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

tracks/mps/solutions/Wander-276/PR_BODY.md

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+<!-- wander-artwork -->
+![Wander — Issue #276](https://raw.githubusercontent.com/JunkaiWang-TheoPhy/quantum.harness/664506886413b68cdbc5a362a6075c78fd5d2c46/docs/showcase/ranger-archive/assets/missions-v2/07-quantum-geometry-curved-state-space-v2.png)
+
+> **EN** The spectrum is silent; the geometry is not.
+>
+> **中文** 能谱沉默，几何仍然说话。
+<!-- /wander-artwork -->
+
+## Team
+
+| Field | Value |
+|---|---|
+| **Team name** | Wander |
+| **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](https://github.com/JunkaiWang-TheoPhy/quantum.harness/blob/codex/issue-276-quantum-geometry/tracks/mps/solutions/Wander-276/README.md)
+- [Research and reproduction guide](https://github.com/JunkaiWang-TheoPhy/quantum.harness/blob/codex/issue-276-quantum-geometry/tracks/mps/solutions/Wander-276/research/README.md)
+- [Letter source](https://github.com/JunkaiWang-TheoPhy/quantum.harness/blob/codex/issue-276-quantum-geometry/tracks/mps/solutions/Wander-276/research/overleaf_sync/cohomological_geometric_eth/main.tex)
+- [Supplement source](https://github.com/JunkaiWang-TheoPhy/quantum.harness/blob/codex/issue-276-quantum-geometry/tracks/mps/solutions/Wander-276/research/overleaf_sync/cohomological_geometric_eth/supplement.tex)
+- [Compiled Letter](https://github.com/JunkaiWang-TheoPhy/quantum.harness/blob/codex/issue-276-quantum-geometry/tracks/mps/solutions/Wander-276/research/01_task_folder/task_05/script/output/response_complex_memory_v7.pdf)
+- [Compiled Supplemental Material](https://github.com/JunkaiWang-TheoPhy/quantum.harness/blob/codex/issue-276-quantum-geometry/tracks/mps/solutions/Wander-276/research/01_task_folder/task_05/script/output/response_complex_memory_supplement_v7.pdf)
+- [Machine-generated result report](https://github.com/JunkaiWang-TheoPhy/quantum.harness/blob/codex/issue-276-quantum-geometry/tracks/mps/solutions/Wander-276/research/01_task_folder/task_05/script/output/susy_hodge_geometric_eth_report_v7.md)
+- [Scientific ceiling and literature memo](https://github.com/JunkaiWang-TheoPhy/quantum.harness/blob/codex/issue-276-quantum-geometry/tracks/mps/solutions/Wander-276/research/docs/2026-08-01-scientific-ceiling-strategy.md)
+- [Public research branch](https://github.com/JunkaiWang-TheoPhy/Chaos-of-Quantum-Geometry/tree/codex/task-05-geometric-chaos-baseline)
+
+## Verification
+
+```bash
+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.

