IndisputableMonolith.NumberTheory.ZeroCompositionInterface
The ZeroCompositionInterface module supplies the abstract composition law for zeta zero locations required by Vector C in Recognition Science. Researchers assembling the Riemann Hypothesis bridge via cost functions would cite it when linking zero defects to the J-functional equation. The module structures an interface of definitions and forcing lemmas drawn from upstream functional equation and discreteness results.
claimThe zero-location composition law states that for zero deviations encoded as $x = e^{2(Re ρ - 1/2)}$, the J-cost satisfies $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$ together with the forcing consequences that this law implies a unique minimum at the critical line.
background
This module sits in the NumberTheory domain and provides the interface for composing zero locations via the J-cost. It imports the functional equation helpers for T5, the discreteness forcing module, and the zero location cost dictionary. The upstream discreteness forcing states that J(x) = ½(x + x⁻¹) - 1 has a unique minimum at x = 1 and in log coordinates J(eᵗ) = cosh(t) - 1 forms a convex bowl centered at t = 0. The zero location cost module defines zeroDeviation ρ = 2 (Re ρ - 1/2) and zeroDefect ρ = defect (exp (zeroDeviation ρ)).
proof idea
This is a definition and interface module. It declares the composition law and related lemmas such as the forcing of cosh equality and unique minimum by reduction to the imported functional equation lemmas and zero defect definitions.
why it matters in Recognition Science
This module feeds the AnalyticTrace interface that assembles the axiom-free RH bridge and the Vector C Symmetry-Only No-Go that certifies symmetry arguments alone do not force the critical line. It supplies the abstract zero composition layer needed for Vector C within the T5 cost uniqueness and discreteness forcing chain.
scope and limits
- Does not prove the Riemann Hypothesis.
- Does not compute explicit zero locations or bounds.
- Does not incorporate analytic continuation or Euler products.
- Does not address numerical verification of the law.
used by (2)
depends on (3)
declarations in this module (8)
-
theorem
cosh_eq_one_iff -
structure
ZeroCompositionLaw -
theorem
zeroCompositionLaw_forces_cosh -
theorem
zeroCompositionLaw_forces_unique_minimum -
theorem
zeroCompositionLaw_forces_eta_zero -
structure
ZeroCompositionWitness -
theorem
zeroCompositionWitness_forces_on_critical_line -
theorem
zeroCompositionWitness_forces_zero_defect