IndisputableMonolith.Gravity.SevenGaps.HKTDynamicTarget

   1import IndisputableMonolith.Gravity.SevenGaps.HypersurfaceDeformation
   2import IndisputableMonolith.Gravity.SevenGaps.DynamicStructureFunctionBlocker
   3
   4/-!
   5# Wave C2 R5/R6 groundwork: widened HKT target with dynamic structure function
   6
   7Definition module (no hard proofs). Codex adjudication rejected folding g into
   8momDensity and selling frozen unit structure as GR. Widened target carries an
   9explicit structureFunction slot with structure_nonconstant.
  10
  11HKTRigidityStatementDyn is DEFINED, neither proved nor assumed. Original
  12HKTRigidityStatement is false as stated at n=1
  13(HKTOneSiteCounterexample.not_HKTRigidityStatement_one).
  14
  15**Unsplit `mom_ham` is uninhabitable for honest nearest-neighbor local
  16profiles against the frozen quadratic Hamiltonian.** The field below keeps the
  17classical unsplit advection form as a falsification-adjacent record. The scoped
  18no-go
  19`HKTPointSplitTarget.unsplit_mom_ham_no_smooth_nearestNeighbor_witness_frozenHam`
  20shows that at `n = 2` no Frechet-smooth local momentum profile
  21`f(d, π_j, π_{j+1})` can satisfy it against the frozen quadratic Hamiltonian
  22(forced singular relation `(π₀+π₁)∂_d f = π₀²+d²`). The HamDyn-level analogue is
  23the open Prop `UnsplitMomHamNoSmoothNearestNeighborWitnessHamDyn`. The
  24schema-only weak sibling is `HKTPointSplitTarget.HKTPointSplitTargetDyn`; the
  25load-bearing class is `HKTPointSplitStrong.HKTPointSplitTargetDynStrong` with
  26binding Prop `HKTRigidityStatementPointSplitDynN2Strong`.
  27-/
  28
  29namespace IndisputableMonolith
  30namespace Gravity
  31namespace SevenGaps
  32namespace HKTDynamicTarget
  33
  34open HypersurfaceDeformation DynamicStructureFunctionBlocker
  35
  36noncomputable section
  37
  38variable {n : ℕ} [NeZero n]
  39
  40/-- OPEN TARGET (deliberately uninhabited). HKT hypotheses with explicit dynamic
  41structure function. structure_nonconstant excludes the frozen/background decoy. -/
  42structure HojmanKucharTeitelboimTargetDyn (n : ℕ) [NeZero n] where
  43  hamDensity : PhaseSpace n → ZMod n → ℝ
  44  momDensity : PhaseSpace n → ZMod n → ℝ
  45  structureFunction : PhaseSpace n → ZMod n → ℝ
  46  ham_differentiable : ∀ N : ZMod n → ℝ,
  47    Differentiable ℝ (fun x : PhaseSpace n => ∑ j : ZMod n, N j * hamDensity x j)
  48  mom_differentiable : ∀ w : ZMod n → ℝ,
  49    Differentiable ℝ (fun x : PhaseSpace n => ∑ j : ZMod n, w j * momDensity x j)
  50  structure_nonconstant : ¬ PhaseSpaceConstant structureFunction
  51  ham_local : ∀ (x y : PhaseSpace n) (j : ZMod n),
  52    x.1 j = y.1 j → x.1 (j + 1) = y.1 (j + 1) → x.2 j = y.2 j →
  53      hamDensity x j = hamDensity y j
  54  ham_covariant : ∀ (x : PhaseSpace n) (a j : ZMod n),
  55    hamDensity (fun i => x.1 (i + a), fun i => x.2 (i + a)) j = hamDensity x (j + a)
  56  structure_local : ∀ (x y : PhaseSpace n) (j : ZMod n),
