[pinterbanget]

Summary

This adds research/forward-turning-point/. It answers open questions 1–2 of my #4408: why the broad-forward phase-cell inverse behind FORWARD-SMALL-A-AUDIT.md's O(α⁵ρ²θ⁻²e^{aρ/3}) grows only like θ^{−1/2}, and how its constant depends on α.

Claim type: a formal (semiclassical) derivation plus numerical confirmation to under 1%. No new κ, and not a proof.

Result

‖K⁻¹‖₂ ≈ e^{a/4}/(2α√π) · θ^{−1/2},   a = πα².

• Derivation. Take an alternating ansatz x_i = (−1)^i u(i) and apply Poisson summation. Near β′ = ±1/2, where the shifted Gaussian symbol vanishes, the block reduces to T = A[(πθ′)y − (π/2b₀)∂_y] with A = 2√(π/b₀)e^{−π²/(4b₀)}/α. Then T*T is a harmonic oscillator, so σ_min = Aπ√(θ′/b₀).
• Numerics. The verifier uses sparse inverse power iteration on the banded block down to θ = 1/102400. The ratio to the prediction reaches 0.99723 at α = 3 (constant 110.44) and 0.99438 at α = 4 (constant 20,223), with an O(θ^{1/2}) gap. A dense SVD and a localisation check serve as controls.
• Consequence. At θ = 1/(4p²) the true conditioning is about 16p²/(α√π). The ceiling loses through e^{aρ/3} in place of e^{a/4} and through θ⁻² in place of θ^{−1/2}. At p = 19 the prediction is 4197, against a measured 4118 and a ceiling of 9.1·10⁹.

Evidence

python3.12 -B scripts/research_package.py --package research/forward-turning-point --check-only
  PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
python3.12 -B scripts/research_package.py --package research/forward-turning-point
  PASS_REGISTERED_RESEARCH_PACKAGE research/forward-turning-point outcome=completed   (~3 s, numpy 1.26.4 + scipy 1.13.1)

Head: d08a3fdd9b840ae79c11d830a53d66586e6a4b8a. Base: canonical main 12fa8c5c, priority revision 7. FORWARD-SMALL-A-AUDIT.md is pinned.

Not covered

• A rigorous upper bound: coercivity away from the turning point, and the boundary layer.
• The narrow kernel.
• Guard changes.

Task and overlap

• Card: uniform-transfer-audit, refined.
• Claim: Handoffs and requests for independent checks #1166 (comment)

Credit

PR1673 (FORWARD-SMALL-A-AUDIT.md), GAUGEFREE-INTERIORS.md (the shifted symbol), and @Matarener's #4356 (the question).

Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance; no independent human review. Apache-2.0. This PR does not merge anything or change the frontier.

🤖 Generated with Claude Code