Conversation
…rvey Now that the function-theoretic count is done (M3a/M3b on main), scopes the remaining operator-level bridge in docs/INDEX_PAIRING.md: - M1: build the weighted L²(L_k) sections + the twisted ∂̄ operator (from scratch; no line-bundle L² in Mathlib). - Route A (Weyl's lemma / elliptic regularity for ∂̄): fully absent in Mathlib — blocked. - Route B (Landau/Hermite decomposition, recommended): L²(L_k) ≅ ⊕ₙ ℂᵏ via Hermite-Gaussian eigenfunctions, D⁺ = lowering, ker = ℂᵏ, ≅ magneticDirac k (transports the formalized model index = k). Gap: Mathlib has only Hermite polynomials + Rodrigues, NOT the L²(ℝ) orthonormal basis — building that basis is the gating, reusable first step. - Connection to Jon's merged Riesz–Schauder (#11): gives Fredholm well-definedness of index(D⁺); the value = k still needs Route B. - Recommended first step: the Hermite L²(ℝ) orthonormal basis. Refreshed the Mathlib inventory (count done; Hermite-basis / Weyl / line-bundle-L² gaps). Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
2 participants
Add this suggestion to a batch that can be applied as a single commit.This suggestion is invalid because no changes were made to the code.Suggestions cannot be applied while the pull request is closed.Suggestions cannot be applied while viewing a subset of changes.Only one suggestion per line can be applied in a batch.Add this suggestion to a batch that can be applied as a single commit.Applying suggestions on deleted lines is not supported.You must change the existing code in this line in order to create a valid suggestion.Outdated suggestions cannot be applied.This suggestion has been applied or marked resolved.Suggestions cannot be applied from pending reviews.Suggestions cannot be applied on multi-line comments.Suggestions cannot be applied while the pull request is queued to merge.Suggestion cannot be applied right now. Please check back later.
Scopes the remaining operator-level bridge in
docs/INDEX_PAIRING.md, now that the function-theoretic count (M3adim ker = k, M3bcoker = 0) is onmain.Key finding from a Mathlib survey — the operator index
= khas a real gap on every route:L²(L_k)sections + twisted∂̄): build from scratch (no line-bundleL²in Mathlib).∂̄: fully absent in Mathlib → blocked.L²(L_k) ≅ ⨁ₙ ℂᵏvia Hermite-Gaussian eigenfunctions ⇒D⁺ ≅ magneticDirac k, transporting the formalized modelindex = k. Gap: Mathlib has only Hermite polynomials + Rodrigues, not theL²(ℝ)orthonormal basis.index(D⁺); the value= kstill needs Route B.Recommended first step: build the Hermite orthonormal basis of
L²(ℝ)— the gating lemma for Route B, a clean reusable Mathlib-gap fill. (Even so, M1 + the magnetic guiding-center reduction remain substantial — this is a multi-step analytic build.)Doc-only. Refreshes the Mathlib inventory.
🤖 Generated with Claude Code