[P2] Automated self-dual gadget and critical-manifold search

[LightChainr]

2026-08-31 P2团队交接：暂停自动扩展代数目录

现有serial composition、作用、理想、幂序列和centralizer等精确资产保留；本轮新增的 #510/#511/#514–#519 也作为P2支持成果保存，不升级为Matching主线推进。它们各自声明尚不涉及可靠性、临界方程或严格界。

本轮不再默认追加相邻代数分类或通用证书任务。下一项值得集中投入的工作应给出两张具体周期图之间的随机比较不等式、可用局部变换，或明确有限类无法提供该比较的反例/定理；相关严格阈值接口在 #14。若尚无这种具体目标，本Issue保持P2储备，廉价独立数学想法可继续记录，不占用 #154 / #334 的主线预算。

团队分工与清理记录。以下保留原文。

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Context reconciliation — 2026-08-31 c / completed work before next assignments

已补读#13全部当前讨论和后续合并状态：#479–483、#488–490及#493–497的有限15态serial代数系列均已完成。最后五项于2026-08-31 14:14合并，不能继续写作新认领待办；本轮不做任何合并。

这些结果可直接复用，但有限局部monoid、power/inverse/stabilizer等不是独立的物理可靠性实验。#438已经构造W5周期primal/dual对象，下一物理问题是明确的概率律比较或law-preserving变换，而非再建一次周期wiring。按用途分组的开放PR保留原标题、head和依赖关系；唯一注意力表不变成许可或锁。原计划与全部历史讨论保留。

Previous body preserved — dated historical context

2026-08-31 上下文恢复（原提案保留在下）

• 终端状态/graph/corpus 与有序 gluing/serial 已完成：29f9716 的 port-aware D4 协变、4baa15a 的15态有序 serial algebra（3375三元组结合，唯一 wire identity）可直接消费，不再列为缺失实现。
• 52a61ff 证明7个 D4 orbit 不是合法 serial quotient：49输入对有26个多值，均匀平均核在343三元组中166个不结合。快照后新增 PR Certify typed serial port reversal #465 / e300609 已完成 port reversal 反对合，225乘积全成立、7固定点+4二循环；它不是 planar/complement dual。
• 真实有限 corpus 已筛过，而非只枚举：branch_only 05514ba 对全部11个 connected 三终端单内点候选求精确 balance-root；最近者为可约路径0.61803399，没有得到新的 square-site 机制。
• 尚需的是明确 law-preserving 周期构造/概率比较或可区分的新机制，而非再做同一代数/11候选筛查。Exact: close the W5 relative-dual state and periodic gluing (#14) #438 的 W5周期对象已建成，但既有 balance roots、port reversal 与图编码均不构成阈值公式或严格界。
• 已补读 2026-08-31 新认领的 Green 关系、最小生成元、自同构、中心与子幺半群五类普查：它们仍可并行，但本线优先产出既有具体图律的比较或明确障碍；不取消、锁定这些任务，也不把普查完成作为概率分析先修。
• 当前总览与下一步见 Draft docs: recover scientific frontier and score production mechanisms #267 的上下文恢复与注意力顺序；原提案全文保留在下。

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Conjecture

A sequence of exactly solvable decorated cells can approximate the four-terminal connectivity law induced by a square-lattice occupied site and generate exact algebraic thresholds converging rapidly to the square-site threshold.

Search space

• Planar three- and four-terminal cells with bounded internal vertices/edges.
• Site, bond, and correlated-bond representations with exact terminal partition probabilities.
• Cells equivalent under graph isomorphism, terminal permutation, series/parallel reduction, and obvious dualities are canonicalized once.
• Retain only constructions that tile periodically and admit a self-dual or otherwise exact critical condition.

Scoring

• Distance of the exact threshold to the current numerical interval.
• Prediction error on another observable or related lattice.
• Convergence rate with cell complexity.
• Structural simplicity and proof transparency.

Deliverables

• Canonical cell encoding and exhaustive enumeration bound.
• Exact terminal-connectivity polynomials.
• Self-duality/critical-manifold checker.
• Ranked candidate sequences with algebraic minimal polynomials.
• Negative-result certificate for the explored search space.

Acceptance

A close decimal match alone is not a result. A retained family must be parameter-free, exactly derived, and predict convergence or another independent quantity.

Resources

8 CPU cores / 16 GB for the first bounded search. GPU is optional for batched evaluation; high-memory CPU may later be more useful.

Tracked as T09 in experiments/tasks.yaml.

[LightChainr]

Claiming a bounded canonical-encoding primitive on branch tools/issue-13-terminal-partition-canonical.

