[E.D.] 🌠 Wander/漫步者: Issue #276 Diagnose Chaos in Exactly Degenerate Quantum Manifolds

[JunkaiWang-TheoPhy]

The spectrum is silent; the geometry is not.

能谱沉默，几何仍然说话。

第三幕 · 在地图尽头重新发问

第十四章 · 当能谱沉默，几何开始说话

精确简并让能谱沉默，简并子空间的投影却仍在参数空间中弯曲、旋转，并沿闭合路径留下几何记忆。

当能量不再区分内部结构，非阿贝尔量子几何便接过了叙述。

← 上一章：玻璃上的雨记得旅程     ·     下一章：没有一粒沙知道河流 →

Team

Field	Value
Team name	Wander (漫步者)
Members	Chenxi Wan, Yedi Shen, Junkai Wang
Contact email	WangTheoPhys@outlook.com

Challenge

Addresses #276: What probes quantum chaos inside an exactly degenerate eigenspace?

This PR gives an executable answer: the probe is the non-Abelian quantum geometry of the protected projector over coupling space. Exact degeneracy silences internal level statistics, while the response amplitudes (X_a=(1-P)\partial_aP) generate the quantum metric, Berry curvature, gauge-invariant four-channel tensors, and global holonomy.

Scientific Advance

Layer	Result
Exact geometry	(\mathcal Q_{ab}=X_a^\dagger X_b), with metric and Berry curvature as its symmetric and antisymmetric parts
Independent mechanism	Generic cubic (\mathcal N=2) SYK supplies a charge-resolved cohomological BPS manifold independent of the Kapit–Mueller/Laughlin parent
Exact structure	Nilpotency produces orthogonal exact/coexact response branches (X=X_-\oplus X_+), (X_-^\dagger X_+=0)
Sealed test	The complete $N=8,10,12$ pilot and separately sealed, held-out $N=14$ four-channel test reject both registered separable covariance nulls
Controls	Decomposable-supercharge curvature atoms and a one-sided Laughlin Gaussian-null regression

The new scientific question is sharper than “is the Hamiltonian chaotic?” It asks whether the protected response tensor is controlled by registered two-point covariance data, or whether gauge-invariant four-point memory survives after that matching.

Outcome-Blind Protocol

The held-out calculation freezes two parameter-free prediction models before opening the outcome sidecars: a collapsed separable covariance null and an exact/coexact Hodge-resolved separable covariance null. Complete disorder realizations—not tangent entries or tensor components—are the uncertainty unit. No scheduler dependency invokes unsealing; seal validation and scoring require a separate explicit action.

Final held-out verdict: independent seal validation passed before explicit unsealing, and the frozen branch is cohomological_non_gaussian_class.

$N=14$ sparse sector	Physical median (95% bootstrap)	Collapsed null (97.5% prediction)	Hodge null (97.5% prediction)
Adjacent	0.301529 [0.291527, 0.312061]	[0.111789, 0.111852]	[0.112344, 0.112513]
Central	0.374993 [0.368980, 0.380473]	[0.111338, 0.111353]	[0.111333, 0.111348]

Both primary physical intervals are disjoint from both sealed prediction intervals. The prediction JSON was frozen at SHA-256 fc300dc7e4bdc1be157919e458ac868d3468533cce31108f23c9fba4f7e9f102; the independently recomputable inference artifact is 177643e07fc6cf210362fc1077070bd1f0ba316b6805042a626de3f96c55a627.

Claim Boundary

• Established: an independent supersymmetric protection mechanism, exact Hodge-resolved response identity, analytic controls, and finite-size $N=8$--$14$ response memory beyond both frozen separable covariance-null families.
• Not established: asymptotic Geometric ETH, thermalization, real-time chaos, or a thermodynamic-limit theorem.
• The selected null-rejection branch means memory beyond the frozen separable Hodge/collapsed covariance models; it does not prove non-Gaussianity relative to every possible entrywise covariance model.

Deliverables

• Harness solution overview
• Research and reproduction guide
• Letter source
• Supplement source
• Compiled Letter
• Compiled Supplemental Material
• Machine-generated result report
• Scientific ceiling and literature memo
• Public research branch

Verification

cd tracks/mps/solutions/Wander-276
bash verify.sh

The final verifier checks exact identities, controls, the complete pilot and held-out grids, the seal state machine, every compact hash, frozen branch recomputation, figure provenance, byte-exact Letter/Supplement PDFs, and 43 focused v7 tests.

@OkongOyangO, please review the protected-response interpretation, the exact/coexact identity, and the outcome-blind four-channel test as Wander's Issue #276 submission.

[JunkaiWang-TheoPhy]

@OkongOyangO This Wander submission provides the executable answer to Issue #276: non-Abelian projector geometry over coupling space remains an operative chaos probe inside an exactly degenerate eigenspace.

Suggested review path:

1. Verify the exact spectral-flatness identity and finite-Jacobi curvature law.
2. Audit the gauge-invariant four-channel tensor and the N=3,4,5 connected-component sequence.
3. Check the fixed-Chern bundle-isomorphism argument and Wilson-holonomy control.
4. Run bash tracks/mps/solutions/Wander-276/verify.sh for the 17 release checks and 38 focused tests.

The fastest entry points are the solution README, Figures 1/6/7, the 17-page article, and release_manifest_v1.json.