IndisputableMonolith.NumberTheory.UniversalCostSpectrum
UniversalCostSpectrum introduces PrimeNormStructure to abstract prime-like objects equipped with a real norm exceeding one. Researchers extending Recognition Science's J-cost to number theory cite it for uniform treatment of primes, polynomials, and ideals. The module supplies definitions together with basic properties of the universal cost function.
claimA type $P$ is a PrimeNormStructure when equipped with a map $\| \cdot \| : P \to \mathbb{R}$ satisfying $\|p\| > 1$ for all $p P$. The universal cost is obtained by applying the J-cost to this norm.
background
Recognition Science derives all physics from one functional equation whose core object is the J-cost. This module sits in the NumberTheory domain and imports the Cost module, which defines the J-cost and the Recognition Composition Law. The central definition abstracts concrete arithmetic structures by requiring only a real-valued norm strictly greater than one on each element.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the number-theoretic interface for the universal cost spectrum, which supports application of the J-cost and Recognition Composition Law to discrete objects. It prepares structures that feed into the forcing chain steps T5 through T8.
scope and limits
- Does not instantiate the norm for integer primes or polynomial rings.
- Does not prove positivity or additivity of universalCost.
- Does not derive the mass formula or alpha band.
- Does not address the Berry creation threshold or phi-ladder rungs.
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