Scope: implement a restricted-growth-string encoding for complete three-/four-terminal set partitions, exact validation, explicit terminal-permutation group validation, orbit canonicalization, and exhaustive Bell-count/orbit regression tests. This is the terminal-partition deduplication layer only.

Out of scope: graph/cell encoding, planarity, series/parallel reduction, terminal-connectivity polynomials, tilings, self-duality, enumeration bounds, ranking, or negative-result certificates. Current main, open PR filenames, and all remote branches show no Issue #13 implementation or overlap. The parent issue will remain open.

[LightChainr]

Merged PR #329 via squash as 7da6013 after the complete CI matrix passed on the rebased head (Python 3.9/3.11/3.13 and C++17).

Exact conclusion delivered by this slice:

• restricted-growth strings now give a unique encoding for every nonempty terminal set partition;
• enumeration returns exactly Bell(3)=5 and Bell(4)=15 encodings;
• terminal permutations use the frozen old-terminal-to-new-terminal convention, and supplied groups fail closed unless they contain a bijective identity and are closed under inverse and composition;
• orbit canonicalization is the lexicographically least RGS under the explicitly supplied group; no full-symmetric-group assumption is implicit;
• under S3 and S4, the audited orbit counts are respectively 3 and 5, with checked-in canonical representatives.

Unresolved boundary: this primitive does not enumerate gadget graphs, test planarity or graph isomorphism, evaluate probability polynomials, checkpoint a search, or establish a replacement-gadget result. Those remaining deliverables keep this parent issue open.

[LightChainr]

Claiming a bounded graph-canonicalization slice on branch tools/issue-13-gadget-graph-canonical.

Scope: define a canonical encoding for finite undirected simple terminal gadgets under arbitrary internal-vertex relabeling and an explicitly validated terminal-permutation group. Add exact validation, isomorphism/orbit invariance tests, and exhaustive small-graph deduplication checks.

Out of scope: terminal probability polynomials, planarity/tiling certification, series-parallel reduction, self-duality, search ranking, or a negative-result certificate. I checked latest main, all open PRs, and remote branches; only the already merged terminal-partition branch exists and no current work overlaps this graph namespace. The parent issue will remain open.

[LightChainr]

Bounded graph-canonicalization slice completed in PR #381, squash-merged as 0fb211d8f20bc6c0d59dd5b42962a0e86c3fdb15 after complete CI passed (C++17 and Python 3.9/3.11/3.13; run 33306059967).

Exact result

• Added a deterministic upper-triangle bit encoding and exact decoder for finite undirected simple terminal gadgets, with terminals fixed as vertices 0..t-1.
• Canonical representatives are lexicographic minima over an explicitly validated terminal-permutation group and all internal-vertex relabelings.
• Restricted terminal groups remain restricted; the implementation does not silently replace them by a full symmetric group.
• Exhaustive audit: all 64 labeled graphs with three terminals plus one internal vertex form exactly 20 S3 orbits; all 1,024 labeled graphs with four terminals plus one internal vertex form exactly 90 S4 orbits. Orbit cardinalities sum back to the complete labeled graph spaces.
• The complete local suite passed 806 tests (1 skipped).

Still unresolved

No terminal-connectivity probability polynomial, planarity/periodic-tiling certificate, graph-isomorphism acceleration beyond bounded relabeling, series/parallel reduction, self-duality/critical-manifold check, ranked search, or negative-result certificate is supplied. Keeping #13 open.

[LightChainr]

Claiming a bounded exact candidate-space census on branch tools/issue-13-bounded-gadget-census.

Scope:

• exhaust every labeled simple graph with three or four terminals and exactly one internal vertex;
• quotient by the full terminal symmetric group using the canonical graph encoding already on main;
• freeze orbit multiplicities and edge-count histograms;
• apply only explicit structural filters: connected carrier, and a stricter degree->=3 internal-vertex filter that removes leaf/series-like one-internal-vertex cases;
• emit a machine-readable exact census and deterministic regression tests, including reconstruction of the complete labeled space.

Out of scope: probability polynomials, planarity/tiling certification, general series-parallel reduction, self-duality, critical-manifold equations, ranking by threshold proximity, optimization, or any new percolation bound. Latest main, all open PRs, and branch searches for gadget/enumeration/series work show no overlapping implementation. The parent issue will remain open.

[LightChainr]

Bounded exact candidate-space census completed in PR #386, squash-merged as e7d63b4c08cfb63a9c027959087c8beaa6228fdf after complete CI passed (C++17 and Python 3.9/3.11/3.13; run 33307690112).

