Enter, I-B-VP-5-Ex1-v2, 2026, pp. 1-43
I-B-VP-5-Ex1 Smooth curvature cores, convergent exterior mode sums, and renormalized apex moments
Ngo Cheung
18 October 2026
Abstract
We study free scalar fields on smooth, rotationally symmetric curvature cores joined to a prescribed conical exterior. The principal analytical advance is a profile-independent reflection estimate that proves absolute convergence of the exterior Euclidean mode sum and every fixed differentiated sum on compact exterior sets. This removes the exterior mode-representation hypothesis of I-B-VP-CORE for the smooth nonnegative-curvature class.
For positive curvature coupling, we derive the complete smooth-core scalar moment, including its interior contribution, and strengthen the logarithmic-channel expansion to a uniform O(M⁻⁴) remainder on explicitly nondegenerate profile families:
c_P^sharp(ε; μ) = 2παslog (1/ε) − ((ξ − 1/6)K)/(8π²)log (με) − 1/(4π)[log M − 1/(2M²) + (2 − ζ(3))/(3M³)] + C_P + O(M⁻⁴), M = log (1/ε) + 1 + d_P⁻¹.
Here K = 4π(1 − α), and the superscript specifies a sharp radial moment. Smooth test cutoffs produce additional, explicit inverse-M terms; thus the absence of a 1/M term is not cutoff-independent.
An independent heat-coefficient and lapse-variation calculation fixes the conformal stress running and its shape dependence. The resulting dilation-invariant projected moment diverges at fixed renormalization scale. A core-adapted scale does not remove its universal log log divergence. After that common divergence is subtracted, an exact profile-rescaling identity obstructs universality on the stated nominal-radius class, while the equal-minimal-radius finite-part problem remains open. We give convergent representations and truncation bounds for the finite constants rather than unexecuted numerical values.
The potential significance is a controlled separation of operator selection, exterior scattering, full scalar contact concentration, and stress-moment normalization. No priority is claimed for cosmic-string core effects, Bessel matching, or heat-kernel renormalization. No full stress-distribution limit, interacting defect theory, or gravitational source equation is constructed.
Bibliographic record
- Title
- I-B-VP-5-Ex1 Smooth curvature cores, convergent exterior mode sums, and renormalized apex moments
- Author
- Cheung, Ngo
- Publication date
- 2026/10/18
- Institution
- Enter
- Report number
- I-B-VP-5-Ex1-v2
- Publisher
- Cheung Ngo Medical Limited
- First page
- 1
- Last page
- 43
- Language
- en