Encyclopedia Gravity Gravity Seven Gaps Wick Four One All Hinges Branch Regular Four One Spacelike Pa

ARTICLE 3 claims 3 theorems

Gravity Seven Gaps Wick Four One All Hinges Branch Regular Four One Spacelike Pa

A machine-checked proof shows that a specific geometric piece of a four-dimensional spacetime model behaves smoothly, but it does not yet connect that piece to a full theory of gravity.

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The spacelike hinge proof

In the Recognition Science framework's study of gravity, a ledger (a discrete record of events) is used to build a four-dimensional simplex, the simplest shape with five points. This shape has ten triangular faces, called hinges. The declaration branchRegular_fourOne_spacelike_pair concerns the four hinges of this simplex that do not touch the special apex point. These are the spacelike hinges, where the geometry is measured by a positive, real number. The declaration proves that for every one of these four hinges, a certain mathematical function, which describes the geometry as it changes, is well-behaved across its entire range of possible values. This means the function does not jump or become undefined; it is smooth and continuous.

This result is part of a larger proof about a process called Wick continuation, which is a standard technique in physics for connecting a model of space (Euclidean geometry) with a model of spacetime (Lorentzian geometry). The proof shows that for these spacelike hinges, the continuation from the spacelike to the timelike regime is regular. It also shows that the continuation ends at a specific, well-defined value. This is a technical but necessary step: it confirms that this particular piece of the geometric model is consistent and does not contain any mathematical singularities where the description would break down.

The declaration is a theorem in the framework's machine-checked library of formal theorems. This means its statement has been verified by a computer program, and no steps in its proof are missing or assumed. The proof relies on explicitly calculating the determinants of small matrices, which are algebraic objects that encode the geometry of the hinges. These calculations are all checked by the computer. The result is a solid, verified fact about this specific geometric construction.

In Recognition Science, this proof is a milestone. It establishes a rigorous foundation for one part of a much larger goal: deriving a full theory of gravity from the framework's basic principles. The proof shows that the framework can handle a concrete, non-trivial piece of geometry with complete mathematical certainty. This is a step toward showing that the framework's abstract ideas can produce the familiar structures of physics.

However, the declaration does not claim to have completed that larger goal. It does not prove that the entire action, the quantity that describes the total energy of the spacetime model, is well-defined. It only proves a property of the individual hinges. The connection from these individual, verified pieces to a complete, continuous theory of gravity remains an open problem. This proof is a necessary brick, but it is not the whole building.

