[PolyOpt] 🌠 Wander/漫步者: Issue #230 Certify Two-Sided XXZ Thermodynamic Energy Bounds

[JunkaiWang-TheoPhy]

Proof is the narrow strip of sunlight that survives between two honest shadows.

证明，是两道诚实阴影之间幸存的那一线阳光。

第三幕 · 在地图尽头重新发问

第十一章 · 两道阴影之间的窄光

当近似值已经足够接近，证明仍要求我们同时守住两侧。严格下界与严格上界投下两道诚实的阴影，

真正能够带走的数字，只存在于它们之间。

← 上一幕：看懂棋盘，而不是绕开它     ·     下一章：过去从未离开这间屋子 →

Team

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

Wander certified frontier for Issue #230

Team	Wander (漫步者)
Challenge	#230: certified thermodynamic energy-density bounds vs. Bethe ansatz
Formal status	Validated certificate; literature-record comparison is the next proof gate

Headline certificate

For the spin-1/2 XXX normalization

$$ h=(XX+YY+ZZ)/4,\qquad e_{\mathrm B}=\frac14-\log 2, $$

the strongest self-contained payload proves

$$ \boxed{-0.443976567\le e_0\le -0.4428702958784947210360110613724028607783}. $$

The exact width is

$$ 0.0011062711215052789639889386275971392217, $$

and the interval contains the independently outward-rounded Bethe enclosure.
The lower endpoint is a depth-47, bond-6, native-U(1)-blocked RG dual; the upper
endpoint is an exact rational bond-32 MPS contraction over a 1,000-site block
with explicit boundaries. The public verifier reconstructs every proof object.

Why this advances the certified stack

• Native U(1) RG: D=6, depth=12 contracts from 93,329 dense variables to
  6,882 charge-block variables — 7.4% retained and 13.6x compression — while
  dense/block objective, derivative, Hessian, lift, and slack equivalence remain
  regression-tested.
• Solver-to-proof recovery: strict/zero-margin interpolation, rational
  reconstruction, exact target repair, and charge-block LDL convert numerical
  duals into independently replayable mathematical witnesses.
• Exact thermodynamic upper engine: integer FLINT contraction extends the
  same rational MPS from 1,000 to 16,000 sites and reduces its independently
  measured upper gap by 3.20x.
• Sprint-extension prototype: fixed-point residuals, RDM Hermiticity/trace/PSD,
  local-spectrum bounds, staged D10/D14 promotion, and saved-dual direct freeze
  concentrate deep certification on physically valid, reproducible candidates.
  These research-workspace prototypes are provenance-documented separately and
  are outside the current formal self-contained certificate package.
• Calibration frontier: 27 compact certificates cover nine anisotropies and
  three levels, allowing endpoint-by-endpoint analysis of symmetry, RG depth,
  and upper construction.

Challenge requirements

Issue #230 defines success as Bethe containment at every level plus improvement
over the best normalization-matched published rigorous Heisenberg-chain bound at
the top computable level. It separately defines valid but wider intervals and
constraint-family profiling as a useful certificate frontier. Every published level
passes containment, and this PR delivers that complete calibration dataset. The
3e-4 sprint target is an internal engineering gate; the official record gate
is the literature comparison above.

Audited deliverables

• Chinese technical report: Markdown, LaTeX, and visually audited PDF;
• exact-decimal certificate-summary.csv and record-gate.json;
• SHA-256 DATA_MANIFEST.txt;
• 1k–16k upper-contraction-frontier.csv with provenance;
• SPRINT_EXTENSION_PROVENANCE.txt defining the research-prototype boundary;
• four paired PDF/PNG evidence figures;
• independent verifier, certificate schema, tests, and reproduction commands.

The fast implementation/equivalence suite completes with 118 passed and five
expected numerical-solver accuracy warnings in candidate-generation tests. All
27 compact certificates audit successfully, with monotone lower and upper
envelopes.

The selected level-47 payload has SHA-256
1f7c684b3c2f62506f9b30f11e80197045f26a3cfcf464b608f1d2cab998e0c7.
Its full clean-checkout verification completed with PASS in 2256.59 seconds,
with peak memory 180,289,536 bytes. Bethe values are reserved for the final
containment test, preserving an auditable separation between search and proof.
DATA_MANIFEST.txt uses a documented four-column format and is validated by
tests/test_delivery.py.

Reproduce

cd tracks/polyopt/solutions/WangTheoPhys/issue230
python3 -m venv .venv
.venv/bin/pip install -e . pytest
.venv/bin/pytest -q --ignore=tests/test_published_outputs.py
.venv/bin/xxzcert verify \
  outputs/final/xxx_best/level_47_rg_d6_mps_d32_block_1000.json

Delivers the certified calibration objective of #230 and establishes a
reusable proof-producing foundation for the normalization-matched record gate.

[JunkaiWang-TheoPhy]

@GiggleLiu the Ranger submission now includes the complete Issue #230 research package for review:

• a self-contained depth-47 XXX certificate with independently replayed PASS;
• 27 verified XXZ calibration certificates across nine anisotropies;
• a Chinese technical report in PDF/LaTeX/Markdown;
• exact-decimal summary, SHA-256 manifest, and record-gate arithmetic;
• four evidence figures and a 1k–16k exact rational-MPS performance frontier;
• native U(1) RG compression (93,329 → 6,882 variables, 13.6x) and solver-to-proof recovery with exact blockwise LDL;
• provenance-documented sprint prototypes for physics-gated staged tensor promotion and saved-dual direct certificate freeze; these are clearly separated from the formal self-contained package.

Every published level contains the independent Bethe enclosure, completing the challenge's hope-signal calibration objective. The official success gate is preserved exactly as written: improvement over the best normalization-matched published rigorous Heisenberg bound at the top computable level. Our 3e-4 sprint target is labeled as an internal engineering gate.

The report and machine-readable artifacts make the formal certificate, the 16k performance frontier, and the next literature-record gate independently distinguishable and auditable. We would welcome review of the normalization, proof-recovery chain, and challenge-gate interpretation.