[MPS] 🌠 Wander/漫步者: Issue #265 Hydrodynamics from Quantum Chains (Self-Proposed Challenge, not for Adjudication)

[JunkaiWang-TheoPhy]

No grain of sand knows the river, yet the desert flows.

没有一粒沙知道河流，沙漠却依然流动。

第三幕 · 在地图尽头重新发问

第十五章 · 没有一粒沙知道河流

系列从一台送出问题卡的机器开始，最后回到机器从量子链数据中发现的一条河流。

每一粒沙都只知道局部运动，而我们必须判断：量子多体中的流体方程近似究竟是渐近定律，还是有限时间留下的形状。

← 上一章：当能谱沉默，几何开始说话     ·     回到开场：清晨，问题机器醒来 ↺

Team

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

Challenge

This PR addresses #265 by classifying the machine-discovered,
constant-coefficient viscous Burgers equation from finite-time,
high-temperature Heisenberg-chain data. The classification asks whether the
equation is a transferable asymptotic law, a chiral-mode law, or a precise
finite-window closure of a trajectory-conditioned field.

For the normalized weak-domain-wall profile (U=\langle S^z\rangle/\mu),
Kharkov et al. discovered

[
\partial_tU+aU\partial_xU=D_{\rm cl}\partial_x^2U,
\qquad
a\approx0.24,
\quad
D_{\rm cl}\approx1.90.
]

The research program resolves four concrete questions:

1. Which physical, sector-labelled, or chiral field does (U) represent?
2. Which local fields, fluctuation currents, and temporal terms constitute the
   effective closure?
3. How broadly do one coefficient law and one field map transfer across new
   conditions and future times?
4. Which model predicts profiles, currents, responses, correlations, and full
   counting statistics through one parameter set?

Result established by the public trajectory

The reproducible audit establishes a high-precision finite-window Burgers
benchmark:

Quantity	Reproducible result	Scientific meaning
Burgers nonlinearity	(a\simeq0.230)	close to the published (0.24) value
Classical viscosity	(D_{\rm cl}\simeq1.97)	close to the published (1.90) value
Integrated profile relative difference	(0.167%)	high-accuracy trajectory compression
Width exponent, (t=80\ldots190)	(0.6802)	near (2/3) in the measured window
Moment-diffusivity exponent	(0.3372)	near (1/3) in the measured window
Width-law amplitude	(A_W=0.741842)	measured quantum broadening
Burgers tangent ratio	(A_B/A_W=0.999154)	quantitative finite-window mechanism

The last row supplies the analytical explanation for the exceptional fit. For
wall width (W), the Burgers moment relation is

[
D_{\rm moment}=D_{\rm cl}+vW,
\qquad
A_W=\frac{2}{3}\frac{dW^{3/2}}{dt},
\qquad
A_B=2\sqrt{D_{\rm cl}v}.
]

The measured ratio differs from unity by approximately (0.085%). Across
the observed interval, the constant-coefficient Burgers constitutive line is
almost exactly tangent to the scale-dependent moment diffusivity sampled by
the wall. The machine therefore found a compact local representation of real
hydrodynamic broadening.

This mechanism produces a sharp prediction. For an underlying
(D_{\rm moment}\propto\sqrt W) law, rolling Burgers fits flow approximately
as

[
a(t_)\propto t_^{-1/3},
\qquad
D_{\rm cl}(t_)\propto t_^{1/3},
]

while preserving the local tangent amplitude. Production A measures this flow
across amplitudes, orientations, shapes, backgrounds, and observables.

Exact field identity from symmetry

At zero magnetic field, spin flip transforms physical magnetization and its
current as

[
m\mapsto-m,
\qquad
j_m\mapsto-j_m.
]

An autonomous local current written directly in terms of physical (m) has
odd parity,

[
j_m(m)=c_1m+c_3m^3+\cdots.
]

The Burgers advective current (am^2/2) has even parity. This algebra assigns
the fitted quadratic term to a field carrying chiral, orientation, sector,
background, or trajectory information. The registered symmetry-aware scalar
competitor makes that assignment quantitative:

[
a_i=2\sigma_i g\mu_i,
\qquad
D_i=D.
]

Both wall orientations and four amplitudes measure whether (g) transfers as
a shared material parameter.

