Harder mathematical progress, 19 October 2026
Sulayman Bowles. Developed with OpenAI Codex.

1. Divisor-word growth (complete partial-result proof)

For the least integer N(w) whose positive-divisor indicator contains a word w,
let M(m) be the maximum of N(w) over words of length m. The paper proves

  log M(m) >= m log m - 2m log log m + (2 log 2 - 1)m + o(m).

The finite obstruction is N(0^p 1^r) >= r*binomial(p^2+r,r), for prime p
and r>=p. The all-word arithmetic mean has the same leading lower growth,
with linear constant log 2 - 1. Matching upper asymptotics remain open.
The paper also proves uniformly

  log N(1^a 0 1^b) = b + max(a,b) + o(a+b).

Read Divisor_Words/Divisor_Word_Growth.pdf. Its editable LaTeX source is
beside it. Build with two pdflatex passes. The source credits Torma's
original question and the classical consecutive-lcm methods.

2. A078121 diagonal coefficients (partial theorem)

Read Diagonal_Coefficients/solution.md for all definitions and proofs.
A finite exact sign certificate proves P(m,j)>0 for every 0<=j<=50 and
EVERY m>=2j+1. Positive integrality is proved for j=0,1,2 for every
admissible m. General integrality and positivity for j>50 remain open.
The logarithm sequence A111814 is prior work by Helms and Hanna (2005).
The finite certificate is computation-assisted mathematics; it is not
proof-assistant formalization.

Reproduce using Python 3.9 or later, standard library only. Assertions
must remain enabled. Run these commands from the extracted package root:

  python3 Divisor_Words/verify.py
  python3 Divisor_Words/independent_single_zero/independent_verify.py
  python3 Divisor_Words/independent_maximum/verify.py
  python3 Diagonal_Coefficients/verify.py
  python3 Diagonal_Coefficients/independent_review/verify_log.py

Each program writes its own results beside the source. Exact integer and
rational assertions are substantive; timing and approximate diagnostics
are not proofs. ROOT_CHECKS.json records the delivery-folder reruns.
SHA256.json hashes the delivered files before later reruns.

Community submission

community_submission/ records the actual A078121 parity-proof submission
for review at https://github.com/Eppie/oeis-proofs/issues/1 . The complete
proof was read back and matched the submitted body. An open issue is not
an acceptance or journal publication. That earlier parity proof is distinct
from the harder diagonal-positivity result in this package.

Attribution and limits

Codex generated proofs, prose and programs at Sulayman Bowles's request.
Separate Codex instances reviewed the arguments and wrote independent
exact programs. All five programs were rerun from the delivery folder after proof review.
No human peer review or formal proof-assistant verification is represented.
Historical priority has not been established. Third-party papers and
repository documents inspected during research are not redistributed.
