Extended arXiv Draft 7 - release/validation notes
Date: 2026-09-03

Purpose
-------
This draft preserves the long/expository arXiv article while porting all
mathematical, computational, reproducibility, and presentation changes made in
the referee-revised journal Version C.  The extended history, reciprocal-period
primality toolbox, Midy/q-Midy material, cyclotomic/Pocklington/BLS discussion,
exploratory low-omega computation, and OEIS section are retained.

Range audit: d<=64 everywhere it should be
--------------------------------------------
All current range-level statements, full tables, cumulative composite counts,
rarity statistics, fixed-base prime-mass calculations, figures, OEIS prefixes,
and certified-search descriptions extend through d=64.  Occurrences of d=33
remain only when mathematically intrinsic or historical: the dyadic-transition
theorem, its explicit line data, or the legacy Julia provenance directory.

Referee-driven computational revisions ported from Version C
-------------------------------------------------------------
- Independent kappa=1 prescribed-Carmichael-index certification for every
  34<=d<=64. Pinch is now an external cross-check, not a proof dependency.
- Exact split certificates for the difficult empty d=34 and d=35 cases.
- kappa>1 C++ primality decisions use the first twelve prime bases 2,...,37,
  certified by the Sorenson-Webster psi_12 threshold.
- The independent kappa=1 factor certificates use psi_12/psi_13 and recursive
  Pocklington fallback above those thresholds.
- Explicit maximum recursive prime bounds, maximum primality inputs, and
  overflow=false status are recorded.
- The d=49 (49,5,3,16,1) branch is highlighted as the principal computational
  application of the last-two-prime divisor-pair lemma; 49,496 prefixes are
  exhausted with no solution.
- A one-page certificate-summary table is incorporated into the manuscript.

Proof/presentation revisions ported
-----------------------------------
- Small d<=8 hand proof now prints the possible smallest primes and states what
  is checked in the finite divisor-pair enumeration.
- The origin of the leading 1/2 in the uniform counting bound is written out.
- The growing-d line-sampling proof explicitly fixes d0 and C and handles the
  finite initial range separately.
- The perfect-power multiplier is tied specifically to Moree (2013), Eq. (8).
- The prime-mass graph labels the lower curve as M_2(D); only its D=64 endpoint
  is claimed to be the proven global minimum.
- Standard literature terms Fermat d-pseudoprime, residual index,
  near-primitive root, and Carmichael index appear in the abstract/keywords/
  introduction.
- The stale introductory 96.7% statement is replaced by the certified universal
  D=64 value 98.2709160681865...%.

Certified d<=64 numerical facts
-------------------------------
- 351 spectral incidences.
- 234 distinct odd composites.
- max e(d)=13, at d=53 and d=62.
- Somer nontrivial-base convention: 231 distinct composites through d=64.
- G_64 = 0.9845291459538847133627846309104729...
- min non-perfect-power mass = M_2(64)
  = 0.9827091606818653790657857435100441...
- unique minimizing squarefree kernel: 2.
- exact 262,143-kernel search and q>=67 tail certificate pass.

Extended-arXiv-only material retained
-------------------------------------
- Low-omega exploratory script validates every omega<=3 incidence against the
  exact d<=64 table before extending through d=500.
- Through d=64 it accounts for 320/351 incidences; the remaining 31 have
  omega=4.
- Four OEIS candidate sequences remain printed/machine-readable through d=64.
- Both cumulative figures extend to D=64.

Fresh reproducibility checks on this archive
--------------------------------------------
- combined verifier: incidences 351; distinct 234; bad []; PASS.
- kappa=1 split-log verifier: d=34 PASS; d=35 PASS; ALL SPLIT LOGS PASS.
- specialized d=49 certificate rerun from packaged source: 49,496 prefixes;
  solutions [].  Its obsolete module import was repaired to use the shipped
  twelve-base search_reference_python module.
- prime-mass exact rerun: DMAX=64; 262143 kernels; minimizer 2; tail PASS.
- low-omega validation: d<=64 PASS.
- both Version C C++ search sources compile; d=34 kappa>1 smoke run returns
  [35,69,341,139231] with overflow=false.

Build validation
----------------
- final PDF: 62 A4 pages.
- clean three-pass pdflatex build: no LaTeX errors, undefined references or
  citations, overfull boxes, or underfull boxes.
- PDF preflight: openable, unencrypted, not scanned, no XFA.
- all pages rendered; modified theorem/table/figure pages visually checked.
- source ZIP is recompiled from a fresh extraction before release.

Archival supplement
-------------------
The supplement is frozen under SHA256SUMS_ARXIV7.txt.  It is intended to be
uploaded unchanged to Zenodo or an institutional repository.  The manuscript
contains [SUPPLEMENT-DOI] until the persistent DOI is assigned.
