FPRD research loop state

Updated 6 September 2026 after the long complementary $f=0$ cancellation closure and stopping audit.

Active program and completed mode

• Program: Selfridge’s question $\lVert2^n\rVert=2n$ for integer complexity.
• Mode completed in this run: ATTACK.
• Active restricted grammar: normalized primitive proper three-addition forks with positive $(2,3)$-smooth displayed operands and exact paid-atom costs.
• Current proof packet: FPRD-Selfridge-f0-cancellation-closure-2026-09-06.md.
• Reproducible evidence: Selfridge-f0-cancellation-closure-code-and-results.zip.
• This program has not solved Selfridge’s unrestricted problem and has not completed the normalized three-addition grammar. The results are ordinary research notes, not Publications. No external review or priority determination is recorded.

New theorem in this run

In the complementary $f=0$ equation

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

the inherited quadratic gate leaves forty ordered canonical factor placements. Exactly twenty tied-minimum strata can cancel in the first nonzero 3-adic digit. The previous run classified the two triple-minimum strata; this run completely classifies the other eighteen, in which exactly two among $v,y,g$ attain the minimum.

The complete quotient

$$Q=7\cdot73\cdot5\cdot13=33215, \qquad \operatorname{ord}_Q(2)=36, \quad \operatorname{ord}_Q(3)=12,$$

makes seven of the eighteen families empty and reduces the rest to 23 exact residue states. The next ternary digit modulo 27 eliminates all but one state. It forces

$$(4+3^m)(1+2^x3^m)+3^g=2^K, \qquad (x,m,g,K)\equiv(3,1,4,8)\pmod{(36,12,12,36)},$$

with $m=1$ exactly. Reducing modulo $2^x$ gives $x=3$, and then

$$(2^{K/2}-3^{g/2})(2^{K/2}+3^{g/2})=175.$$

The three positive factor pairs leave the unique solution

$$\boxed{(4+3)(1+8\cdot3)+3^4=2^8}.$$

Its displayed cost is $29=2K+13$, so it is a harmless control.

Together with the preceding triple-minimum classification, every possible lowest-digit cancellation in $f=0$ is now resolved. The exceptional locus consists exactly of the new control above and

$$(2+3)(1+8\cdot3)+3=2^7,$$

whose cost is $18=2K+4$. Every other $f=0$ solution therefore satisfies

$$\boxed{\min(v,y,g)=1+v_3(K-r-u)},$$

where $2^r$ and $2^u$ are the constant binary terms of the two ordered factors.

Evidence and reconstruction

• The primary Python certificate reconstructs the 40/20/18 canonical counts, all 23 quotient states, the sole modulo-27 survivor, the terminal factorization, and the exact cost.
• A separately written C++20 implementation independently reconstructs the canonical table and every conclusion.
• No bounded exponent search is used for completeness.

Useful failure and stopping boundary

The obvious continuation was tested rather than assumed. Across all forty $f=0$ placements, the same quotient leaves 149 residue states. Lifting to modulo $3^6=729$ creates 2,032 low/high patterns. Even after imposing the new universal valuation law and retaining its two exceptions, 1,953 patterns remain across twenty placements. Thus the next-digit mechanism branches in the general remainder rather than freezing it.

This diagnostic is not an existence result. It is an exact failure of the specific proposed finite-closure route. Larger digits or additional aligned primes enlarge coordinate periods before any available height cutoff makes the enumeration finite.

The Evertse–Schlickewei–Schmidt theorem proves that the normalized complementary family has finitely many solutions, but supplies no height bound. Bajpai–Bennett effectively treats homogeneous five-term S-unit equations for at most three places. The expanded complementary equation here has six homogeneous terms–five positive monomials and the negative output power–so that theorem does not directly provide the missing bound.

Retained completed results

The following earlier results remain active inputs. Consult the public claim ledger and their packets for 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 all exterior layers.
• The normalized positive-type-II tower for $a\ge1$ is complete and contains exactly ten controls, all costing at least $2N+4$.
• Quadratic reciprocity reduces the normalized complementary orientation and eliminates the double-even $f=2$ split.
• The surviving $f=1$ and $f=2$ branches obey exact universal valuation laws.
• Their complete minimum layers $m=1,2$ contain exactly seventeen proper controls, all costing at least $2N+6$.
• The normalized complementary family is finite by the established Evertse–Schlickewei–Schmidt theorem, with the previously recorded crude solution-count bound; this is not effective in exponent height.
• The full $f=0$ exceptional cancellation locus is now classified and contains exactly two harmless controls.

The public claim sequence should now run through FPRD-IC-T30:

• FPRD-IC-T30: complete $f=0$ cancellation-locus classification and the resulting universal valuation law outside two explicit controls.

What remains open

1. Nonexceptional $f=0$ strata governed by the new valuation law.
2. Higher minimum layers $m\ge3$ in the valuation-controlled $f=1,2$ branches.
3. A cost theorem for the full normalized three-addition grammar.
4. A reduction from arbitrary integer-complexity expression trees to that restricted grammar. Without this, even a complete grammar theorem would not settle Selfridge.

These are structural obligations, not a short residue-table tail.

Recommended next mode and reopening criterion

NEXT MODE: HUNT.

The promised eighteen-stratum attack is finished, and the phase it belonged to is now complete. Preserve Selfridge as a paused program and conduct a genuinely broad hunt across mathematics. Do not continue by enumerating the next minimum layer or merely increasing a modular/search height.

Reopen Selfridge only if a hunt finds one of the following:

• an effective reduction for the product-structured six-term S-unit equation;
• a uniform dominance or freezing theorem that turns it into lower-term equations;
• a minimal-counterexample normal form that connects arbitrary formulas to the normalized grammar; or
• another structural bridge of comparable scope.

Site checkpoint

• Site: https://fprd-lab.joshgay.chatgpt.site
• Site update: version 225, with claim FPRD-IC-T30, the theorem exposition, research-program frontier, subject-area summary, journal entry, and downloadable evidence archive.
• The six-item Publications collection must remain untouched.
• The proof packet, code, results, and this named handoff are bundled in the evidence archive so the state remains recoverable even if standalone persistent-file transfer fails again.