Encyclopedia Gravity Gravity Seven Gaps Wick Hinge Data Complete Wick Product Form Kills Memorialized

ARTICLE 4 claims 4 theorems

Gravity Seven Gaps Wick Hinge Data Complete Wick Product Form Kills Memorialized

A machine-checked theorem records two exact numerical failures in a proposed transcription of a Wick rotation, preserving them as permanent warnings rather than erasing them.

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The product-form kill certificates

In quantum gravity research, a Wick rotation is a mathematical bridge that lets physicists work with a Euclidean (positive-definite) version of a problem and then continue the results back to the Lorentzian (real-time) setting. The Recognition Science framework's machine-checked library of formal theorems contains a declaration called wick_product_form_kills_memorialized. It records two exact numerical failures in a proposed transcription of that bridge.

The two failures are precise. At an interior point of the continuation arc, a certain diagonal-cofactor product lands exactly on the branch cut of the complex square root, with value -40. At another interior point, the same product equals -48, again sitting on the cut. These are not approximations; the theorem states them as exact equalities. The declaration's name reflects its purpose: it memorializes, or permanently records, these two failures so that no future work can silently reintroduce the flawed transcription.

The theorem is a kill certificate: it certifies that a particular single-sqrt product transcription is dead. The repaired convention, a split-sqrt form, is what the framework uses instead. The declaration does not itself prove that the repaired convention works; it only documents the failure of the old one.

What the declaration does not claim is as important as what it does. It does not claim that the action-level continuation, the full Regge action itself, is complete. That remains open, requiring a genuine three-or-more-pent interior-hinge simplicial complex. It does not change any FullTheoryLedger flag, a framework-internal record of which theoretical results are closed. The endpoint values are exact because each split cosine is proved continuous on the closed interval, but no unrestricted equality with the real Lorentzian formula is claimed.

