IndisputableMonolith.Gravity.LedgerToGeometryBridge

   1import Mathlib
   2import IndisputableMonolith.Gravity.RecognitionLedger
   3import IndisputableMonolith.Gravity.TensorShearSector
   4
   5namespace IndisputableMonolith
   6namespace Gravity
   7
   8/-!
   9# Ledger-to-Geometry Bridge: Honest Status
  10
  11This module records the honest, machine-checked status of the connection
  12between the discrete recognition-ledger substrate and the effective
  13geometric (hinge) description.
  14
  15**Key findings:**
  16
  171. The bridge from ledger deficits to geometric hinge deficits is an
  18   EXPLICIT ASSUMPTION, not a theorem derived from the ledger axioms.
  19   The map `x_sigma` and the deficit-matching condition are recorded as
  20   fields of `LedgerToHingeBridge`, tagged as assumed.
  21
  222. The conformal edge ansatz is INSUFFICIENT for the transverse-traceless
  23   gravitational-wave sector. This follows from the rectangle/shear
  24   obstruction proved in `TensorShearSector`: a nontrivial rectangle
  25   shear mode (`h ≠ v`) has no vertex-conformal potential realization.
  26   Since gravitational waves require transverse-traceless (shear) degrees
  27   of freedom, the conformal route cannot recover them.
  28-/
  29
  30/-- A bridge from a recognition ledger on substrate `Λ` to geometric
  31hinge deficits on a hinge type `H`.
  32
  33The field `x_sigma` is the substrate-to-hinge comparison map: it assigns
  34to each substrate cell the hinge whose deficit is to be compared with the
  35ledger deficit at that cell.
  36
  37The field `bridge_assumed` is an EXPLICIT ASSUMPTION (not derived from the
  38recognition-ledger axioms) that the ledger deficit at each cell equals the
  39geometric deficit at the corresponding hinge. This assumption is the
  40load-bearing bridge between the discrete ledger substrate and the effective
  41geometry; it is tagged as assumed because the ledger axioms (symmetry,
  42diagonal zero, non-negativity, RCL subadditivity) do not by themselves
  43force any particular relation to geometric deficits. -/
  44structure LedgerToHingeBridge
  45    {Λ : Type*} [Fintype Λ] [DecidableEq Λ]
  46    (H : Type*)
  47    (L : RecognitionLedger.RecognitionLedger Λ) where
  48  /-- The substrate-to-hinge comparison map `x_σ : Λ → H`. -/
  49  x_sigma : Λ → H
  50  /-- The geometric deficit function on hinges. -/
  51  geometricDeficit : H → ℝ
  52  /-- EXPLICIT ASSUMPTION (not derived): the ledger deficit at each cell
  53  `i` equals the geometric deficit at the hinge `x_sigma i`. -/
  54  bridge_assumed : ∀ i : Λ,
  55    RecognitionLedger.deficit L i = geometricDeficit (x_sigma i)
  56
  57/-- **Conformal ansatz cannot recover gravitational waves.**
  58
  59The conformal edge ansatz assigns one scalar potential to each vertex and
  60induces edge-length variations by averaging endpoint potentials. This is
  61exactly the vertex-conformal log-strain map. The rectangle/shear obstruction
  62from `TensorShearSector` proves that a nontrivial rectangle shear mode
  63(with horizontal strain `h ≠ v` vertical strain) has no vertex-conformal
  64potential realization.
  65
  66Since transverse-traceless (TT) gravitational-wave modes are pure shear
  67modes, and the conformal ansatz cannot represent any nontrivial shear, the
  68conformal route is insufficient for the gravitational-wave sector. This is
  69exactly why the conformal edge ansatz cannot serve as the actual connection
  70between the ledger substrate and the effective geometry. -/
  71theorem conformal_ansatz_cannot_recover_gravitational_waves
  72    (h v : ℝ) (hne : h ≠ v) :
  73    ¬ ∃ ξa ξb ξc ξd : ℝ,
  74      (ξa + ξb) / 2 = h ∧
  75      (ξc + ξd) / 2 = h ∧
  76      (ξb + ξc) / 2 = v ∧
  77      (ξd + ξa) / 2 = v :=
  78  TensorShearSector.nontrivial_rectangle_shear_not_vertexConformal h v hne
  79
  80/-- Status flags recording the honest state of the ledger-to-geometry bridge. -/
  81structure LedgerToGeometryBridgeStatus where
  82  /-- The bridge condition is an explicit assumption, not derived from
  83  the recognition-ledger axioms. -/
  84  bridge_is_assumed_not_derived : Bool
  85  /-- The conformal edge ansatz is insufficient for the transverse-traceless
  86  gravitational-wave sector. -/
  87  conformal_route_insufficient_for_gw : Bool
  88
  89/-- The canonical status: the bridge is assumed (not derived), and the
  90conformal route is insufficient for gravitational waves. -/
  91def ledgerToGeometryBridgeStatus : LedgerToGeometryBridgeStatus where
  92  bridge_is_assumed_not_derived := true
  93  conformal_route_insufficient_for_gw := true
  94
  95/-- **Status flags theorem.** Both status flags are `true`: the bridge
  96condition is assumed (not derived), and the conformal route is insufficient
  97for gravitational waves. -/
  98theorem ledgerToGeometryBridgeStatus_flags :
  99    ledgerToGeometryBridgeStatus.bridge_is_assumed_not_derived = true ∧
 100    ledgerToGeometryBridgeStatus.conformal_route_insufficient_for_gw = true :=
 101  ⟨rfl, rfl⟩
 102
 103end Gravity
 104end IndisputableMonolith