pith. sign in

arxiv: 2603.25896 · v3 · submitted 2026-03-26 · 🧮 math.NT

On nonconvex constellations among primes I

Fred B. Holt This is my paper

Pith reviewed 2026-05-14 23:54 UTC · model grok-4.3

classification 🧮 math.NT
keywords prime constellationsk-tuple conjectureEngelsma counterexamplesnonconvex constellationsprimorial coordinatesEratosthenes sieveasymptotic population
0
0 comments X

The pith

None of the 58 Engelsma (459,3242) counterexamples occur before 9.7 × 10^73.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends methods developed for the k-tuple conjecture to a specific family of 58 narrow prime constellations of length 459 and span 3242. These constellations, called Engelsma counterexamples, are tracked from their origins in inadmissible driving terms within small primorial gap cycles up through G(211#). Exhaustive breadth-first searches followed by depth-first strategies are used to check whether any instance survives the sieve and appears among actual primes. The calculations establish that none of these counterexamples occur before 9.7 × 10^73 and also determine the asymptotic relative population of each among other length-459 constellations.

Core claim

Extending prior work on the k-tuple conjecture, none of the 58 Engelsma (459,3242)-counterexamples occur before 9.7 × 10^73, and each has a calculable asymptotic relative population among constellations of length J=459.

What carries the argument

Primorial coordinates for admissible instances, developed via breadth-first exhaustive search through G(211#) and followed by depth-first strategies to test survival under Eratosthenes sieve.

If this is right

  • These particular narrow constellations remain absent from the primes at least up to 9.7 × 10^73.
  • The asymptotic relative populations can be used to compare the expected frequency of these counterexamples against other length-459 constellations.
  • The tracking of evolution from small primorial cycles to G(211#) provides a template for studying other Engelsma counterexamples.
  • Depth-first search strategies developed here become available for checking still larger constellations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the search is complete, the k-tuple conjecture holds for these specific constellations at least through the range checked.
  • Similar exhaustive searches could be applied to other narrow constellations to produce comparable bounds.
  • The absence at such large scales suggests that any actual occurrence would require correspondingly large driving terms or special sieving properties.

Load-bearing premise

The breadth-first exhaustive search through G(211#) combined with later depth-first strategies fully captures all possible instances without omissions from computational truncation or incomplete sieving of inadmissible driving terms.

What would settle it

Discovery of any single (459,3242)-counterexample among primes below 9.7 × 10^73.

read the original abstract

Extending our work on the $k$-tuple conjecture, we apply those methods to the Engelsma counterexamples (narrow constellations) of length $J=459$ and span $|s|=3242$. We track the evolution of these $58$ counterexamples from inadmissible driving terms starting in the cycle of gaps ${\mathcal G}(11^\#)$ up through their first appearance in ${\mathcal G}(113^\#)$. We continue developing primorial coordinates for each admissible instance through a breadth-first exhaustive search through ${\mathcal G}(211^\#)$, at which point we need to develop strategies for depth-first searches for an instance that would survive Eratosthenes sieve. Our calculations show that {\em none} of the $(459,3242)$-counterexamples occur before $9.7\,E73$. For each of the $58$ Engelsma $(459,3242)$-counterexamples we calculate its asymptotic relative population, among other constellations of length $J=459$, and we study how these counterexamples work. In this version (9 April) we have completed the calculations in Table 6 to include all of the terms in primorial expansion for the smallest initial generator and corrected a typographical error on page 6.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript extends k-tuple methods to the 58 Engelsma (459,3242)-counterexamples. It tracks their inadmissible driving terms through the gap cycles G(11#) to G(113#), performs exhaustive breadth-first enumeration through G(211#), and then applies depth-first continuation with pruning of inadmissible terms to conclude that none of these constellations occur before 9.7 × 10^73. It also computes the asymptotic relative population of each instance among length-459 constellations.

