[JoyboyBrian]

Result

Proves the two-edge operator-norm limit for the original untruncated broad Gaussian phase cell, for each fixed alpha>0 and N=floor(1/theta). Under an explicit step-function embedding, the checkerboard inverse scaled by sqrt(theta) converges to a direct sum of two half-line Volterra operators. The proof includes the reflected endpoint offset, finite boundary rows, diagonal strip, middle and cross-edge tails, and a quantitative remainder.

Writing b=pi/alpha² and P(b)=product_(r>=1)(1-exp(-2br)), the exact norm limit is alpha exp(b/4)/(2 sqrt(b) P(b)^3). The full positive q-LDU sum supplies all three product factors. The half-line norm follows from the actual Dirichlet adjoint domain, rather than an assumed full-line boundary condition.

This resolves the boundary-layer question left formal in Ryanson Jonathan #4435 and open after Nick Petalas #5150. Credit JoyboyBrian #5103 for the full-cell upper bound, Jose Manuel #4193, Vlado Vojdanovski #1673 and the retained classical q-LDU lineage. Prepared by JoyboyBrian with substantial OpenAI Codex assistance. No new kappa.

Validation

Registered normal and optimized checks pass. The exact finite certificate covers12 rational matrices/261 inverse entries,7906 original source-expression and gauge pairs including alpha3/theta1/400/N400,216 compact exponent comparisons, geometric tail bounds and step-embedding controls. These corroborate the written universal proof; they do not establish it by fitting norms.

Committed-byte/source-history preflight and official export pass. Fresh Git-free, PATH-empty optimized replay passes in0.327 seconds on CPython3.12.7, preserving all17 exported files including the manifest. Eight actual mutants reject: missing RESULT/SOURCES, source drift, typed float/int receipt alias, dropped positive tail, omitted reflected offset, missing h scaling and wrong q power.

Reproduce: python3.12 -B scripts/research_package.py --package research/gaussian-half-line-operator-limit.

Scope

Fixed alpha and this full-cell section only. No uniform growing-alpha conclusion, production guard substitution, rounded-prefix bound, coefficient-preparation/inverse-application algorithm, physical storage or movement tariff, native admission, outer recovery or selected-frontier change. Independent root-agent proof review passes at the exact proof hash; this is AI-assisted review, not formal or human acceptance. Final documentation-only commit preserves every cold-tested scientific byte.

Canonical claim: #4202 (comment)