`BoundaryApproaching`

definition

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module: IndisputableMonolith.Unification.UnifiedRH
domain: Unification
line: 255 · github
papers citing: none yet

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IndisputableMonolith.Unification.UnifiedRH on GitHub at line 255.

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formal source

 252
 253/-- A realizable ledger is boundary-approaching if its scalar proxy state can
 254be driven arbitrarily close to `0`. -/
 255def BoundaryApproaching (sensor : DefectSensor)
 256    [PhysicallyRealizableLedger sensor] : Prop :=
 257  ∀ ε > 0, ∃ N : ℕ,
 258    PhysicallyRealizableLedger.scalarState (sensor := sensor) N < ε
 259
 260/-- T1 forbids a physically realizable ledger from approaching the boundary
 261`x = 0`: a uniform T1 defect bound contradicts `nothing_cannot_exist`. -/
 262theorem physicallyRealizableLedger_not_boundaryApproaching
 263    (sensor : DefectSensor) [PhysicallyRealizableLedger sensor] :
 264    ¬ BoundaryApproaching sensor := by
 265  intro hboundary
 266  obtain ⟨K, hK⟩ := PhysicallyRealizableLedger.scalarDefectBounded (sensor := sensor)
 267  obtain ⟨ε, hεpos, hε⟩ := t1_cost_barrier K
 268  obtain ⟨N, hN⟩ := hboundary ε hεpos
 269  have hlt :
 270      K <
 271        IndisputableMonolith.Foundation.LawOfExistence.defect
 272          (PhysicallyRealizableLedger.scalarState (sensor := sensor) N) := by
 273    exact hε
 274      (PhysicallyRealizableLedger.scalarState (sensor := sensor) N)
 275      (PhysicallyRealizableLedger.scalarStatePos (sensor := sensor) N)
 276      hN
 277  exact not_lt_of_ge (hK N) hlt
 278
 279/-
 280The auxiliary Euler stiffness proxy `eulerScalarProxy` is the actual
 281T1-bounded realizability scalar for the Euler ledger.  It stays away from the
 282boundary and therefore supports the `PhysicallyRealizableLedger` interface.
 283
 284The realized collapse observable extracted from the canonical defect family is
 285defined separately below. It captures the divergence-to-boundary mechanism, but

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