Contribution Classes and Horizon Contribution

Contribution Classes and Horizon Contribution introduces a least–prime–factor (LPF) partition of the integers—contribution classes—as a structural alternative to the usual parity split. Instead of grouping numbers as “even vs. odd,” the paper partitions integers n>1n>1n>1 by their least prime divisor ppp, giving a disjoint LPF-class decomposition that treats every prime as a “contributor,” with 222 distinguished only by its structural weight.

The paper then moves from global density to a horizon-local viewpoint by working in the quadratic window

W(P)=(P,P2)(integers n with P<n<P2),W(P)=(P,P^2)\quad(\text{integers }n\text{ with }P<n<P^2),W(P)=(P,P2)(integers n with P<n<P2),

and restricting attention to composites \Comp(P)=W(P)∩C\Comp(P)=W(P)\cap\mathsf{C}\Comp(P)=W(P)∩C. For each prime p≤Pp\le Pp≤P, it defines the composite LPF class

Cp(P)={n∈\Comp(P):LPF⁡(n)=p},δP(p)=∣Cp(P)∣∣\Comp(P)∣,\mathcal{C}_p(P)=\{n\in \Comp(P):\operatorname{LPF}(n)=p\}, \qquad \delta_P(p)=\frac{|\mathcal{C}_p(P)|}{|\Comp(P)|},Cp(P)={n∈\Comp(P):LPF(n)=p},δP(p)=∣\Comp(P)∣∣Cp(P)∣,

so that {Cp(P)}p≤P\{\mathcal{C}_p(P)\}_{p\le P}{Cp(P)}p≤P partitions the composite field inside the horizon.

Main result (Horizon Contribution Law). The paper proves that the LPF–2 class holds a strict majority of the composites in every horizon:

δP(2)>12(for primes P≥3),\delta_P(2)>\tfrac12 \quad (\text{for primes }P\ge 3),δP(2)>21(for primes P≥3),

equivalently ∑p≥3δP(p)<12\sum_{p\ge 3}\delta_P(p)<\tfrac12∑p≥3δP(p)<21. It also gives an explicit identity for the majority gap Δ(P)=δP(2)−12\Delta(P)=\delta_P(2)-\tfrac12Δ(P)=δP(2)−21 in terms of the number of primes lying in W(P)W(P)W(P), making the strictness transparent and tying the decay Δ(P)→0+\Delta(P)\to 0^+Δ(P)→0+ to standard prime-counting input.

Computation and verification. A sieve-based procedure is provided to compute LPF assignments over W(P)W(P)W(P), extract composite-only LPF counts, and recover δP(p)\delta_P(p)δP(p) numerically. This makes the theorem directly testable and supplies a practical template for large-scale verification or extensions.

v2 (2025-12-30): Consistency and correctness pass focused on horizon-share normalization, empirical verification, and LPF-class notation. Standardized the horizon-share definition δP(p)\delta_P(p)δP(p) throughout as composites-normalized δP(p)=∣Cp(P)∣/∣\Comp(P)∣\delta_P(p)=|\mathcal{C}_p(P)|/|\Comp(P)|δP(p)=∣Cp(P)∣/∣\Comp(P)∣ with \Comp(P)=W(P)∩C\Comp(P)=W(P)\cap\mathsf{C}\Comp(P)=W(P)∩C, and clarified the horizon window as the integer interval P<n<P2P<n<P^2P<n<P2 (open endpoints). Repaired the key logical dependency: the strict majority δP(2)>12\delta_P(2)>\tfrac12δP(2)>21 is now proved directly from horizon geometry and the removal of (odd) primes when restricting to composites, and an explicit identity for the gap Δ(P)=δP(2)−12\Delta(P)=\delta_P(2)-\tfrac12Δ(P)=δP(2)−21 in terms of the prime count in W(P)W(P)W(P) is given; “Uniformity/Occupancy” hypotheses are retained only as optional input for quantitative refinement (decay rate), not for the strict inequality itself. Corrected the global density table values for δ(7)\delta(7)δ(7) and δ(11)\delta(11)δ(11) to match the stated product formula. Recomputed and updated the sample horizon table/figure (P = 5, 11, 17) so the reported LPF-2 shares agree with the composites-normalized definition (eliminating the prior mismatch between theorem and numerics). Aligned the computational walkthrough/pseudocode with the stated window endpoints and composite-only normalization (unassigned entries correspond to primes in the horizon and are excluded from the denominator). No changes to the conceptual claim of HCL or to AZ’s core axioms; updates are definitional, proof-correctness, computation-alignment, and expository clarity fixes to prevent reader ambiguity.

This note is part of the Axiom Zero (AZ) series of papers. Readers who are new to AZ should first consult the main foundational paper:

• Axiom Zero: Structural Irreducibility and the Unpredictability of Primes — DOI: 10.5281/zenodo.16998285

• Axiom Zero: Laws, Principles, and Rules of Structure — DOI: 10.5281/zenodo.17728204

• Deck Limit Laws in Axiom Zero — DOI: 10.5281/zenodo.17728647

• No Third Mechanism in Axiom Zero: Structural Completeness and Non–Existence Schemes — DOI: 10.5281/zenodo.18097942

• Dyadic Rigidity and the Non-Existence of Odd Perfect Numbers in Axiom Zero — DOI: 10.5281/zenodo.17886421

• Channel Non-Extinction in Axiom Zero — DOI: 10.5281/zenodo.18099516

• Twin Primes in Axiom Zero: Horizon Contribution, Uniformity, and No Third Mechanism — DOI: 10.5281/zenodo.17903043

• Goldbach in Axiom Zero: A Deterministic Additive-Sieve — DOI: 10.5281/zenodo.17728745

• Conservativity of Axiom Zero over N — DOI: 10.5281/zenodo.17073211

• Analytic Conservativity of Axiom Zero — DOI: 10.5281/zenodo.17073257

• Stress Testing Axiom Zero: Computational Falsification Attempts and Classical Consistency Checks — DOI: 10.5281/zenodo.17728602

• Euclid–Euler in Axiom Zero: Even Perfect Numbers from Structural Arithmetic — DOI: 10.5281/zenodo.17728314

• Lagrange in Axiom Zero: Four Squares from Structural Arithmetic — DOI: 10.5281/zenodo.17728407

Together, these works position the present paper as part of a broader AZ toolkit for structural treatments of additive questions, gap phenomena, and local patterns in the primes, under a carefully labeled package of structural laws and conjectures rather than hidden analytic imports.

Contact: axiomzero.math@gmail.com