slug: jaffe-primitive-frobenius-successor-ratio
bibkey: jaffe2026denominators
doi: null
url: https://arxiv.org/abs/2610.17267v1
triage: theorem
motivation_gids:
  - D5/S3/ArithUnits/JaffeSuccessorRatioRefutation

Jaffe’s unrestricted primitive Frobenius successor-ratio conjecture

Problem

The target is exactly Conjecture 7.3 in arXiv:2610.17267v1:

For all a,b >= 3 relatively prime and for c(a,b) the minimum integer satisfying both c(a,b)>b and gcd(c(a,b),a)=1, we have g(a,c(a,b))/g(a,b) <= 5.

Section 3 defines [ R(a,b)={ax+by:x,y\in\mathbb Z_{\geq0},\ \gcd(x,y)=1}, \quad g(a,b)=\max(\mathbb Z\setminus R(a,b)), ] with infinity when a greatest missing integer does not exist. Zero coefficients are allowed. There is no (a<b) assumption. The successor is the least integer above (b) coprime to (a). The same conjecture is numbered 8.3 in the author’s 2023 dissertation; this dossier anchors one assertion, not two. Conjecture 7.2 is a different target.

The Lean definition fullClaim : Prop uses precisely this integer domain, representation predicate, greatest-missing convention, successor and quotient bound. The designated result : Not fullClaim is unconditional.

Motivation

The source asks for a uniform bound when the second generator advances to its next coprime integer. The primitive missing-number scaling obstruction allows a large successor gap to violate that uniform bound. Exchange symmetry of the represented set does not exchange this successor operation.

Gap

The registered target #20244 is the whole universal claim. The bounded source and literature qualification found no inspected whole answer. The formal gap is an admissible pair with an exact least successor and actual finite positive greatest missing integers whose quotient violates the bound. Finite searches for missing values or an assumed finiteness hypothesis do not settle that gap.

Route

Take (a=87841397512641630), (b=7). The integer (a) is coprime to 7 and 49, and every integer strictly between them is noncoprime to (a). Thus (c(a,7)=49).

For arbitrary positive coprime (A,B), count all nonnegative solutions by (H(n)) and primitive ones by (P(n)). Residues modulo (B) give (\lfloor n/(AB)\rfloor\leq H(n)\leq\lfloor n/(AB)\rfloor+1). Partitioning by the coefficient gcd and applying Möbius inversion yields (ABP(n)\geq\varphi(n)-AB\tau(n)). Prime-power estimates give (n\tau(n)^2\leq16^7\varphi(n)^2), hence every (n>16^7(AB)^2) has a primitive representation. For (A,B\geq3), the missing integer 1 and this universal tail prove that the integer complement has a positive maximum.

If (n) is missing for ((a,7)), then (7n) is missing for ((a,49)). A primitive representation of (7n) would force its first coefficient to be divisible by 7; dividing it gives a primitive representation of (n). Apply this to the actual maximum for ((a,7)). The maxima are finite and positive, and [ \frac{g(a,c(a,7))}{g(a,7)} =\frac{g(a,49)}{g(a,7)}\geq7>5. ]

Evidence

The mathematical source is D5/S3/ArithUnits/JaffeSuccessorRatioRefutation.lean, and the designated declaration is D5/S3/ArithUnits/JaffeSuccessorRatioRefutation.result. The claim definition’s type is literally Prop; the result is written literally as theorem result : ¬ fullClaim. Its utility basis is certified-instance / refutes, and its owning Scribe carries OpenProblemResolutionClaim(Refuted) for this slug.

The proof includes the complete primitive tail, actual greatest-element construction in (\mathbb Z), least-successor argument and quotient transport. It uses no assumed maximum, truncated maximum or conclusion hypothesis. The source’s existing finiteness and scaling ingredients are attributed to Theorems 4.3 and 7.1. The Library note identifies the source versions and qualification limits.

Falsifier

A mismatch in coefficient nonnegativity or coprimality, integer complement, source ordering, successor minimality, tail count, maximum construction or division by a positive finite value would invalidate this route. An ordered-only statement or a conditional tail would not refute this target.

Triage

theorem, under open-problem-resolution with registration #20244. This is the complete unfrozen StageA candidate. Independent source-mirror review, freezing, required integration checks and acceptance remain pending. The constant-five claim is the single target and carries at most one accepted credit. The general prime-indexed unbounded-ratio family is not a separate formal result here. The ordered variant remains unresolved; the source’s earlier proved results are unaffected.

ASSUMED-UNVERIFIED

Worldwide priority and exhaustive absence of an older equivalent theorem are unverified. The checked thesis states the same conjecture; the checked Shimizu manuscript addresses 7.2. Incomplete citation indexing and bounded searches do not establish originality. The Library note gives the qualified source and reuse account. This dossier makes no final review, CI, merge, KPI or host-continuation completion claim.