[qmc] Genshin_Impact: polyhedral sign-free generator supports beyond fixed metrics

[Mao-Kexiang]

Polyhedral sign-free generator supports beyond fixed metrics

Team

Team name	Genshin_Impact
Members	Kexiang Mao (@Mao-Kexiang)

Challenge

Row
Challenge	Find structured Gaussian-vertex sets with det(I+T)>=0 for arbitrary words beyond split-orthogonal and fixed-metric sufficient principles, then realize one as an interacting QMC weight.
Catalog issue	QuantumBFS/quantum.harness issue #121
Pull request	PR #261
Track	tracks/qmc/

Bottom line

The algebraic construction, determinant/Fock oracles, proof, reduction audit, and small-instance physical benchmark are complete. The original four-site thermodynamic benchmark alone does not establish a useful many-body Monte Carlo system; the new arbitrary-L two-dimensional realization below is being validated under a separate staged protocol.

The result is an open two-orbit family of real 3 x 3 generators. A common polyhedral norm proves positivity at every depth. Exact certificates exclude every common quadratic contraction metric and the stated fixed, same-dimensional complex-CAR Wei classes. An S3 Fock-space twirl maps the same twelve vertices to a local Hermitian interacting Hamiltonian with an exact positive continuous-time Gaussian-vertex expansion.

The formal run at executable commit 9fe85a6 passed 224/224 preregistered cells and every exact, twirl, and physical stage. This meets the literal issue submission standard. It does not imply maintainer acceptance, publication priority, exclusion of every alternative representation, a finite-density breakthrough, or publication readiness.

Large-lattice status (2026-07-30)

The physical construction is now extended to an extensive two-dimensional system: one spinless orbital per site on a periodic triangular L by L lattice, with the interacting S3-twirled triangle vertices on every up and down triangle. It is defined for arbitrary L and the frozen grid reaches L=16, N=256, so it is not an isolated three-mode or four-site model.

The frozen v1 small-lattice validation at executable commit 382e64e completed 32/32 chains and 8/8 exact-diagonalization fixtures. All eight cells had zero negative determinants and zero zero-weight failures; seven cells passed every ED observable. One real-space Green-function leaf and three upper acceptance-cut leaves failed, so v1 remains INCONCLUSIVE and no large-production result is claimed.

Commit 5bf5905 froze v2 before any v2 data. The completed v2 audit has 32/32 chains, 8/8 positivity cells, zero negative or zero-weight events, and the old real-space discrepancy now passes strongly. One narrowly localized momentum-space one-body leaf at L=2, beta=4 exceeds its frozen allowance, so v2 also remains INCONCLUSIVE. A post-hoc reconstruction from the stored real-space traces finds a large autocorrelation correction, but it is diagnostic only and does not regrade v2. See the v2 preregistration and the completed v2 audit comment. The frozen v3 G1 run at commit ec62842 now passes all 32 chains, all 8 positivity cells, and all 160 ED comparisons. The G2 pilot completed 12 chains through N=144 with zero negative or zero weights, but L=8 and L=12 failed the frozen mixing and warmup gates, so G2 remains INCONCLUSIVE and G3/G4 are correctly blocked. N=256 has a passing rank-3 kernel benchmark and an advisory resource projection, not completed production observables. See the final v3 audit comment and the illustrated Chinese report. Finite-density and publication claims remain open.

1. Main theorem

For epsilon>0 and kappa>0, define

A(epsilon,kappa) =
[ -1-epsilon-kappa     1          -epsilon ]
[       0          -1-kappa          1     ]
[       2              0          -2-kappa ]

S=diag(1,1,-1),  B=SAS,
C=S3 orbit of A union S3 orbit of B.

