwritten by this command. +""" +from __future__ import annotations + +import argparse +import json +from pathlib import Path +from mpmath import mp +from p398_double_pulse_exact import build_model + + +def report() -> dict: + with mp.workdps(45): + model = build_model(4) + n = len(model['states']) + g, h = mp.matrix(model['G']), mp.matrix(model['H']) + source = mp.matrix([[mp.mpf(x) / n for x in model['sources_scaled'][2]]]) + readout = mp.matrix([x[2] for x in model['F']]) + tau, duration = mp.mpf(1) / 4, mp.mpf(1) / 32 + waiting = mp.expm(tau * g) + # An independent Frechet derivative series, not a finite difference: + # D_k = d/d epsilon (G+epsilon H)^k |0. + power, derivative = mp.eye(n), mp.zeros(n) + window_derivative, weight = mp.zeros(n), mp.mpf(1) + for k in range(1, 36): + derivative = g * derivative + h * power + power = g * power + weight *= duration / k + window_derivative += weight * derivative + truth = (source * window_derivative * waiting * window_derivative * readout)[0] + ideal = (source * h * waiting * h * readout)[0] + rows = [] + for denominator in (8, 16, 32): + epsilon = mp.mpf(1) / denominator + plus = mp.expm(duration * (g + epsilon * h)) + minus = mp.expm(duration * (g - epsilon * h)) + pp = (source * plus * waiting * plus * readout)[0] + pm = (source * plus * waiting * minus * readout)[0] + mp_value = (source * minus * waiting * plus * readout)[0] + mm = (source * minus * waiting * minus * readout)[0] + mixed = (pp - pm - mp_value + mm) / (4 * epsilon**2) + rows.append(dict( + epsilon=f'1/{denominator}', + centered_mixed_difference=mp.nstr(mixed, 35), + finite_window_derivative_error=mp.nstr(mixed - truth, 25), + parity_pair_max_error=mp.nstr(max(abs(pp - mm), abs(pm - mp_value)), 15), + )) + return dict( + schema='matching-one.p398-double-pulse-numerical-control.v1', + standing='Numerical independent matrix-exponential control, not an exact certificate or simulation.', + dps=45, source='delta_wrapped_pair', readout='wrap', tau='1/4', + pulse_duration='1/32', frechet_series_terms=35, + finite_window_mixed_derivative=mp.nstr(truth, 35), + short_pulse_kernel=mp.nstr(ideal, 35), + finite_window_derivative_divided_by_delta_squared=mp.nstr(truth / duration**2, 35), + ladder=rows, + ) + + +if __name__ == '__main__': + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--out', type=Path) + args = parser.parse_args() + text = json.dumps(report(), 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 handle: + handle.write(text) + else: + print(text, end='') diff --git a/scripts/verify_p398_double_pulse_certificate.py b/scripts/verify_p398_double_pulse_certificate.py new file mode 100644 index 00000000..edc7a82f --- /dev/null +++ b/scripts/verify_p398_double_pulse_certificate.py @@ -0,0 +1,118 @@ +#!/usr/bin/env python3 +"""Check the stored Hankel rank witnesses by integer, fraction-free elimination. + +Rebuild each selected word coefficient before computing its determinant. This +checks the claim-bearing ranks, not document wording or a repository registry. +Requires only the standard library. Output paths use exclusive creation. +""" +from __future__ import annotations +import argparse +import json +from pathlib import Path +from p398_double_pulse_exact import (build_model, matvec, multiply, parity_basis, transpose) + +ROOT = Path(__file__).resolve().parents[1] +DEFAULT = ROOT / 'results/research-control-20260912/p398-double-pulse-exact.json' + + +def bareiss(matrix): + m = [row[:] for row in matrix] + sign, previous, n = 1, 1, len(m) + if not n: + return 1 + for k in range(n - 1): + pivot = next((i for i in range(k, n) if m[i][k]), None) + if pivot is None: + return 0 + if pivot != k: + m[k], m[pivot] = m[pivot], m[k] + sign = -sign + value = m[k][k] + for i in range(k + 1, n): + for j in range(k + 1, n): + numerator = value * m[i][j] - m[i][k] * m[k][j] + if numerator % previous: + raise