From 4fe94cf63a877f3ff191e103650e08e0c00c5021 Mon Sep 17 00:00:00 2001 From: "R. J. Mathews" Date: Thu, 14 May 2026 18:32:54 -0400 Subject: [PATCH] Add 84a: finite-jet norm-equivalence constant for Leray-projected point jets on T^d Adds C_{84} = C_{3,5,3}, the smallest positive eigenvalue reciprocal-sqrt of the Fourier Gram for cubic point jets on the 3-torus at Sobolev exponent s = 5. Includes: * a kernel theorem fixing the quotient dimension as 50 = 60 - 10 (projected-gradient relations); * a W(B_3) sector decomposition reducing the smallest positive eigenvalue to a piecewise closed form: a 2x2 quadratic below the sector crossing at s* ~= 4.7892803103 and a 3x3 trigonometric Cardano above it; * trace and orbit identities reducing the below-crossing closed form to three Sobolev-zeta shifts plus two anisotropic moments; * classical certified bracket [6.53538338, 6.59846649] and engineering-rigorous mpmath Ewald bracket [6.576432995516946, 6.576432995517075]; * cross-domain applications (parabolic controllability with point-supported terminal data, divergence-free FEM inf-sup, sparse divergence-free measure recovery, LDDMM). Companion scripts at https://github.com/kantrarian/finite-jet-constants. Submitted under the "particular mathematical interest" exception clause of CONTRIBUTING.md. Co-Authored-By: Claude Opus 4.7 (1M context) --- constants/84a.md | 142 +++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 142 insertions(+) create mode 100644 constants/84a.md diff --git a/constants/84a.md b/constants/84a.md new file mode 100644 index 0000000..b3c4dd8 --- /dev/null +++ b/constants/84a.md @@ -0,0 +1,142 @@ +# The Finite-Jet Norm-Equivalence Constant + +## Description of constant + +For integers $m \geq 0$, $d \geq 2$, and $s > m + d/2$, consider the family +of Leray-projected point jets + +$$F\_{(\beta,j)} = \mathbb{P}\,\partial^\beta\bigl(\delta\_0 \, e\_j\bigr) \in H^{-s}(\mathbb{T}^d), +\qquad \lvert\beta\rvert \leq m,\ j = 1,\ldots,d,$$ + +on the torus $\mathbb{T}^d = \mathbb{R}^d/(2\pi\mathbb{Z})^d$, where $\mathbb{P}$ +is the Leray projector onto divergence-free vector fields. The raw family is +overcomplete: the projected gradient relations + +$$\mathbb{P}\,\nabla \partial^\gamma \delta\_0 = 0,\qquad \lvert\gamma\rvert \leq m-1,$$ + +give an exact kernel of dimension $\binom{m+d-1}{d}$ in the raw coefficient space. +The *finite-jet norm-equivalence constant* $C\_{m,s,d}$ is defined by the +quotient norm equivalence + +$$C\_{m,s,d} = \sup\_{F \neq 0} \frac{\lVert A \rVert\_{\ell^2\_{\min}}}{\lVert F \rVert\_{H^{-s}(\mathbb{T}^d)}}, +\qquad F = \sum\_{\beta,j} A\_{(\beta,j)} F\_{(\beta,j)},$$ + +where $\lVert A \rVert\_{\ell^2\_{\min}}$ is the minimum Euclidean norm among +coefficient vectors representing $F$ (equivalently, the Euclidean norm of the +canonical representative in the quotient by the gradient kernel). The reciprocal +of $C\_{m,s,d}^2$ is the smallest positive eigenvalue of the Fourier-side Gram +matrix + +$$G\_{(\beta,j),(\beta',j')} + = \sum\_{k \in \mathbb{Z}^d \setminus \{0\}} + (1+\lvert k\rvert^2)^{-s} + (ik)^\beta \overline{(ik)^{\beta'}} + \bigl(\delta\_{jj'} - k\_j k\_{j'} / \lvert k\rvert^2\bigr).$$ + +The threshold $s = m + d/2$ is not admissible: at this value the top-order +lattice sums diverge logarithmically. + +This page treats the *quintessential* case + +$$C\_{84} := C\_{3,5,3},$$ + +which is the natural representative for the 3D incompressible Navier-Stokes +and Oseen-controllability application: $(m, d) = (3, 3)$ captures cubic point +jets in physical dimension three while leaving a $50$-dimensional quotient +small enough for explicit $W(B\_3)$ sector decomposition, and $s = 5$ is the +smallest stable benchmark with a half-derivative buffer above the threshold +$s = 9/2$. Variants for other $(m, s)$ are recorded in Additional comments +below. + +## Known upper bounds + +| Bound | Reference | Comments | +| ----- | --------- | -------- | +| $6.66760093$ | [Mathews2026] | Cube truncation at $\lvert k\_i\rvert \leq K = 50$; rigorous in Loewner order. | +| $6.59846649$ | [Mathews2026] | Cube truncation at $K = 200$. | +| $6.576432995517075$ | [Mathews2026] | mpmath Ewald/Mellin evaluator with analytic truncation bounds at precision 80, treating mpmath's quadrature as exact at the working precision (validated by precision-doubling and the $m = 0$ closed form). | + +## Known lower bounds + +| Bound | Reference | Comments | +| ----- | --------- | -------- | +| $6.53538338$ | [Mathews2026] | Cube + Rayleigh-tail certificate (AM-GM pointwise polynomial bound on the $K = 50$ cube minimizer, Euclidean inflated-cube tail integral). | +| $6.576432995516946$ | [Mathews2026] | mpmath Ewald lower bound (same engineering-rigorous evaluator as above). | + +## Additional comments and links + +**Quotient structure.