tracks/mps/solutions/Wander-276/README.md

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+# Wander — Issue #276: Chaos in Exactly Degenerate Quantum Manifolds
+
+![Quantum geometry over an exactly degenerate manifold](https://raw.githubusercontent.com/JunkaiWang-TheoPhy/quantum.harness/664506886413b68cdbc5a362a6075c78fd5d2c46/docs/showcase/ranger-archive/assets/missions-v2/07-quantum-geometry-curved-state-space-v2.png)
+
+> **EN** The spectrum is silent; the geometry is not.
+>
+> **中文** 能谱沉默，几何仍然说话。
+
+## Team
+
+| Field | Value |
+|---|---|
+| **Team name** | Wander |
+| **Members** | Chenxi Wan, Yedi Shen, Junkai Wang |
+| **Contact email** | WangTheoPhys@outlook.com |
+
+## Challenge and Answer
+
+This submission addresses [QuantumBFS/quantum.harness#276](https://github.com/QuantumBFS/quantum.harness/issues/276): **What probes quantum chaos inside an exactly degenerate eigenspace?**
+
+The executable answer is the **non-Abelian quantum geometry of the protected projector over coupling space**. If (P(\lambda)) is the degenerate projector and (X_a=(1-P)\partial_aP), then
+
+$$\mathcal Q_{ab}=X_a^\dagger X_b,\qquad g_{ab}=\frac{\mathcal Q_{ab}+\mathcal Q_{ba}}{2},\qquad F_{ab}=i(\mathcal Q_{ab}-\mathcal Q_{ba}).$$
+
+Ordinary level statistics is undefined inside a flat multiplet, but (g), (F), gauge-invariant four-channel contractions, Chern numbers, and Wilson holonomy remain nontrivial.
+
+## Independent Protection Mechanisms
+
+The release no longer rests on one fractional-quantum-Hall example.
+
+| Mechanism | Protected manifold | Exact response structure | Role |
+|---|---|---|---|
+| Kapit–Mueller/Laughlin parent (H=B^\dagger B) | Frustration-free, gapped zero modes | One-sided response (X=-H_\perp^+B^\dagger\delta BP) | Local topological benchmark, Jacobi curvature, fixed-Chern holonomy |
+| Generic cubic (\mathcal N=2) SYK (H=\{Q,Q^\dagger\}) | Charge-resolved harmonic cohomology | Orthogonal exact/coexact response (X=X_-\oplus X_+) | Independent supersymmetric mechanism and sealed covariance test |
+
+Nilpotency gives (X_-^\dagger X_+=0) exactly. The two-sided Hodge decomposition is therefore a physical response-complex structure, not a basis rewrite of the Laughlin calculation.
+
+## Registered Scientific Test
+
+For eight coupling tangents, the code whitens the channel covariance and evaluates the gauge-invariant tensor
+
+$$\mathcal T_{abcd}=\frac1D\operatorname{Tr}(\widehat X_a^\dagger\widehat X_b\widehat X_c^\dagger\widehat X_d).$$
+
+The (N=8,10,12) pilot uses complete disorder realizations as the uncertainty unit. A held-out (N=14) calculation freezes two prediction models before any four-channel outcome is opened:
+
+1. a collapsed separable covariance Gaussian null;
+2. an exact/coexact Hodge-resolved separable covariance Gaussian null.
+
+Safe covariates and predictions are SHA-256 sealed. Unsealing is an explicit, separate command and is never launched by the scheduler dependency chain. The seal was independently checked before opening, and the frozen result is:
+
+| $N=14$ 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 nulls miss both primary sectors, selecting `cohomological_non_gaussian_class`.
+
+## Claim Boundary
+
+This submission establishes an **independent model/operator class** for protected quantum-geometric chaos and finite-size $N=8$--$14$ four-channel response memory beyond two preregistered separable covariance nulls. It does **not** claim asymptotic Geometric ETH, conventional energy-resolved ETH, real-time chaos, or a thermodynamic-limit theorem. Null rejection does not by itself prove intrinsic non-Gaussianity after matching every entrywise nonseparable covariance.
+
+## Deliverables
+
+- [Research and reproduction guide](research/README.md)
+- [Task-level guide](research/01_task_folder/task_05/README.md)
+- [Letter source](research/overleaf_sync/cohomological_geometric_eth/main.tex)
+- [Supplement source](research/overleaf_sync/cohomological_geometric_eth/supplement.tex)
+- [Compiled Letter](research/01_task_folder/task_05/script/output/response_complex_memory_v7.pdf)
+- [Compiled Supplemental Material](research/01_task_folder/task_05/script/output/response_complex_memory_supplement_v7.pdf)
+- [Exact-data result report](research/01_task_folder/task_05/script/output/susy_hodge_geometric_eth_report_v7.md)
+- [Scientific ceiling and literature memo](research/docs/2026-08-01-scientific-ceiling-strategy.md)
+- [Public research branch](https://github.com/JunkaiWang-TheoPhy/Chaos-of-Quantum-Geometry/tree/codex/task-05-geometric-chaos-baseline)
+
+## Verification
+
+```bash
+cd tracks/mps/solutions/Wander-276
+bash verify.sh
+```
+
+The verification path checks the exact Hodge identity, analytic controls, pilot-grid completeness, outcome-blind sealing, artifact hashes, frozen branch selection, figure provenance, and byte-exact Letter/Supplement delivery.
+
+## Next Scientific Gate
+
+The finite-size result is designed to expose the next theorem rather than hide it. The two high-ceiling targets are:
+
+- an asymptotic concentration/scaling law for the covariance-controlled response tensor;
+- a spatially local nilpotent-supercharge model with a stable zero-mode count, open gap, and moving cohomological projector.
+
+These are journal-ceiling gates, not assumptions built into the current claim.