THEOREM branchRegular_fourOne_allHinges · IndisputableMonolith/Gravity/SevenGaps/WickFourOneAllHinges.lean
branchRegular_fourOne_allHinges · IndisputableMonolith/Gravity/SevenGaps/WickFourOneAllHinges.lean:1011 
/-- THEOREM (B1 headline, branch certificates): for EVERY hinge of the
fourOne causal 4-simplex (every unordered opposite vertex pair, both
orientations), the split-form continuation is branch-regular on the FULL
open arc interior at the physical point `a = 1`, `alpha = 1`.  The ten
hinges: (0,1,2)|(3,4), (0,1,3)|(2,4), (0,1,4)|(2,3), (0,2,3)|(1,4),
(0,2,4)|(1,3), (0,3,4)|(1,2), (1,2,3)|(0,4), (1,2,4)|(0,3),
(1,3,4)|(0,2), (2,3,4)|(0,1). -/
theorem branchRegular_fourOne_allHinges :
    ∀ p q : Fin 5, p ≠ q →
      BranchRegularOn
        (fun t => continuationEdgesC CausalPentType.fourOne 1 1 t) p q
        (Set.Ioo 0 1) := by
  intro p q hpq
  fin_cases p <;> fin_cases q
  · exact absurd rfl hpq
  · exact branchRegular_pair01
  · exact branchRegular_pair02
  · exact branchRegular_pair03
  · exact branchRegular_pair04
  · exact branchRegularOn_symm branchRegular_pair01
  · exact absurd rfl hpq
  · exact branchRegular_pair12
  · exact branchRegular_pair13
  · exact branchRegular_pair14
  · exact branchRegularOn_symm branchRegular_pair02
  · exact branchRegularOn_symm branchRegular_pair12
  · exact absurd rfl hpq
  · exact branchRegular_pair23
  · exact branchRegular_pair24
  · exact branchRegularOn_symm branchRegular_pair03
  · exact branchRegularOn_symm branchRegular_pair13
  · exact branchRegularOn_symm branchRegular_pair23
  · exact absurd rfl hpq
  · exact branchRegular_pair34
  · exact branchRegularOn_symm branchRegular_pair04
  · exact branchRegularOn_symm branchRegular_pair14
  · exact branchRegularOn_symm branchRegular_pair24
  · exact branchRegularOn_symm branchRegular_pair34
  · exact absurd rfl hpq
THEOREM wick_boundary_continuation_fourOne_allHinges · IndisputableMonolith/Gravity/SevenGaps/WickFourOneAllHinges.lean
wick_boundary_continuation_fourOne_allHinges · IndisputableMonolith/Gravity/SevenGaps/WickFourOneAllHinges.lean:1111 
/-- THEOREM (B1 headline, boundary continuation): every split-form cosine
path of the fourOne type is continuous on the CLOSED interval `[0, 1]` and
ends at the Euclidean regular-4-simplex value `-(1/4)` (`+C_pq` numerator
convention of the landed module; textbook `-C` interior cosine `+1/4`). -/
theorem wick_boundary_continuation_fourOne_allHinges :
    ∀ p q : Fin 5, p ≠ q →
      ContinuousOn (fourOneCosPath p q) (Set.Icc 0 1)
        ∧ fourOneCosPath p q 1 = -(1 / 4 : ℂ) := by
  intro p q hpq
  fin_cases p <;> fin_cases q
  · exact absurd rfl hpq
  · exact ⟨boundary_pair01.1, boundary_pair01.2.2⟩
  · exact ⟨boundary_pair02.1, boundary_pair02.2.2⟩
  · exact ⟨boundary_pair03.1, boundary_pair03.2.2⟩
  · exact ⟨boundary_pair04.1, boundary_pair04.2.2⟩
  · exact boundary_symm ⟨boundary_pair01.1, boundary_pair01.2.2⟩
  · exact absurd rfl hpq
  · exact ⟨boundary_pair12.1, boundary_pair12.2.2⟩
  · exact ⟨boundary_pair13.1, boundary_pair13.2.2⟩
  · exact ⟨boundary_pair14.1, boundary_pair14.2.2⟩
  · exact boundary_symm ⟨boundary_pair02.1, boundary_pair02.2.2⟩
  · exact boundary_symm ⟨boundary_pair12.1, boundary_pair12.2.2⟩
  · exact absurd rfl hpq
  · exact ⟨boundary_pair23.1, boundary_pair23.2.2⟩
  · exact ⟨boundary_pair24.1, boundary_pair24.2.2⟩
  · exact boundary_symm ⟨boundary_pair03.1, boundary_pair03.2.2⟩
  · exact boundary_symm ⟨boundary_pair13.1, boundary_pair13.2.2⟩
  · exact boundary_symm ⟨boundary_pair23.1, boundary_pair23.2.2⟩
  · exact absurd rfl hpq
  · exact ⟨boundary_pair34.1, boundary_pair34.2.2⟩
  · exact boundary_symm ⟨boundary_pair04.1, boundary_pair04.2.2⟩
  · exact boundary_symm ⟨boundary_pair14.1, boundary_pair14.2.2⟩
  · exact boundary_symm ⟨boundary_pair24.1, boundary_pair24.2.2⟩
  · exact boundary_symm ⟨boundary_pair34.1, boundary_pair34.2.2⟩
  · exact absurd rfl hpq
THEOREM branchRegular_fourOne_allHinges · IndisputableMonolith/Gravity/SevenGaps/WickFourOneAllHinges.lean
branchRegular_fourOne_allHinges · IndisputableMonolith/Gravity/SevenGaps/WickFourOneAllHinges.lean:1011 
/-- THEOREM (B1 headline, branch certificates): for EVERY hinge of the
fourOne causal 4-simplex (every unordered opposite vertex pair, both
orientations), the split-form continuation is branch-regular on the FULL
open arc interior at the physical point `a = 1`, `alpha = 1`.  The ten
hinges: (0,1,2)|(3,4), (0,1,3)|(2,4), (0,1,4)|(2,3), (0,2,3)|(1,4),
(0,2,4)|(1,3), (0,3,4)|(1,2), (1,2,3)|(0,4), (1,2,4)|(0,3),
(1,3,4)|(0,2), (2,3,4)|(0,1). -/
theorem branchRegular_fourOne_allHinges :
    ∀ p q : Fin 5, p ≠ q →
      BranchRegularOn
        (fun t => continuationEdgesC CausalPentType.fourOne 1 1 t) p q
        (Set.Ioo 0 1) := by
  intro p q hpq
  fin_cases p <;> fin_cases q
  · exact absurd rfl hpq
  · exact branchRegular_pair01
  · exact branchRegular_pair02
  · exact branchRegular_pair03
  · exact branchRegular_pair04
  · exact branchRegularOn_symm branchRegular_pair01
  · exact absurd rfl hpq
  · exact branchRegular_pair12
  · exact branchRegular_pair13
  · exact branchRegular_pair14
  · exact branchRegularOn_symm branchRegular_pair02
  · exact branchRegularOn_symm branchRegular_pair12
  · exact absurd rfl hpq
  · exact branchRegular_pair23
  · exact branchRegular_pair24
  · exact branchRegularOn_symm branchRegular_pair03
  · exact branchRegularOn_symm branchRegular_pair13
  · exact branchRegularOn_symm branchRegular_pair23
  · exact absurd rfl hpq
  · exact branchRegular_pair34
  · exact branchRegularOn_symm branchRegular_pair04
  · exact branchRegularOn_symm branchRegular_pair14
  · exact branchRegularOn_symm branchRegular_pair24
  · exact branchRegularOn_symm branchRegular_pair34
  · exact absurd rfl hpq

What this page does not claim

The declaration does not prove the full action-level continuation of the entire spacetime model. It does not claim that the framework has derived a complete theory of gravity. It does not claim that the spacelike hinge result extends to the timelike hinges without further proof.

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/SevenGaps/WickFourOneAllHinges.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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