module module high

`IndisputableMonolith.Unification.UnifiedRH`

The UnifiedRH module unifies the Riemann Hypothesis with Recognition Science by defining cost-divergence for defect sensor traces under uniform-charge sampling. Researchers deriving RH from the Law of Existence cite it as the bridge that forces annular cost past any bound via Θ(m² log N) growth for m ≠ 0. The module achieves this by importing LawOfExistence and EulerInstantiation then deriving the divergence predicate from the Euler product representation.

claimA defect sensor trace is cost-divergent when the topological floor of its annular cost grows as $\Theta(m^2 \log N)$ for charge $m \neq 0$, forcing the total cost to exceed every finite bound at large refinement depth $N$.

background

The module operates in the setting supplied by LawOfExistence, whose core statement is that x exists if and only if defect(x) = 0. It further imports EulerInstantiation, whose doc states that the Euler product of ζ(s) naturally instantiates the abstract RS carrier/sensor framework.

Defect sensors are equipped with annular cost measures that detect topological defects; uniform-charge sampling is the probe that reveals divergence. The module therefore supplies the concrete cost-divergence predicate that converts the abstract existence law into a number-theoretic obstruction.

proof idea

This is a unification module whose argument proceeds by importing the two upstream modules, introducing the CostDivergent predicate, and proving the key lemma nonzero_charge_cost_divergent via reduction to the Θ(m² log N) growth estimate already available from the Euler product instantiation.

why it matters in Recognition Science

The module supplies the cost-divergence bridge imported by RH_Certificate to complete the RS chain for the Riemann Hypothesis. It is also used by AnalyticTrace, DirectedEulerLedger, EulerCarrierRealizability, T1BoundaryExclusion, and ZetaLedgerBridge to connect the abstract DefectSensor framework to concrete zeta realizations. It fills the formalization gap between the proved T1 boundary exclusion and the analytic trace infrastructure.

scope and limits

Does not prove the Riemann Hypothesis outright.
Does not treat non-uniform charge distributions.
Does not extend the divergence claim beyond the Euler product setting.
Does not address continuous refinements outside the discrete N ladder.

used by (6)

From the project-wide theorem graph. These declarations reference this one in their body.

IndisputableMonolith.NumberTheory.AnalyticTrace module_import
IndisputableMonolith.NumberTheory.DirectedEulerLedger module_import
IndisputableMonolith.NumberTheory.EulerCarrierRealizability module_import
IndisputableMonolith.NumberTheory.RH_Certificate module_import
IndisputableMonolith.NumberTheory.T1BoundaryExclusion module_import
IndisputableMonolith.NumberTheory.ZetaLedgerBridge module_import

depends on (2)

Lean names referenced from this declaration's body.

IndisputableMonolith.Foundation.LawOfExistence module_import
IndisputableMonolith.NumberTheory.EulerInstantiation module_import

declarations in this module (64)

def CostDivergent L77
theorem nonzero_charge_cost_divergent L83
structure EulerTraceAdmissible L100
theorem euler_trace_admissible L111
theorem t1_cost_barrier L123
def eulerCarrierRadius L132
theorem eulerCarrierRadius_pos L136
def eulerScalarGap L150
theorem eulerScalarGap_pos L155
def eulerScalarProxy L173
theorem eulerScalarProxy_pos L177
theorem defect_one_div_one_add L187
theorem eulerScalarProxy_defect_bounded L197
class PhysicallyRealizableLedger L245
def BoundaryApproaching L255
theorem physicallyRealizableLedger_not_boundaryApproaching L262
theorem annularCost_nonneg L293
def realizedDefectCollapseScalar L305
theorem realizedDefectCollapseScalar_pos L310
def RealizedCollapseBoundaryApproaching L322
theorem not_costDivergent_of_charge_zero L327
theorem costDivergent_charge_ne_zero L347
theorem realizedDefectCollapseBoundaryApproaching_of_nonzero_charge L356
theorem defect_realizedDefectCollapseScalar L402
theorem realizedDefectCollapseScalar_defect_unbounded L415
theorem not_realizedDefectCollapseScalar_t1_bounded L434
def eulerLedgerScalarState L449
theorem eulerLedgerScalarState_eq_one L454
theorem eulerLedgerScalarState_eq_collapse L461
theorem eulerLedgerScalarState_pos L467
instance eulerPhysicallyRealizableLedger L477
def euler_physically_realizable L485
theorem eulerScalarProxy_not_boundaryApproaching L492
class DivergenceWitnessesBoundary L504
class CollapseBoundaryTransport L513
theorem euler_collapse_boundary_transport L523
def EulerBoundaryBridgeAssumption L535
theorem euler_divergence_witnesses_boundary L544
def euler_boundary_bridge L556
theorem ontological_exclusion_of_realizable L565
theorem ontological_exclusion L578
theorem unified_rh L597
structure UnifiedRHCert L616
def unified_rh_cert_of_bridge L635
theorem bridge_contradiction_core L680
theorem obstruction_structural_asymmetry L690
theorem single_scalar_obstruction L703
theorem EBBA_of_rh L724
theorem EBBA_iff_rh L736
theorem costDivergent_iff_charge_ne_zero L775
theorem rh_from_cost_finiteness L803
theorem cost_finiteness_iff_rh L810
theorem EBBA_of_cost_finiteness L820
theorem cost_finiteness_of_EBBA L826
theorem direct_t1_exclusion L851
theorem charge_zero_cost_scalar_t1_bounded L859
theorem ontological_dichotomy L873
def PhysicallyExists L921
theorem charge_zero_iff_physicallyExists L927
def PhysicalSensor L934
theorem physical_sensor_charge_zero L947
theorem rh_from_ontological_dichotomy L953
theorem all_physicallyExist_of_rh L961
theorem rh_iff_all_physical L971