Cost Ndim Scalar Certificates Nabla P000 Ne Zero

A certificate of non-parallelism

In differential geometry, a vector field is parallel along a curve when its derivative, taken with respect to the geometry's connection, vanishes everywhere on that curve. Parallelism is a strong symmetry: it means the field does not twist or shear as you move. The declaration nablaP000_ne_zero in the framework's machine-checked library of formal theorems establishes that a specific object, the projector P, fails this property decisively. On a two-dimensional slice of the framework's cost geometry, the covariant derivative of one component of P is never zero, for any nonzero value of the slice parameter t. The proof does not rely on a single numerical check. It derives a closed-form scalar expression for the derivative, and then proves that this expression is nonzero for all t in the allowed range.

The closed form itself is a rational function of hyperbolic functions: nablaP000(t) = -sinh(t) * (cosh(t) + 3) / (2 * (cosh(t) + 1)^2 * (cosh(t) + 2)). The numerator is nonzero whenever t is nonzero, because sinh(t) is nonzero for all nonzero t. The denominator is always positive, since cosh(t) + 1 and cosh(t) + 2 are strictly positive for all real t. The theorem therefore holds universally on the slice, not just at an isolated point. This universality is the point of the certificate: it avoids the need to construct a specific point inside the proof assistant, and it rules out parallelism everywhere at once.

The statement is scoped to a particular setting. It concerns the Levi-Civita connection of a metric h that depends on a parameter λ, evaluated at λ = 1 and at a slice point (t, 0). The companion result dP00_ne_zero proves the same nonvanishing for the derivative with respect to the flat connection D. The library also proves a generalized version, nablaP000Gen_ne_zero, for arbitrary nonzero parameters a and b, and a separate theorem R0101Closed_neg showing that the metric's Riemann tensor component is strictly negative, meaning the geometry is non-flat. These results together support a structural claim in the framework's cost theory: the golden and metallic family of potentials is not parallel and not flat in the relevant sense.

What the declaration does not claim is just as important as what it proves. It does not claim that the projector P is non-parallel with respect to every possible connection, only the two named ones. It does not claim that the geometry is non-flat in all components, only that one specific Riemann tensor component is negative. And it does not claim that the closed-form expression is derived from first principles inside the proof assistant. The identification of the scalar formula with the actual tensor component is a separate, hypothesis-level bridge, recorded in the module docstring and not re-derived in this file. The file is the algebraic core: given the closed form, it proves the nonvanishing.

/-- **Theorem 1b** (non-parallelism of `P_λ` w.r.t. the Levi-Civita connection `∇^λ`):
    the covariant derivative of the `(0,0)` entry of `P_λ` is nonzero for every `t ≠ 0`.
    Hence `∇^λ P_λ ≠ 0`, i.e. `P_λ` is not `∇^λ`-parallel. -/
theorem nablaP000_ne_zero (t : ℝ) (ht : t ≠ 0) : nablaP000 t ≠ 0 := by
  unfold nablaP000
  have hsinh_ne : Real.sinh t ≠ 0 := Real.sinh_ne_zero.mpr ht
  have hc3_ne : Real.cosh t + 3 ≠ 0 := ne_of_gt (by linarith [Real.cosh_pos t])
  apply div_ne_zero
  · exact mul_ne_zero (neg_ne_zero.mpr hsinh_ne) hc3_ne
  · have h1 : (0:ℝ) < Real.cosh t + 1 := by linarith [Real.cosh_pos t]
    have h2 : (0:ℝ) < Real.cosh t + 2 := by linarith [Real.cosh_pos t]
    positivity

/-- `(∇^λ)_0 P^0_0` at the slice point, `λ = 1`: the covariant derivative of `P_λ`
    w.r.t. the Levi-Civita connection of `h_λ`. -/
def nablaP000 (t : ℝ) : ℝ :=
  -(Real.sinh t) * (Real.cosh t + 3) / (2 * (Real.cosh t + 1) ^ 2 * (Real.cosh t + 2))

/-- **Theorem 1b** (non-parallelism of `P_λ` w.r.t. the Levi-Civita connection `∇^λ`):
    the covariant derivative of the `(0,0)` entry of `P_λ` is nonzero for every `t ≠ 0`.
    Hence `∇^λ P_λ ≠ 0`, i.e. `P_λ` is not `∇^λ`-parallel. -/
theorem nablaP000_ne_zero (t : ℝ) (ht : t ≠ 0) : nablaP000 t ≠ 0 := by
  unfold nablaP000
  have hsinh_ne : Real.sinh t ≠ 0 := Real.sinh_ne_zero.mpr ht
  have hc3_ne : Real.cosh t + 3 ≠ 0 := ne_of_gt (by linarith [Real.cosh_pos t])
  apply div_ne_zero
  · exact mul_ne_zero (neg_ne_zero.mpr hsinh_ne) hc3_ne
  · have h1 : (0:ℝ) < Real.cosh t + 1 := by linarith [Real.cosh_pos t]
    have h2 : (0:ℝ) < Real.cosh t + 2 := by linarith [Real.cosh_pos t]
    positivity

/-- **Theorem 1b** (non-parallelism of `P_λ` w.r.t. the Levi-Civita connection `∇^λ`):
    the covariant derivative of the `(0,0)` entry of `P_λ` is nonzero for every `t ≠ 0`.
    Hence `∇^λ P_λ ≠ 0`, i.e. `P_λ` is not `∇^λ`-parallel. -/
theorem nablaP000_ne_zero (t : ℝ) (ht : t ≠ 0) : nablaP000 t ≠ 0 := by
  unfold nablaP000
  have hsinh_ne : Real.sinh t ≠ 0 := Real.sinh_ne_zero.mpr ht
  have hc3_ne : Real.cosh t + 3 ≠ 0 := ne_of_gt (by linarith [Real.cosh_pos t])
  apply div_ne_zero
  · exact mul_ne_zero (neg_ne_zero.mpr hsinh_ne) hc3_ne
  · have h1 : (0:ℝ) < Real.cosh t + 1 := by linarith [Real.cosh_pos t]
    have h2 : (0:ℝ) < Real.cosh t + 2 := by linarith [Real.cosh_pos t]
    positivity

What this page does not claim

The projector P is non-parallel with respect to every possible connection, only the flat and Levi-Civita connections named in the file. The geometry is non-flat in all components, only that one specific Riemann tensor component is strictly negative. The closed-form expression is derived from first principles inside the proof assistant; the identification with the tensor component is a separate hypothesis.

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/Cost/Ndim/ScalarCertificates.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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