ValueError('Nonexact Bareiss division') + m[i][j] = numerator // previous + m[i][k] = 0 + previous = value + return sign * m[-1][-1] + + +def word_minor(operators, f, src, cert): + rows, columns = [], [] + for item in cert['row_words']: + vector = src[item['seed']][:] + for name in item['word']: + vector = matvec(transpose(operators[name]), vector) + rows.append(vector) + for item in cert['column_words']: + vector = [row[item['seed']] for row in f] + for name in reversed(item['word']): + vector = matvec(operators[name], vector) + columns.append(vector) + return [[sum(x*y for x, y in zip(row, col)) for col in columns] for row in rows] + + +def verify(payload): + checks = [] + for row in payload['widths']: + width = row['width'] + model = build_model(width) + for key, other in [('G','G'), ('H','H'), ('sources_scaled','sources_scaled'), ('readouts','F')]: + if row[key] != model[other]: + raise ValueError(f'Model mismatch at width {width}: {key}') + g, h, f, src = (model[k] for k in ('G','H','F','sources_scaled')) + u, reps = parity_basis(model, 1) + gp = [multiply(g, u)[i] for i in reps] + fp, sp = [f[i] for i in reps], multiply(src, u) + odd = row['double_pulse'] + u, reps = parity_basis(model, -1) + gm = [multiply(g, u)[i] for i in reps] + bm, cm = [multiply(h, f)[i] for i in reps], multiply(multiply(src,h),u) + for got, expected in [(odd['G_odd'],gm),(odd['B_odd'],bm),(odd['C_odd_scaled'],cm)]: + if got != expected: + raise ValueError('Odd sector mismatch') + contrast = [[src[i][j]-src[3][j] for j in range(len(g))] for i in range(3)] + cases = [ + ('baseline', {'G':gp}, fp, sp, row['baseline']), + ('double_pulse', {'G':gm}, bm, cm, odd['rank_certificate']), + ('controlled', {'G':g,'H':h}, f, src, row['arbitrary_controlled_word']), + ('contrasts', {'G':g,'H':h}, f, contrast, row['controlled_source_contrasts']), + ] + for name, operators, inputs, outputs, cert in cases: + matrix = word_minor(operators, inputs, outputs, cert) + if matrix != cert['integer_hankel_minor']: + raise ValueError(f'Word coefficient mismatch: {width}, {name}') + det = bareiss(matrix) + if not det or det % cert['prime'] != cert['hankel_minor_determinant_mod_prime']: + raise ValueError(f'Rank witness failed: {width}, {name}') + checks.append(dict(width=width, object=name, dimension=cert['minimal_order'], + exact_integer_determinant=str(det), modular_residue_matches=True)) + scalar = odd['scalar_witness'] + d = len(scalar['hankel_minor']) + matrix = [[scalar['moments'][i+j] for j in range(d)] for i in range(d)] + if matrix != scalar['hankel_minor']: + raise ValueError('Scalar Hankel mismatch') + det = bareiss(matrix) + if not det or det % 1_000_000_007 != scalar['hankel_determinant_mod_prime']: + raise ValueError('Scalar rank witness failed') + checks.append(dict(width=width, object='scalar_pulse', dimension=d, + exact_integer_determinant=str(det), modular_residue_matches=True)) + return dict(schema='matching-one.p398-double-pulse-bareiss-check.v1', + method='Independent fraction-free Bareiss elimination of stored integer Hankel matrices, with exact division at every step.', + checks=checks) + + +if __name__ == '__main__': + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--input', type=Path, default=DEFAULT) + parser.add_argument('--out', type=Path) + args = parser.parse_args() + text = json.dumps(verify(json.loads(args.input.read_text(encoding='utf-8'))), indent=2) + '\n' + if args.out: + args.out.parent.mkdir(parents=True, exist_ok=True) + with args.out.open('x', encoding='utf-8') as handle: + handle.write(text) + else: + print(text, end='') diff --git a/tests/test_p398_double_pulse_exact.py b/tests/test_p398_double_pulse_exact.py new file mode 100644 index 00000000..3d041898 --- /dev/null +++ b/tests/test_p398_double_pulse_exact.py @@ -0,0 +1,97 @@ +"""Small checks that prevent a false zero, false rank, or a pulse normalizer