The tribonacci and four-Bonacci local Dold observations

Chuysurichay, Jaidee, Panraksa and Ward, The local Dold congruence for Fibonacci and Lucas sequences, arXiv:2609.24544v1, section 4, PDF page 11:

Another natural problem from the results here is to try and characterize these sets of primes for linear recurrence sequences in general, starting with the intriguing observation that the classical tribonacci sequence (1,1,2,4,...) and 4-bonacci sequence (1,1,1,2,5,...) seem (computationally) to have G_D = ∅.

The quotation normalizes mathematical notation and whitespace. It is an empirical open observation, without a numbered conjecture. Its positive sequences are interpreted by their preceding-three and preceding-four recurrences: T1=T2=1, T3=2, and B1=B2=B3=1, B4=2. Thus B=(1,1,1,2,5,9,17,33,...).

For a positive natural sequence a and prime p, put b(n)=p^valuation_p(a(n)). The assertion is that neither sequence has any prime p for which every positive n divides the integer sum of mu(n/d)b(d) over positive divisors d of n. Both sequences and every prime are included.

The local Dold condition tests whether arithmetic support of a recurrence can satisfy orbit-count congruences. Here the recurrence is a coordinate of an integral invertible action. Its actual inverse coordinate vanishes, providing a residue class on which that arithmetic support is obstructed.

The first-tier recent-source observation and its exact quantifiers were preregistered in issue 19439. The bounded literature readings collected on 2026-10-20 cover the target v1 and version history, predecessor arXiv:2508.11054v2, the Dold survey arXiv:2007.04031v4, Rajs arXiv:2509.09847v1, and bounded metadata, Crossref and repository searches. No resolution was found in that scope. The survey's tribonacci trace sequence has different initial data; a trace-sequence theorem supplies no identification with the nonlinear p-part sequence here.

The pinned Mathlib supplier SModEq.pow_pow_add_one supplies the quantified weak congruence lift. Its commutative polynomial-ring formulation can be evaluated in the noncommutative integer matrix ring. This transfers the existing lift to the integer matrix ring.

Reduce the integral invertible matrix A modulo p to obtain a positive period T. Write A^T=I+pC. In integer polynomials, the ideal-power lift of 1+pX evaluated at C gives A^(T*p^j)=I+p^(j+1)D for every natural j.

Set r=pT-1. The inverse coordinate zero implies p divides a(r). Positive-index positivity gives a(r)>0, so e=valuation_p(a(r)) is at least one. The residue r is positive, smaller than M=T*p^e, and coprime to M, including when p divides T. Congruence modulo p^(e+1) preserves valuation e throughout the positive class r modulo M. Dirichlet gives prime q in that class above any prescribed H and p^e-1. Since a(1)=1, the prime-index Mobius sum is exactly p^e-1, strictly between zero and q.

The native companion time at u=1 uses reversed states. Tribonacci uses (1,1,0), with inverse (1,0,0) and observed coordinate 2. Four-Bonacci uses (1,1,1,-1), with inverse (1,1,-1,0) and observed coordinate 3. The integer extension B0=-1 is solely a companion-model value; the natural definition's unused zero placeholder is zero. All coordinate identifications hold at every positive index.

A prime p satisfying all Dold divisibilities for either literal positive sequence would contradict the assertion. A different four-Bonacci initialization or a finite list of unsuccessful primes does not test this assertion. Exact valuation requires modulus p^(e+1), not just p^e.

D5/S3/Arith/LocalDold/TribonacciFourBonacci.result states, for every a equal to either literal sequence, that goodPrimes(a) is empty and that for every prime p and every natural H there exists a prime q>H whose integer Mobius sum S satisfies 0<S<q and q does not divide S. This includes p=2 and H=0. Its hypotheses select the source sequence; they assume neither prime existence nor valuation classes nor Dold divisibility.

The proof uses D5/S3/Arith/LocalDold/UnimodularObstruction.unimodular_coordinate_obstruction, the native finite companion time, and the existing Tribonacci positivity. The axiom closure is contained in propext, Classical.choice and Quot.sound.

What the settlement shows

Proved, by D5/S3/Arith/LocalDold/TribonacciFourBonacci.result: both printed sequences have empty local-Dold good-prime sets. At every prime there are prime-index failures beyond every natural bound, not merely one failing index.

Proved, by unimodular_coordinate_obstruction: the decisive conditions are an integral invertible action, positive-index coordinate identification and positivity, a(1)=1, and the vanishing actual inverse coordinate. The finite mod-p period need not be coprime to p. No restriction to odd primes or assertion about positivity at index zero is needed.

Open: characterize good-prime sets for arbitrary linear recurrences, as requested in the same source paragraph. This proof addresses the two specified observations. The source's separate Fibonacci and Lucas characterizations have different initial data; no dependency on these observations was identified in the inspected source.

The bounded literature search does not certify worldwide novelty, first-publication priority or official uniqueness. The source prints names and prefixes; the preceding-k recurrence interpretation is stated explicitly above. No claim concerns another tetranacci convention.