FPRD research loop state

Updated 6 September 2026 after the uniform-finiteness audit and complementary $f=0$ triple-minimum attack.

Contents

Active program and mode

Program: Selfridge’s question $\lVert2^n\rVert=2n$ for integer complexity.
Current mode: ATTACK.
Active restricted grammar: normalized primitive proper three-addition forks with positive $(2,3)$ -smooth displayed operands and exact paid-atom costs.
Current packet: FPRD-Selfridge-uniform-finiteness-and-f0-triple-minimum-2026-09-06.md.
Reproducible evidence: Selfridge-uniform-finiteness-and-f0-triple-minimum-code-and-results.zip.
This track has not solved Selfridge’s unrestricted problem. Its results are ordinary research notes, not Publications. No external review or priority determination is recorded.

New result in this run

1. Uniform finiteness, but not a uniform height bound

Every normalized complementary presentation under study expands to

$2^{r+u}+2^{r+w}3^y+2^{s+u}3^v+2^{s+w}3^{v+y}+2^f3^g=2^{K+f}.$

After division by the right side, five positive S-units sum to one in a multiplicative group of rank at most six. The parameter map is injective for a fixed factor shape. Evertse–Schlickewei–Schmidt therefore gives at most

$8\exp(7\cdot30^{15})$

presentations across the four $f=0$ , two $f=1$ , and two surviving $f=2$ factor patterns.

This is a direct classical corollary. Crucially, the theorem counts solutions but does not bound their heights. The proposed comparison between the growing valuation period and one Matveev cutoff is therefore not available before a dominance/freezing argument reduces the five-term equation. Do not resume that route without a genuinely effective multi-term height reduction.

2. Complementary f=0 triple-minimum classification

For

$RS+3^g=2^K,$

where each factor is $1+2^x3^v$ or $2^x+3^v$ , the inherited quadratic gate and complete modulo-24 factor table leave exactly 40 ordered canonical placements. The first 3-adic digit permits exactly 20 tied-minimum cancellation strata. Exactly two have $v=y=g=m$ .

Family A: $(1+4\cdot3^m)(1+2^x3^m)+3^m=2^K,$ with $x,K$ even and $m$ odd. Complete periods modulo 7 and 73 have no CRT-compatible rows, so this family is empty.
Family B: $(2+3^m)(1+2^x3^m)+3^m=2^K,$ with $x,m,K$ odd. Modulo $2^x$ and $v_2(1+3^m)=2$ force $x=3$ . Completing the square gives $(8\cdot3^m+9-2^{(K+3)/2})(8\cdot3^m+9+2^{(K+3)/2})=65.$ The unique positive solution is $(m,x,K)=(1,3,7),\qquad (2+3)(1+8\cdot3)+3=2^7.$ Its displayed cost is $18=2K+4$ , so it is a harmless control.

The exact certificate has two independent implementations:

Python reconstructs the 40/20/2 counts, residue tables, CRT incompatibility, factorization, and cost.
C++20 independently reconstructs the same objects and conclusions.

Retained completed results

The following results remain active inputs. Consult the corresponding public claims and earlier packets for their exact hypotheses.

The two-successor family is completely classified using an elementary modular exclusion and Bennett’s theorem.
The normalized positive-type-I branch is complete, including the exceptional $x=4$ lift and all exterior layers.
The normalized positive-type-II tower for $a\ge1$ is complete and contains exactly ten controls, all costing at least $2N+4$ .
In the complementary $a=0<g$ orientation, quadratic reciprocity leaves six residue pairings and eliminates the double-even $f=2$ split.
The surviving $f=1$ and split- $(2,0)$ $f=2$ branches satisfy exact valuation laws because every tied-minimum cancellation is excluded.
Their complete minimum layers $m=1,2$ contain exactly seventeen proper controls, all costing at least $2N+6$ ; no other solution occurs in those layers.

Public claim sequence now runs through FPRD-IC-T29. The new claims are:

FPRD-IC-T28: uniform finiteness of the normalized complementary family by the Evertse–Schlickewei–Schmidt theorem.
FPRD-IC-T29: complete $f=0$ triple-minimum classification.

Unresolved obligations

Classify the 18 remaining $f=0$ pair-minimum cancellation strata. In each, exactly two among $v,y,g$ equal $m$ and the third is larger.
Derive the next nonzero 3-adic digit for each canonical parity type before adding primes. Combine that digit with complete periods modulo 7 and 73.
Separate genuine small controls from formal or improper boundary roots.
Seek a finite quotient or descent. Do not report a larger search height as closure.
Preserve exact construction costs and the distinction between normalized presentations and unrestricted integer complexity.
If pair-minimum graphs survive, exploit their centered equation; the S-unit corollary proves there cannot be an actual infinite solution family, but it does not locate the last point.

Site checkpoint

Site: https://fprd-lab.joshgay.chatgpt.site
Live version: 224 (evidence-preservation refresh of version 223 mathematics).
The exact source revision is recorded in the Site version ledger.
New public claims: FPRD-IC-T28, FPRD-IC-T29.
Updated surfaces: Selfridge program and results, number-theory area, research journal, claim/maturity/theorem indexes, and evidence archive.
Production checks returned HTTP 200 for the result page, both claim pages, and the downloadable archive. The rendered result page contains both new claim anchors.
Build, structural tests, changed-surface validation, lint (one pre-existing warning only), and mathematical certificate checks passed. Full archive validation still reports eight pre-existing 404s for unrelated historical reading-edition/paper assets absent from this checkout; no changed Selfridge route failed.
The Publications collection was untouched.
Standalone persistent-file transfer failed explicitly. The proof packet, code, results, and this named handoff are therefore bundled together in the verified public evidence archive as the durable recovery path. The older standalone handoff remains stale and must not supersede this copy.