Cost Ndim Scalar Certificates Nabla P000 Gen Ne Zero

A certificate of non-parallelism

In differential geometry, a projection is a map that sends every point to a chosen subspace, like dropping a shadow onto a flat floor. The declaration nablaP000Gen_ne_zero concerns a particular family of projections built from a potential function Φλ(t₀,t₁) = cosh t₀ + cosh t₁ + λ(cosh(t₀+t₁) − 1). On a two-dimensional slice of the space, the projection Pλ sends vectors onto the diagonal line spanned by (1,1). The theorem states that, for every nonzero parameter t and every positive λ, the covariant derivative of this projection along the slice is never zero. In plain language, the projection is never parallel to the space it projects onto; it always tilts or twists in a way that the derivative detects.

The proof works by computing a single scalar expression, nablaP000Gen a b lam t, that captures the relevant derivative component. The theorem shows this expression is nonzero whenever a and b are nonzero, λ is positive, and t is nonzero. The key ingredients are that the hyperbolic sine sinh t vanishes only at t = 0, and the denominator involves sums of positive terms like cosh t, which are always strictly positive. This makes the certificate universal on the slice: it rules out parallelism at every point at once, rather than checking one isolated location. The same file also proves the companion fact that the metric hλ is non-flat for positive λ, using a similar scalar certificate for the Riemann curvature component.

In Recognition Science, this result supports the structural claim that the golden and metallic family of potentials has genuine geometric content: the projection is not a trivial artifact but a genuinely non-parallel object. The machine-checked library of formal theorems records this as nablaP000Gen_ne_zero, with the proof verified by the kernel. The declaration does not claim that the projection is parallel or flat, nor does it establish anything about the full space beyond the two-dimensional slice. It also does not prove the geometric bridge that identifies the scalar expression with the actual tensor component; that identification is recorded separately as a hypothesis-level fact, not re-derived in this file.

/-- **Theorem 1b, general `α`**: the covariant derivative of the `(0,0)` entry of `P_λ`
    is nonzero for every `t ≠ 0`, `a ≠ 0`, `b ≠ 0`, `λ > 0`. -/
theorem nablaP000Gen_ne_zero (a b lam t : ℝ) (ha : a ≠ 0) (hb : b ≠ 0) (hlam : 0 < lam)
    (ht : t ≠ 0) : nablaP000Gen a b lam t ≠ 0 := by
  unfold nablaP000Gen
  have hkap : 0 < kappaGen a b lam t := kappaGen_pos a b lam t ha hlam
  have hc : 0 < Real.cosh t := Real.cosh_pos t
  have ha2 : (0:ℝ) < a ^ 2 := by positivity
  have hb2 : (0:ℝ) < b ^ 2 := by positivity
  have hden_block : (0:ℝ) < a ^ 2 + b ^ 2 * Real.cosh t := by
    have : (0:ℝ) ≤ b ^ 2 * Real.cosh t := by positivity
    linarith
  apply div_ne_zero
  · apply mul_ne_zero
    · exact neg_ne_zero.mpr
        (mul_ne_zero (mul_ne_zero (ne_of_gt ha2) (ne_of_gt hb2)) (Real.sinh_ne_zero.mpr ht))
    · exact ne_of_gt (by linarith)
  · exact ne_of_gt (by positivity)

/-- `(∇^λ)_0 P^0_0` at the slice point, general `α = (a, b)`, general `λ`. -/
def nablaP000Gen (a b lam t : ℝ) : ℝ :=
  -(a ^ 2 * b ^ 2 * Real.sinh t) * (kappaGen a b lam t + Real.cosh t) /
    (2 * (a ^ 2 + b ^ 2 * Real.cosh t) ^ 2 * kappaGen a b lam t)

/-- **Theorem 2, general `α`**: the Riemann tensor component `R^0_{1,0,1}` is strictly
    negative for every `a ≠ 0`, `b ≠ 0`, `λ > 0`, `t ≠ 0`. Hence `h_λ` is non-flat for
    the whole `α = (a, b)` family, not just `α = (1,1)`. Specializes to
    `R0101Closed_neg` at `a = b = 1` (checked symbolically to agree with `R0101Closed`
    there). -/
theorem R0101Gen_neg (a b lam t : ℝ) (ha : a ≠ 0) (hb : b ≠ 0) (hlam : 0 < lam)
    (ht : t ≠ 0) : R0101Gen a b lam t < 0 := by
  unfold R0101Gen
  have hkap : 0 < kappaGen a b lam t := kappaGen_pos a b lam t ha hlam
  have hden : 0 < 4 * kappaGen a b lam t ^ 2 := by positivity
  have hb2 : (0:ℝ) < b ^ 2 := by positivity
  have hfac1 : 0 < b ^ 2 * lam * (b ^ 2 * lam * Real.cosh (a * t) + 1) := by
    have hcoshpos : (0:ℝ) < Real.cosh (a * t) := Real.cosh_pos _
    have : (0:ℝ) < b ^ 2 * lam * Real.cosh (a * t) + 1 := by positivity
    positivity
  have hfac2 : 0 < Real.sinh t * (a * Real.sinh (a * t)) := sinh_cross_pos a t ha ht
  have hnum : -(b ^ 2 * lam * (b ^ 2 * lam * Real.cosh (a * t) + 1)) *
      (Real.sinh t * (a * Real.sinh (a * t))) < 0 := by
    have := mul_pos hfac1 hfac2
    linarith
  exact div_neg_of_neg_of_pos hnum hden

What this page does not claim

The declaration does not prove that the projection is parallel or flat. The declaration does not establish anything about the full space beyond the two-dimensional slice. The declaration does not prove the geometric bridge identifying the scalar expression with the actual tensor component.

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)

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