Encyclopedia Gravity Gravity Seven Gaps Three Pent Causal Consistency Three Pent Causal Assignment

ARTICLE 5 claims 5 theorems

Gravity Seven Gaps Three Pent Causal Consistency Three Pent Causal Assignment

A machine-checked proof shows three standard causal building blocks can be glued around a shared hinge, closing one gap in a larger quantum-gravity program.

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The causal assignment

In causal set theory and related discrete approaches to quantum gravity, spacetime is built from simple pieces, and the rules for gluing them must be consistent. The declaration threePent_causal_assignment is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It proves that a specific geometric configuration, three pentagonal building blocks arranged around a common interior hinge, can be assigned edge lengths that make every block a valid causal simplex, a standard discrete analogue of a Lorentzian spacetime region.

The configuration has six vertices. Three form the central hinge triangle on one time slice; the other three form a separate link cycle on the next time slice. The assignment is a single global rule: edges within the same slice get a positive squared length a², and edges crossing between slices get a negative squared length -αa². The negative sign is what makes the cross-slice edges timelike, the discrete version of lying inside a light cone. The theorem states that on this assignment, each of the three pents is exactly the standard Lorentzian (3,2) simplex, meaning it has three spacelike edges and two timelike edges in the expected pattern.

The proof also verifies the geometric conditions that make each pent physically admissible. Before a Wick rotation, a standard calculation gives a negative Cayley-Menger determinant, the expected signature for a Lorentzian simplex. After the Wick rotation, which turns timelike edges into spacelike ones, the same determinant becomes positive for α > 7/12, the exact range used in the framework's causal simplex layer. This is the condition that the Euclideanized pent is a genuine nondegenerate 4-simplex. At the specific point a = 1 and α = 1, each pent rotates to the regular unit 4-simplex, a concrete check that the assignment is not vacuous.

In Recognition Science, this result closes the existential reading of a specific gap in the seven-gaps campaign, a program to build a discrete theory of gravity. The gap was whether any consistent causal edge-length assignment existed on this minimal hinge complex. The theorem provides one explicitly, so the branch of the program that would have certified non-existence does not fire. The action-level continuation, which would compute the actual gravitational action around the hinge, remains a separate open target.

The theorem does not claim to classify all possible assignments. It realizes the symmetric standard slab, where lengths depend only on whether an edge is within a slice or crosses one. Asymmetric assignments around the hinge cycle, and any obstruction theory for non-standard per-pent data, are not addressed. The dihedral angle values around the hinge, which would be needed for the action, are also explicitly out of scope. The proof itself is axiom-clean, using only the standard trio of axioms from the ambient type theory, with no additional assumptions.

