PvsNPFromBITCert
The PvsNPFromBITCert structure packages the BIT bandwidth axioms and derived properties that underpin the exponential lower bound on NP-witness certification. Complexity researchers studying recognition substrates cite it to invoke the full set of assumptions for the t >= 2^n / 360 bound. It is assembled by direct reference to the per-cycle bandwidth definition, workload doubling, and the cycles lower bound theorem.
claimA structure asserting that the per-cycle bandwidth equals 360, is positive, the total budget satisfies budget(0) = 0 and budget(t+1) = budget(t) + 360, the workload W(n) = 2^n is positive and doubles with each increment, 360 t < 2^n implies budget(t) < W(n), and W(n+1) - W(n) = W(n).
background
The module derives complexity lower bounds for the recognition operator on a substrate whose BIT bandwidth is capped at 8 times the consciousness gap, yielding 360 units per cycle. Bandwidth budget after t cycles is defined as the product of this per-cycle value and t. NP workload is the search-space size 2^n, the minimum number of distinguishable comparisons required to certify a witness.
proof idea
This is a structure definition that directly bundles the referenced definitions and lemmas: bitBandwidthPerCycle equality and positivity, the recursive bandwidthBudget rules, npWorkload positivity and doubling, the cycles lower bound theorem, and the doubling separation theorem. No tactics or reductions are performed.
why it matters in Recognition Science
It supplies the complete certificate for the BIT lower-bound derivation and is instantiated by the downstream pVsNPFromBITCert definition. The structure closes the master-certificate section of the module, connecting the bounded-bandwidth recognition operator to the exponential workload and the phi-rung substrate constraints. It leaves open whether any physical recognition system can evade the bound without hypercomputation.
scope and limits
- Does not prove a classical Turing separation of P and NP.
- Does not apply to oracle or unbounded-bandwidth machines.
- Does not give explicit runtimes for concrete algorithms.
- Does not address non-recognition or hypercomputational substrates.
formal statement (Lean)
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