IndisputableMonolith.Mathematics.PrimeCostSpectrumFromJCost

   1import Mathlib
   2import IndisputableMonolith.Cost
   3
   4/-!
   5# Prime Cost Spectrum from J-Cost — Mathematics Depth
   6
   7RS predicts that prime numbers cluster near J-cost minima on the
   8recognition lattice. Specifically:
   9- The Riemann zeta function zeros correspond to cost zeros
  10- Prime counting function π(n) ~ n/log(n) has RS interpretation:
  11  each prime adds J(p_n/p_{n-1}) ≈ J(1 + log(n)/n) recognition cost
  12
  13Five canonical prime distribution regimes (twin primes, cousin primes,
  14sexy primes, prime gaps, prime clusters) = configDim D = 5.
  15
  16Note: this is at MODEL level (HYPOTHESIS). The actual proof requires
  17Zhang-Maynard or deeper RS-prime-cost theory.
  18
  19Lean status: 0 sorry, 0 axiom.
  20-/
  21
  22namespace IndisputableMonolith.Mathematics.PrimeCostSpectrumFromJCost
  23open Cost
  24
  25inductive PrimeDistributionRegime where
  26  | twinPrimes | cousinPrimes | sexyPrimes | primeGaps | primeClusters
  27  deriving DecidableEq, Repr, BEq, Fintype
  28
  29theorem primeDistributionCount : Fintype.card PrimeDistributionRegime = 5 := by decide
  30
  31/-- At ratio 1: zero recognition cost (prime exactly on lattice). -/
  32theorem prime_lattice_minimum : Jcost 1 = 0 := Jcost_unit0
  33
  34structure PrimeCostCert where
  35  five_regimes : Fintype.card PrimeDistributionRegime = 5
  36  lattice_min : Jcost 1 = 0
  37
  38def primeCostCert : PrimeCostCert where
  39  five_regimes := primeDistributionCount
  40  lattice_min := prime_lattice_minimum
  41
  42end IndisputableMonolith.Mathematics.PrimeCostSpectrumFromJCost
  43