** The kernel theorem $\ker \Phi\_m = \operatorname{span}\{\mathbb{P}\nabla \partial^\gamma \delta\_0 : \lvert\gamma\rvert \leq m-1\}$ has a Fourier-side proof: the constraint $\mathbb{P} P\_A = 0$ at every nonzero lattice point gives $P\_A(k) \parallel k$ on $\mathbb{Z}^d \setminus \{0\}$, and Zariski density of $\mathbb{Z}^d \setminus \{0\}$ in $\mathbb{C}^d$ upgrades this to a polynomial identity. For $(m, d) = (3, 3)$ the raw dimension is $60$, the kernel dimension is $\binom{5}{3} = 10$, and the quotient dimension is $50$. + +**Parity splitting.** Central inversion $k \to -k$ commutes with the Gram, giving an exact even-odd parity decomposition. For $(m, d) = (3, 3)$ the quotient blocks have dimensions $18$ (even, grades $0$ and $2$) and $32$ (odd, grades $1$ and $3$); the minimizer at $s = 5$ lies in the odd block, so the absence of grades $0$ and $2$ from the minimizer is forced by symmetry rather than numerical roundoff. + +**$W(B\_3)$ sector decomposition and sector crossing.** The full hyperoctahedral group $W(B\_3) = (\mathbb{Z}\_2)^3 \rtimes S\_3$ of signed permutations of $\mathbb{R}^3$ also commutes with the Gram. The $50$-dimensional quotient decomposes into nine $W(B\_3)$ isotypic components, two of which can contain the smallest positive eigenvalue depending on $s$: + +- *Block 0* (above crossing): the isotypic component of the $S\_3$ standard $2$D irrep lifted trivially to $(\mathbb{Z}\_2)^3$, with multiplicity $3$ inside the quotient. Its $3 \times 3$ multiplicity-space Gram has eigenvalues given in closed form by the trigonometric Cardano formula. Six independent lattice-sum entries enter. + +- *Block 1* (below crossing): the isotypic component of the irrep "natural $3$D rep tensor $\operatorname{sgn}(\sigma)$" with multiplicity $2$. Its $2 \times 2$ multiplicity-space Gram has eigenvalues in closed form + + $$\lambda\_-^{(1)} = \tfrac{1}{6}W(s) - 3D(s) - \tfrac{1}{2}\sqrt{\bigl(\tfrac{W(s)}{3} - 4B(s)\bigr)^2 + 4\bigl(3D(s) - B(s)\bigr)^2},$$ + + with + + $$W(s) = Z(s-2) - 2Z(s-1) + Z(s) = \sum\_{k \neq 0} \lvert k\rvert^4 (1+\lvert k\rvert^2)^{-s},$$ + $$B(s) = \sum\_{k \neq 0} k\_1^2 k\_2^2 (1+\lvert k\rvert^2)^{-s},$$ + $$D(s) = \sum\_{k \neq 0} k\_1^4 k\_2^2 (1+\lvert k\rvert^2)^{-s} / \lvert k\rvert^2,$$ + $$Z(s) = \sum\_{k \neq 0} (1+\lvert k\rvert^2)^{-s}.$$ + + Two structural identities collapse the original four lattice-sum entries to this minimal form: the trace identity $3A + 6B = W$ (from $\sum \lvert k\rvert^2 (1+\lvert k\rvert^2)^{-s} = Z(s-1) - Z(s)$) and the orbit identity $2D + C = B$ (from $k\_1^2 k\_2^2 \lvert k\rvert^2 = k\_1^4 k\_2^2 + k\_1^2 k\_2^4 + k\_1^2 k\_2^2 k\_3^2$), where $A = \sum k\_1^4 w$ and $C = \sum k\_1^2 k\_2^2 k\_3^2 w / \lvert k\rvert^2$ are eliminated. The anisotropic content of the constant below the crossing is therefore exactly two scalar lattice moments $B$, $D$ plus three Sobolev-zeta values $Z(s-2)$, $Z(s-1)$, $Z(s)$. + +- *Sector crossing.* The two lead blocks switch dominance at + + $$s^\* \approx 4.7892803103, + \qquad \lambda\_{\min}^+(G\_{3,s^\*,3}) \approx 0.0383102507, + \qquad C\_{3,s^\*,3} \approx 5.109077647.$$ + + For $s \in (9/2, s^\*)$ the constant equals $1/\sqrt{\lambda\_-^{(1)}(s)}$ (block 1, $2 \times 2$ quadratic form). For $s > s^\*$ it equals $1/\sqrt{\lambda\_-^{(0)}(s)}$ (block 0, $3 \times 3$ trigonometric Cardano form). The crossing exponent itself is determined by the transcendental equation $\lambda\_-^{(0)}(s) = \lambda\_-^{(1)}(s)$. + +**$m = 0$ closed form.** For $m = 0$, the Gram is isotropic, $G = \tfrac{2}{3} Z\_d(s) I\_d$, and + +$$C\_{0,s,d} = \sqrt{d / ((d-1)Z\_d(s))}.