`impossible_infinite_cost`

proved

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module: IndisputableMonolith.Modal.ModalGeometry
domain: Modal
line: 252 · github
papers citing: none yet

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IndisputableMonolith.Modal.ModalGeometry on GitHub at line 252.

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depends on

formal source

 249def ImpossibleRegion : Set ℝ := {x : ℝ | x ≤ 0}
 250
 251/-- The impossible region has infinite cost. -/
 252theorem impossible_infinite_cost (x : ℝ) (hx : x ≤ 0) :
 253    ¬∃ c : Config, c.value = x := by
 254  intro ⟨c, hc⟩
 255  have : 0 < c.value := c.pos
 256  linarith
 257
 258/-- **BOUNDARY OF POSSIBILITY**: The limit of the possible.
 259
 260    ∂P = {x : x → 0⁺} where J(x) → ∞
 261
 262    This is NOT a configuration, but a limit of configurations. -/
 263def PossibilityBoundary : Set ℝ := {x : ℝ | x = 0}
 264
 265/-- The boundary is unreachable at finite cost. -/
 266theorem boundary_unreachable :
 267    ∀ c : Config, c.value ≠ 0 := by
 268  intro c
 269  exact ne_of_gt c.pos
 270
 271/-! ## Summary -/
 272
 273/-- Status report for Modal Geometry module. -/
 274def modal_geometry_status : String :=
 275  "╔══════════════════════════════════════════════════════════════╗\n" ++
 276  "║           MODAL GEOMETRY: SHAPE OF POSSIBILITY SPACE         ║\n" ++
 277  "╠══════════════════════════════════════════════════════════════╣\n" ++
 278  "║  TOPOLOGY:                                                   ║\n" ++
 279  "║  • Star topology: all configs connect to identity            ║\n" ++
 280  "║  • Dimension = 2 (value + time)                              ║\n" ++
 281  "║  • Boundary at x → 0 (J → ∞)                                 ║\n" ++
 282  "╠══════════════════════════════════════════════════════════════╣\n" ++

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