  57    x.1 j = y.1 j → structureFunction x j = structureFunction y j
  58  mom_mom : ∀ (v w : ZMod n → ℝ) (x : PhaseSpace n),
  59    bracket (fun y => ∑ j : ZMod n, v j * momDensity y j)
  60      (fun y => ∑ j : ZMod n, w j * momDensity y j) x = 0
  61  mom_ham : ∀ (w N : ZMod n → ℝ) (x : PhaseSpace n),
  62    bracket (fun y => ∑ j : ZMod n, w j * momDensity y j)
  63      (fun y => ∑ j : ZMod n, N j * hamDensity y j) x
  64      = ∑ j : ZMod n, (w j * (N (j + 1) - N j)) * hamDensity x j
  65  ham_ham : ∀ (N M : ZMod n → ℝ) (x : PhaseSpace n),
  66    bracket (fun y => ∑ j : ZMod n, N j * hamDensity y j)
  67      (fun y => ∑ j : ZMod n, M j * hamDensity y j) x
  68      = ∑ j : ZMod n,
  69          (N j * M (j + 1) - M j * N (j + 1)) *
  70            (structureFunction x j * momDensity x j)
  71
  72/-- DEFINED, neither proved nor assumed. Repaired GR-strength rigidity target. -/
  73def HKTRigidityStatementDyn (n : ℕ) [NeZero n] : Prop :=
  74  ∀ T : HojmanKucharTeitelboimTargetDyn n,
  75    ∃ cKin cGrad cVac : ℝ, ∀ (x : PhaseSpace n) (j : ZMod n),
  76      T.hamDensity x j
  77        = cKin * (x.2 j * x.2 j)
  78          + cGrad *
  79              (T.structureFunction x j *
  80                ((x.1 (j + 1) - x.1 j) * (x.1 (j + 1) - x.1 j)))
  81          + cVac
  82
  83def unitStructureHamHamRHS (momDensity : PhaseSpace n → ZMod n → ℝ)
  84    (N M : ZMod n → ℝ) (x : PhaseSpace n) : ℝ :=
  85  ∑ j : ZMod n, (N j * M (j + 1) - M j * N (j + 1)) * ((1 : ℝ) * momDensity x j)
  86
  87theorem unitStructure_recovers_original_ham_ham_RHS
  88    (momDensity : PhaseSpace n → ZMod n → ℝ) (N M : ZMod n → ℝ)
  89    (x : PhaseSpace n) :
  90    unitStructureHamHamRHS momDensity N M x
  91      = ∑ j : ZMod n, (N j * M (j + 1) - M j * N (j + 1)) * momDensity x j := by
  92  unfold unitStructureHamHamRHS
  93  refine Finset.sum_congr rfl fun j _ => ?_
  94  ring
  95
  96theorem unitStructure_is_phaseSpaceConstant :
  97    PhaseSpaceConstant (fun (_x : PhaseSpace n) (_j : ZMod n) => (1 : ℝ)) := by
  98  intro x y j
  99  rfl
 100
 101structure HKTDynamicTargetStatus where
 102  dynTargetDefined : Bool
 103  dynRigidityDefined : Bool
 104  unitStructureDecoyExcluded : Bool
 105  dynInhabitantBanked : Bool
 106  gap5ConstraintRecovery : Bool
 107
 108def hktDynamicTargetStatus : HKTDynamicTargetStatus where
 109  dynTargetDefined := true
 110  dynRigidityDefined := true
 111  unitStructureDecoyExcluded := true
 112  dynInhabitantBanked := false
 113  gap5ConstraintRecovery := false
 114
 115theorem hktDynamicTargetStatus_flags :
 116    hktDynamicTargetStatus.dynTargetDefined = true ∧
 117      hktDynamicTargetStatus.dynRigidityDefined = true ∧
 118        hktDynamicTargetStatus.unitStructureDecoyExcluded = true ∧
 119          hktDynamicTargetStatus.dynInhabitantBanked = false ∧
 120            hktDynamicTargetStatus.gap5ConstraintRecovery = false := by
 121  decide
 122
 123end
 124end HKTDynamicTarget
 125end SevenGaps
 126end Gravity
 127end IndisputableMonolith
 128