THEOREM threePent_causal_assignment · IndisputableMonolith/Gravity/SevenGaps/ThreePentCausalConsistency.lean
threePent_causal_assignment · IndisputableMonolith/Gravity/SevenGaps/ThreePentCausalConsistency.lean:310 
/-- **GAP6-A HEADLINE (THEOREM): an explicit admissible causal
edge-length assignment on the minimal three-pent interior-hinge complex
EXISTS.**  On the exact 4d CDT range (`0 < a`, `alpha > 7/12`), the ONE
global assignment `causalSqLength` presents all three pents
simultaneously as standard Lorentzian (3,2) simplices (consistency
core), members of the Lorentzian causal class, with strict Lorentzian
CM negativity and Euclidean CM admissibility after Wick, per pent.
EXISTENCE-ONLY SCOPE: this realizes the symmetric standard CDT slab
assignment; it does not classify asymmetric assignments or hinge-cycle
monodromy.  The certified-non-existence branch of the W3-2 lane does
not fire; gap6-b may proceed against this concrete object. -/
theorem threePent_causal_assignment (a alpha : ℝ) (ha : 0 < a)
    (halpha : 7 / 12 < alpha) :
    (inducedSqEdges pentAVert a alpha
          = lorentzianSqEdges CausalPentType.threeTwo a alpha
        ∧ inducedSqEdges pentBVert a alpha
          = lorentzianSqEdges CausalPentType.threeTwo a alpha
        ∧ inducedSqEdges pentCVert a alpha
          = lorentzianSqEdges CausalPentType.threeTwo a alpha)
      ∧ (inducedSqEdges pentAVert a alpha
            ∈ LorentzianClass CausalPentType.threeTwo
        ∧ inducedSqEdges pentBVert a alpha
            ∈ LorentzianClass CausalPentType.threeTwo
        ∧ inducedSqEdges pentCVert a alpha
            ∈ LorentzianClass CausalPentType.threeTwo)
      ∧ (cm4 (inducedSqEdges pentAVert a alpha) < 0
        ∧ cm4 (inducedSqEdges pentBVert a alpha) < 0
        ∧ cm4 (inducedSqEdges pentCVert a alpha) < 0)
      ∧ (0 < cm4 (wick CausalPentType.threeTwo
            (inducedSqEdges pentAVert a alpha))
        ∧ 0 < cm4 (wick CausalPentType.threeTwo
            (inducedSqEdges pentBVert a alpha))
        ∧ 0 < cm4 (wick CausalPentType.threeTwo
            (inducedSqEdges pentCVert a alpha))) := by
  have halpha0 : 0 < alpha := lt_trans (by norm_num) halpha
  exact ⟨⟨induced_pentA_eq a alpha, induced_pentB_eq a alpha,
      induced_pentC_eq a alpha⟩,
    threePent_lorentzian_class a alpha ha halpha0,
    threePent_lorentzian_cm4_neg a alpha ha halpha0.le,
    threePent_euclidean_admissible a alpha ha halpha⟩
THEOREM threePent_causal_assignment · IndisputableMonolith/Gravity/SevenGaps/ThreePentCausalConsistency.lean
threePent_causal_assignment · IndisputableMonolith/Gravity/SevenGaps/ThreePentCausalConsistency.lean:310 
/-- **GAP6-A HEADLINE (THEOREM): an explicit admissible causal
edge-length assignment on the minimal three-pent interior-hinge complex
EXISTS.**  On the exact 4d CDT range (`0 < a`, `alpha > 7/12`), the ONE
global assignment `causalSqLength` presents all three pents
simultaneously as standard Lorentzian (3,2) simplices (consistency
core), members of the Lorentzian causal class, with strict Lorentzian
CM negativity and Euclidean CM admissibility after Wick, per pent.
EXISTENCE-ONLY SCOPE: this realizes the symmetric standard CDT slab
assignment; it does not classify asymmetric assignments or hinge-cycle
monodromy.  The certified-non-existence branch of the W3-2 lane does
not fire; gap6-b may proceed against this concrete object. -/
theorem threePent_causal_assignment (a alpha : ℝ) (ha : 0 < a)
    (halpha : 7 / 12 < alpha) :
    (inducedSqEdges pentAVert a alpha
          = lorentzianSqEdges CausalPentType.threeTwo a alpha
        ∧ inducedSqEdges pentBVert a alpha
          = lorentzianSqEdges CausalPentType.threeTwo a alpha
        ∧ inducedSqEdges pentCVert a alpha
          = lorentzianSqEdges CausalPentType.threeTwo a alpha)
      ∧ (inducedSqEdges pentAVert a alpha
            ∈ LorentzianClass CausalPentType.threeTwo
        ∧ inducedSqEdges pentBVert a alpha
            ∈ LorentzianClass CausalPentType.threeTwo
        ∧ inducedSqEdges pentCVert a alpha
            ∈ LorentzianClass CausalPentType.threeTwo)
      ∧ (cm4 (inducedSqEdges pentAVert a alpha) < 0
        ∧ cm4 (inducedSqEdges pentBVert a alpha) < 0
        ∧ cm4 (inducedSqEdges pentCVert a alpha) < 0)
      ∧ (0 < cm4 (wick CausalPentType.threeTwo
            (inducedSqEdges pentAVert a alpha))
        ∧ 0 < cm4 (wick CausalPentType.threeTwo
            (inducedSqEdges pentBVert a alpha))
        ∧ 0 < cm4 (wick CausalPentType.threeTwo
            (inducedSqEdges pentCVert a alpha))) := by
  have halpha0 : 0 < alpha := lt_trans (by norm_num) halpha
  exact ⟨⟨induced_pentA_eq a alpha, induced_pentB_eq a alpha,
      induced_pentC_eq a alpha⟩,
    threePent_lorentzian_class a alpha ha halpha0,
