Gravity Seven Gaps Wick Action Complex First Wick Boundary Continuation Four One

The boundary continuation theorem

In the framework's study of discrete gravity, a ledger (a discrete record of geometric events) tracks the squared lengths of edges in a four-dimensional simplex. The declaration wick_boundary_continuation_fourOne_hinge concerns one specific quantity: the cosine of the dihedral angle at a triangular hinge, the shared face of two tetrahedra, within a particular causal 4-simplex. The result proves that this cosine, when viewed as a function of a complex parameter moving along a semicircular arc in the upper half-plane, is continuous on the entire closed interval from the Lorentzian endpoint to the Euclidean endpoint.

The arc is parameterized by a real number t from 0 to 1, with t=0 giving the Lorentzian squared edge length -1 and t=1 giving the Euclidean value +1. The result establishes the path connects the value -3/8 at t=0 to -1/4 at t=1. The proof works by showing the cosine equals a simple rational function, (1-2z)/(6z-2), along the whole arc, except at a single point where the denominator vanishes. This rational form is continuous everywhere the denominator is nonzero, and the result verifies the endpoints and interior behavior explicitly.

The importance of this result lies in what it makes possible: a well-defined, branch-free continuation of a hinge quantity between two physical regimes. The proof is machine-checked, meaning every step is verified by a computer proof assistant, so the continuity claim and endpoint values are not subject to human calculation error. This is a necessary technical step for the larger program of defining a full quantum gravity action through a complex continuation.

What the result does not claim is equally precise. It does not establish the existence of a full action-level continuation for the entire simplicial complex, which would require additional interior hinges and deficit angles. That remains an open problem in the framework. The result also does not claim an unrestricted equality with the real Lorentzian formula at the endpoint; a sign factor of -1 appears due to the complex square root convention, and this is documented separately. The result is a boundary continuation for a single hinge, not the whole action.

/-- THEOREM: Lorentzian endpoint value of the SPLIT form: `-(3/8)`.
(The real product-form formula gives `+3/8` here; see
`lorentzian_endpoint_sign_factor` for the documented sign factor.) -/
theorem hingeCosPath_zero : hingeCosPath 0 = -(3 / 8 : ℂ) := by
  rw [hingeCosPath_eq_moebius, zArc_zero]
  norm_num

/-- THEOREM: Euclidean endpoint value: `-(1/4)`, the regular unit
4-simplex value in the `+C_pq` convention (textbook `-C` interior dihedral
cosine `+1/4`; RESULTS.txt endpoint table). -/
theorem hingeCosPath_one : hingeCosPath 1 = -(1 / 4 : ℂ) := by
  rw [hingeCosPath_eq_moebius, zArc_one]
  norm_num

/-- The split-form cosine path of the traced hinge along the physical
arc: `t ↦ C_pq / (csqrt C_pp * csqrt C_qq)` at hinge `(0,1,4)`, opposite
pair `(2,3)`, type fourOne, `a = alpha = 1`. -/
noncomputable def hingeCosPath (t : ℝ) : ℂ :=
  dihedralCosSplitC (continuationEdgesC CausalPentType.fourOne 1 1 t) 2 3

What this page does not claim

The result does not establish the existence of a full action-level continuation for the entire simplicial complex. The result does not claim an unrestricted equality with the real Lorentzian formula at the endpoint. The result does not prove the continuity of the cosine path for all possible hinge types in the 4-simplex.

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

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