Gravity Restricted Incidence Recovery

Recovering gravity from sparse probes

In a discrete model of gravity, the geometry of a three-dimensional lattice is encoded in edge lengths, and the physics is encoded in how those lengths respond to changes at vertices. The natural question is whether the edge-length data can be recovered from vertex-level measurements alone. In a bulk lattice there are generally more edges than vertices, so full recovery is impossible. The framework's Recognition Science (a framework that derives physical structure from a forced cost of recognition events) therefore states a restricted version: recovery and separation hold on an explicitly declared geometric deficit subspace, not on the whole space of edge variables.

The classical problem this addresses is the inverse problem for discrete gravity: given vertex potentials, can one reconstruct the edge deficits that drive curvature? The unrestricted version asks too much. The restricted version, proved in the framework's machine-checked library of formal theorems, shows that if the deficit vector lies in a declared subspace, then a recovery matrix exists that reconstructs it exactly from vertex-basis probes. The proof also establishes that recovery implies separation: if two deficit vectors in the subspace produce the same vertex measurements, they are identical. This is the mathematically valid form of the claim that vertex data determine edge geometry, and it is what the module RestrictedIncidenceRecovery establishes.

The key definitions are precise. A deficit subspace is a predicate on edge-deficit vectors, declaring which ones are admissible. Recovering means there exists a matrix that maps vertex-basis probes back to the original deficit for every admissible vector. Separating means that if two admissible vectors agree on all vertex probes, they are equal. The theorem restrictedIncidenceDeficitSeparating_of_recovering proves that recovering implies separating. The library then constructs the natural subspace: the image of the directional length coefficient operator, which maps vertex potentials to edge-length directions. For this subspace, separation is proved directly, giving a concrete geometric condition under which recovery works.

The payoff is a bridge to the vacuum Einstein equations. The module defines discreteVacuumEinsteinInput_of_restrictedRecovery, which packages the restricted recovery assumptions into the input type for the discrete vacuum Einstein equations. This means that if the actual deficit vector lies in the declared subspace, and the separation condition holds, then a critical point of the Regge action with zero deficit at flat space follows. In plain language: under the right geometric restriction, vertex measurements can certify that a lattice is vacuum, with no curvature. This is a machine-checked theorem, not a heuristic.

What this changes is the practical reach of discrete gravity. Full recovery is impossible in bulk lattices, but the restricted version gives a precise, checkable condition under which the inverse problem is solvable. The framework's library proves the implication from recovery to separation, and the construction of the directional-length image subspace gives a natural geometric candidate for the admissible deficits. The remaining physical question is whether the actual deficit vector in a real lattice lies in that subspace, a condition the module explicitly leaves as a hypothesis to be checked for each lattice.

def RestrictedIncidenceDeficitRecovering
    (K : Triangulation3D) (S : DeficitSubspace K) : Prop :=
  ∃ recover : Fin K.nE → Fin K.nV → ℝ,
    ∀ δ : Fin K.nE → ℝ, S δ →
      ∀ e : Fin K.nE,
        δ e =
          ∑ i : Fin K.nV,
            recover e i *
              (∑ e' : Fin K.nE,
                δ e' * directionalLengthCoefficient K
                  (fun j : Fin K.nV => if j = i then (1 : ℝ) else 0) e')

theorem restrictedIncidenceDeficitSeparating_of_recovering
    (K : Triangulation3D) (S : DeficitSubspace K)
    (hRecover : RestrictedIncidenceDeficitRecovering K S) :
    RestrictedIncidenceDeficitSeparating K S := by
  rcases hRecover with ⟨recover, hrecover⟩
  intro δ hS hpair
  funext e
  rw [hrecover δ hS e]
  apply Finset.sum_eq_zero
  intro i _
  rw [hpair (fun j : Fin K.nV => if j = i then (1 : ℝ) else 0)]
  ring

/-- The geometric deficit subspace generated by conformal vertex-potential
edge-length directions.  This is the natural image of the incidence operator
`η ↦ directionalLengthCoefficient K η`. -/
def DirectionalLengthImageSubspace (K : Triangulation3D) : DeficitSubspace K :=
  fun δ => ∃ η : VertexPotential K,
    ∀ e : Fin K.nE, δ e = directionalLengthCoefficient K η e

theorem directionalLengthImageSubspace_separating
    (K : Triangulation3D) :
    RestrictedIncidenceDeficitSeparating K
      (DirectionalLengthImageSubspace K) := by
  intro δ hS hpair
  rcases hS with ⟨η, hη⟩
  have hsum :
      (∑ e : Fin K.nE,
        directionalLengthCoefficient K η e *
          directionalLengthCoefficient K η e) = 0 := by
    simpa [hη] using hpair η
  have hzero_all :
      ∀ e ∈ (Finset.univ : Finset (Fin K.nE)),
        directionalLengthCoefficient K η e *
          directionalLengthCoefficient K η e = 0 := by
    have hnonneg :
        ∀ x ∈ (Finset.univ : Finset (Fin K.nE)),
          0 ≤ directionalLengthCoefficient K η x *
            directionalLengthCoefficient K η x := by
      intro x _
      exact mul_self_nonneg _
    exact (Finset.sum_eq_zero_iff_of_nonneg hnonneg).mp hsum
  funext e
  have hsq := hzero_all e (Finset.mem_univ e)
  have hcoeff : directionalLengthCoefficient K η e = 0 := by
    exact mul_self_eq_zero.mp hsq
  change δ e = 0
  rw [hη e, hcoeff]

def discreteVacuumEinsteinInput_of_restrictedRecovery
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (h_flat : FlatConfiguration K hK)
    (hFirst : ReggeActionFirstVariationInput K hK h_flat)
    (S : DeficitSubspace K)
    (hDeficitInS : S (fun e : Fin K.nE => deficitAngle K (zeroPotential K) e))
    (hFormula : ReggeFirstVariationFormula K hK)
    (hSep : RestrictedIncidenceDeficitSeparating K S) :
    DiscreteVacuumEinsteinInput K hK h_flat where
  critical_iff_zero_deficit := by
    constructor
    · intro hCrit
      exact zero_deficit_of_critical_of_restrictedVariationFormula
        K hK S hDeficitInS hFormula hSep hCrit
    · intro _hZero
      exact hFirst.firstVariation_zero

What this page does not claim

Full recovery of arbitrary edge deficits from vertex probes is possible. The actual deficit vector for a physical lattice is always in the directional-length image subspace. The module proves the discrete vacuum Einstein equations hold; it only constructs an input type under additional hypotheses.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/RestrictedIncidenceRecovery.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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