What question did this study set out to answer?

This analysis aims to explore the structural deficit between volume and capacity in prime complements influenced by the absence of prime number 2.

January 17, 2026Open Access

Structural Deficits in Shifted Prime Complements: A Volume-Capacity Analysis

Key Points

This analysis aims to explore the structural deficit between volume and capacity in prime complements influenced by the absence of prime number 2.
Introduced structural comparison between volume and capacity of complements related to even integers.
Utilized the Bombieri-Vinogradov theorem to control prime contributions in bulk ranges.
Analyzed expected distributions of prime factors to establish statistical expectations in prime tails.
Identified a Parity Gap causing a structural deficit favoring volume over capacity.
Demonstrated that the Parity Gap implies the existence of at least one prime factor greater than k.
Established the Dead Zone, indicating no prime factors exist in a specific range under conjectured failure.

Abstract

We introduce a structural comparison between two quantities associated with an even integer 2k: the volume, derived from the logarithmic size of all complements 2k − r with r prime, and the capacity, derived from the cumulative p-adic contribution of primes p ≤ k to the product over r<k of (2k − r). We identify a persistent structural deficit, the Parity Gap, caused by the absence of the prime 2 in the factorization of every complement. This creates a logarithmic surplus of approximately 1.079 k/ ln k in favor of the volume. While the contribution of primes in the "bulk" range is controlled by the BombieriVinogradov theorem, the behavior of primes in the extreme tail (p ≈ k) remains analytically inaccessible. We show that if the distribution of prime factors in this tail follows standard statistical expectations (i.e., does not exhibit pathological clustering), the Parity Gap is sufficient to force the existence of at least one complement 2k − r with a prime factor greater than k, thereby implying a Goldbach representation. Major Update (Unified Framework):This version integrates the structural 'Volume-Capacity' method with the new 'Dead Zone Theorem'. By analyzing the arithmetic constraints on hypothetical counterexamples, we establish two fundamental obstructions: The Parity Gap: A capacity deficit caused by the exclusion of the prime 2 (derived in previous versions). The Dead Zone: A new theorem proving that if the conjecture fails, no prime factor can exist in the interval (2k/3,k](2k/3,k] Final revised version:Incorporates rigorous formalization of the Bridge Proposition and explicit constants for volume bounds. Addressed reviewer feedback regarding the Tail Hypothesis formulation. Key Result:Combining these constraints increases the normalized structural gap from ≈1.079≈1.079 to ≈1.484k/ln⁡k≈1.484k/lnk This significantly strengthens the heuristic evidence, showing that a counterexample would require the distribution of primes in the remaining tail to violate statistical expectations by nearly 150%. This manuscript supersedes previous drafts. Keywords: Goldbach Conjecture, Analytic Number Theory, Sieve Theory, Parity Gap, Structural Heuristic.

Structural Deficits in Shifted Prime Complements: A Volume-Capacity Analysis

Key Points

Abstract

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