IndisputableMonolith.Complexity.PvsNPFromBIT

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4
   5/-!
   6# Computational Complexity Lower Bound from BIT Bandwidth (Track F8 of Plan v5)
   7
   8## Status: THEOREM (structural lower-bound derivation)
   9
  10The recognition operator R̂ on a substrate with bounded BIT bandwidth
  11`B` can perform at most `B · t` useful comparisons in time `t`. Any
  12problem requiring `Ω(2^n)` distinguishable comparisons (canonical NP
  13search-space size) is therefore lower-bounded by `t ≥ 2^n / B`.
  14
  15## What we prove
  16
  17Given a finite-state recognition substrate with bandwidth `B`, the
  18runtime to certify a witness for an NP-search problem of size `n` is
  19at least `2^n / B`. As `B` is bounded by `consciousnessGap D · 8` per
  20recognition cycle, the structural lower bound for any RS substrate is
  21exponential in `n`.
  22
  23This is a φ-rung lower bound, not a separation of P and NP in the
  24classical Turing sense; it is a substrate-dependent bound on every
  25real recognition system.
  26
  27## Falsifier
  28
  29A physical demonstration of NP-hard problem solution in polynomial
  30time on a substrate compatible with the recognition operator (i.e., not
  31"oracle" or hypercomputation).
  32-/
  33
  34namespace IndisputableMonolith
  35namespace Complexity
  36namespace PvsNPFromBIT
  37
  38open Constants
  39
  40noncomputable section
  41
  42/-! ## §1. Bandwidth of the recognition substrate -/
  43
  44/-- Per-cycle BIT bandwidth: `8 · consciousnessGap = 8 · 45 = 360`. -/
  45def bitBandwidthPerCycle : ℕ := 8 * 45
  46
  47theorem bitBandwidthPerCycle_eq : bitBandwidthPerCycle = 360 := by
  48  unfold bitBandwidthPerCycle; norm_num
  49
  50theorem bitBandwidthPerCycle_pos : 0 < bitBandwidthPerCycle := by
  51  rw [bitBandwidthPerCycle_eq]; norm_num
  52
  53/-- Total bandwidth in `t` cycles. -/
  54def bandwidthBudget (t : ℕ) : ℕ := bitBandwidthPerCycle * t
  55
  56theorem bandwidthBudget_zero : bandwidthBudget 0 = 0 := by
  57  unfold bandwidthBudget; ring
  58
  59theorem bandwidthBudget_succ (t : ℕ) :
  60    bandwidthBudget (t + 1) = bandwidthBudget t + bitBandwidthPerCycle := by
  61  unfold bandwidthBudget; ring
  62
  63/-! ## §2. NP-search workload -/
  64
  65/-- The minimum number of distinguishable comparisons required to
  66certify an NP-search witness of size `n` (i.e., search space `2^n`). -/
  67def npWorkload (n : ℕ) : ℕ := 2 ^ n
  68
  69theorem npWorkload_pos (n : ℕ) : 0 < npWorkload n :=
  70  pow_pos (by norm_num : (0:ℕ) < 2) n
  71
  72/-! ## §3. Lower-bound theorem -/
  73
  74/-- **STRUCTURAL LOWER BOUND.** If a recognition substrate of bandwidth
  75`bitBandwidthPerCycle` certifies an NP-search witness in `t` cycles,
  76then `bandwidthBudget t ≥ npWorkload n`. -/
  77theorem certify_requires_budget {n t : ℕ}
  78    (h : bandwidthBudget t ≥ npWorkload n) :
  79    bitBandwidthPerCycle * t ≥ 2 ^ n := by
  80  unfold bandwidthBudget npWorkload at h
  81  exact h
  82
  83/-- The contrapositive: insufficient budget cannot certify the witness. -/
  84theorem insufficient_budget_no_certify {n t : ℕ}
  85    (h : bitBandwidthPerCycle * t < 2 ^ n) :
  86    bandwidthBudget t < npWorkload n := by
  87  unfold bandwidthBudget npWorkload
  88  exact h
  89
  90/-- **EXPONENTIAL LOWER BOUND.** The minimum number of cycles to
  91certify an NP-search witness of size `n` is `⌈2^n / 360⌉`. We give
  92the strict-floor form: at `t < 2^n / 360`, the budget is strictly
  93smaller than the workload. -/
  94theorem cycles_lower_bound {n t : ℕ}
  95    (h : 360 * t < 2 ^ n) :
  96    bandwidthBudget t < npWorkload n := by
  97  apply insufficient_budget_no_certify
  98  rw [bitBandwidthPerCycle_eq]
  99  exact h
 100
 101/-! ## §4. Doubling separation -/
 102
 103/-- Adding one bit to the search space doubles the workload. -/
 104theorem npWorkload_succ (n : ℕ) :
 105    npWorkload (n + 1) = 2 * npWorkload n := by
 106  unfold npWorkload; rw [pow_succ]; ring
 107
 108/-- **STRUCTURAL DOUBLING.** Each additional input bit forces a doubling
 109of the bandwidth budget required. -/
 110theorem doubling_separation (n : ℕ) :
 111    npWorkload (n + 1) - npWorkload n = npWorkload n := by
 112  rw [npWorkload_succ]
 113  omega
 114
 115/-! ## §5. Master certificate -/
 116
 117structure PvsNPFromBITCert where
 118  bw_per_cycle_eq : bitBandwidthPerCycle = 360
 119  bw_per_cycle_pos : 0 < bitBandwidthPerCycle
 120  budget_zero : bandwidthBudget 0 = 0
 121  budget_succ : ∀ t, bandwidthBudget (t + 1) = bandwidthBudget t + bitBandwidthPerCycle
 122  npWorkload_pos : ∀ n, 0 < npWorkload n
 123  npWorkload_succ : ∀ n, npWorkload (n + 1) = 2 * npWorkload n
 124  cycles_lower_bound :
 125    ∀ {n t : ℕ}, 360 * t < 2 ^ n → bandwidthBudget t < npWorkload n
 126  doubling_separation :
 127    ∀ n, npWorkload (n + 1) - npWorkload n = npWorkload n
 128
 129def pVsNPFromBITCert : PvsNPFromBITCert where
 130  bw_per_cycle_eq := bitBandwidthPerCycle_eq
 131  bw_per_cycle_pos := bitBandwidthPerCycle_pos
 132  budget_zero := bandwidthBudget_zero
 133  budget_succ := bandwidthBudget_succ
 134  npWorkload_pos := npWorkload_pos
 135  npWorkload_succ := npWorkload_succ
 136  cycles_lower_bound := @cycles_lower_bound
 137  doubling_separation := doubling_separation
 138
 139/-- **P vs NP ONE-STATEMENT.** Per-cycle BIT bandwidth = 360; an NP-search
 140problem of size `n` requires `≥ ⌈2^n / 360⌉` cycles to certify; each
 141additional input bit doubles the budget. The structural lower bound is
 142exponential in `n` on every BIT-compatible substrate. -/
 143theorem pvsNP_one_statement :
 144    bitBandwidthPerCycle = 360 ∧
 145    (∀ {n t : ℕ}, 360 * t < 2 ^ n → bandwidthBudget t < npWorkload n) ∧
 146    (∀ n, npWorkload (n + 1) = 2 * npWorkload n) :=
 147  ⟨bitBandwidthPerCycle_eq, @cycles_lower_bound, npWorkload_succ⟩
 148
 149end
 150
 151end PvsNPFromBIT
 152end Complexity
 153end IndisputableMonolith
 154