Six-spinon sampling geometry¶

The internal bethe::heisenberg::detail::SpinonJointProposal<Real> in bethe/detail/xxx_spinon_proposal.hpp generates four free rapidities for a fixed-energy, fixed-momentum six-spinon integration. It supplies proposal densities and balanced coarea weights. Physical form factors and the solution of the two conservation equations are separate components.

Fresh draws¶

The constructor takes a rapidity cutoff R, a soft fraction strictly between zero and one, and positive mixture weights that sum to one. Runtime-precision types require an explicit finite precision. Five open uniform coordinates select a component and its four continuous coordinates. No saved sample file, Python process or global random-number state is needed.

The mixture contains a background, independent Cauchy rapidities, one soft pair, or two soft pairs. A pair uses its centre B and separation d with unit Jacobian:

\[ \beta_i=B+d/2,\qquad \beta_j=B-d/2,\qquad |B|\lt R,\quad |d|\lt 2(R-|B|). \]

Both variables have normalized truncated Cauchy distributions. One-pair draws choose uniformly from all six free-label pairs; two-pair draws choose among the three matchings. The background provides full support. Each returned log_density evaluates the complete mixture, including components that could also have produced the draw. Its measure is product(d beta), not product(dp). The parameters use the convention p=2 atan(exp(beta)).

Random-number generation, independent replicates and sample sharing across frequencies belong to the caller. A spectral estimator must preserve those replicate identities when reporting uncertainties and covariances.

bethe::MonteCarloStream<Dimension> in bethe/sampling.hpp keys an ordinary Monte Carlo stream by master seed, replicate id and a declared uniform bit count. It uses standardized seed_seq and mt19937_64 engines. A draw retains its binary integer words; UniformSample::at_precision constructs the exact midpoint (integer+1/2)/2^b at the requested precision. At least b+1 significand bits are required. A retry reuses those words and never redraws or silently lowers the grid resolution. This stream is not the historical campaign's scrambled quasi-Monte Carlo rule.

The background inverse CDF uses stable lower-tail probabilities. Its stopping criterion accounts for both CDF roundoff and the probability change associated with one representable rapidity step. A small uniform coordinate does not imply that an arbitrarily small relative CDF residual is attainable at fixed precision. Open-support and convergence failures remain explicit; no input is redrawn.

bethe::ReplicateStatistics<Real> collects complete, equal-size replicate means. It reduces them in replicate-id order and supplies the full covariance of the mean, preserving correlations between frequencies or moment orders. One replicate supplies no sampling-error estimate. Missing or failed replicates leave the result incomplete/failed with no successful mean over the survivors. Genuine support zeros count as observations; numerical failures do not. Underflow of a nonzero variance is an explicit numerical failure, not a zero sampling error. These statistical diagnostics do not include cutoff or numerical integration bias, and the uniform-grid resolution is a separate input choice.

Energy and momentum constraints¶

SpinonEnergySurface<Real> in bethe/detail/xxx_spinon_surface.hpp accepts the spinon count (2, 4 or 6), cutoff, momentum in [0,2 pi), momentum-section index and positive frequency. Its solve method takes n-2 free rapidities and returns both labelled solutions for the remaining pair. The section fixes the total spinon momentum to k+2 pi s; sum every admissible section.

The map's log_coarea_jacobian is the inverse velocity difference per root, before proposal or free-rapidity Jacobians. Use either that unbalanced chart measure with the complete proposal PDF or the balanced measure below, not their product. The map has no physical form-factor normalization.

Resolved support exclusions return inside=false. The spectral surface keeps signed endpoint distances separate from integer multiples of pi, and retains their exponential scale when those distances cannot be represented directly. A cutoff of 1040 therefore no longer imposes a significand precision proportional to the cutoff. The endpoint-preserving pilot completes at 256 geometry bits with the same sample counts as the archived 1536-bit calculation. Geometry tests also compare 128 and 256 bits against 2048 bits; the scaled-endpoint replay checks native formats separately.

Ambiguous boundaries, poorly conditioned solutions and numerical collisions still fail explicitly and can require higher precision or a separate limiting calculation. These checks do not establish density, sampling or cutoff convergence; successful proposal evaluation alone is not a geometry check. The fixed-momentum moments proposal uses a different geometry construction and still requires much wider precision at cutoff 1040.

Balanced coarea charts¶

For each possible solved pair (i,j), let q_ij be the complete proposal density of the four remaining rapidities. At J=1 its pushforward intensity is

\[ h_{ij}=q_{ij}\prod_{a\ne i,j}\cosh\beta_a\, \frac{\pi}{2}\frac{|\sinh(\beta_i-\beta_j)|} {\cosh\beta_i\cosh\beta_j}. \]

This convention emits both labelled solutions of the conservation equations. For a permutation-invariant physical density and integration domain, symmetry allows replacing the original chart's inverse intensity by the inverse mean of all fifteen intensities:

\[ J_{\rm balanced}=\frac{15}{\sum_{i\lt j}h_{ij}}. \]

balanced_log_jacobian returns its logarithm per labelled solution. The caller sums both solutions and all allowed momentum sections. The physical density retains its single state-counting factorial; this balancing factor does not change that normalization. An ordered-domain or hardest-pair restriction is a different estimator and must not be stacked on this full-domain balance.

Logarithmic assembly keeps the separate cosh and proposal factors from overflowing. It does not establish that a subsequent momentum map or energy constraint is resolved at the same precision. Unresolved draws fail explicitly; callers must recover them or report an incomplete estimate rather than discard them. A vanishing individual chart is permitted; an entirely degenerate chart orbit requires a separate limiting calculation.

Bounded density recovery checks the physical weight at independently strengthened arithmetic and contour settings. The integrator must preserve the same generated point throughout recovery and keep its diagnostic failures distinct from geometric support zeros.

Validation¶

test_xxx_spinon_proposal.cpp checks independent high-precision density, coarea and draw references, all pair choices, permutation symmetry and the literal 720-label orbit measure. It also checks invalid inputs, inverse CDFs, deep-soft configurations and explicit precision growth beyond binary128. The optional tests/reference_xxx_spinon_proposal.py regenerates the small fixtures with mpmath using direct positive formulas and a bisection inverse.

test_xxx_spinon_surface.cpp independently checks the inverse map against forward kinematics and velocity differences at prescribed rapidities, including all three six-spinon momentum sections. Its optional reference generator uses 720 decimal digits, sufficient to retain the deep-soft inputs tested here.

The six-spinon spectral-point interface combines these pieces with W6, bounded recovery and Uni20 Async execution. It retains complete replicate records and separates sampling errors from the numerical density diagnostics and omitted rapidity tails.

The shared Monte Carlo controller executes these physical strategies and retains covariance and work records. Proposal design, complete mixture probabilities and balanced chart factors remain in this model layer.