Add 84a: finite-jet norm-equivalence constant for Leray-projected point jets on T^d

[kantrarian]

Summary

Adds constants/84a.md for the finite-jet norm-equivalence constant $C_{84} := C_{3,5,3}$ on $\mathbb{T}^3$. The constant is the reciprocal-square-root of the smallest positive eigenvalue of the Fourier-side Gram matrix for Leray-projected cubic point jets at Sobolev exponent $s = 5$. The page treats the quintessential $(m,d,s) = (3,3,5)$ case and lists variants for $s \in {4.6, 4.75}$ in the Additional comments, per the family-with-variants guidance in CONTRIBUTING.md.

Key structural results recorded:

• Kernel theorem fixing the quotient dimension as $50 = 60 - 10$ via the projected-gradient relations $\mathbb{P}\nabla\partial^\gamma \delta_0 = 0$.
• $W(B_3)$ hyperoctahedral sector decomposition: the smallest positive eigenvalue admits a piecewise closed form — a $2 \times 2$ quadratic below the sector crossing at $s^* \approx 4.7892803103$ and a $3 \times 3$ trigonometric Cardano above it.
• Trace identity $3A + 6B = Z(s-2) - 2Z(s-1) + Z(s)$ and orbit identity $2D + C = B$ that reduce the below-crossing closed form to three Sobolev-zeta shifts plus two anisotropic moments.
• Classical certified bracket $[6.53538338, 6.59846649]$ (cube + Rayleigh-tail, fully classical except for double-precision linear algebra).
• Engineering-rigorous mpmath Ewald bracket $[6.576432995516946, 6.576432995517075]$ with explicit analytic truncation bounds and precision-doubling cross-validation.

Companion scripts (cube truncation, Rayleigh-tail certificate, mpmath Ewald, $W(B_3)$ block decomposition, closed-form eigenvalue extraction, sector-crossing bisection) are at https://github.com/kantrarian/finite-jet-constants.

Why this is in scope

The constant depends on three parameters $(m, s, d)$, which would ordinarily fail the no-additional-parameters criterion. I am submitting it under the "particular mathematical interest" exception clause for the following reasons:

1. Parabolic controllability with point-supported terminal data (Imanuvilov, Coron, Le Rousseau, Fernandez-Cara line): $1/C_{m,s,d}$ controls the terminal-symbol constant $c_{\text{term}}$ in adjoint observability estimates.
2. Inf-sup / Babuska-Brezzi analysis for divergence-free FEM (Bramble-Pasciak, Boffi-Brezzi-Fortin, Schoberl, Falk, Neilan): discrete analogs of $C_{m,s,d}$ control well-posedness; the continuous-side benchmark is what existing literature uses implicitly without pinning down values.
3. Off-the-grid sparse divergence-free measure recovery / atomic-norm minimization (Candes-Fernandez-Granda, De Castro-Gamboa, Duval-Peyre, Bredies-Pikkarainen): $C_{m,s,d}$ is a coherence parameter for divergence-free atomic dictionaries.
4. $H^{-s}$ norm equivalence on point-supported momenta in LDDMM (Trouve, Younes, Miller, Pennec): registration energy is $|p|{H^{-s}}^2$, controlled by $C{m,s,d}^2$.

The kernel theorem, the certified two-sided bracket, the $W(B_3)$ sector-crossing observation, and the piecewise closed form for $\lambda_{\min}^+$ are, to the best of my knowledge, not yet recorded in any of these four lines. If the maintainers prefer literature to exist before this is recorded, or prefer an issue-first discussion, I am happy to convert or withdraw.

Test plan

• File placed at constants/84a.md, follows template.md six-section structure.
• Constant numbered 84a: highest existing was 83a at time of this PR.
• All inline LaTeX uses \_ and \lvert/\rvert (no _ or | inside $...$).
• Companion scripts at https://github.com/kantrarian/finite-jet-constants compile (python -m py_compile) and reproduce the numerical brackets in the file.
• AI assistance disclosed in Contribution notes (Anthropic Claude Code, Opus 4.7).

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