Gravity Seven Gaps Wick Three Two Hinges Product Form Crossing Three Two Upper

A crossing that rules out a shortcut

In attempts to quantize gravity, physicists often work with a 4-simplex, the four-dimensional analog of a triangle, and try to compute the angles between its faces. A natural shortcut is to multiply two simpler quantities together, a product form, hoping it gives the correct angle. The declaration product_form_crossing_threeTwo_upper is a machine-checked proof that this shortcut fails for a specific class of these simplices, at a specific, exactly computable point.

The object in question is the "three-two" causal 4-simplex, a configuration with three points in one slice of time and two in another. The proof shows that at a parameter value corresponding to a time fraction of exactly 2/3, the product of two intermediate expressions equals -48. This is a negative result: it demonstrates that the product form crosses a branch cut, a discontinuity in the complex mathematical structure, and therefore cannot be a valid universal formula. The value -48 is not an arbitrary number; it is the precise point where the shortcut provably breaks down.

This result is part of a broader campaign to understand how angles in quantum gravity behave. The declaration is a kill certificate, a formal record that a particular approach is ruled out. It complements a similar finding for a different class of hinges, where the product equals -40 at a different point. Together, these certificates map out the boundaries of what is mathematically possible, preventing future work from pursuing a known dead end.

What the declaration does not claim is equally important. It does not claim that the entire approach to quantum gravity is invalid, only that this one product-form shortcut fails. It does not provide an alternative formula that works. The proof is silent on the deeper question of whether the angles themselves are physically meaningful; it only establishes a fact about the intermediate mathematical expressions. The action-level continuation, which would connect these angles to a full physical theory, remains explicitly open and unproven.

/-- THEOREM (exact crossing value, upper-pair class): at `t = 2/3` the
product-form denominator argument `(8z - 4)^2 = 16 (2z - 1)^2` equals
`-48` EXACTLY. -/
theorem product_form_crossing_value_upper :
    (8 * zArc (2 / 3) - 4) * (8 * zArc (2 / 3) - 4) = -48 := by
  have hz : zArc (2 / 3) = ((1 / 2 : ℝ) : ℂ)
      + ((Real.sin (Real.pi * (1 - 2 / 3)) : ℝ) : ℂ) * Complex.I := by
    rw [zArc_eq_exp, Complex.exp_mul_I, ← Complex.ofReal_cos,
      ← Complex.ofReal_sin, cos_arg_twoThirds]
  have h84 : 8 * zArc (2 / 3) - 4
      = 8 * ((Real.sin (Real.pi * (1 - 2 / 3)) : ℝ) : ℂ) * Complex.I := by
    rw [hz]
    push_cast
    ring
  have hprod : (8 * zArc (2 / 3) - 4) * (8 * zArc (2 / 3) - 4)
      = -(64 : ℂ) * ((Real.sin (Real.pi * (1 - 2 / 3)) : ℝ) : ℂ) ^ 2 := by
    rw [h84]
    linear_combination
      (64 * ((Real.sin (Real.pi * (1 - 2 / 3)) : ℝ) : ℂ) ^ 2)
        * Complex.I_sq
  have hs2 : Real.sin (Real.pi * (1 - 2 / 3)) ^ 2 = 3 / 4 := by
    rw [Real.sin_sq, cos_arg_twoThirds]
    norm_num
  have hcast : ((Real.sin (Real.pi * (1 - 2 / 3)) : ℝ) : ℂ) ^ 2
      = ((3 / 4 : ℝ) : ℂ) := by
    rw [← Complex.ofReal_pow, hs2]
  rw [hprod, hcast]
  push_cast
  norm_num

/-- THEOREM (exact crossing value, upper-pair class): at `t = 2/3` the
product-form denominator argument `(8z - 4)^2 = 16 (2z - 1)^2` equals
`-48` EXACTLY. -/
theorem product_form_crossing_value_upper :
    (8 * zArc (2 / 3) - 4) * (8 * zArc (2 / 3) - 4) = -48 := by
  have hz : zArc (2 / 3) = ((1 / 2 : ℝ) : ℂ)
      + ((Real.sin (Real.pi * (1 - 2 / 3)) : ℝ) : ℂ) * Complex.I := by
    rw [zArc_eq_exp, Complex.exp_mul_I, ← Complex.ofReal_cos,
      ← Complex.ofReal_sin, cos_arg_twoThirds]
  have h84 : 8 * zArc (2 / 3) - 4
      = 8 * ((Real.sin (Real.pi * (1 - 2 / 3)) : ℝ) : ℂ) * Complex.I := by
    rw [hz]
    push_cast
    ring
  have hprod : (8 * zArc (2 / 3) - 4) * (8 * zArc (2 / 3) - 4)
      = -(64 : ℂ) * ((Real.sin (Real.pi * (1 - 2 / 3)) : ℝ) : ℂ) ^ 2 := by
    rw [h84]
    linear_combination
      (64 * ((Real.sin (Real.pi * (1 - 2 / 3)) : ℝ) : ℂ) ^ 2)
        * Complex.I_sq
  have hs2 : Real.sin (Real.pi * (1 - 2 / 3)) ^ 2 = 3 / 4 := by
    rw [Real.sin_sq, cos_arg_twoThirds]
    norm_num
  have hcast : ((Real.sin (Real.pi * (1 - 2 / 3)) : ℝ) : ℂ) ^ 2
      = ((3 / 4 : ℝ) : ℂ) := by
    rw [← Complex.ofReal_pow, hs2]
  rw [hprod, hcast]
  push_cast
  norm_num

What this page does not claim

This result does not invalidate the broader approach to quantum gravity. This result does not provide a working alternative formula for the angles. This result does not establish the physical meaning of the angles themselves.

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

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