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785c727 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 | import Mathlib.Analysis.Normed.Module.RCLike.Basic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
import Lean.Elab.Tactic.Omega
namespace OAI
/-!
# Algebraic final steps for positively curved Einstein four-manifolds
These are the linear-algebra and characteristic-number deductions used in
Sections 1 and 8 of *Positively curved Einstein four-manifolds* (2026).
The geometric hypotheses needed to apply them are kept explicit.
-/
namespace PositiveEinsteinFour
/-- An involution with no nonzero fixed vector is the antipodal linear map.
The proof uses the fixed vector `A x + x`; orthogonality and a spectral theorem
are unnecessary once the sphere action has been linearized. -/
theorem linear_involution_eq_neg_id
{V : Type*} [AddCommGroup V] [Module ℝ V]
(A : V →ₗ[ℝ] V)
(hinvolution : ∀ x, A (A x) = x)
(hfixed : ∀ x, A x = x → x = 0) :
A = -LinearMap.id := by
ext x
have hsum : A (A x + x) = A x + x := by
rw [map_add, hinvolution, add_comm]
have hzero := hfixed (A x + x) hsum
simpa using eq_neg_of_add_eq_zero_left hzero
/-- A linear involution acting freely on the unit sphere is antipodal. -/
theorem linear_involution_eq_neg_id_of_sphere_free
{V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
(A : V →ₗ[ℝ] V)
(hinvolution : ∀ x, A (A x) = x)
(hfree : ∀ x, ‖x‖ = 1 → A x ≠ x) :
A = -LinearMap.id := by
apply linear_involution_eq_neg_id A hinvolution
intro x hfixed
by_contra hx
apply hfree ((‖x‖⁻¹ : ℝ) • x) (norm_smul_inv_norm hx)
rw [map_smul, hfixed]
/-- The characteristic identities, the Weyl gap, and the strict lower volume
bound in the exceptional moment-three case force the sole surviving harmonic
count to be one. Volume is divided by `π²`; `moment` denotes the normalized
second moment of the nonzero block. This is Proposition 8.2's numerical step. -/
theorem remaining_harmonic_count
(n : ℕ) (volume moment : ℝ)
(hvolume_pos : 0 < volume)
(hgap : 1 ≤ moment)
(hmoment_lt : moment < 4)
(hcharacteristic : volume = (8 - 2 * (n : ℝ)) / 3)
(hsignature : moment * volume = 2 * (n : ℝ))
(hvolume_exceptional : moment = 3 → 4 / 3 < volume) :
n = 1 ∧ moment = 1 ∧ volume = 2 := by
have hn_pos : 0 < (n : ℝ) := by nlinarith
have hn_nat_pos : 0 < n := by exact_mod_cast hn_pos
have hn_nat_lt : n < 3 := by
by_contra hnot
have hn_ge : 3 ≤ n := by omega
have hn_real_ge : 3 ≤ (n : ℝ) := by exact_mod_cast hn_ge
have hmoment_volume : moment * volume < 4 * volume :=
mul_lt_mul_of_pos_right hmoment_lt hvolume_pos
nlinarith
have hn_cases : n = 1 ∨ n = 2 := by omega
have hn : n = 1 := by
rcases hn_cases with hn | hn
· exact hn
· subst n
norm_num at hcharacteristic hsignature
have hmoment : moment = 3 := by nlinarith
have := hvolume_exceptional hmoment
linarith
subst n
norm_num at hcharacteristic hsignature
refine ⟨rfl, ?_, ?_⟩
· nlinarith
· linarith
/-- The normalized eigenvalues of a trace-free symmetric three-by-three
matrix have the parameter range and cubic invariant used in Section 2.
Here `-r` is the least eigenvalue and the squared Hilbert--Schmidt norm is six.
This elementary form needs neither eigenvector choices nor a square root. -/
theorem normalized_eigenvalue_parameter
(r s t : ℝ)
(htrace : -r + s + t = 0)
(hnorm : r ^ 2 + s ^ 2 + t ^ 2 = 6)
(hs : -r ≤ s) (ht : -r ≤ t) :
1 ≤ r ∧ r ≤ 2 ∧ s * t = r ^ 2 - 3 ∧
((-r) ^ 3 + s ^ 3 + t ^ 3) / 6 = (3 * r - r ^ 3) / 2 := by
have ht_eq : t = r - s := by linarith
subst t
have hr_nonneg : 0 ≤ r := by linarith
have hproduct : s * (r - s) = r ^ 2 - 3 := by nlinarith
have hr_lower : 1 ≤ r := by
have hpositive_product : 0 ≤ (s + r) * (r - s + r) :=
mul_nonneg (by linarith) (by linarith)
nlinarith
have hr_upper : r ≤ 2 := by
nlinarith [sq_nonneg (s - (r - s))]
refine ⟨hr_lower, hr_upper, hproduct, ?_⟩
have hcube : (-r) ^ 3 + s ^ 3 + (r - s) ^ 3 = -3 * r * (s * (r - s)) := by
ring
rw [hcube, hproduct]
ring
end PositiveEinsteinFour
end OAI
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