IndisputableMonolith.NumberTheory.UniversalCostSpectrum
The UniversalCostSpectrum module defines PrimeNormStructure as any type carrying a real norm strictly greater than 1 on every element, abstracting primes across integers, polynomials, and ideals, then builds universalCost and its basic properties from the J-cost supplied by the Cost module. Number theorists and physicists working on Recognition Science mass formulas would cite these definitions when mapping arithmetic factorizations onto the phi-ladder. The module consists entirely of definitions and elementary lemmas establishing non-negality,
claimA type $P$ carries a PrimeNormStructure when equipped with a norm function $P$ to the reals satisfying $||p|| > 1$ for all $p$. The universal cost spectrum is the function that assigns to each natural number the sum of prime J-costs over its prime factors (with multiplicity), satisfying universalCost(0) = 0, universalCost(1) = 0, additivity under multiplication of coprime arguments, and non-negativity.
background
The module imports IndisputableMonolith.Cost, which supplies the J-cost function J(x) = (x + x^{-1})/2 - 1 used throughout Recognition Science. PrimeNormStructure abstracts the concrete arithmetic data (integer primes, monic irreducibles, prime ideals) so that the same cost machinery applies uniformly. Sibling declarations then define primeJcost as the J-cost of the norm, universalCost as the total cost accumulated over a factorization, and the supporting lemmas universalCost_zero, universalCost_add, universalCost_nonneg.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the number-theoretic cost layer that feeds the mass formula yardstick * phi^(rung - 8 + gap(Z)) and the phi-ladder constructions in the Recognition framework. It realizes the J-uniqueness (T5) inside arithmetic structures, enabling later derivations of constants inside the alpha band (137.030, 137.039). No downstream theorems are recorded in the current dependency graph, but the universalCost functions are the direct input to any cost-based spectrum calculation.
scope and limits
- Does not construct explicit PrimeNormStructure instances for the integers or polynomial rings.
- Does not compute numerical values of universalCost for specific integers.
- Does not relate the cost spectrum to the eight-tick octave or to D = 3.
- Does not address the Berry creation threshold or Z_cf = phi^5.
depends on (1)
declarations in this module (15)
-
class
PrimeNormStructure -
def
primeJcost -
theorem
primeJcost_pos -
theorem
primeJcost_nonneg -
def
universalCost -
theorem
universalCost_zero -
theorem
universalCost_single -
theorem
universalCost_single_one -
theorem
universalCost_add -
theorem
universalCost_nonneg -
theorem
universalCost_pos -
theorem
universalCost_smul_single -
theorem
universal_cost_certificate -
instance
natPrimesInstance -
def
funcFieldNormStructure