`realizedDefectCollapseScalar_pos`

proved

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module: IndisputableMonolith.Unification.UnifiedRH
domain: Unification
line: 310 · github
papers citing: none yet

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IndisputableMonolith.Unification.UnifiedRH on GitHub at line 310.

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formal source

 307  1 / (1 + annularCost ((canonicalDefectSampledFamily sensor hm).mesh N))
 308
 309/-- The realized defect collapse scalar is always positive. -/
 310theorem realizedDefectCollapseScalar_pos
 311    (sensor : DefectSensor) (hm : sensor.charge ≠ 0) (N : ℕ) :
 312    0 < realizedDefectCollapseScalar sensor hm N := by
 313  unfold realizedDefectCollapseScalar
 314  apply one_div_pos.mpr
 315  have hcost_nonneg :
 316      0 ≤ annularCost ((canonicalDefectSampledFamily sensor hm).mesh N) :=
 317    annularCost_nonneg _
 318  linarith
 319
 320/-- Boundary-approach predicate for the concrete realized-defect collapse
 321scalar. -/
 322def RealizedCollapseBoundaryApproaching
 323    (sensor : DefectSensor) (hm : sensor.charge ≠ 0) : Prop :=
 324  ∀ ε > 0, ∃ N : ℕ, realizedDefectCollapseScalar sensor hm N < ε
 325
 326/-- Cost divergence is impossible for charge `0`. -/
 327theorem not_costDivergent_of_charge_zero (sensor : DefectSensor)
 328    (hzero : sensor.charge = 0) :
 329    ¬ CostDivergent sensor := by
 330  intro hdiv
 331  obtain ⟨N, hN⟩ := hdiv 1
 332  let mesh : AnnularMesh N := uniformChargeMesh N sensor.charge
 333  have hcost : 1 < annularCost mesh := by
 334    exact hN mesh (by intro n; rfl)
 335  have hexcess_zero : annularExcess mesh = 0 := by
 336    simpa [mesh, hzero] using uniformChargeMesh_excess_zero N sensor.charge
 337  have hfloor_zero : annularTopologicalFloor N mesh.charge = 0 := by
 338    simpa [mesh, hzero] using (annularTopologicalFloor_zero N)
 339  have hcost_zero : annularCost mesh = 0 := by
 340    unfold annularExcess at hexcess_zero

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