Significance. If the search completeness holds, the explicit upper bound of 9.7E73 supplies concrete computational evidence on the first-occurrence scale of these narrow constellations, complementing theoretical work on admissible k-tuples and prime-gap distributions. The population calculations further quantify relative densities, which could inform future heuristics.

major comments (2)
  1. [depth-first search strategy and Eratosthenes-sieve continuation] The central bound of 9.7E73 rests on the completeness of the depth-first search after G(211#) together with the sieving rules that discard inadmissible driving terms. The manuscript describes the strategy but supplies neither pseudocode for the pruning decisions, a formal statement of the residue-class tracking modulo primes >211, nor a machine-checkable certificate. Any gap in these rules would allow an undetected instance to appear earlier than reported.
  2. [computational results and Table 6] No error analysis, truncation bounds, or independent verification is provided for the reported threshold 9.7E73 or for the claim that the breadth-first enumeration through G(211#) captured all 58 instances without omission. The soundness of the population figures in Table 6 likewise depends on the same exhaustive enumeration.
minor comments (1)
  1. [abstract and Table 6] The abstract mentions a typographical correction on page 6 and completion of Table 6; verify that all primorial-expansion terms for the smallest initial generator are now correctly listed and that no other transcription errors remain in the gap-cycle tables.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their thorough review and valuable feedback on our work extending k-tuple methods to the Engelsma counterexamples. Below we respond point-by-point to the major comments, outlining the clarifications and revisions we will make.

read point-by-point responses

  1. Referee: [depth-first search strategy and Eratosthenes-sieve continuation] The central bound of 9.7E73 rests on the completeness of the depth-first search after G(211#) together with the sieving rules that discard inadmissible driving terms. The manuscript describes the strategy but supplies neither pseudocode for the pruning decisions, a formal statement of the residue-class tracking modulo primes >211, nor a machine-checkable certificate. Any gap in these rules would allow an undetected instance to appear earlier than reported.

    Authors: We acknowledge the need for more explicit documentation of our search procedures. In the revised manuscript, we will append pseudocode for the depth-first search strategy, detailing the pruning decisions for inadmissible driving terms. Additionally, we will include a formal description of the residue-class tracking modulo primes exceeding 211, grounded in the primorial coordinate system. While a complete machine-checkable certificate would require extensive formal verification infrastructure not available in our current setup, the underlying sieving rules follow directly from the Eratosthenes sieve and are designed to preserve all potentially admissible terms. This should mitigate concerns about undetected instances appearing earlier. revision: partial

  2. Referee: [computational results and Table 6] No error analysis, truncation bounds, or independent verification is provided for the reported threshold 9.7E73 or for the claim that the breadth-first enumeration through G(211#) captured all 58 instances without omission. The soundness of the population figures in Table 6 likewise depends on the same exhaustive enumeration.

    Authors: The breadth-first enumeration up to G(211#) is complete by its exhaustive nature, ensuring all 58 counterexamples are accounted for as they emerge earlier in G(113#). The 9.7E73 bound arises from continuing the depth-first search until no admissible paths remain. We will incorporate an error analysis section, including truncation bounds derived from the primorial growth and estimates of the unexplored search space. Independent verification is possible through reimplementation using the forthcoming pseudocode. The population figures in Table 6 are computed from the fully enumerated admissible set, and we will add bounds reflecting any potential truncation effects. revision: yes

standing simulated objections not resolved
  • Providing a machine-checkable certificate for the depth-first search completeness

Circularity Check

0 steps flagged

No circularity; bound obtained from direct exhaustive search

full rationale

The paper reports a computational bound on the first occurrence of specific (459,3242)-counterexamples, obtained by tracking admissible instances through primorial gap cycles via breadth-first enumeration up to G(211#) and subsequent depth-first continuation with sieving. This process applies prior k-tuple methods to independent instances without fitting any parameter to the reported bound, without defining the target result in terms of itself, and without reducing the final claim to a self-citation chain or ansatz. The derivation remains self-contained against external primality and sieving benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The work rests on the applicability of prior k-tuple conjecture methods to these large inadmissible cases and on the completeness of the described search procedure; no new free parameters or invented entities are introduced.

free parameters (1)
  • search depth threshold
    Choice of 211# as the switch point from breadth-first to depth-first search is a computational cutoff chosen by the authors.
axioms (1)
  • domain assumption Methods developed for the k-tuple conjecture extend without modification to Engelsma counterexamples of length 459
    Stated as extension of prior work in the abstract.

pith-pipeline@v0.9.0 · 5510 in / 1233 out tokens · 42211 ms · 2026-05-14T23:54:24.746771+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On nonconvex constellations among primes II: (458,3240)

    math.NT 2026-05 unverdicted novelty 4.0

    Extends prior analysis of Engelsma counterexamples to 116 constellations of length 458 and span 3240, tracking their first appearances in gap cycles up to G(211#) and computing asymptotic relative populations.