Section 11 9 October 2026

The quantitative cost of the anisotropic continuity route

Jeremy Rodgers · Independent Researcher
doi:10.5281/zenodo.23259569

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11 The quantitative cost of the anisotropic continuity route

Proposition 11.1 (A flat embedding bound with the physical factors retained)

Let a localized extension $u$ on one radial and seven tangent Euclidean coordinates satisfy $\|u\|_2\leq e$ and $\|u\|_{H^2_rH^4_{\rm tan}}\leq M$ . With unitary Fourier convention and $\theta=15/16$ ,

\|u\|_\infty\leq C_\theta e^{1/16}M^{15/16},\qquad C_\theta^2\leq\frac{15}{616\pi^5}<10^{-4}.

(11.1)

On physical charts the localization, extension, coordinate Jacobian and any half-density constants must multiply this flat bound.

Proof

Fourier Hölder bounds the $H^{2\theta}_rH^{4\theta}_{\rm tan}$ norm by $e^{1-\theta}M^\theta$ . Cauchy in the evaluation integral gives

C_\theta^2=(2\pi)^{-8}I_1(2\theta)I_7(4\theta),\qquad I_m(s)=\int_{\mathbb R^m}(1+|\xi|^2)^{-s}d\xi .

The strict thresholds are $2\theta>1/2$ and $4\theta>7/2$ . For the displayed rational upper bound no special-function evaluation is needed. Splitting each integral at radius one,

I_1(15/8)\leq2(1+4/11)=30/11,\qquad I_7(15/4)\leq |S^6|(1/7+2)=16\pi^3/7.

Here $|S^6|=16\pi^3/15$ . Multiplying and dividing by $(2\pi)^8$ gives $15/(616\pi^5)$ ; the elementary lower bound $\pi>3.14$ proves it is below $10^{-4}$ .

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For two waves, the density difference obeys $\|\rho-\widetilde\rho\|_\infty \leq(\|\psi\|_\infty+\|\widetilde\psi\|_\infty) \|\psi-\widetilde\psi\|_\infty$ . Thus the genuine product graphs can supply a common positive-density tube for Theorem 5.1. They supply only the slow error exponent $1/16$ in the eight-coordinate case. The compact COM radius enters the tangent ellipticity constant; physical source widths enter the profile derivatives; chart and extension factors remain; and the propagated graph constants can be large. The flat number below $0.01$ is consequently not a numerical record-error certificate or a small physical stability coefficient. It is the explicit analytic link between an attained common-form/graph comparison and the positive-tube hypothesis of the whole-history theorem.