IndisputableMonolith.NumberTheory.PrimePhaseDistribution
The module defines prime phase distributions over square-budget divisor boxes for trapped residuals. Researchers extending Recognition Science arithmetic functions to Erdős-Straus residuals cite it to link cost spectra with balanced phase gates. Its content assembles the combinatorial box representation with the prime cost extension to assert admissible gate existence.
claimFor every residual trapped $n$, there exists a bounded admissible gate $c$ such that a divisor box point $(d,e)$ with $d e = N^2$ hits the required balanced phase, where the phase is determined by the arithmetic cost function $c(n) = sum_p v_p(n) J(p)$.
background
The module sits inside the Recognition Science number theory layer that extends the J-cost from positive reals to natural numbers. Upstream PrimeCostSpectrum states: 'For each n ≥ 1 we define c(n) := Σ_{p^k ‖ n} k · J(p) = Σ_p v_p(n) · J(p)'. The companion ErdosStrausBoxPhase isolates the finite part: 'For a square budget N², a divisor exponent box is represented by a complementary pair (d,e) with d*e = N²'.
proof idea
This is a definition module, no proofs. It assembles the distribution by importing the divisor-box combinatorics and the prime cost spectrum, then states the existence claim for balanced phase points under the residual trapping condition.
why it matters in Recognition Science
The module supplies the exact prime-distribution input required by EffectivePrimePhaseInput for the residual Erdős-Straus proof. It feeds the implication that the stated distribution yields PrimePhaseBoxDistribution. The construction closes the combinatorial bridge between the J-cost spectrum and the square-budget phase constraints.
scope and limits
- Does not prove the full Erdős-Straus conjecture.
- Does not compute explicit numerical values for the phase distribution.
- Does not address non-residual or infinite-budget cases.
- Does not link to the forcing chain T0-T8 or phi-ladder mass formulas.