Exact census

• Three terminals plus one internal vertex: all 64 labeled simple graphs form exactly 20 full-terminal-symmetry orbits; 11 have connected carriers, and 4 additionally have internal degree at least 3.
• Four terminals plus one internal vertex: all 1,024 labeled simple graphs form exactly 90 orbits; 58 have connected carriers, and 27 additionally have internal degree at least 3.
• Exact orbit multiplicity histograms reconstruct the complete labeled spaces (sums 64 and 1,024), and full/connected/degree-filtered edge-count histograms are frozen.
• The stricter filter removes isolated, leaf, and degree-two one-internal-vertex cases; it is explicitly not presented as a general series-parallel irreducibility test.

Validation: focused tests 7/7, locally available combined regression tests 64/64, exact artifact validation and Python compilation passed, then the complete repository CI matrix passed.

Still unresolved

No probability polynomial, planarity or periodic-tiling certificate, general series-parallel reduction, self-duality/critical-manifold equation, candidate ranking, optimization, algebraic threshold sequence, negative-result certificate beyond this bound, or percolation bound is supplied. Keeping #13 open.

[LightChainr]

Claiming the first of three bounded follow-ups: an independent Burnside orbit-count certificate on branch exact/issue-13-gadget-burnside.

Scope: for three/four terminals plus one fixed internal vertex, derive the induced permutation action on edge slots; compute fixed-graph counts and edge-refined orbit polynomials by conjugacy class; average with Burnside's lemma; and require exact agreement with the canonical census counts and edge histograms already on main.

This checker does not call graph canonicalization when deriving its Burnside result, so it is an independent counting route. Out of scope remain connectivity filtering, probability polynomials, planarity/tiling, self-duality, ranking, critical thresholds, or bounds. No open PR or branch matches Burnside work; #13 remains open.

[LightChainr]

Independent Burnside subtask completed in PR #387, squash-merged as 75da8839446afff472a3d96ec572a96e55707278 after full CI passed (C++17 and Python 3.9/3.11/3.13; run 33308542558).

Exact result: induced edge-cycle fixed-graph polynomials give fixed-point sums 120 for S3 and 2160 for S4; division by group orders 6 and 24 yields exactly 20 and 90 graph orbits. Coefficient-wise Burnside averaging gives edge-count histograms [1,2,4,6,4,2,1] and [1,2,5,11,17,18,17,11,5,2,1], exactly matching the canonical census through an independent derivation.

Validation: focused plus adjacent tests 14/14, artifact validator and compilation passed, complete CI green.

Unresolved boundary: no connectivity filter, probability, planarity/tiling, self-duality, ranking, threshold, or bound result. Keeping #13 open.

[LightChainr]

Claiming the second bounded follow-up: an exact complement-involution certificate on branch exact/issue-13-gadget-complement.

Scope: prove computationally and encode that graph complementation commutes with the explicit terminal permutation action; map every canonical one-internal-vertex orbit to the canonical orbit of its complement; certify involution, edge-count reversal, the palindromic orbit histogram, and exact self-complementary versus paired-orbit counts for t=3,4.

Out of scope: connected-filter preservation (complement need not preserve it), probability duality, planar duality, periodic tilings, self-dual critical manifolds, ranking, thresholds, or bounds. No open PR or branch matches complement work. #13 remains open.

[LightChainr]

Production use of the bounded census: exact three-terminal balance-root screen

Branch analysis/p13-three-terminal-balance-screen-20260830; protocol freeze c0bbe9c; result 05514ba. No PR or merge.

The terminal-reliability evaluator was applied to all 11 connected three-terminal plus one-internal-vertex canonical orbits. Every candidate now has exact Bernstein counts, a primitive integer power polynomial, and a 192-bit dyadic isolation of the unique formal root of P(all connected)=P(all separate).

The four frozen primary candidates, with internal degree at least three, have roots:

• three-spoke star: 0.6527036446661393;
• star plus one terminal edge: 0.5;
• star plus two terminal edges: 0.3745431865507393;
• complete graph: 0.2928932188134525.

The closest of all 11 is the reducible degree-two path, with exact equation p^2+p-1=0 and root 0.6180339887498948; it is still 0.0252879379602948 from the square-site reference. The closest primary candidate is the familiar star, distance 0.0599575938765393.

Decision: this bounded homogeneous independent-bond space has now been used rather than only enumerated, and it contains no close or structurally new square-site mechanism. Further work should change mechanism: correlated cells, more internal vertices with an irreducibility reduction, or an actual periodic critical-polynomial basis.

Boundary: no tiling, dual-cell map, square-site law preservation, self-dual embedding, critical manifold, bound, convergence sequence, or exact square-site threshold is certified. Issue #13 remains open. Focused tests: 18/18 across the new score, bounded census, and reliability evaluator; stored checksums pass.