[QMC] 🌠 Wander/漫步者: Issue #158 Classify the Marginal 1/r⁴ XY Phase

[JunkaiWang-TheoPhy]

At first every tree stood alone; then the whole forest chose a direction.

起初每棵树都独自站立，后来整座森林选择了同一个方向。

第二幕 · 让计算穿过物质的风暴

第八章 · 森林选择方向之前

定理说森林不能永久朝向同一侧，有限尺寸数据却仿佛看见了秩序。我们沿着证明、模型约定与热力学外推逐层返回，

寻找枝干开始共同倾斜的真正原因。

← 上一章：用一把误差尺裁定未来     ·     下一章：同一阵风吹向两种手性 →

Team

Team	Wander (漫步者)
Members	Chenxi Wan, Yedi Shen, Junkai Wang
Contact email	WangTheoPhys@outlook.com
Track	qmc
Challenge	Addresses #158

Headline result

For the exact two-dimensional size-normalized minimum-image 1/r^4 interaction, every finite hard-spin O(n) model with n>=2, fixed T>0, and sufficiently large even L satisfies

$$ \boxed{\lim_{L\to\infty}\langle|\mathbf M_L|^2\rangle_{L,0}=0}, \qquad \boxed{\langle|\mathbf M_L|^2\rangle_{L,0} \le\frac{C_{n,T}}{\sqrt{\log\log L}}}. $$

For XY at sufficiently low temperature, Ginibre comparison adds

$$ \liminf_{L\to\infty}C_L(x)\ge\frac1{8|x|}, $$

which establishes a rigorous zero-magnetization massless phase at the marginal boundary.

Closed proof chain

1. Exact torus convention

   $$
   |J_L^{\mathrm{MI}}-J_\infty|{\ell^1}=O(L^{-2}),
   \qquad
   E(q)=\frac{\pi c\infty}{2}|q|^2\log\frac1{|q|}+O(|q|^2).
   $$

2. Finite-volume hard-spin inequality

   $$
   1\ge(n-1)T,m_{L,h}^2\frac1{L^2}
   \sum_q\frac1{h+E_L(q)}.
   $$

3. Observable bridge

   $$
   m_{L,h}\ge\frac{\tanh(\beta hL^2)}{n}
   \langle|\mathbf M_L|^2\rangle_{L,0}.
   $$

4. Quantitative shell estimate

   With h_L=T/L^2, centered max-norm shells contain exactly 8m momenta and give

   $$
   A_L(h_L)\ge a_T\log\log L.
   $$

The four statements combine directly into the quantitative zero-field theorem.

Numerical and algorithmic contribution

• Exact minimum-image kernel arithmetic through L=8192 reaches a four-path relative coefficient discrepancy below 9.1e-7.
• Locked finite-size comparison covers AICc, BIC, goodness of fit, holdouts, L_min scans, 2,000-replica identifiability maps, covariance profiling, residual sensitivity, and a Gaussian known-zero benchmark.
• FFT energy and gradient evaluation scales as O(L^2 log L), compared with the O(L^4) direct pair reference.
• FFT-HMC, replica exchange, paired q1/q2 covariance, delete-one-block jackknife, orthogonal orientation, rank-normalized split R-hat, ESS, and vortex-sector excursions form the compact-model validation stack.
• Sixteen resumable Slurm workers partition deterministic (L,g,chain) rows and aggregate complete four-chain records.

Deliverables

• Solution overview
• Numerical report
• XY proof audit
• O(n) proof audit
• Manuscript sources
• Public research repository
• PDF and report release v1.0.0

Validation

• 5 passed original solution checks.
• 23 passed theorem-extension checks.
• 249 passed complete public research suite.
• Fresh 6-page XY article, 19-page supplement, and 11-page O(n) manuscript.
• All 36 official manuscript pages visually inspected.
• LaTeX citations, cross-references, equation layout, and figures verified.
• Zenodo provenance, checksums, deterministic artifacts, and Git whitespace verified.

Next registered measurement

The remaining endpoint selection compares

$$ g_R(\ell)\to0 \quad\text{and}\quad g_R(\ell)\to g_*>0. $$

The production target is a converged physical-g=1 dyadic sequence of

$$ g_{\mathrm{step}}(L)=\frac{\mathcal F(q_1)-\mathcal F(q_2)}{\log2},$$

paired with the connected second cumulant, mixed compact/vortex feedback, four-chain diagnostics, and vortex-sector excursions. This extends the established massless-phase classification into a complete infrared endpoint map.

@hz-xiaxz the package is ready for independent theorem-scope and claim audit. Maintainer workflow approval is ready for a single click.

[JunkaiWang-TheoPhy]

@yjdeng The completed Issue #158 analysis identifies the exact marginal structure of the two-dimensional size-normalized minimum-image 1/r^4 XY model.

The direct classical finite-volume proof gives

$$ \lim_{L\to\infty}\langle|\mathbf M_L|^2\rangle_{L,0}=0, $$

and the hard-spin O(n) extension strengthens this to

$$ \langle|\mathbf M_L|^2\rangle_{L,0} \le C_{n,T}/\sqrt{\log\log L} $$

for every finite n>=2, fixed T>0, and sufficiently large even L. The proof uses h_L=T/L^2, a uniform finite-L kernel bound, exact 8m max-norm shell counting, and the exponential-tilt bridge.

At sufficiently low temperature, Ginibre comparison gives

$$ \liminf_{L\to\infty}C_L(x)\ge1/(8|x|), $$

so the XY phase has zero magnetization and massless correlations. The exact kernel coefficient is pi c_infinity/2 = 1.0425387859782584..., reproduced on four momentum paths through L=8192.

The public package contains the proofs, locked finite-size analysis, FFT-HMC pipeline, machine-readable certificates, 249 tests, and four PDF manuscripts:
https://github.com/JunkaiWang-TheoPhy/issue-158-long-range-mermin-wagner/releases/tag/v1.0.0

The next registered target is the physical-g=1 transverse step-coupling sequence that selects between BKT power-law and logarithmic massless infrared orders.

[JunkaiWang-TheoPhy]

@hz-xiaxz Final review handoff at ef55e21.

Three theorem-level contributions are ready for independent audit:

1. the direct zero-field statement lim_{L->infinity} <|M_L|^2>_{L,0}=0;
2. the hard-spin O(n) envelope <|M_L|^2> <= C_{n,T}/sqrt(log log L) from h_L=T/L^2, the uniform finite-L kernel estimate, exact 8m shell counting, and the tilt bridge;
3. the XY low-temperature classification from the zero-field theorem plus liminf C_L(x) >= 1/(8|x|).

Review map:

• theorem_extensions/paper/main_on.tex, Secs. VI.D and the conclusion;
• theorem_extensions/ON_PROOF_AUDIT.md, Sec. 10.4;
• theorem_extensions/PROOF_AUDIT.md, massless comparison section;
• public release: https://github.com/JunkaiWang-TheoPhy/issue-158-long-range-mermin-wagner/releases/tag/v1.0.0

Validation: 5 original checks, 23 theorem-extension checks, 249 public-suite checks, and 36 visually inspected official manuscript pages. The workflow approval is ready for a maintainer click.