Closure-Normalized Branchwise Critical-Haar Recurrence, Complete Sector Packing, Gaussian Sign Localization, and an End-to-End PIHR Proof Candidate for the Riemann Hypothesis (PIHR-RH L)

PIHR–RH XXXXIX repaired the principal structural defect discovered in PIHR–RH XXXXVIII: square summability of individual top-prime arithmetic updates had incorrectly been followed by squaring their global vector sum. The corrected architecture kept every primary top-prime update inside its own Split–Merge branch and applied Young’s inequality before summation over primary primes.

A subsequent referee-style audit isolated the remaining tasks with greater precision. They were not a new global Carleson theorem or a new arithmetic positivity principle. Rather, the argument required: an explicit common Critical-Haar Hilbert normalization; a proof that every ordinary, fractional, annular, Bridge, proper-power, collision, and shell-refinement update is square-summable in that same normalization; a precise once-only recurrence avoiding duplicate error allocation; a shell-to-cumulative Critical-Charge Carleson argument; and a fully quantitative formulation of the Gaussian localization, phase-matching, tail, and one-sided Laplace–Landau steps.

The present manuscript incorporates those corrections directly.