error.""" +from fractions import Fraction +from pathlib import Path +import sys +import unittest +sys.path.insert(0, str(Path(__file__).resolve().parents[1] / 'scripts')) +import p398_double_pulse_exact as p + + +def determinant_fraction(a): + b = [[Fraction(x) for x in row] for row in a] + result = Fraction(1) + for k in range(len(b)): + pivot = next((i for i in range(k, len(b)) if b[i][k]), None) + if pivot is None: + return Fraction(0) + if pivot != k: + b[k], b[pivot] = b[pivot], b[k] + result = -result + v = b[k][k] + result *= v + for i in range(k + 1, len(b)): + factor = b[i][k] / v + for j in range(k + 1, len(b)): + b[i][j] -= factor * b[k][j] + return result + + +class DoublePulseTests(unittest.TestCase): + @classmethod + def setUpClass(cls): + cls.rows = {w: p.analyse(w) for w in (4, 5)} + + def test_existing_model_counts_and_parity(self): + for w, n in ((4, 14), (5, 42)): + row = self.rows[w] + self.assertEqual(len(row['states']), n) + self.assertEqual(row['double_pulse']['odd_dimension'], n - row['baseline']['minimal_order']) + # H is NOT a stochastic generator on its own; only G+epsilon H is. + self.assertTrue(any(x < 0 for i, r in enumerate(row['H']) for j, x in enumerate(r) if i != j)) + + def test_exact_first_order_zero_and_second_order_nonzero(self): + for row in self.rows.values(): + src, g, h, f = (row[k] for k in ('sources_scaled', 'G', 'H', 'readouts')) + first = p.multiply(p.multiply(src, h), f) + self.assertTrue(all(x == 0 for r in first for x in r)) + second = p.multiply(p.multiply(p.multiply(src, h), h), f) + self.assertEqual(second[0][0], 2 * row['source_denominator']) + self.assertEqual(second[1], [0, 0, 0]) + + def test_width4_scalar_resolvent_and_independent_determinant(self): + scalar = self.rows[4]['double_pulse']['scalar_witness'] + self.assertEqual(scalar['moments'][:8], [0, 2, -20, 156, -1122, 7822, -53932, 371172]) + self.assertEqual(scalar['laplace_numerator_high_first'], [2, 22, 54]) + self.assertEqual(scalar['laplace_denominator_high_first'], [1, 21, 159, 513, 598]) + self.assertEqual(determinant_fraction(scalar['hankel_minor']), -16) + + def test_all_odd_modes_accessible(self): + for w, d in ((4, 4), (5, 16)): + data = self.rows[w]['double_pulse'] + cert = data['rank_certificate'] + self.assertEqual(cert['minimal_order'], d) + self.assertNotEqual(p.determinant_mod(cert['integer_hankel_minor']), 0) + self.assertNotEqual(p.determinant_mod(data['scalar_witness']['hankel_minor']), 0) + + def test_controlled_language_requires_more_than_baseline(self): + for w, base, total in ((4, 10, 14), (5, 26, 42)): + row = self.rows[w] + self.assertEqual(row['baseline']['minimal_order'], base) + self.assertEqual(row['arbitrary_controlled_word']['minimal_order'], total) + self.assertEqual(row['controlled_source_contrasts']['minimal_order'], total - 1) + for name in ('baseline', 'arbitrary_controlled_word', 'controlled_source_contrasts'): + cert = row[name] + self.assertEqual(p.determinant_mod(cert['integer_hankel_minor']), + cert['hankel_minor_determinant_mod_prime']) + + def test_sign_not_fixed_by_parity(self): + data = self.rows[4]['double_pulse'] + n = data['source_denominator'] + self.assertEqual(data['markov_moments_scaled'][0][0][0], 2 * n) + self.assertEqual(data['markov_moments_scaled'][1][0][2], -2 * n) + self.assertEqual(data['markov_moments_scaled'][1][2][2], 2 * n) + + def test_archived_half_strength_gives_quarter_kernel(self): + row = self.rows[4] + h = [[Fraction(x, 2) for x in r] for r in row['H']] + for lag_power in range(3): + right = p.multiply(h, row['readouts']) + for _ in range(lag_power): + right = p.multiply(row['G'], right) + got = p.multiply(p.multiply(row['sources_scaled'], h), right) + expected = row['double_pulse']['markov_moments_scaled'][lag_power] + self.assertEqual(got, [[Fraction(x, 4) for x in r] for r in expected]) + + +if __name__ == '__main__': + unittest.main()