| import OAI.NumberTheory.PrimePackets.ZetaDerivatives | |
| import OAI.NumberTheory.DukePrimeDegree.Geometry.Convergence | |
| namespace OAI | |
| /-! | |
| # Conditional prime-degree packet equidistribution | |
| Complete volume-weighted local-homothety packets converge to Haar | |
| probability and form a tight family under a residue-relative uniform | |
| zeta scale bound and prime-degree positive-entropy rigidity. The | |
| arithmetic and dynamical hypotheses concern the actual homogeneous | |
| lattice space and its diagonal action. A subsequential packet limit | |
| supplies Haar probability. | |
| -/ | |
| open MeasureTheory ProbabilityTheory Filter Set | |
| open scoped ENNReal Topology | |
| namespace DukePrimeDegree | |
| open PrimeDegreePackets | |
| universe u | |
| /-- The published rigidity input, on the actual lattice space and with | |
| actual diagonal ergodicity and measure-theoretic entropy. -/ | |
| def EKLPrimeInput (d : ℕ) : Prop := | |
| Nat.Prime (d + 1) → 3 ≤ d + 1 → | |
| ∀ ν : Measure (X (d + 1)), IsProbabilityMeasure ν → | |
| ErgodicSMul (Multiplicative (LogCoordinates d)) (X (d + 1)) ν → | |
| 0 < measureEntropy ν | |
| (fun x => rightAction x (diagonalOneParameter (regularDiagonalWeights d) | |
| (regularDiagonalWeights_sum d) 1)) → IsHaarProbability ν | |
| /-- Every weak packet limit is Haar under the analytic scale bound and | |
| published prime-degree rigidity. -/ | |
| theorem packet_limit_isHaar_of_scale_and_EKL | |
| (d : ℕ) (hprime : Nat.Prime (d + 1)) (hthree : 3 ≤ d + 1) (hodd : Odd (d + 1)) | |
| (Hscale : UniformZetaScaleBound.{u} (d + 1)) (ekl : EKLPrimeInput d) | |
| (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)] | |
| [∀ i, NumberField.IsTotallyReal (K i)] | |
| (M : ∀ i, PrimeDegreePackets.FullLattice (d + 1) (K i)) | |
| (σ : ∀ i, OrderedEmbeddings (d + 1) (K i)) | |
| (hn : ∀ i, Module.finrank ℚ (K i) = d + 1) | |
| (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop) | |
| (ν : ProbabilityMeasure (X (d + 1))) | |
| (hlim : Tendsto (fun i => packetProbability (M i) (σ i) (hn i) hodd) | |
| atTop (𝓝 ν)) : IsHaarProbability (ν : Measure (X (d + 1))) := by | |
| obtain ⟨m, hm, hdec, hcomp⟩ := | |
| packet_limit_positive_ergodic_components_of_zeta_scale d hodd Hscale K M σ hn hD ν hlim | |
| let : MeasurableSpace (X (d + 1)) := xMeasurableSpace (d + 1) | |
| let : @IsProbabilityMeasure (X (d + 1)) m | |
| ((ν : Measure (X (d + 1))).trim hm) := by | |
| constructor | |
| rw [trim_measurableSet_eq hm (@MeasurableSet.univ _ m)] | |
| exact measure_univ | |
| have hHaar : ∀ᵐ z ∂(ν : Measure (X (d + 1))).trim hm, | |
| IsHaarProbability (condExpKernel (ν : Measure (X (d + 1))) m z) := by | |
| filter_upwards [hcomp] with z hz | |
| exact ekl hprime hthree _ inferInstance hz.1 hz.2 | |
| obtain ⟨z, hz⟩ := hHaar.exists | |
| have heq := measure_eq_haar_of_condExpKernel_invariance | |
| (ν : Measure (X (d + 1))) | |
| (condExpKernel (ν : Measure (X (d + 1))) m z) hz m hm hdec | |
| (hHaar.mono fun _ h => h.2) | |
| exact heq.symm ▸ hz | |
| /-- The full volume-weighted packet conclusion, including no escape of | |
| mass, conditional on the explicitly separated analytic and published | |
| rigidity inputs. The family may vary through arbitrary full lattices and | |
| arbitrary local homothety types. -/ | |
| theorem prime_degree_packet_equidistribution_of_scale_and_EKL | |
| (d : ℕ) (hprime : Nat.Prime (d + 1)) (hfive : 5 ≤ d + 1) | |
| (Hscale : UniformZetaScaleBound.{u} (d + 1)) (ekl : EKLPrimeInput d) | |
| (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)] | |
| [∀ i, NumberField.IsTotallyReal (K i)] | |
| (M : ∀ i, PrimeDegreePackets.FullLattice (d + 1) (K i)) | |
| (σ : ∀ i, OrderedEmbeddings (d + 1) (K i)) | |
| (hn : ∀ i, Module.finrank ℚ (K i) = d + 1) | |
| (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop) : | |
| ∃ ν : ProbabilityMeasure (X (d + 1)), | |
| IsHaarProbability (ν : Measure (X (d + 1))) ∧ | |
| Tendsto (fun i => packetProbability (M i) (σ i) (hn i) | |
| (hprime.odd_of_ne_two (by omega))) atTop (𝓝 ν) ∧ | |
| IsTightMeasureSet (Set.range (fun i => packetMeasure (M i) (σ i))) := by | |
| let hodd : Odd (d + 1) := hprime.odd_of_ne_two (by omega) | |
| let p : ℕ → ProbabilityMeasure (X (d + 1)) := | |
| fun i => packetProbability (M i) (σ i) (hn i) hodd | |
| have hQ := packetScale_tendsto_of_multiplierDiscriminant_tendsto | |
| (Nat.succ_pos d) K M hn hD | |
| have htight := packet_tight_of_zeta_scale d hodd Hscale K M σ hn hQ | |
| obtain ⟨ν, φ, hφ, hsub⟩ := | |
| packet_probability_subsequence_of_zeta_scale d hodd Hscale K M σ hn hD | |
| have hHaar := packet_limit_isHaar_of_scale_and_EKL d hprime (by omega) hodd Hscale ekl | |
| (fun i => K (φ i)) (fun i => M (φ i)) (fun i => σ (φ i)) | |
| (fun i => hn (φ i)) (hD.comp hφ.tendsto_atTop) ν hsub | |
| refine ⟨ν, hHaar, ?_, htight⟩ | |
| have htight' : IsTightMeasureSet {μ : Measure (X (d + 1)) | ∃ i, μ = p i} := by | |
| convert htight using 1 | |
| ext μ | |
| simp only [Set.mem_ofPred_eq, Set.mem_range] | |
| constructor <;> rintro ⟨i, hi⟩ <;> exact ⟨i, hi.symm⟩ | |
| apply tendsto_of_tight_of_unique_subsequential_limit p ν htight' | |
| intro ψ hψ ξ hξ | |
| have hHaarξ := packet_limit_isHaar_of_scale_and_EKL d hprime (by omega) hodd Hscale ekl | |
| (fun i => K (ψ i)) (fun i => M (ψ i)) (fun i => σ (ψ i)) | |
| (fun i => hn (ψ i)) (hD.comp hψ.tendsto_atTop) ξ hξ | |
| exact Subtype.ext (isHaarProbability_unique hHaarξ hHaar) | |
| end DukePrimeDegree | |
| end OAI | |