Direct six-spinon moments¶

See momentum input near endpoints for distance and beta inputs, conversion warnings, and the current scalar-geometry limit.

bethe/xxx_six_spinon_moments.hpp calculates the longitudinal six-spinon intensity and first moment directly, without integrating a displayed frequency grid. ThermodynamicSixSpinonMoments<Geometry,Low,High> reuses the same physical W6, bounded recovery and soft-pair proposal as the spectral-point calculation. It uses the shared Monte Carlo engine to accumulate both moments and their covariance from each configuration.

Both fixed-momentum and global moments are available through the installed bethe-xxx-moments-thermo app. The calculation uses a finite absolute-rapidity cutoff, not a claim of unrestricted convergence.

Measure and importance proposal¶

Let q4 be the complete four-variable joint proposal and g its native marginal. A draw chooses one of the fifteen unordered pairs uniformly, samples that pair from g, and samples the other four rapidities from q4. Its actual density is

Every term includes all overlapping channels of q4. The mixture contains the background and the requested soft-pair/cluster components. Seven open uniform coordinates select the native pair, joint component and continuous coordinates. The complete density determines the weight; the selected component alone does not. This importance strategy remains an explicit part of the physical adapter.

For labelled rapidities restricted by max(abs(beta)) less than R, a contribution to the order-r global moment is

The Jacobian converts rapidities to momenta. W6 already contains its single state-counting factorial; there is no additional permutation correction. Both moment measures are included. The unrestricted first-moment momentum law is not imposed on this global integral.

The same cutoff box is used for spectra and these global moments, allowing matched-domain comparisons. It excludes soft configurations throughout the spectrum. A gap cutoff with an analytically integrated common shift is a different domain and needs its own comparison. The common-shift reduction is a useful alternative; its five-gap importance proposal requires separate validation.

Fixed-momentum geometry¶

detail/xxx_spinon_moment_proposal.hpp supplies the direct fixed-momentum geometry used by fixed_moments(k). It chooses uniformly among thirty labelled charts: eliminate momentum i, draw a distinct rapidity j from g, and draw the other four from q4. It solves the momentum constraint in every section and retains the solution only inside the same absolute cutoff box. A draw with resolved empty support contributes zero and is not redrawn. Numerically ambiguous support fails explicitly.

The chart intensity against the constrained labelled momentum measure is

Its reciprocal mixture weight is 30 divided by the sum of all thirty intensities. Eliminating a momentum has unit Jacobian; the cosh factors convert the five free proposal densities to momentum densities. The moment observation is this weight times W6 and the sampled energy to power r, divided by 2*pi. No further factorial or normalization adjustment is needed. Chart selection is actually sampled, so the measure also applies to nonsymmetric labelled observables.

Geometry checks cover the full mixture and label orbit, all chart choices, momentum conservation, analytic periodic box-spline volume and labelled-momentum integrals. R=1040 conservation is checked at 1800 and 2048 bits; insufficient 384-bit geometry is rejected. These establish the geometric measure, not the efficiency of this proposal for physical W6 or a converged moment estimate.

Interface and result¶

Construct the model with a SpinonJointProposal<Geometry>, a SpinonRecoveryPolicy and positive exchange J, as in the spectral example. global_async(options, execution) returns an owned Async result; global(options, execution) waits using the same scheduler. A retained UniformSample<7> can also be passed to global_observation for exact replay. The scheduler remains borrowed until completion.

For fixed momentum, use model.fixed_moments(k).evaluate_async(options, execution) or the synchronous evaluate wrapper. The query owns the model state through completion. observation(UniformSample<6>, stop) replays one exact sample. The result has the same six integral columns and covariance, with m0(k) and m1(k) replacing I0 and I1. It records the requested momentum and an optional first_moment_denominator and first_moment_fraction. The denominator is the full-response first moment, (2J/3)(log(2)-1/4)(1-cos(k)); the numerator is integrated from the sampled energy and is not forced to follow that law.

SixSpinonMomentStatus distinguishes an estimate, an exact physical zero, no sampled support, numerical failure and cancellation. At k=0 or 2*pi both moments are exactly zero without sampling; their first-moment fraction is undefined. At k=pi, unrestricted_static_divergence warns that the finite-R zeroth moment is not a converged static structure factor. No exact zeroth-moment denominator is invented at other momenta. A complete sample stream that finds no support has no aggregate estimate; its zero observations remain in the replicate records for diagnosis.

The result records J, R, proposal parameters and recovery settings. Its integral contains the complete numerical-engine result, with columns:

The 6 by 6 integral.covariance is the covariance of the mean, in row-major order. Entries 0 and 7 give the two moment variances; entry 1 gives their covariance. Retained replicate means and maxima expose sampling scatter and unusually influential observations. All numerical quantities remain at the explicit geometry/accumulation precision.

fraction, when present, divides the absolute moments by the full-response sum rules I0=1/4 and I1=(2J/3)(log(2)-1/4). These are reference denominators, not fitted multipliers. Fractions share the same cutoff and sampling limitations as the absolute moments. No fraction or mean is returned after an unrecovered sample failure or cancellation. Increasing J leaves I0 unchanged and scales I1 by J; both fractions are invariant under that exchange change.

xxx_spinon_observation.hpp shares the checked density weighting between spectra and moments. Observable adapters supply logarithms of all physical weights; the helper imports the same rapidities into the declared recovery contexts and returns values, contour diagnostics and arithmetic diagnostics separately. It supplies no missing measure or normalization factor.