On the open region

epsilon>0,  kappa>0,  40 epsilon+59 kappa<2,

main_theorem.md proves:

1. Every X in C has logarithmic infinity norm mu_infinity(X)=-kappa. Every positive-time word is a strict contraction on the isolated three-mode space, so det(I+T)>0; locally embedded words obey det(I+T)>=0 at arbitrary depth.
2. No common H>0 satisfies X^T H+HX<=0 for all X in C. At epsilon=1/100 and kappa=1/1000, A requires r<=-1541/24791 and B requires r>=1541/42609 after permutation averaging.
3. The twelve generators span M_3(R).
4. The number-conserving Nambu support cannot enter the compared fixed, same-dimensional complex-CAR Wei semigroups under one fixed allowed basis change. The statement includes sufficiently small alternate logarithms of the same discrete support.
5. Section 6 gives a seven-parameter signed directed-triangle design cone, so the displayed rational matrix is not an isolated fitted point.

The determinant step is short: eigenvalues of a real contraction lie in the closed unit disk; real eigenvalues contribute 1+lambda>=0 and nonreal conjugate pairs contribute |1+lambda|^2.

2. Interacting physical realization

For X=A or B, define

M_X(s)=(1/6) sum_(sigma in S3)
       exp[s c^dagger(P_sigma X P_sigma^T)c].

The complete twirl is Hermitian although a resolved Gaussian vertex need not be. At the rational interior point and sufficiently small s>0, both twirls are interacting and non-Gaussian. Positive couplings on overlapping triples define

H_bar=sum_(Delta,X) g_(Delta,X)[I-M_(Delta,X)].

Resolving the continuous-time expansion of exp(-beta H_bar) gives nonnegative scalar activities and configuration weight det(I+U_m...U_1)>=0 at every order. The formal four-site benchmark used overlapping triples (1,2,3) and (2,3,4), s=1/10, g_A=g_B=1/4, mu=0, and beta=1/4,1/2,1,2.

All 16,384 sampled physical configurations were nonnegative. Exact diagonalization and the positive Poisson estimator agreed within every preregistered allowance; the minimum sampled determinant was 4.330910819303328. See physical_realization.md and verification_record.md.

This is a local, spinless, number-conserving, genuinely interacting model and an implementable continuous-time expansion. It is not a standard two-body Hubbard auxiliary-field decomposition. Its mu=0 ground state is the vacuum; positive chemical potential and canonical finite-density positivity remain open.

3. Novelty boundary

Established ingredients include logarithmic norms, polyhedral Lyapunov functions, contraction semigroups, compound matrices, cone preservation, and group twirling. The candidate contribution is their QMC combination:

• an explicit nonzero-volume arbitrary-depth determinant-positive family;
• exact separation from every common ellipsoidal contraction certificate;
• a full-span odd-dimensional obstruction for the same support against the stated fixed complex-CAR classes;
• an interacting Hermitian realization using only certified vertices.

A targeted primary-source audit found no direct QMC use of this common-polyhedral construction. Absence of a search hit is not proof of priority. PR #259's total-nonnegative/Jordan-Wigner route and split-orthogonal, Kramers, Majorana, pseudo-unitary, and Wei results are treated as prior or competing mechanisms. See novelty_audit.md and reduction_checklist.md.

4. Formal verification result

The compact immutable record is verification_record.md. Full row-level artifacts remain on t02-server at

tracks/qmc/results/Genshin_Impact-121/20260730-021845-9fe85a6/

That directory also contains the standalone challenge report at
challenge-report/report.html and scheduler provenance at slurm_jobs.json.
A compact public copy of the formal evidence, excluding the 9.6 MB row-level
sample table and per-cell files, is tracked in
formal_run_snapshot/.

Check	Result
Protocol	ae54430bfb17790c197fabed523138ed6ba3a632881978b5665367e1517a2e20
Cells / core random words	224/224 pass; 40,320
Candidate A/B words	35,840/35,840 positive
Direct Fock checks	336; maximum absolute error 5.400124791776761e-13
High-precision rebuilds	672
Exact-zero controls	448 expected; 0 unexpected inconclusive
Physical Poisson words	16,384/16,384 nonnegative
Total randomized words	56,704
Exact certificates / twirl / physical benchmark	pass
Regression suite	39 tests pass

Sampling audits implementation and boundary behavior; it is not the proof. The exact common-norm theorem proves arbitrary-depth positivity, and exact rational certificates audit the separation claims.