Open-system mean evolution

For a stochastic Burgers mode, ensemble averaging gives

[
\langle u^2\rangle=\bar u^2+\operatorname{Var}(u).
]

The mean equation therefore contains a variance current proportional to
(\partial_x\operatorname{Var}(u)). In the two-field description,

[
\partial_t\langle m\rangle+
\partial_x\langle m\phi\rangle
=D_m\partial_x^2\langle m\rangle,
]

and the mode covariance contributes to the physical current. This makes
current, connected response, correlation, and FCS measurements direct probes
of the closure encoded by the mean profile.

Chiral two-mode bridge

On the equal-coupling two-mode manifold,

[
j_m=g m\phi,
\qquad
j_\phi=\frac{g}{2}(m^2+\phi^2).
]

The fields

[
u_+=m+\phi,
\qquad
u_-=m-\phi
]

diagonalize algebraically into opposite-chirality Burgers currents. Physical
magnetization is their symmetric combination. This supplies a
symmetry-compatible microscopic route to Burgers structure and motivates the
registered comparison between independent chiral modes and a coupled
stochastic two-mode system.

Long-time model signature

Deterministic continuation of the fitted scalar equation approaches the
viscous Burgers rarefaction law (W\propto t). Its local width exponent moves
from approximately (0.665) near (t=200) to approximately (0.851) at
(t=5000). The quantum chain supplies an independently generated long-time
trajectory. Their comparison directly measures future-time transfer and
separates the tangent regime from the scalar equation's own rarefaction flow.

Joint observable panel

The registered comparison uses one parameter set to predict:

• mean magnetization profiles;
• complete physical-cut spin currents;
• connected (C^{zz}) data;
• opposite-sign local-pulse responses;
• the complex transfer characteristic function (\log Z(\gamma,t));
• cumulants (\kappa_1) through (\kappa_4).

Spin-flip pairing organizes odd cumulants, while even cumulants and the full
characteristic function resolve mode coupling and distributional shape. The
joint panel therefore turns field identity and open-system closure into
observable predictions.

Preregistered confirmation experiment

Frozen time partition

Stage	Executable mask	Purpose
Training	(50\le t\le150)	estimate shared parameters
Validation	(150<t\le200)	time, condition, and orientation holdouts
Sealed confirmation	(200<t\le400)	one future-time score of the frozen forecast

The executable masks assign each shared endpoint to the earlier interval,
creating three disjoint scoring sets. Preprocessing, model classes, parameter
bounds, thresholds, and solver budgets are frozen before the sealed interval.
Explicit human authorization opens the single confirmation transaction.

Registered physical conditions

• amplitudes (\mu=0.02,0.05,0.10,0.20), with both orientations;
• tanh widths (1,2,4,8);
• erf, double-wall, Gaussian, and two sinusoidal profiles;
• backgrounds (m_0=\pm0.05);
• equilibrium and opposite-sign local-pulse response conditions;
• environment controls at (\Delta=0.8), (\Delta=1.2), and
  (\Delta=1,J_2=0.1).

The primary isotropic rows determine the restricted (\Delta=1)
classification. The environment panel measures mechanism and scope.

Numerical convergence ladder

Level	(L)	(\Delta t)	(\chi_{\max})	cutoff
coarse	256	0.05	256	(10^{-8})
medium	384	0.025	512	(10^{-10})
fine	512	0.0125	1024	(10^{-11})

A representative condition advances when the medium-to-fine profile relative
(L^2) difference is below (0.002) and the maximum relative width
difference is below (0.003). The resulting artifact selects the production
resolution ahead of model scoring.

Frozen model hierarchy

The same held-out folds compare:

1. Gaussian diffusion;
2. shared and condition-specific scalar Burgers;
3. the symmetry-aware sector and amplitude scalar law;
4. independent opposite-chirality Burgers modes;
5. a coupled stochastic two-mode open system;
6. a memory or additional-mode extension.

Two-mode selection requires at least (30%) held-out improvement over the
leading scalar competitor, a positive paired-bootstrap (95%) lower
endpoint, exact symmetry checks, and one parameter set for the joint observable
panel. The coupled model additionally requires at least (10%) improvement
over the independent manifold and (\Delta\mathrm{BIC}\ge10).

Uncertainty uses 2,000 paired blocks of physical duration 10. The stochastic
solver budget is frozen from solver-convergence targets: 1,024 trajectories
for screening and at least 2,048 for the final ensemble.

