Encyclopedia Gravity Gravity Seven Gaps Wick Four One All Hinges Wick Boundary Continuation Four One
ARTICLE 5 claims 5 theorems
Gravity Seven Gaps Wick Four One All Hinges Wick Boundary Continuation Four One
A machine-checked proof shows that all ten hinges of a special curved spacetime building block connect smoothly to its Euclidean counterpart, with no gaps at the boundary.
The all-hinge boundary theorem
In the Recognition Science framework's study of quantum gravity, a fundamental object is the four-dimensional simplex, the simplest curved spacetime cell. The declaration wick_boundary_continuation_fourOne_allHinges proves a precise property about one specific type of this cell, called the fourOne causal 4-simplex. This cell has a special vertex, the apex, with timelike edges to the other four vertices, and spacelike edges among those four. The theorem concerns a procedure called Wick continuation, which rotates time into imaginary time to connect a Lorentzian spacetime (with its familiar time and space separation) to a Euclidean one (where all directions are on equal footing). This rotation is a standard tool in physics for making sense of quantum amplitudes.
The theorem states that for every one of the ten triangular hinges of this simplex, the mathematical path that describes this rotation is continuous on the entire closed interval from the starting point to the end point. More specifically, each path ends at the value -(1/4), which is the known value for the regular Euclidean 4-simplex. This means there is no singularity or discontinuity in the rotation process for any of the hinges. The proof is fully formal and machine-checked, meaning it has been verified by a computer to follow logically from the definitions, with no unproven assumptions. This extends an earlier result that covered only a single hinge to all ten, at a specific physical configuration where the scale factor and a coupling constant are both set to 1.
The theorem also establishes the endpoint values separately for the two classes of hinges. For the six hinges that contain the apex vertex, the Lorentzian endpoint value is -(3/8). For the four hinges that do not contain the apex, the endpoint is the purely imaginary number (sqrt 2 / 8) * i. This imaginary value is a known feature of the geometry at a spacelike hinge in a Lorentzian setting. The proof also provides closed-form expressions for the areas squared of all ten hinges, showing they avoid a problematic branch cut in the complex plane during the interior of the path, with a specific allowed contact at the endpoint.
In Recognition Science, this result is a step toward a larger goal: defining a full quantum gravity action that sums over all possible spacetime configurations. The theorem proves the boundary behavior of the building blocks is well-behaved. However, it does not claim to have completed that larger action. The action-level continuation, which would involve the full interior structure of the simplex, including deficit angles and the continued Regge action itself, remains an open problem. The theorem is a certificate about the boundary of a single cell, not a complete theory of quantum gravity.
THEOREM wick_boundary_continuation_fourOne_allHinges · IndisputableMonolith/Gravity/SevenGaps/WickFourOneAllHinges.lean
/-- 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 wick_boundary_continuation_fourOne_allHinges · IndisputableMonolith/Gravity/SevenGaps/WickFourOneAllHinges.lean
/-- 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 wick_boundary_continuation_fourOne_allHinges · IndisputableMonolith/Gravity/SevenGaps/WickFourOneAllHinges.lean
/-- 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 det_minor41_45C · IndisputableMonolith/Gravity/SevenGaps/WickFourOneAllHinges.lean
theorem det_minor41_45C (z : ℂ) : Matrix.det (minor41_45C z) = 1 := by
unfold minor41_45C
simp [Matrix.det_succ_row_zero, Fin.sum_univ_succ, Fin.succAbove]
ring
THEOREM det_minor41_45C · IndisputableMonolith/Gravity/SevenGaps/WickFourOneAllHinges.lean
theorem det_minor41_45C (z : ℂ) : Matrix.det (minor41_45C z) = 1 := by
unfold minor41_45C
simp [Matrix.det_succ_row_zero, Fin.sum_univ_succ, Fin.succAbove]
ring
What this page does not claim
The theorem does not claim to have completed the action-level continuation for the full interior of the simplex. The theorem does not claim to have defined the continued Regge action itself, only the boundary behavior of the cosine paths. The theorem does not claim any physical prediction about quantum gravity, only a mathematical property of a geometric building block.
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.
Derived articles
This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red:
- What is the full continued Regge action for the fourOne simplex, including all interior hinges and deficit angles?
- How does the boundary continuation result extend to other types of causal simplices beyond the fourOne type?
- What physical predictions follow from the imaginary Lorentzian dihedral datum at a spacelike hinge?
- How does the all-hinge result contribute to the larger goal of defining a complete quantum gravity path integral in the framework?
MACHINE LAYER · GROUNDED CLAIM TABLE · CLICK TO EXPAND
THEOREM wick_boundary_continuation_fourOne_allHinges · IndisputableMonolith/Gravity/SevenGaps/WickFourOneAllHinges.lean
/-- 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 hpqThe declaration wick_boundary_continuation_fourOne_allHinges proves a precise property about one specific type of this cell, called the fourOne causal 4-simplex. wick_boundary_continuation_fourOne_allHinges · IndisputableMonolith/Gravity/SevenGaps/WickFourOneAllHinges.leanTHEOREM wick_boundary_continuation_fourOne_allHinges · IndisputableMonolith/Gravity/SevenGaps/WickFourOneAllHinges.lean
/-- 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 hpqThe theorem states that for every one of the ten triangular hinges of this simplex, the mathematical path that describes this rotation is continuous on the entire closed interval from the starting point to the end point. wick_boundary_continuation_fourOne_allHinges · IndisputableMonolith/Gravity/SevenGaps/WickFourOneAllHinges.leanTHEOREM wick_boundary_continuation_fourOne_allHinges · IndisputableMonolith/Gravity/SevenGaps/WickFourOneAllHinges.lean
/-- 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 hpqMore specifically, each path ends at the value -(1/4), which is the known value for the regular Euclidean 4-simplex. wick_boundary_continuation_fourOne_allHinges · IndisputableMonolith/Gravity/SevenGaps/WickFourOneAllHinges.leanTHEOREM det_minor41_45C · IndisputableMonolith/Gravity/SevenGaps/WickFourOneAllHinges.lean
theorem det_minor41_45C (z : ℂ) : Matrix.det (minor41_45C z) = 1 := by unfold minor41_45C simp [Matrix.det_succ_row_zero, Fin.sum_univ_succ, Fin.succAbove] ringFor the six hinges that contain the apex vertex, the Lorentzian endpoint value is -(3/8). det_minor41_45C · IndisputableMonolith/Gravity/SevenGaps/WickFourOneAllHinges.leanTHEOREM det_minor41_45C · IndisputableMonolith/Gravity/SevenGaps/WickFourOneAllHinges.lean
theorem det_minor41_45C (z : ℂ) : Matrix.det (minor41_45C z) = 1 := by unfold minor41_45C simp [Matrix.det_succ_row_zero, Fin.sum_univ_succ, Fin.succAbove] ringFor the four hinges that do not contain the apex, the endpoint is the purely imaginary number (sqrt 2 / 8) * i. det_minor41_45C · IndisputableMonolith/Gravity/SevenGaps/WickFourOneAllHinges.lean