    threePent_lorentzian_cm4_neg a alpha ha halpha0.le,
    threePent_euclidean_admissible a alpha ha halpha⟩
THEOREM induced_pentA_eq · IndisputableMonolith/Gravity/SevenGaps/ThreePentCausalConsistency.lean
induced_pentA_eq · IndisputableMonolith/Gravity/SevenGaps/ThreePentCausalConsistency.lean:186 
/-- **THEOREM (consistency core, pent A): the induced tuple of pent A is
EXACTLY the standard CDT Lorentzian (3,2) tuple.** -/
theorem induced_pentA_eq (a alpha : ℝ) :
    inducedSqEdges pentAVert a alpha =
      lorentzianSqEdges CausalPentType.threeTwo a alpha := by
  funext e
  fin_cases e <;> rfl
THEOREM threePent_euclidean_admissible · IndisputableMonolith/Gravity/SevenGaps/ThreePentCausalConsistency.lean
threePent_euclidean_admissible · IndisputableMonolith/Gravity/SevenGaps/ThreePentCausalConsistency.lean:235 
/-- THEOREM (per-pent Euclidean admissibility): on the EXACT 4d CDT
range `alpha > 7/12` (with `0 < a`), the Wick image of every pent
satisfies the Cayley-Menger positivity criterion `cm4 > 0` — all three
pents Euclideanize to nondegenerate 4-simplices simultaneously. -/
theorem threePent_euclidean_admissible (a alpha : ℝ) (ha : 0 < a)
    (halpha : 7 / 12 < alpha) :
    0 < cm4 (wick CausalPentType.threeTwo (inducedSqEdges pentAVert a alpha))
      ∧ 0 < cm4 (wick CausalPentType.threeTwo
          (inducedSqEdges pentBVert a alpha))
      ∧ 0 < cm4 (wick CausalPentType.threeTwo
          (inducedSqEdges pentCVert a alpha)) := by
  rw [induced_pentA_eq, induced_pentB_eq, induced_pentC_eq]
  have h := wick_lorentzian_nondegenerate CausalPentType.threeTwo a alpha ha
    (by rw [alphaMin_threeTwo]; exact halpha)
  exact ⟨h, h, h⟩
THEOREM threePent_causal_assignment · IndisputableMonolith/Gravity/SevenGaps/ThreePentCausalConsistency.lean
threePent_causal_assignment · IndisputableMonolith/Gravity/SevenGaps/ThreePentCausalConsistency.lean:310 
/-- **GAP6-A HEADLINE (THEOREM): an explicit admissible causal
edge-length assignment on the minimal three-pent interior-hinge complex
EXISTS.**  On the exact 4d CDT range (`0 < a`, `alpha > 7/12`), the ONE
global assignment `causalSqLength` presents all three pents
simultaneously as standard Lorentzian (3,2) simplices (consistency
core), members of the Lorentzian causal class, with strict Lorentzian
CM negativity and Euclidean CM admissibility after Wick, per pent.
EXISTENCE-ONLY SCOPE: this realizes the symmetric standard CDT slab
assignment; it does not classify asymmetric assignments or hinge-cycle
monodromy.  The certified-non-existence branch of the W3-2 lane does
not fire; gap6-b may proceed against this concrete object. -/
theorem threePent_causal_assignment (a alpha : ℝ) (ha : 0 < a)
    (halpha : 7 / 12 < alpha) :
    (inducedSqEdges pentAVert a alpha
          = lorentzianSqEdges CausalPentType.threeTwo a alpha
        ∧ inducedSqEdges pentBVert a alpha
          = lorentzianSqEdges CausalPentType.threeTwo a alpha
        ∧ inducedSqEdges pentCVert a alpha
          = lorentzianSqEdges CausalPentType.threeTwo a alpha)
      ∧ (inducedSqEdges pentAVert a alpha
            ∈ LorentzianClass CausalPentType.threeTwo
        ∧ inducedSqEdges pentBVert a alpha
            ∈ LorentzianClass CausalPentType.threeTwo
        ∧ inducedSqEdges pentCVert a alpha
            ∈ LorentzianClass CausalPentType.threeTwo)
      ∧ (cm4 (inducedSqEdges pentAVert a alpha) < 0
        ∧ cm4 (inducedSqEdges pentBVert a alpha) < 0
        ∧ cm4 (inducedSqEdges pentCVert a alpha) < 0)
      ∧ (0 < cm4 (wick CausalPentType.threeTwo
            (inducedSqEdges pentAVert a alpha))
        ∧ 0 < cm4 (wick CausalPentType.threeTwo
            (inducedSqEdges pentBVert a alpha))
        ∧ 0 < cm4 (wick CausalPentType.threeTwo
            (inducedSqEdges pentCVert a alpha))) := by
  have halpha0 : 0 < alpha := lt_trans (by norm_num) halpha
  exact ⟨⟨induced_pentA_eq a alpha, induced_pentB_eq a alpha,
      induced_pentC_eq a alpha⟩,
    threePent_lorentzian_class a alpha ha halpha0,
    threePent_lorentzian_cm4_neg a alpha ha halpha0.le,
    threePent_euclidean_admissible a alpha ha halpha⟩

What this page does not claim

This theorem does not compute the dihedral angle values around the hinge, which are needed for the gravitational action. This theorem does not classify all possible causal edge-length assignments on the complex, only the symmetric standard one. This theorem does not prove the classical equivalence between the Cayley-Menger determinant condition and embeddability in R^4. This theorem does not touch any FullTheoryLedger flag or the gap6-b action-level continuation.

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

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