Validation and practical limits¶

Independent controls check the complete fifteen-component PDF, all native-pair choices, permutation invariance, the labelled momentum volume, total energy and a nonsymmetric labelled observable. Deep-soft log weights are tested at R=1040 with 160/256-bit MPFR; that geometric test is not a physical R=1040 integration.

An independent 85-decimal-digit Python construction checks a complete physical I0/I1 observation and a fixed-k m0/m1 observation, including all mixture terms and moment measures. Physical integration tests check serial/concurrent agreement, covariance, exchange scaling, owned lifetimes and explicit failure/cancellation. The optional tests/reference_xxx_six_spinon_moments.py regenerates the physical reference. Use --fixed for the fixed-k reference. The compact sampling record retains the following pilot settings, replicate summaries, timings and comparisons.

An accurate density does not ensure an accurate integral. Production work must vary sample budgets, independent streams, proposal parameters, uniform-grid resolution, geometry precision and rapidity cutoff as appropriate. The numerical density diagnostics exclude sampling and omitted-tail uncertainty. The R=8 settings used for interface tests and small pilots do not resolve the full thermodynamic six-spinon weight.

A fresh R=8, J=1 execution pilot used 8 replicates of 16 draws at 384-bit geometry precision, with 128/192-bit density controls and a declared 224/320-bit recovery stage. All 128 configurations completed, requiring 256 density-control attempts. It gave I0=0.00012085 with sampling standard error 0.00007627 and I1=0.00016657 with standard error 0.00011297. These large errors (about 63% and 68%) show that the pilot budget is insufficient for precise integration. It used 48.73 seconds wall time and 97.03 seconds CPU time on two workers. The tiny sampled density diagnostics do not reduce those sampling errors or quantify the omitted rapidity region. These are execution measurements, not estimates of the unrestricted sector fractions.

Compare proposals on new samples¶

At k=pi/2 and R=8, a 512-draw calibration suggested emphasizing the two-pair component. A separate comparison froze both proposals, then used 16 new replicates of 256 draws each (seed 42, replicate IDs 100--115). The same identities were used between proposals, so their difference has a paired sampling error. The calibration used IDs 0--7 and was not reused as validation data.

The joint-mixture weights for background, one pair and two pairs were (0.5, 0.25, 0.25) and (0.1, 0.2, 0.7). Both used native soft fraction 0.5, one-pair scale/gap 4/0.5 and two-pair scale/gap 8/1, with the precision and density policy above. These are explicit physical proposal choices, not adaptation performed by the Monte Carlo engine.

Each calculation used two workers on an Intel Core i7-12700KF; the two processes ran concurrently. Timings describe this pilot, not a portable performance target.

Errors are between-replicate sampling standard errors. The two means agree within 0.51 and 0.54 paired standard errors. The second proposal's observed variance times CPU cost is 0.24 and 0.19 of the original for m0 and m1. Its largest individual draws nevertheless account for 6.8% and 5.3% of the corresponding sums, compared with 18.8% and 17.7% for the original proposal. The comparison supports this proposal for further finite-R pilots; it does not establish a universally efficient mixture or exclude unseen rare regions. No library default is changed by this measurement.

The earlier 512-draw calculations at pi/4, pi/2 and 3pi/4 also compared the independently measured first-moment fractions. All pairwise differences are less than 0.51 paired standard errors, but individual fractional uncertainties are 33--36%. There is no resolved momentum dependence in this small sample, and no numerical verification of the unrestricted momentum law. A finite absolute-rapidity box can itself violate that law. The record also includes a separate 1024-draw global integral; it is not reconstructed from these three momenta or from a spectral grid.

Separate cutoff sensitivity from sampling error¶

For nested rapidity boxes, one outer-box observation can estimate several cutoffs without repeating the density evaluation. Retain the complete outer proposal density and multiply each weighted moment by the indicator max(abs(beta)) < R. Do not replace that density by the proposal for the smaller box: the draw still came from the outer proposal. Accumulating all cutoffs together supplies the covariance needed for their differences.

A second pi/2 pilot used the pair-heavy proposal with outer cutoff 16 and 16 replicates of 128 draws (seed 43, IDs 200--215). Geometry remained at 384 bits with the same recovery policy. All 2048 draws completed; 1009 supported configurations required 2018 density-control attempts. Two workers used 394.65 seconds wall and 769.92 seconds CPU time. The nested-indicator method was first checked against analytic momentum-box volumes at cutoffs 1, 2 and 3.

The R=8 estimates agree with the independent proposal comparison above. The first-moment increases from 8 to 12 and from 12 to 16 are 0.001366 +/- 0.000353 and 0.001204 +/- 0.000449, using their paired covariance. These finite-domain increments show why the R=8 execution examples cannot estimate the full response. They do not establish convergence at R=16 or determine its remaining tail. The largest draws still contribute about 12--16% of these sums, so further sampling and proposal checks are needed alongside a larger-cutoff comparison.