THEOREM wick_product_form_kills_memorialized · IndisputableMonolith/Gravity/SevenGaps/WickHingeDataComplete.lean
wick_product_form_kills_memorialized · IndisputableMonolith/Gravity/SevenGaps/WickHingeDataComplete.lean:149 
/-- THEOREM (B3, kill certificates memorialized): the two product-form
gate FAIL events as one kernel statement: at interior arc parameters the
diagonal-cofactor PRODUCT sits ON the `csqrt` branch cut with the exact
values `-40` (mixed class, `tStarMixed`) and `-48` (upper-pair class,
`t = 2/3` exactly).  The single-sqrt product transcription stays KILLED;
the split-sqrt form is the repaired convention. -/
theorem wick_product_form_kills_memorialized :
    (tStarMixed ∈ Set.Ioo (0 : ℝ) 1
      ∧ cmCofactorC (continuationEdgesC CausalPentType.threeTwo 1 1 tStarMixed) 1 1
          * cmCofactorC (continuationEdgesC CausalPentType.threeTwo 1 1 tStarMixed) 4 4
          = -40
      ∧ (-40 : ℂ) ∉ Complex.slitPlane)
      ∧ ((2 / 3 : ℝ) ∈ Set.Ioo (0 : ℝ) 1
      ∧ cmCofactorC (continuationEdgesC CausalPentType.threeTwo 1 1 (2 / 3)) 1 1
          * cmCofactorC (continuationEdgesC CausalPentType.threeTwo 1 1 (2 / 3)) 2 2
          = -48
      ∧ (-48 : ℂ) ∉ Complex.slitPlane) :=
  ⟨product_form_crossing_threeTwo_mixed, product_form_crossing_threeTwo_upper⟩
THEOREM wick_product_form_kills_memorialized · IndisputableMonolith/Gravity/SevenGaps/WickHingeDataComplete.lean
wick_product_form_kills_memorialized · IndisputableMonolith/Gravity/SevenGaps/WickHingeDataComplete.lean:149 
/-- THEOREM (B3, kill certificates memorialized): the two product-form
gate FAIL events as one kernel statement: at interior arc parameters the
diagonal-cofactor PRODUCT sits ON the `csqrt` branch cut with the exact
values `-40` (mixed class, `tStarMixed`) and `-48` (upper-pair class,
`t = 2/3` exactly).  The single-sqrt product transcription stays KILLED;
the split-sqrt form is the repaired convention. -/
theorem wick_product_form_kills_memorialized :
    (tStarMixed ∈ Set.Ioo (0 : ℝ) 1
      ∧ cmCofactorC (continuationEdgesC CausalPentType.threeTwo 1 1 tStarMixed) 1 1
          * cmCofactorC (continuationEdgesC CausalPentType.threeTwo 1 1 tStarMixed) 4 4
          = -40
      ∧ (-40 : ℂ) ∉ Complex.slitPlane)
      ∧ ((2 / 3 : ℝ) ∈ Set.Ioo (0 : ℝ) 1
      ∧ cmCofactorC (continuationEdgesC CausalPentType.threeTwo 1 1 (2 / 3)) 1 1
          * cmCofactorC (continuationEdgesC CausalPentType.threeTwo 1 1 (2 / 3)) 2 2
          = -48
      ∧ (-48 : ℂ) ∉ Complex.slitPlane) :=
  ⟨product_form_crossing_threeTwo_mixed, product_form_crossing_threeTwo_upper⟩
THEOREM wick_product_form_kills_memorialized · IndisputableMonolith/Gravity/SevenGaps/WickHingeDataComplete.lean
wick_product_form_kills_memorialized · IndisputableMonolith/Gravity/SevenGaps/WickHingeDataComplete.lean:149 
/-- THEOREM (B3, kill certificates memorialized): the two product-form
gate FAIL events as one kernel statement: at interior arc parameters the
diagonal-cofactor PRODUCT sits ON the `csqrt` branch cut with the exact
values `-40` (mixed class, `tStarMixed`) and `-48` (upper-pair class,
`t = 2/3` exactly).  The single-sqrt product transcription stays KILLED;
the split-sqrt form is the repaired convention. -/
theorem wick_product_form_kills_memorialized :
    (tStarMixed ∈ Set.Ioo (0 : ℝ) 1
      ∧ cmCofactorC (continuationEdgesC CausalPentType.threeTwo 1 1 tStarMixed) 1 1
          * cmCofactorC (continuationEdgesC CausalPentType.threeTwo 1 1 tStarMixed) 4 4
          = -40
      ∧ (-40 : ℂ) ∉ Complex.slitPlane)
      ∧ ((2 / 3 : ℝ) ∈ Set.Ioo (0 : ℝ) 1
      ∧ cmCofactorC (continuationEdgesC CausalPentType.threeTwo 1 1 (2 / 3)) 1 1
          * cmCofactorC (continuationEdgesC CausalPentType.threeTwo 1 1 (2 / 3)) 2 2
          = -48
      ∧ (-48 : ℂ) ∉ Complex.slitPlane) :=
  ⟨product_form_crossing_threeTwo_mixed, product_form_crossing_threeTwo_upper⟩
THEOREM action_level_closed_by_v2_elsewhere · IndisputableMonolith/Gravity/SevenGaps/WickHingeDataComplete.lean
action_level_closed_by_v2_elsewhere · IndisputableMonolith/Gravity/SevenGaps/WickHingeDataComplete.lean:168 
/-- Documentation theorem (by `rfl`): hinge-data continuation is not the
action-level closer. The CausalSimplex4D action-level open bit was cleared
2026-07-23 by the V2 terminal (`wick_action_continuation_4d_v2_holds`),
not by this module. -/
theorem action_level_closed_by_v2_elsewhere :
    causalSimplex4DStatus.action_level_continuation_open = false := rfl

What this page does not claim

The declaration does not prove that the action-level continuation is complete. It does not establish the repaired split-sqrt convention as correct. It does not assert any unrestricted equality with the real Lorentzian formula.

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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/WickHingeDataComplete.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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