$$ + +For $(d, s) = (3, 5)$ this gives $C\_{0,5,3} = \sqrt{3/(2 Z\_3(5))} \approx 2.439\ldots$. The cube truncation converges to this closed form at rate $O(K^{d - 2s})$, providing a calibration of the truncation methodology. + +**Variants.** The same construction with $s \neq 5$ gives a one-parameter family of related constants. Selected mpmath-Ewald values: + +| $(m,d,s)$ | $C\_{m,s,d}$ (engineering-rigorous, ${\sim}10^{-13}$ width) | Lead $W(B\_3)$ block | +| --- | --- | --- | +| $(3,3,5.0)$ | $6.576432995516\ldots517$ | block 0 (above crossing) | +| $(3,3,4.75)$ | $4.923777057473\ldots$ | block 1 (below crossing) | +| $(3,3,4.6)$ | $4.249762005328\ldots329$ | block 1 (below crossing) | + +The above-crossing single-grade vs below-crossing mixed-grade distinction explains a sharp gap collapse from ${\sim}1\%$ to ${\sim}0.002\%$ in the cube + Rayleigh-tail certificate as $s$ moves from $5$ toward the threshold $9/2^+$; the slowest tail integrand changes from $t^{8 - 2s}$ (mixed-grade) to $t^{6 - 2s}$ (pure-grade-2). + +**Cross-domain applications.** The constant has natural interpretations in several active research lines: + +- Parabolic controllability with point-supported terminal data: the terminal-symbol factor in adjoint observability estimates with finite-jet terminal data has the form $\lVert F\_\alpha \rVert\_{H^{-s}} \geq c\_{\mathrm{term}} \lvert\alpha\rvert$, with $c\_{\mathrm{term}}$ controlled by $1/C\_{m,s,d}$. +- Inf-sup (Babuska-Brezzi) analysis for divergence-free finite element methods: discrete analogs of $C\_{m,s,d}$ control the well-posedness of divergence-constrained variational forms. +- Off-the-grid sparse divergence-free measure recovery via atomic-norm minimization: $C\_{m,s,d}$ is a coherence parameter for divergence-free atomic dictionaries built from point-supported jets. +- $H^{-s}$ norm equivalence on point-supported momenta in LDDMM-style diffeomorphic image registration: the registration energy is $\lVert p \rVert^2\_{H^{-s}}$ with $p$ a point-momentum measure, and the squared coefficient norm is controlled by $C\_{m,s,d}^2$. + +Existing literature in each of these uses related quantities implicitly without pinning down values; the closed-form expressions, kernel theorem, and $W(B\_3)$ sector-crossing observation in this submission are not yet recorded in any of them. + +**Live optimization targets.** + +1. Strict Arb-rigor for the engineering-rigorous Ewald evaluator: replace mpmath's quadrature with a provable Gauss-Legendre / tanh-sinh scheme with rigorous Lagrange remainder bounds. The mpmath Ewald row above is engineering-rigorous (every truncation has an explicit analytic bound, mpmath quadrature treated as exact at the working precision and validated by precision-doubling), but not yet strictly Arb-rigorous. +2. Higher-precision two-sided certified brackets approaching the threshold regime $s \to 9/2^+$, where the closed-form $W(B\_3)$ structure simplifies but the cube convergence rate degrades. +3. A closed-form analytic eigenvalue inside one of the $W(B\_3)$ multiplicity-space Gram blocks would eliminate the trigonometric Cardano root (in the above-crossing block) or further reduce the two-anisotropic-moment dependence (in the below-crossing block) to elementary expressions in known Sobolev/Epstein zetas. + +This constant is recorded here under the "particular mathematical interest" exception clause of the contributing guidelines. The four cross-domain applications above and the explicit closed-form $W(B\_3)$ sector structure are the basis. + +## References + +- [Mathews2026] R. J. Mathews. Finite-jet norm-equivalence constants for Leray-projected point jets on $\mathbb{T}^d$: closed-form $W(B\_3)$ sector decomposition. Preprint and companion scripts (2026). [Code repository](https://github.com/kantrarian/finite-jet-constants). + +## Contribution notes + +Claude (Anthropic Claude Code, Opus 4.7) was used to assist with the +representation-theoretic decomposition, the closed-form $W(B\_3)$ block-0 and +block-1 multiplicity-Gram eigenvalue extraction, the moment-reduction identities, +and the drafting of this submission. All mathematical content (kernel theorem, +sector-crossing exponent, closed-form expressions, numerical brackets) was +reviewed and verified by the human contributor against independent cube and +Ewald computations.