5. Reproduce

From the repository root:

python -m pip install -r tracks/qmc/solutions/Genshin_Impact-121/requirements.txt
python -m pytest -q \
  tracks/qmc/solutions/Genshin_Impact-121/test_sign_problem_hunter.py \
  tracks/qmc/solutions/Genshin_Impact-121/test_issue121_verification.py

python tracks/qmc/solutions/Genshin_Impact-121/issue121_verification.py \
  --manifest tracks/qmc/solutions/Genshin_Impact-121/issue121_full_run.json \
  --validate-only

output=tracks/qmc/results/Genshin_Impact-121/REPRODUCE-$(date -u +%Y%m%d-%H%M%S)
python tracks/qmc/solutions/Genshin_Impact-121/issue121_verification.py \
  --manifest tracks/qmc/solutions/Genshin_Impact-121/issue121_full_run.json \
  --output "$output"
sha256sum "$output"/{manifest.json,report.json,report.md,samples.csv,COMPLETE}

The runner writes cells atomically, binds resume state to the protocol and environment, and emits COMPLETE only after every stage passes. On t02-server, submit the full command as a 1-CPU, 2-GB, no-GPU Slurm job.

6. File map

File	Role
main_theorem.md	Open A/B theorem, exact separation, fixed-CAR boundary, and seven-parameter cone.
physical_realization.md	S3-twirl Hamiltonian, positive CT expansion, sampler, and benchmark.
verification_record.md	Formal-run hashes, counts, environment, results, and claim boundary.
formal_run_snapshot/	Tracked compact copy of the materialized manifest, formal reports, exact/twirl/physical evidence, Slurm metadata, COMPLETE, and challenge report; the source manifest remains issue121_full_run.json and row-level samples remain remote by hash.
issue121_full_run.json	Preregistered manifest.
issue121_verification.py	Protocol-bound exact, randomized, Fock, twirl, and physical verifier.
test_issue121_verification.py	Formal verifier tests.
sign_problem_hunter.py	Independent split-orthogonal/Wang-2015 determinant and Fock oracle.
test_sign_problem_hunter.py	Baseline oracle tests.
novelty_audit.md	Primary-source novelty audit and priority boundary.
reduction_checklist.md	Known-mechanism and physical-realizability filter.
external_review_draft.md	Expert-review draft with permanent source links.
finite_density_extension.md	Optional Perron-compound extension and non-itinerant limitation.
wang2015_run_template.json	Earlier Wang-2015 baseline manifest.

7. Issue #121 completion audit

Explicit issue gate	Status	Evidence
1. Oracle plus known positive/negative anchors	Completed	Independent determinant/Fock identity; O(n,n), four O(1,1) components, exact -4/3 negative anchor, and 336 Fock checks.
2. State-of-art and reduction checklist	Completed	Targeted primary-source audit and split-orthogonal, Kramers, Majorana, Wei, one-dimensional, stoquastic, and physical filters.
3. New arbitrary-depth structure, tests, and proof	Completed	Open A/B theorem; exact proof; 35,840 candidate words over d=3,4,6,8,12 and depths 1-64; exact separation certificates.
4. Interacting physical determinant weight	Completed	Local overlapping-triple H_bar, positive CT expansion, ED/Poisson benchmark, and 16,384 nonnegative physical configurations.
5. Full reporting and reproducibility	Completed	Protocol-bound manifest, hashes, atomic resume, COMPLETE, tracked verification record, and 39 tests.
6. Public endgame draft	Completed as a draft	external_review_draft.md is ready for expert circulation; it has not been posted to MathOverflow or arXiv.

These rows mean PR #261 supplies every artifact explicitly requested by issue #121. They do not prejudge maintainer review. Finite-density physics is special credit rather than a hard gate and remains open; priority and independent specialist proof review also remain open.

8. Suggested reading order

1. main_theorem.md, Sections 1-3: common polyhedral positivity and the no-ellipsoid certificate.
2. main_theorem.md, Section 5: the precise fixed-CAR Wei separation and its limits.
3. physical_realization.md, Sections 1-4: Fock twirl, CT expansion, sampler, and benchmark.
4. verification_record.md, then issue121_full_run.json: registered checks, counts, and hashes.
5. novelty_audit.md and reduction_checklist.md: prior art and defensible novelty.
6. finite_density_extension.md: optional next research direction, not part of the solved hard gate.

