Exact identifiability controls: Jordan closure and a C4-protected five-state quotient

[LightChainr]

2026-08-31 上下文恢复（原提案保留在下）

• 本 PR 的有限噪声 ordinary→Jordan 非分离定理与 C4 五态条件障碍保留（head b5c761a）；它们是声明模型类的精确结果，不是已应用于 P250 的物理 Jordan 识别。
• 正文 R2“已选定”仅对应早期 radius5 受限零线 11130ae。后续 radius6 2646e8f 中 rank8只是首个相容截断阶数，并未得到 flat extension。
• 再后的 93eaab1 已拒绝完整 rank8 R2 kernel-plane，df57b69 的 old4+fresh5 联合 annihilation/extension 拒绝五个预列映射（R2 p=3.55e−8）。因此“固定 R2 下继续同一 flat-extension”不是当前已选路径；原阶段结果不删除。
• 应用本 PR 的 C4 定理仍需真实满足 Laurent action/source/flatness 假设；应比较当前完整谱或明确新机制，而非把合成五态模型当 P250 状态。
• 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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Summary

Related: #218, #249, #250, #370; interpretation boundary for PR #273. No dependency on an unmerged code branch.

This adds two complementary, self-contained model results rather than another fit or a new production campaign:

1. Finite-noise nonseparation. The common ordinary generator G_epsilon=[[0,1],[epsilon^2,0]] has two distinct real eigenvalues, while exp(t G_epsilon) converges to a nontrivial Jordan semigroup at order epsilon^2. Exact commutation and composition survive throughout. A finite positive-radius confidence set containing an interior Jordan prediction therefore cannot exclude all arbitrarily near-colliding ordinary alternatives, provided the same hard constraints/covariance support are respected. This makes [P2 method/no-go] Proof-carrying model elimination for low-rank algebraic realizations #370's spectral-separation/constraint requirement explicit; it does not invalidate Reveal frozen norm-4 q=2 versus Jordan production score #273's frozen q=2 rejection.

2. A symmetry-protected escape. A reduced cyclic length-five Laurent quotient with an invariant cyclic source and R^4=I, R U R^-1=V, R V R^-1=U^-1 must have C4 character (5,1,1,1), or multiplicities (2,1,1,1). The proof is an exact spectral-orbit count. The explicit nonreduced algebra C[x,y]/((x,y)^3,x^2+y^2) realizes the same Laurent symmetry with character (5,-1,1,-1), or multiplicities (1,1,2,1). It cannot be smoothed to a reduced five-point quotient while preserving those exact symmetry/source hypotheses.

The P250 radius-five result at 11130ae is explicitly acknowledged: R2+conjugation is already selected under the frozen alpha=.01 rule, and that vote is not reopened. Applying the new character test requires a higher-degree flat extension, a cyclic source and a verified internal Laurent C4 action; none is silently inferred from truncated rank-five compatibility.

Change class and claim level

• Conditional mathematical theorems with complete proofs and exact rational controls.
• C1 executable controls; independently checkable proof/certificate content concerns only the declared model classes.
• Application to Matching One remains C0 until flatness/source/symmetry hypotheses are established.
• Not a claim of an identified CFT field, physical Jordan block, exponent x=21/4, or a value of p_c.

Files

• notes/jordan-identifiability-and-c4-five-state-obstruction.md
• Two standard-library oracle scripts, each with a separate stored-certificate verifier.
• Two unittest modules, 28 tests in total.
• results/jordan-nonseparation/: exact witnesses, C4 controls, verification outputs, test log, source hashes and metadata.

Validation actually run

In the isolated new contribution bundle on CPython 3.13.5:

python -m compileall -q scripts tests
python scripts/jordan_nonseparation.py --verify results/jordan-nonseparation/latest.json
python scripts/c4_five_state_obstruction.py --verify results/jordan-nonseparation/c4-five-state.json
python -m unittest discover -s tests -p 'test_*.py' -v

Results: 28/28 tests passed, six ordinary witnesses across 63 finite-word contexts each, 378 exact error inequalities, and 20 exact C4 construction checks. Both verifiers reject deliberately corrupted inputs. No floating-point matrix diagonalization, optimizer, Monte Carlo, or repository production data are used.

The complete repository test suite was not run. After integration the two targeted unittest patterns in the note reproduce the new tests without implying a full-suite result.

Provenance and chronology

Initial research baseline: 4d70c1787ff97dbb98cb5e96022f947bb8fad97e. Integration parent: 0dc4430ed3a5633bc95320378dea59104cfc7bd8 (2026-08-30 10:25:43 UTC). The late P250 note at 11130ae and issue comment 5467890317 were read before this contribution.

All 12 uploaded file blob hashes were checked against the locally tested bundle by rebuilding the same tree from the expected blob SHAs; tree SHA remained 85cc3ba9cfb440c1196b1817ba68132d396c72d5.

No existing source, frozen protocol, raw result, status ledger or historical prediction is overwritten. The new results are synthetic model controls, not additional independent evidence from any existing random block.

Limitations / next discriminator

The C4 obstruction assumes a global Laurent action, not the invalid substitution of negative translation by minus a matrix. The source character and cyclicity matter; multi-source/noncyclic or differently charged modules require a separately typed theorem. Matching the allowed character does not prove reducedness. The constructed nonreduced control has nilpotency order three and is not asserted to be the actual P250 state.

The proposed next step is the existing higher-degree flat-extension/shared-translation calculation in the now-fixed R2 gauge, followed by a character-feasibility test where the premises hold—not another scalar logarithmic fit.