Published January 29, 2026 | Version 1
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Cousin Primes as a Persistence Phenomenon in a Canonically Indexed Sieve

Authors/Creators

  • 1. Independent Researcher, Argentina

Description

We reformulate the cousin prime problem within a fully deterministic and
indexed framework, where divisibility and primality are encoded exactly by
congruence conditions on a single index axis.
Using the canonical decomposition $n=6t+r$ with $r\in\{1,5\}$, cousin prime
candidates are represented as ordered indexed pairs at equal height,
corresponding to the integer pairs $(6t+1,6t+5)$.

For each prime $p\ge5$, primality induces exactly two canonically anchored
congruence obstructions acting on the same index variable.
These obstructions are rigid, admit no translational freedom, and never
coincide.
We analyze their exact periodic structure and their interaction at finite
levels via Chinese Remainder Theorem assembly.
Composite moduli introduce no additional freedom beyond prime intersections,
and higher prime powers do not generate new constraints below their natural
index scales.

The decisive step occurs at the Archimedean level.
Extinction of cousin primes would require complete coverage of finite index
intervals by canonically anchored obstruction sets.
We show that such coverage is structurally impossible: the cousin obstruction
system satisfies bounded local multiplicity and therefore falls within the
scope of a general Archimedean non--coverage principle.
As a consequence, admissible indices persist beyond every finite scale, and
infinitely many cousin prime pairs exist.

All arguments are exact, finite, and deterministic.
No analytic estimates, density heuristics, or probabilistic methods are used.

Abstract (Spanish)

Reformulamos el problema de los primos \emph{cousin} dentro de un marco
completamente determinista e indexado, en el cual la divisibilidad y la
primalidad se codifican de manera exacta mediante condiciones de congruencia
sobre un único eje de índices.
Usando la descomposición canónica $n=6t+r$ con $r\in\{1,5\}$, los candidatos a
primos \emph{cousin} se representan como pares indexados ordenados a la misma
altura, correspondientes a los pares enteros $(6t+1,6t+5)$.

Para cada primo $p\ge5$, la primalidad induce exactamente dos obstrucciones de
congruencia canónicamente ancladas que actúan sobre la misma variable índice.
Estas obstrucciones son rígidas, no admiten libertad traslacional y nunca
coinciden.
Analizamos su estructura periódica exacta y su interacción a niveles finitos
mediante el ensamblaje del Teorema Chino del Resto.
Los módulos compuestos no introducen nueva libertad más allá de las
intersecciones primas, y las potencias de primos no generan nuevas restricciones
por debajo de sus escalas naturales de índice.

El paso decisivo ocurre a nivel arquimediano.
La extinción de los primos \emph{cousin} requeriría la cobertura completa de
intervalos finitos del eje de índices por obstrucciones canónicamente ancladas.
Mostramos que tal cobertura es estructuralmente imposible: el sistema de
obstrucciones asociado a los primos \emph{cousin} satisface multiplicidad local
acotada y, por lo tanto, cae dentro del alcance de un principio general de
no--cobertura arquimediana.
Como consecuencia, los índices admisibles persisten más allá de toda escala
finita y existen infinitos pares de primos \emph{cousin}.

Todos los argumentos son exactos, finitos y deterministas.
No se utilizan estimaciones analíticas, heurísticas de densidad ni métodos
probabilísticos.

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Additional details

Additional titles

Translated title (Spanish)
Los primos cousin como un fenómeno de persistencia en una criba indexada canónicamente

References

  • Hardy, G. H.; Littlewood, J. E. Some problems of Partitio Numerorum; III: On the expression of a number as a sum of primes, Acta Mathematica 44 (1923), 1–70.
  • de Polignac, A. Recherches nouvelles sur les nombres premiers, Comptes Rendus de l'Académie des Sciences de Paris 29 (1849), 397–401.
  • Halberstam, H.; Richert, H.-E. Sieve Methods, Academic Press, London, 1974.
  • Maynard, J., Small gaps between primes, Annals of Mathematics (2) 181 (2015), no. 1, 383–413.
  • Zhang, Y. Bounded gaps between primes, Annals of Mathematics (2) 179 (2014), no. 2, 1121–1174.
  • D. A. Jorge Zafaranich, An Indexed Large Sieve and the Exact Structure of Divisibility, Zenodo preprint, 2025.
  • D. A. Jorge Zafaranich, An Indexed CRT Assembly Lemma and Exact Admissibility in the Index Axis, Zenodo preprint, 2026.
  • D. A. Jorge Zafaranich, SALEN: Structural Archimedean Non–Coverage in Canonically Anchored Obstruction Systems, Zenodo preprint, 2026
  • D. A. Jorge Zafaranich, Twin Primes as a Persistence Phenomenon in a Canonically Indexed Sieve, Zenodo preprint, 2026