[Mao-Kexiang]

Large-lattice interacting-system update (2026-07-30)

This PR now contains a genuinely extensive two-dimensional realization, not an isolated three-mode benchmark: one spinless orbital per site on a periodic triangular L by L lattice, with the local S3-twirled interacting triangle vertices placed on every up and down triangle (2N overlapping vertices). The construction is defined for arbitrary L; the frozen production grid reaches L=16, N=256. It remains a finite-temperature, mu=0 result, not a finite-density ground-state solution.

Frozen v1 evidence. Executable commit 382e64e; result root tracks/qmc/results/Genshin_Impact-121/20260730-large-lattice-382e64e. G0 passed; all 32/32 G1 chains and all 8 ED fixtures completed. Positivity passed in 8/8 cells with zero negative determinants and zero zero-weight failures. Seven of eight cells passed every ED observable. The sole ED leaf failure was L=3, beta=1/2, G(1,0): QMC 0.0032190527258243243 versus ED 0.003530271749985057. Three additional leaves were only upper acceptance-cut failures (rates 0.90196, 0.90739, 0.93531). Therefore the immutable v1 G1 verdict remains INCONCLUSIVE; no pilot or large production result is claimed from v1. Gate SHA256: 8d110898750476dde97f5ecfb605c16b9f301b4642e7e76e0c11063827e4e3ff.

Prospective v2, frozen before generating v2 data. Commit 5bf5905 adds an independent seed base, 300000 steps with 30000 warmup per G1 chain, complete real/imaginary per-measurement traces for every N<=9 displacement, rank-split R-hat plus bulk/tail ESS and autocorrelation-aware MCSE, and treats high acceptance as an efficiency diagnostic rather than a correctness failure. The final ED error bar is the maximum of correlated, between-chain, and naive estimates. Validation is green: 67 non-slow tests plus the slow CTQMC/ED test passed. See the non-retroactive v2 preregistration.

Claim boundary. This update closes the experts isolated-three-body objection at the level of the model and executable protocol: the Hamiltonian is a nontrivial overlapping 2D many-body system and the determinant certificate composes at arbitrary volume. It does not yet turn v1 into a pass, prove rapid mixing, establish finite-density physics, or supply completed N=256 Monte Carlo evidence. Those claims remain gated on prospective v2 G1, pilot scaling, and the full grid.

[Mao-Kexiang]

Completed v2 G1 audit: still INCONCLUSIVE, with one narrowly localized leaf

Slurm jobs: G0 43478; G1 probes 43482 and 43485; remaining arrays 43486 and 43501; independent audit 43516. All 32/32 independent 300000-step chains and all 8/8 ED fixtures completed. Every cell again has zero negative determinants and zero zero-weight failures; acceptance, convergence, rebuild residual, R-hat, bulk ESS, and tail ESS checks pass. The old v1 real-space G(1,0) discrepancy now passes strongly: absolute error 2.9723583887392158e-06 versus allowance 2.83145741373099e-04.

The only false leaf in v2 is L=2, beta=4, momentum q=(1,0), one_body: error 1.8844035057746789e-04, recorded MCSE 3.608558929286638e-05, and frozen allowance 1.8042794646433192e-04. Thus the immutable v2 G1 status is INCONCLUSIVE; pilot and production remain blocked.

A strictly post-hoc diagnostic, not a regrade, reconstructs that momentum trace exactly from the preregistered complete real-space traces. It reproduces the four recorded chain means and gives tau_int 42.49-46.32, bulk ESS 10625.98, split R-hat 1.000223, and correlated MCSE 3.6379896767951657e-04. This shows the remaining failure is consistent with the same omitted-autocorrelation problem in a Fourier-transformed observable, but v2 explicitly applied correlated MCSE only to real-space comparisons, so its verdict cannot be changed retroactively.

