Elimination Geometry II: Integrability, Representation, and Structural Limits
Abstract
Elimination geometry studies the exact nonnegative defect left after an auxiliary variable is optimized out. Part~II audits this certified-elimination interface through three theorem families. For integrability, local gaps form a repeated partial-elimination tower precisely under nonnegativity, touching, and zero elementary-square circulation. Hodge projection constructs nearest finite-space repairs and separates curvature, certification, and global period taxes. For representation and price, a descent gate characterizes surjective refinements and rules out strictly convex separable, duplication-invariant weight penalties. Dummy and target-calling lifts show that unrestricted lift design is vacuous, whereas the Fidelity-coherence frontier isolates irreducible cost. A modal Gram block classifies unconstrained quadratic lift-coupling germs and yields aligned Gaussian risk ledgers. Bernstein spectral functions preserve graph-Laplacian structure. Graph-universal Markov resolvents exist exactly up to first-order fractional qualification, and universal polynomial spectral penalties reduce to nonnegative Laplacian multiples. This linear limit does not preclude Part~I's nonlinear CDF repair and sharp CRPS theory. For converses, finite-order target-jet preservation uniquely requires subtraction of the corresponding defect jet, up to flat terms; approximate jets yield quasi-descent and stationary limits. Fixed-marginal coupling obeys a dependence-benefit frontier. Reversing certified KL in entropy-regularized control eliminates dimension-free guarantees, while a fixed measurement preserves all Umegaki defects of a faithful Gibbs family exactly when that family commutes.
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BibTeX
@misc{huang2026elimination,
title = {{Elimination Geometry II: Integrability, Representation, and Structural Limits}},
author = {Mian Huang and Xueqin Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30145},
eprint = {2610.30145},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30145},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Elimination Geometry II: Integrability, Representation, and Structural Limits
AU - Mian Huang
AU - Xueqin Wang
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.30145
UR - https://arxiv.org/abs/2610.30145
AB - Elimination geometry studies the exact nonnegative defect left after an auxiliary variable is optimized out. Part~II audits this certified-elimination interface through three theorem families. For integrability, local gaps form a repeated partial-elimination tower precisely under nonnegativity, touching, and zero elementary-square circulation. Hodge projection constructs nearest finite-space repairs and separates curvature, certification, and global period taxes. For representation and price, a descent gate characterizes surjective refinements and rules out strictly convex separable, duplication-invariant weight penalties. Dummy and target-calling lifts show that unrestricted lift design is vacuous, whereas the Fidelity-coherence frontier isolates irreducible cost. A modal Gram block classifies unconstrained quadratic lift-coupling germs and yields aligned Gaussian risk ledgers. Bernstein spectral functions preserve graph-Laplacian structure. Graph-universal Markov resolvents exist exactly up to first-order fractional qualification, and universal polynomial spectral penalties reduce to nonnegative Laplacian multiples. This linear limit does not preclude Part~I's nonlinear CDF repair and sharp CRPS theory. For converses, finite-order target-jet preservation uniquely requires subtraction of the corresponding defect jet, up to flat terms; approximate jets yield quasi-descent and stationary limits. Fixed-marginal coupling obeys a dependence-benefit frontier. Reversing certified KL in entropy-regularized control eliminates dimension-free guarantees, while a fixed measurement preserves all Umegaki defects of a faithful Gibbs family exactly when that family commutes.
ER -