| 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 |
|
|
| / |
| 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 |
|
|
| / |
| 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] |
|
|
| / |
| 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 |
|
|
| / |
| matrix have the parameter range and cubic invariant used in Section 2. |
| Here `-r` is the least eigenvalue and the squared Hilbert |
| 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 |
|
|
