import Erdos1135.Tao.Probability.GatedSubmassComplement import Erdos1135.Tao.Section5.AffineIdealSumRegroup import Erdos1135.Tao.Section5.EndpointCoefficientBound import Erdos1135.Tao.Section5.PrefixTypicalityTransport /-! # Section 5 Affine Typicality Removal This leaf compares the fixed-length chronological typical tuple sum produced by the affine regrouping with the unrestricted iid Geom(2) affine-residue expectation. The rejected contribution costs exactly `5 * delta`. Tuple reversal and the Syracuse source-law identification remain downstream. -/ open scoped BigOperators namespace Erdos1135 namespace Tao noncomputable section /-- The fixed-`q` chronological typical tuple sum from the ideal affine regrouping. -/ def taoSection5TypicalAffineCoefficientSum (B q : ℕ) (E : Set ℕ) : ℝ := ∑ as : {as // as ∈ taoSection5TypicalTuples B q}, geom2PNatListMass as.1 * taoSection5PayloadFreeCoefficient q (taoSection5EPrime B E) (taoAffineOffsetZMod q as.1) /-- The accepted gated version of the chronological typical tuple sum. -/ def taoSection5AcceptedAffineCoefficientExpectation (B q : ℕ) (E : Set ℕ) : ℝ := ∑ X : ZMod (3 ^ q), taoSection5PayloadFreeCoefficient q (taoSection5EPrime B E) X * taoGatedSubmass (geom2PNatListPMF q) (taoSection5TypicalTuple B q) (taoAffineOffsetZMod q) X /-- The unrestricted affine-residue expectation before tuple reversal. -/ def taoSection5FullAffineCoefficientExpectation (B q : ℕ) (E : Set ℕ) : ℝ := ∑ X : ZMod (3 ^ q), taoSection5PayloadFreeCoefficient q (taoSection5EPrime B E) X * (((geom2PNatListPMF q).map (taoAffineOffsetZMod q)) X).toReal /-- The weighted contribution rejected by the chronological typicality gate. -/ def taoSection5RejectedAffineCoefficientExpectation (B q : ℕ) (E : Set ℕ) : ℝ := ∑ X : ZMod (3 ^ q), taoSection5PayloadFreeCoefficient q (taoSection5EPrime B E) X * taoGatedSubmass (geom2PNatListPMF q) (fun as => ¬ taoSection5TypicalTuple B q as) (taoAffineOffsetZMod q) X /-- The finite tuple sum is exactly the accepted gated iid expectation. -/ theorem taoSection5TypicalAffineCoefficientSum_eq_accepted (B q : ℕ) (E : Set ℕ) : taoSection5TypicalAffineCoefficientSum B q E = taoSection5AcceptedAffineCoefficientExpectation B q E := by classical let I : Finset (List ℕ+) := taoSection5TypicalTuples B q let f : List ℕ+ → ZMod (3 ^ q) := taoAffineOffsetZMod q let c : ZMod (3 ^ q) → ℝ := taoSection5PayloadFreeCoefficient q (taoSection5EPrime B E) let massTerm : List ℕ+ → ℝ := fun as => geom2PNatListMass as * c (f as) let atomTerm : List ℕ+ → ℝ := fun as => (geom2PNatListPMF q as).toReal * c (f as) have hfinite : (∑ as : {as // as ∈ I}, massTerm as.1) = ∑ as ∈ I, atomTerm as := by rw [Finset.sum_coe_sort I massTerm] apply Finset.sum_congr rfl intro as has have htyp : taoSection5TypicalTuple B q as := mem_taoSection5TypicalTuples_iff.mp has have hmass := geom2PNatListPMF_apply_length_toReal as rw [htyp.1] at hmass dsimp only [massTerm, atomTerm] rw [hmass] have hweighted := sum_mul_taoGatedSubmass_mem_eq (geom2PNatListPMF q) I f c have hgate : (fun as : List ℕ+ => as ∈ I) = taoSection5TypicalTuple B q := by funext as apply propext exact mem_taoSection5TypicalTuples_iff unfold taoSection5TypicalAffineCoefficientSum taoSection5AcceptedAffineCoefficientExpectation change (∑ as : {as // as ∈ I}, massTerm as.1) = _ rw [hfinite] simpa only [atomTerm, hgate, f, c] using hweighted /-- The unrestricted expectation splits exactly into accepted and rejected chronological contributions. -/ theorem taoSection5FullAffineCoefficientExpectation_eq_typical_add_rejected (B q : ℕ) (E : Set ℕ) : taoSection5FullAffineCoefficientExpectation B q E = taoSection5TypicalAffineCoefficientSum B q E + taoSection5RejectedAffineCoefficientExpectation B q E := by rw [taoSection5TypicalAffineCoefficientSum_eq_accepted] unfold taoSection5FullAffineCoefficientExpectation taoSection5AcceptedAffineCoefficientExpectation taoSection5RejectedAffineCoefficientExpectation calc (∑ X : ZMod (3 ^ q), taoSection5PayloadFreeCoefficient q (taoSection5EPrime B E) X * (((geom2PNatListPMF q).map (taoAffineOffsetZMod q)) X).toReal) = ∑ X : ZMod (3 ^ q), taoSection5PayloadFreeCoefficient q (taoSection5EPrime B E) X * (taoGatedSubmass (geom2PNatListPMF q) (taoSection5TypicalTuple B q) (taoAffineOffsetZMod q) X + taoGatedSubmass (geom2PNatListPMF q) (fun as => ¬ taoSection5TypicalTuple B q as) (taoAffineOffsetZMod q) X) := by apply Finset.sum_congr rfl intro X _hX rw [taoGatedSubmass_map_apply_split] _ = (∑ X : ZMod (3 ^ q), taoSection5PayloadFreeCoefficient q (taoSection5EPrime B E) X * taoGatedSubmass (geom2PNatListPMF q) (taoSection5TypicalTuple B q) (taoAffineOffsetZMod q) X) + ∑ X : ZMod (3 ^ q), taoSection5PayloadFreeCoefficient q (taoSection5EPrime B E) X * taoGatedSubmass (geom2PNatListPMF q) (fun as => ¬ taoSection5TypicalTuple B q as) (taoAffineOffsetZMod q) X := by rw [← Finset.sum_add_distrib] apply Finset.sum_congr rfl intro X _hX ring /-- The rejected weighted expectation is nonnegative. -/ theorem taoSection5RejectedAffineCoefficientExpectation_nonneg {B q : ℕ} {E : Set ℕ} (lost : TaoSection5PassLostWindowFacts B) (hindex : taoSection5M0 B + q ≤ taoSection5N0 B) : 0 ≤ taoSection5RejectedAffineCoefficientExpectation B q E := by unfold taoSection5RejectedAffineCoefficientExpectation apply Finset.sum_nonneg intro X _hX exact mul_nonneg (taoSection5PayloadFreeCoefficient_ePrime_mem_Icc_five lost hindex).1 (taoGatedSubmass_nonneg _ _ _ _) /-- The rejected weighted expectation costs at most `5 * delta`. -/ theorem taoSection5RejectedAffineCoefficientExpectation_le_five_mul_delta {B q : ℕ} {E : Set ℕ} (lost : TaoSection5PassLostWindowFacts B) (failure : TaoSection5PassTypicalFailureFacts B) (hindex : taoSection5M0 B + q ≤ taoSection5N0 B) : taoSection5RejectedAffineCoefficientExpectation B q E ≤ 5 * taoSection5PowerInteriorDelta B := by have hq : q ≤ taoSection5N0 B := by omega have hfailure := failure.short_notTypical_le_delta hq unfold taoSection5RejectedAffineCoefficientExpectation calc (∑ X : ZMod (3 ^ q), taoSection5PayloadFreeCoefficient q (taoSection5EPrime B E) X * taoGatedSubmass (geom2PNatListPMF q) (fun as => ¬ taoSection5TypicalTuple B q as) (taoAffineOffsetZMod q) X) ≤ ∑ X : ZMod (3 ^ q), 5 * taoGatedSubmass (geom2PNatListPMF q) (fun as => ¬ taoSection5TypicalTuple B q as) (taoAffineOffsetZMod q) X := by apply Finset.sum_le_sum intro X _hX exact mul_le_mul_of_nonneg_right (taoSection5PayloadFreeCoefficient_ePrime_mem_Icc_five lost hindex).2 (taoGatedSubmass_nonneg _ _ _ _) _ = 5 * ∑ X : ZMod (3 ^ q), taoGatedSubmass (geom2PNatListPMF q) (fun as => ¬ taoSection5TypicalTuple B q as) (taoAffineOffsetZMod q) X := by rw [Finset.mul_sum] _ = 5 * taoGatedRejectedMass (geom2PNatListPMF q) (taoSection5TypicalTuple B q) (taoAffineOffsetZMod q) := by rw [taoGatedSubmass_complement_sum_eq_rejectedMass] _ = 5 * ((geom2PNatListPMF q).toOuterMeasure {as | ¬ taoSection5TypicalTuple B q as}).toReal := by rw [taoGatedRejectedMass_eq_toOuterMeasure] _ ≤ 5 * taoSection5PowerInteriorDelta B := mul_le_mul_of_nonneg_left hfailure (by norm_num) /-- The full-minus-typical defect lies in the exact local error interval. -/ theorem taoSection5FullAffineCoefficientExpectation_sub_typical_mem_Icc {B q : ℕ} {E : Set ℕ} (lost : TaoSection5PassLostWindowFacts B) (failure : TaoSection5PassTypicalFailureFacts B) (hindex : taoSection5M0 B + q ≤ taoSection5N0 B) : taoSection5FullAffineCoefficientExpectation B q E - taoSection5TypicalAffineCoefficientSum B q E ∈ Set.Icc 0 (5 * taoSection5PowerInteriorDelta B) := by have hsplit := taoSection5FullAffineCoefficientExpectation_eq_typical_add_rejected B q E have hnonneg := taoSection5RejectedAffineCoefficientExpectation_nonneg (E := E) lost hindex have hupper := taoSection5RejectedAffineCoefficientExpectation_le_five_mul_delta (E := E) lost failure hindex constructor <;> linarith /-- Paper-facing absolute comparison between the chronological typical sum and the unrestricted iid affine expectation. -/ theorem abs_taoSection5TypicalAffineCoefficientSum_sub_full_le_five_mul_delta {B q : ℕ} {E : Set ℕ} (lost : TaoSection5PassLostWindowFacts B) (failure : TaoSection5PassTypicalFailureFacts B) (hindex : taoSection5M0 B + q ≤ taoSection5N0 B) : |taoSection5TypicalAffineCoefficientSum B q E - taoSection5FullAffineCoefficientExpectation B q E| ≤ 5 * taoSection5PowerInteriorDelta B := by have hdefect := taoSection5FullAffineCoefficientExpectation_sub_typical_mem_Icc (E := E) lost failure hindex rw [abs_of_nonpos (by linarith [hdefect.1])] linarith [hdefect.2] end end Tao end Erdos1135