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ForceFreeStates - BUGFIX - Preserve the symplectic structure across ideal singular-surface crossings #481
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Activity
Findings, proposed fix and draft diff: https://claude.ai/artifact/MaMkwYpbRxALD99cT6GAWB
The non-Hermiticity in W after ideal crossings is truncation error from the crossing's jump. It is first order in the jump half-width, so DIIID n=1 et₁ is 0.34% off at today's
singfac_min= 1e-4.Lowering
singfac_minalone doesn't work, for two reasons:- Forward: after a high-α surface every column grows alike, so the
ucritratio test never fires and the basis collapses (cond up to 1e15). - Riccati:
singfac_minis also the Δ′ matching distance, and Δ′ loses precision as it shrinks.
Proposed fix, in three parts:
- ForceFreeStates - BUGFIX - ⚠️ Remove the resonant solution at ideal crossings regardless of reduction timing #489's resonant-row crossing.
- A Gaussian reduction also triggered when the solution columns become nearly parallel. The ucrit test only fires when growth rates differ; here every column grows at the same rate because each is dominated by the same solution. The prototype detects this with a condition number (cond > 1e6), but that is one option among several, including cheaper ones (see the page).
- A separate jump half-width (default 1e-6), reached by ODE legs inside the crossings.
singfac_minkeeps its other roles.
With all three, energies converge with the jump and forward and Riccati agree, while the Δ′ matrix is unchanged. Today they disagree by up to 2.7× (DIIID n=3).
Input wanted on:
- the parameter design;
- crossing in asymptotic-coefficient space as an alternative;
- which test to use for detecting the collapse.
- Forward: after a high-α surface every column grows alike, so the
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Summary
Once the forward Euler-Lagrange integration has a per-column absolute tolerance (#480), W⁻¹ = U₁U₂⁻¹ stays Hermitian to roundoff between singular surfaces. The non-Hermiticity that remains is added at each ideal singular-surface crossing. This is the residual left over from #93 after that fix.
Figure: https://claude.ai/artifact/8VJSoYZRUPDZvKqryqY5aP
Evidence
DIIID-like_ideal_example, n = 1, fixed axis start (frobenius_psi_max = 0),ucrit = 1e3, on the #480 branch. Values are the stability check's ‖anti-Hermitian‖/‖Hermitian‖ of W⁻¹ at every saved step:The jump happens exactly at the first crossing, and each later crossing leaves a similar level. The flagged steps are not at the surfaces: they are where the ~1e-4 defect is amplified between them.
Likely source
cross_ideal_singular_surf!zeroes one column, takes a two-point trapezoidal step across2·dpsi, and substitutes the asymptotic small solution. That is the one operation in the integration that does not preserve U₁†U₂ − U₂†U₁ = 0.Candidate fixes, from the discussion on #93:
sing_get_ca/sing_get_ua) in extended precision.Related