Every selected registered predictive family receives the independent
(200<t\le400) confirmation. The memory or additional-mode destination
initiates a separately preregistered extension centered on the measured
structured remainder.

Implementation and validation

This PR contributes:

• infinite-temperature purification TEBD in TeNPy;
• magnetization, complete current, connected (C^{zz}), and genuine
  two-measurement transfer FCS;
• dense (L=6) evolution, spin flip, continuity, grouped-(J_2), and
  checkpoint/resume checks;
• scalar, independent-chiral, coupled-two-mode, and joint-observable selection;
• source-hash, reuse, convergence, Production-A, authorization, and
  Production-B gates;
• 34 logical Production-A and 34 logical Production-B conditions;
• an independently converged stochastic solver-budget artifact.

Key implementation evidence:

• spin-flip agreement within (2\times10^{-15}) of exact;
• total-magnetization conservation within (10^{-14});
• lattice-continuity agreement within (4.17\times10^{-4});
• dense original-observable agreement at (10^{-9}) scale or finer;
• HDF5 checkpoint/resume with bit-identical arrays;
• (J_2=0.1) dense agreement within (8.3\times10^{-10});
• grouped and ordinary (J_2=0) agreement within (1.0\times10^{-8}),
  inside the frozen (2\times10^{-7}) gate;
• solver-budget SHA-256
  c8973d0d8b922c0ee357f7467253bdd477d19a5bb7f47626e1ab6dda8c427465.

SCNet job 23015027 completed the (J_2) compute-node qualification with
exit code 0:0 in 48 seconds. Exact, symmetry, FCS, grouped-equivalence, and
checkpoint checks all reached their registered thresholds.

Current execution stage

Research branch	Current evidence
finite-window scalar closure	established by the public trajectory
moment-tangent mechanism	established quantitatively
shared scalar transfer	assigned to Production A
independent and coupled two-mode classification	assigned to the joint observable panel
future-time transfer	assigned to sealed Production B

Twelve preregistered convergence jobs cover four representative conditions at
three resolutions. The committed launch audit records jobs
23009466–23009477 with initial checkpoints and controller 23009668
linked to their completion. The completed datasets and generated convergence
summary form the next decisive evidence artifact.

The execution sequence is:

1. complete the convergence audit;
2. run Production A through (t=200);
3. fit on (50\le t\le150);
4. score time, condition, orientation, current, response, correlation, and FCS
   holdouts on (150<t\le200);
5. record the frozen model, parameters, seeds, hashes, and uncertainty;
6. authorize the one-time confirmation;
7. score the frozen forecast on (200<t\le400).

Quantum-computing endpoint

The MPS campaign supplies a controlled reference regime with numerical floors,
exact small-system checks, and auditable convergence. Entanglement growth
defines the frontier where a quantum processor extends the benchmark.

A quantum processor can prepare the registered walls, pulses, backgrounds,
and equilibrium ensembles at later times, then measure profiles, currents,
correlators, and characteristic functions. The classical analysis performs
symmetry checks, cross-condition model selection, uncertainty estimation, and
sealed confirmation.

The endpoint is a quantum-assisted certification of which effective equation
the dynamics support, for which hydrodynamic field, in which regime, and with
which quantified additional-mode contribution.

Detailed research package

• Scientific case: derivation, literature, evidence, and decision logic
• Reviewer entry point and package map
• Dated status and evidence ledger
• Frozen universality protocol
• Closed-loop hydrodynamic synthesis
• Machine-readable conditions and thresholds
• Frozen stochastic solver budget

Validation

• focused suite: 170 checks passed;
• Python compilation: successful;
• committed JSON records: parsed successfully;
• local Markdown links and display-math pairs: verified;
• GitHub GFM rendering: verified;
• solver-budget artifact and source/config hashes: verified;
• scoped diff integrity: verified.

Claim boundary

The machine-learned deterministic Burgers equation is a quantitatively
accurate, trajectory-conditioned finite-window closure for the public
weak-domain-wall profile. Exact field symmetry, nonlinear stochastic averaging,
deterministic rarefaction, and higher-order observables define the registered
tests that identify a transferable scalar or two-mode hydrodynamic law. Those
tests are preregistered, implemented, numerically qualified, and progressing
through the convergence evidence gate.

Addresses #265.