Research Article · Preprint Mathematics
Zero geometry and sharp compensated permanent derivatives on a complex Gram stratum
- Primary subject
- Mathematics
- Subject subfields
- Linear Algebra and Matrix Inequalities
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 1 · Permanent link
Overview
Abstract
We study a homogeneous compensated derivative arising from permanent inequalities for centered complex positive semidefinite matrices of order six. On the conjugate-disjoint locus K K̄ = 0, the ordinary omission variance has exactly forty trace-normalized zeros, all sparse rank-one cube-root configurations. On the entire flat, equal-spectrum rank-two stratum, including all compatible complex phases, we determine the compensated derivative explicitly and prove its sharp lower bound 18912 − 140√2. The proof retains the common Gram structure: a real balanced-frame complement gives a complete complex-structure chart, and threefold permutation symmetry eliminates the remaining phase from the paired permanent derivative. An explicit Frobenius tube of radius 10⁻⁶ has a positive uniform margin and contains full-dimensional families of centered rank-five matrices, whose positive rank-one anchors have rank six. A conditioning-free positive-part construction also reduces small conjugate overlap to the exact locus. Its nonflat, unequal-spectrum part and the unrestricted compensation inequality remain open.
Keywords permanent · positive semidefinite matrix · Gram matrix · compensated derivative · Fischer norm · balanced frame
Scope of the work
Exactly forty omission-variance zeros on the centered conjugate-disjoint trace-one locus; sharp phase-complete bound F₄ ≥ 18912 − 140√2 on the flat equal-spectrum rank-two stratum; explicit full-rank neighborhood and positive-part reduction. Generic nonflat and unequal-spectrum compensation and the full Chollet conjecture remain open.
Reproduction notes
The supplementary ZIP includes the manuscript PDF, complete editable LaTeX source, a standard-library Python exact verifier, exact output, reader instructions, citation metadata and SHA-256 member manifest. The verifier computes a symbolic coefficient and checks an independent 720-permutation implementation; the universal geometry and phase argument are proved in the paper.
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Author disclosures
AI assistance was used in research, drafting and checking. The author is responsible for the mathematical statements and presentation. This author-published preprint does not claim external refereeing or formal proof-assistant verification.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Zero geometry and sharp compensated permanent derivatives on a complex Gram stratum (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/zero-geometry-sharp-compensated-permanent-derivatives-complex-gram-stratum/v/1
BibTeX
@techreport{luminary_zero_geometry_sharp_compensated_permanent_derivatives_complex_gram_stratum_v1,
title = {Zero geometry and sharp compensated permanent derivatives on a complex Gram stratum},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/zero-geometry-sharp-compensated-permanent-derivatives-complex-gram-stratum/v/1}
}Loading discussion…