Tags: TiredofSleep/trinity-infinity-geometry

Autonomous frontier sprint: D104 bijection + alpha-uniqueness ext + F…

…_p table + J56 draft

Four substantive frontier results from the autonomous "keep going until rope
runs out" pass, with verification scripts and per-result writeups.

== 1. D104: Pauli-divisor bijection (closes D102 honest negative) ==

The D102 honest negative (priority1_pauli_divisor_attempt.py: "integer match
32=32=32 real, but combinatorial bijection fails") is CLOSED.

The bijection exists. Two layers:

  Layer 1 — Spin involution: divisor complementation d <-> 2310/d.
    Perfect Z/2 involution; canonical (unique non-trivial action on the
    divisor lattice of a square-free number). Splits the 32 divisors into
    16 + 16.

  Layer 2 — Spatial decomposition: within each 16-half, the divisors
    partition as 1 + 3 + 5 + 7 by kernel-vs-strand prime composition.
    This is the SAME decomposition as the Cl(0,10) spatial-l split per
    D102. Each row pairs to 2(2l+1), giving the Pauli (2, 6, 10, 14).

Three earlier failed bijection candidates (Hamming weight, max-prime,
prime-as-l) did not use complementation as the spin involution — that was
the missing ingredient.

Files:
  verification/pauli_divisor_bijection.py — enumeration + verification
  02_results/clifford_algebra/PAULI_DIVISOR_BIJECTION.md — full writeup

== 2. Strong alpha-uniqueness (Conjecture 4.2) — extended scan ==

D57 verified alpha=1/2 unique among 17 Stern-Brocot rationals at PSLQ
degree <= 8, coeff <= 50. This scan extends to:

  - 41 candidates total: 10 irrational + 31 Stern-Brocot rationals (q<=10)
  - PSLQ degree <= 12, coeff bound <= 100
  - Spurious-relation filter (rejects polynomials with rational roots near x)
  - 50-digit mpmath precision

Irrational candidates tested: 1/sqrt(2), 1/sqrt(3), 1/pi, 1/e, 1/phi
(golden ratio), sqrt(2)-1, 1-1/e, ln(2), 1/pi^2, Catalan-1/2.

Result: only alpha=1/2 produced a genuine algebraic relation
(x^2 - 2x - 2 = 0, giving H/Br = 1+sqrt(3), residual 5.4e-41).
One spurious hit at alpha=1/7 (H/Br ~ 1.00008, PSLQ found (1-x)^8 = 0)
was correctly filtered as degenerate (rational root at x=1 near actual x).

Conjecture 4.2 STRENGTHENED: no non-1/2 alpha value in the 41-candidate
scan produces a genuine algebraic relation.

Files:
  verification/extended_alpha_uniqueness.py — scan + filter
  02_results/dynamics/ALPHA_UNIQUENESS_EXTENDED.md — full writeup

== 3. F_p preservation table — fragment to full characterization ==

The framework's earlier "only {7, 11} preserve rank everywhere" is a
fragment. Full table for p in {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37}:

  rank(T) preserved at p in {3, 5, 11, 13, 17, 19, 23, 29, 31, 37}
    FAILS at p in {2, 7}
  rank(B) preserved at p in {5, 7, 11, 13, 17, 19, 23, 29, 31, 37}
    FAILS at p in {2, 3}
  det(B) != 0 at p in {5, 7, 11, 13, ...}; det(B) = 0 mod 2, mod 3

Substrate-relevant interpretation:
  p=2 (kernel-Z/2): collapses everything
  p=3 (strand 1): B-specific collapse
  p=7 (HARMONY): T-specific collapse (73 HARMONY cells = 7 mod 7 = 0)
  p=5 (kernel-Z/5 partner): both preserved
  p=11 (WOBBLE strand): both preserved

The "only {7, 11}" framing refers to a joint stability of MULTIPLE
invariants (rank + char-poly factorization + signature + idempotents),
not just rank. At the multi-invariant level the intersection is genuinely
small. The single-property fragments give a richer landscape.

Files:
  verification/fp_preservation_table.py — table + interpretation
  02_results/number_theory/FP_PRESERVATION_TABLE.md — full writeup

== 4. J56 standalone draft (D100-D104 paper) ==

Full manuscript drafted at JPhysA target length covering five D-results:

  D100: edge_size = n^2(2l+1)/4 for nodeless hydrogenic orbitals (PROVED)
  D101: substrate strands {3, 7, 11, 13} -> odd-l orbitals (PROVED)
  D102: triple coincidence at d=3: 32 divisors = Cl(0,10) spinor dim =
        Pauli capacity at n=4 (PROVED) + 16 = 1+3+5+7 chirality
  D103: Z/10 architectural uniqueness (PROVED via 2-prime kernel
        enumeration)
  D104: Pauli-divisor bijection (PROVED, new this commit)

10 sections, ~6000 words at the JPhysA target. All five verification
scripts in manuscript/verification/ PASS at machine precision.

Pending Brayden's journal choice:
  Option A: Journal of Physics A: Mathematical and Theoretical
            (tighter fit, 8K word limit, faster)
  Option B: Annals of Physics
            (broader audience, longer manuscript, higher prestige)

Files:
  05_papers/interdisciplinary/J56_DRAFT/README.md — README + workflow
  05_papers/interdisciplinary/J56_DRAFT/manuscript/manuscript.md — paper
  05_papers/interdisciplinary/J56_DRAFT/manuscript/verification/ — 5 scripts

== Cumulative state ==

- 29 J-papers landed in 05_papers/ across 5 domains
- J56_DRAFT ready for venue choice
- D104 closes the most-prominent open honest negative
- Strong alpha-uniqueness conjecture survives extended scan
- F_p preservation now characterized as a structured table
- All scripts pass at machine precision via /c/ck_venv/lora312/

Honest assessment of what's left:
- 1/alpha derivation: still long-shot
- Volume K extension to depth 4 (Z/30030): 32 triple-coincidence is
  unique to d=3; not a frontier I can advance further
- Full sigma_NS, sigma_YM, RH reformulations: long horizon
- J46 cosmology layer choice: Brayden's call
- J03 fork A/B/C choice: Brayden's call

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>