module module moderate

`IndisputableMonolith.Complexity.PvsNPFromBIT`

The module fixes per-cycle BIT bandwidth at 360 units via eight times the consciousness gap. It defines recursive bandwidth budgets and NP workloads, then proves that certification demands sufficient budget while insufficient budget blocks certification and yields explicit cycle lower bounds. Researchers analyzing computational hardness inside Recognition Science cite these relations to tie bit costs to physical constants. The module is a short chain of definitions followed by elementary positivity and implication lemmas.

claimPer-cycle BIT bandwidth equals $8 times 45 = 360$. Bandwidth budget for workload size $n$ is defined recursively from this constant. NP workload is positive and closed under successor. Certification of an NP instance requires budget at least the workload size; insufficient budget precludes certification. The number of cycles is bounded below by the ratio of workload to per-cycle bandwidth.

background

Constants supplies the RS time quantum tau_0 equal to one tick. Cost supplies the underlying cost functions that measure information processing in J-cost units. The module specializes these to BIT bandwidth, setting the per-cycle value to 360 by direct multiplication with the consciousness gap of 45. It then builds bandwidthBudget as a recursive accumulator and npWorkload as the demand side for nondeterministic problems. All constructions remain inside the Complexity domain and inherit the eight-tick octave structure from upstream.

proof idea

This is a definition module. It opens with the constant definition bitBandwidthPerCycle set to 360, followed by the recursive bandwidthBudget and npWorkload. Positivity and equality lemmas are one-line wrappers. The key implications certify_requires_budget and insufficient_budget_no_certify are proved by direct unfolding and arithmetic. cycles_lower_bound follows by dividing workload size by the fixed bandwidth.

why it matters in Recognition Science

The module supplies the quantitative bridge from BIT bandwidth to NP certification cost, positioning Recognition Science to derive cycle lower bounds that separate P from NP. It feeds the broader P-versus-NP argument by grounding workload size in the phi-ladder and consciousness gap. No downstream theorems are listed yet, but the construction directly extends the T7 eight-tick octave and the mass-formula rung structure into computational complexity.

scope and limits

Does not prove P not equal to NP outright.
Does not treat quantum or relativistic extensions.
Does not give explicit numerical bounds for concrete NP-complete problems.
Does not address decidability outside the NP class.

depends on (2)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import
IndisputableMonolith.Cost module_import

declarations in this module (16)

def bitBandwidthPerCycle L45
theorem bitBandwidthPerCycle_eq L47
theorem bitBandwidthPerCycle_pos L50
def bandwidthBudget L54
theorem bandwidthBudget_zero L56
theorem bandwidthBudget_succ L59
def npWorkload L67
theorem npWorkload_pos L69
theorem certify_requires_budget L77
theorem insufficient_budget_no_certify L84
theorem cycles_lower_bound L94
theorem npWorkload_succ L104
theorem doubling_separation L110
structure PvsNPFromBITCert L117
def pVsNPFromBITCert L129
theorem pvsNP_one_statement L143