Result root: tracks/qmc/results/Genshin_Impact-121/20260730-large-lattice-5bf5905. G1 SHA256: d865048b10a88b51d9caefff158c80961c19bce9b5b453c86074a20084c7db73. The next defensible protocol must prospectively apply trace-based autocorrelation errors to every ED-compared observable, not selectively to one representation.

[Mao-Kexiang]

Chinese illustrated system report and prospectively frozen v3

Commit a6b091c adds a self-contained Chinese HTML introduction at system_introduction_zh.html. It includes inline SVG lattice, proof-flow, error-bar, and scaling diagrams; the local Fock decomposition; v1/v2 evidence; and explicit finite-density/publication boundaries. The file is 18.3 kB, contains three inline SVG figures and eight sections, and has SHA256 bcd82f51b7fd6f63c30f8916db010dc9dc4b9a5a95239a1e748163f06609b986.

The same commit freezes G1 v3 before any v3 data. v3 uses seed base 321000000, retains 300000 steps and the fixed 32-chain design, stores real/imaginary traces for every real-space and momentum ED observable (one_body, density_raw, density_mode), applies the same correlated/between-chain/naive maximum MCSE universally, and streams audit cells to bound memory. The frozen prior evidence and non-retroactivity rule are in g1_v3_preregistration.md. Validation before compute: 68 non-slow tests plus the slow CTQMC/ED test pass.

[Mao-Kexiang]

Final v3 validation and large-system pilot (2026-07-30)

The self-contained illustrated Chinese report is now tracked at system_introduction_zh.html, commit ec62842, SHA256 7383a71ee07b9656b71ff5513067f104022505d8b7498bf418ed5f6ee6625d49.

G1: PASS under the frozen v3 protocol

• 32/32 independent chains, 300000 steps per chain, four cold/hot chains per cell.
• 8/8 L=2,3 cells passed positivity; negative determinants=0, zero weights=0, zero-determinant failures=0.
• 160/160 ED complex-observable comparisons passed with the universal max of correlated, between-chain, and naive MCSE.
• Worst split R-hat=1.006396; minimum bulk ESS=1127.738; minimum tail ESS=3350.969.
• Acceptance range 0.661609 to 0.934882.
• Gate: tracks/qmc/results/Genshin_Impact-121/20260730-large-lattice-a6b091c/gates/G1.json
• Gate SHA256: bef1f76d7d63ccdcb14d2c061490762a000fe02e2c76b187e6d827a10d4fbd91.
• Slurm: probes 43744, remaining array 43800, independent audit 43821.

G2 large-system pilot: INCONCLUSIVE, with positivity intact

• 12/12 pilot chains completed at N=16,64,144; every pilot cell again had negative determinants=0, zero weights=0, and zero-determinant failures=0.
• L=4 passed convergence. L=8 and L=12 did not pass the frozen mixing and warmup gates.
• L=8 worst R-hat=1.010135 and minimum bulk ESS=465.908. L=12 worst R-hat=1.103364 and minimum bulk ESS=130.787.
• The L=12 beta=4 maximum tau_int was 408.09 measurements. The frozen 10000-step warmup is below the preregistered 50 tau requirement of about 204045 steps.
• Gate SHA256: 458dd04f985d75005587410230f375122cebd6b2f0a66ea74f186feafc3a9cd8.
• Slurm: pilot 43908, kernel benchmark 43917, audit 43922.

N=256 kernel evidence

The rank-3 update benchmark passed its 1e-9 correctness threshold through N=256. At N=256, median insertion and deletion speedups over full rebuild were 34.08x and 31.21x; maximum observed numerical error was below 7.8e-15 and fallback count was zero. The pilot resource model projects about 396 s and 104 MiB for one 100000-step N=256 chain, but this is an advisory projection, not a completed production observable run.

Claim boundary

This is strong evidence that the construction extends beyond an isolated three-body example to a genuine two-dimensional interacting lattice system and remains sign-free through the N=144 Monte Carlo pilot. It is not a completed G3/G4 large-grid result: slow mixing, not sign, blocks production. The never-satisfied downstream jobs were removed without stopping any healthy computation. Finite-density physics, literature priority, and publication readiness remain open.