Numerical witnesses for mixed phantom states¶

XXZ family · Supported models · Phantom reduction · Dressing map · Odd-ring scan

check_phantom_lift<Real> in xxz_phantom_check.hpp combines endpoint deflation, finite-root regularity, root recovery, and coordinate dressing. Its purpose is to find a numerically resolved nonzero amplitude, not to assume that the phantom map has no kernel. It does not allocate a spin basis or diagonalize a Hamiltonian.

The input includes an explicit phantom count p and chirality. The reduction must first pass its factorization, commensurability, and finite twisted-equation checks. The reduced root recovery must also return resolved. Neither the input coefficients nor Delta are changed.

Resolution, not merely a nonzero floating-point value¶

For a recovered root x in affine coordinates z=center+scale*x, use its recovery diagnostic's largest root-uncertainty estimate, transformed to z, plus a native rounding allowance. For a radius rho around z, the momentum factor \(v =-(1-i \,z)/(1+i \,z)\) has the variation estimate

The denominator must be positive. Conversion roundoff is added separately. This formula is an exact perturbation inequality for a supplied disk in exact arithmetic; the input disks here are numerical estimates, not certified root enclosures.

The coordinate evaluator now accepts optional momentum radii. It propagates variation through the plane powers, pair factors, and subset recurrence. For two uncertain factors, the product radius is abs(a)*radius_b + abs(b)*radius_a + radius_a*radius_b. For the pair factor,

Divide by the nominal pair normalization, holding that common global scale fixed. Powers use repeated uncertain multiplication, avoiding the cancellation in (abs(v)+radius)^x-abs(v)^x. This propagates nonlinear variation, not just the first derivative. Exact arithmetic would give an envelope for all momentum factors in the supplied disks; the implementation uses ordinary rounding, not outward-rounded interval arithmetic.

The dressing sum accumulates the finite-amplitude variations and all absolute permutation terms. A separate heuristic arithmetic allowance is

The default amplitude_tolerance is \(128\,\epsilon\). This allowance accounts for the extent of phase powers and finite sums at a conservative numerical scale; it is not a derived interval error bound. Delta and the nominal phantom phase are fixed in the input-variation calculation. None of these diagnostics rigorously bounds continuation error, deflation error, or uncertainty in the requested coupling.

A witness is recorded only if its magnitude exceeds the sum of propagated input variation and arithmetic allowance. The absolute-term scale must be normal and the total allowance positive, so underflow to a zero allowance cannot produce a spurious infinite-resolution success. This is stronger than testing amplitude != 0, but remains a numerical state check, not a proof about an exact Bethe vector or a sector minimum.

Work budgets and outcomes¶

Ordered configurations are tried lexicographically until a witness is found, the basis is exhausted, or a budget prevents further work. Defaults are 16 configurations and 1000000 total subset updates. Each configuration costs binomial(M,p)*2^r updates, with O(r²) arithmetic per update; the complete cost is checked before starting that configuration. A partial amplitude is never accepted. The finite-degree and iteration limits in root_options apply separately to root recovery. Counters include attempted configurations and charge a complete subset table for each attempted finite-amplitude evaluation, even if arithmetic fails before that evaluation completes.

Statuses distinguish:

• nonzero_witness: a tested amplitude passes the resolution comparison.
• reduction_unresolved: the specified endpoint/phase/finite-state hypothesis did not pass; its detailed reduction result is retained.
• roots_unresolved: root recovery did not resolve distinct finite roots.
• resolution_unresolved: conversion or amplitude resolution is insufficient, including exhaustive searches with no resolved witness.
• work_limit: a degree, amplitude, or subset budget prevents completion.
• nonfinite: the reduction or amplitude arithmetic is nonfinite.

In particular, none of the failure statuses means the vector is zero. The result retains the best tested configuration and its complex amplitude, absolute-term sum, variation and arithmetic allowances, resolution ratio, and work counters. all_configurations_tested distinguishes actual exhaustion from a truncated search, without needing to form binomial(N,M).

Use in the internal sector scan¶

The odd-ring candidate scan tries this diagnostic only after the regular Gaudin criterion and explicit helix match are unresolved. It tries mixed multiplicities and both chiralities, retaining every attempted result in sector.phantom_lifts. Configuration and subset budgets are shared across these hypotheses per sector; root-iteration limits apply per hypothesis. phantom_work_limited records whether any attempt was prevented by a budget. The fifth scan argument supplies PhantomLiftOptions; a zero amplitude budget disables this extra work.

state_checks_complete now includes numerical phantom witnesses alongside regular states and helix matches. This flag is explicitly not a rigorous state certificate. It also remains separate from equation convergence, Wronskian consistency, and candidate-energy comparison. Disabling witnesses does not change continuation coefficients, energies, Newton iteration counts, the selected index, or the near-degeneracy report.

Tests cover continued mixed branches through N=13, both chiralities, precisely exhausted budgets, deliberately unattainable resolution demands, and rejected off-shell hypotheses. The variation recurrence is checked against independently enumerated, perturbed permutation sums. Existing direct-Hamiltonian tests remain independent checks of the constructed vectors. All run in fp64, long double, and fp128. Public odd-ring ground-state support still needs reliable sector-minimum tracking and coverage of the remaining singular families.