import Erdos1135.ND.Band.A6PhysicalInterior import Mathlib.Analysis.Real.Pi.Bounds /-! # A6 Physical Native Quadrature Rate This finite leaf bounds the checked native strict-fiber quadrature remainder by the half-power scale consumed by the later fixed-`C` A6 interior wrapper. -/ namespace Erdos1135 namespace ND noncomputable section set_option maxHeartbeats 1500000 in /-- Under the physical moderate-window guard, the native Gaussian remainder is already in a uniform inverse-square-root class. -/ theorem ndA6PhysicalGaussianQuadratureError_le_invSqrtN0 {B j : ℕ} {branch : Tao.TaoSection5SourceBranch} {C M : ℝ} (hlogB : (300000 : ℝ) ≤ Real.log B) (hTube : NDA5InteriorTubeFacts B j branch C M) (hmoderate : ndA6PhysicalModerateBudget B j branch C M ≤ 1) : ndA6PhysicalGaussianQuadratureError B j branch C M ≤ 42 / Real.sqrt (Tao.taoSection5N0 B : ℝ) := by classical let A := ndA5PhysicalPhase (ndA5BandLower B branch j) M let W := ndA5TubeWidth B C let lo := ndA5TubeLo A W let S := ndA6PhysicalCenter B j branch M let N : ℝ := Tao.taoSection5N0 B let R : ℝ := ndA5TubeRadius W have hne : (ndA5FullTube A W).Nonempty := by simpa [A, W] using hTube.nonempty have hWpos : 0 < W := by obtain ⟨nu, hnu⟩ := hne have hstrict := (Finset.mem_filter.mp hnu).2 exact (abs_nonneg (ndA5StrictAffineSweep A nu : ℝ)).trans_lt hstrict have hEq := ndA5FullTube_eq_Icc hWpos hne have hlohi : ndA5TubeLo A W ≤ ndA5TubeHi A W := by rw [← Finset.nonempty_Icc, ← hEq] exact hne have hloMem : lo ∈ ndA5FullTube A W := by rw [hEq] exact Finset.left_mem_Icc.mpr (by simpa [lo] using hlohi) have hcell : NDA5InteriorCellFacts B j lo branch C M := hTube.pointwise lo (by simpa [A, W] using hloMem) have hloPos : (0 : ℝ) < lo := by exact_mod_cast hTube.lo_pos have hsub : lo + Tao.taoSection5M0 B - Tao.taoSection5M0 B = lo := by omega have hratioLowerRaw := hcell.t2.n0_div_nu_lower have hratioUpperRaw := hcell.t2.n0_div_nu_upper rw [hsub] at hratioLowerRaw hratioUpperRaw have hratioLower : (1374 / 100 : ℝ) ≤ N / (lo : ℝ) := by simpa [N] using hratioLowerRaw have hratioUpper : N / (lo : ℝ) ≤ (4244 / 100 : ℝ) := by simpa [N] using hratioUpperRaw have hNpos : 0 < N := by have hmul := (le_div_iff₀ hloPos).mp hratioLower nlinarith have hNnonneg : 0 ≤ N := hNpos.le have hS : 0 < S := by simpa [S] using (ndA6PhysicalCenter_pos_and_close_of_interior hlogB hTube).1 have hW71 : (71 : ℝ) ≤ W := by simpa [W] using ndA5TubeWidth_ge_seventy_one hcell.one_half_le_C hlogB have hceilReal : (71 : ℝ) ≤ (Nat.ceil W : ℝ) := hW71.trans (Nat.le_ceil W) have hceilNat : 71 ≤ Nat.ceil W := by exact_mod_cast hceilReal have hRadiusNat : 70 ≤ ndA5TubeRadius W := by unfold ndA5TubeRadius omega have hRpos : 0 < R := by dsimp [R] exact_mod_cast (show 0 < ndA5TubeRadius W by omega) have hRadiusSucc : ndA5TubeRadius W + 1 = Nat.ceil W := by unfold ndA5TubeRadius omega have hRlower : W - 1 ≤ R := by have h := Nat.le_ceil W rw [← hRadiusSucc] at h push_cast at h simpa [R] using (sub_le_iff_le_add.mpr h) have hRW : (70 / 71 : ℝ) * W ≤ R := by have hfrac : (70 / 71 : ℝ) * W ≤ W - 1 := by nlinarith exact hfrac.trans hRlower have hcrossing : ndA5StrictAffineSweep A lo = (ndA5TubeRadius W : ℤ) := by simpa [A, W, lo] using hTube.endpoint_crossings.1 have hresidual := ndA5StrictAffineSweep_sub_affine_mem_Ioc A lo rw [hcrossing] at hresidual have hcenterDiff : A - ndA5PhaseDelta * (lo : ℝ) = ndA5PhaseDelta * (S - (lo : ℝ)) := by dsimp [A, S, ndA6PhysicalCenter] field_simp [ndA5PhaseDelta_mem_Ioo.1.ne'] rw [hcenterDiff] at hresidual change 0 < R - ndA5PhaseDelta * (S - (lo : ℝ)) ∧ R - ndA5PhaseDelta * (S - (lo : ℝ)) ≤ 1 at hresidual have hloLtS : (lo : ℝ) < S := by have hRlarge : (1 : ℝ) < R := by dsimp [R] exact_mod_cast (show 1 < ndA5TubeRadius W by omega) by_contra hnot have hdiff : S - (lo : ℝ) ≤ 0 := sub_nonpos.mpr (le_of_not_gt hnot) have hmul : ndA5PhaseDelta * (S - (lo : ℝ)) ≤ 0 := mul_nonpos_of_nonneg_of_nonpos ndA5PhaseDelta_mem_Ioo.1.le hdiff nlinarith [hresidual.2] have hlogLe : ndA5LogFourThirds ≤ (1 / 3 : ℝ) := by have h := Real.log_le_sub_one_of_pos (show (0 : ℝ) < 4 / 3 by norm_num) norm_num [ndA5LogFourThirds] at h ⊢ exact h have hcoef : (12 / 25 : ℝ) ≤ (4 / 25 : ℝ) / ndA5LogFourThirds := by apply (le_div_iff₀ ndA5LogFourThirds_pos).2 nlinarith have hshiftTerm : (12 / 25 : ℝ) * W ≤ ((4 / 25 : ℝ) * W) / ndA5LogFourThirds := by calc (12 / 25 : ℝ) * W ≤ ((4 / 25 : ℝ) / ndA5LogFourThirds) * W := mul_le_mul_of_nonneg_right hcoef hWpos.le _ = ((4 / 25 : ℝ) * W) / ndA5LogFourThirds := by ring have hmargin := ndA5InteriorContainmentMargin hcell.one_half_le_C hlogB have hDle : (W + 1) / ndA5PhaseDelta ≤ (63 / 25 : ℝ) * W := by nlinarith have hcenterW : S - (lo : ℝ) < (63 / 25 : ℝ) * W := by have hphysical : lo ∈ ndA5FullTube (ndA5PhysicalPhase (ndA5BandLower B branch j) M) (ndA5TubeWidth B C) := by simpa [A, W] using hloMem have hclose := abs_natCast_sub_ndA6PhysicalCenter_lt_of_mem_fullTube hphysical have hsigned : S - (lo : ℝ) ≤ |(lo : ℝ) - S| := by rw [abs_sub_comm] exact le_abs_self _ exact (hsigned.trans_lt hclose).trans_le hDle let a := W / (lo : ℝ) let b := W ^ 3 / (lo : ℝ) ^ 2 have hH : 16 * (a + b) ≤ 1 := by simpa [ndA6PhysicalModerateBudget, ndA6PhysicalLinearRatio, ndA6PhysicalCubicRatio, A, W, lo, a, b] using hmoderate have ha : 0 ≤ a := div_nonneg hWpos.le hloPos.le have hb : 0 ≤ b := div_nonneg (by positivity) (sq_nonneg _) have haSmall : a ≤ 1 / 16 := by nlinarith have hSratio : S / (lo : ℝ) < (463 / 400 : ℝ) := by have hcenterDiv : (S - (lo : ℝ)) / (lo : ℝ) < (63 / 25 : ℝ) * a := by apply (div_lt_iff₀ hloPos).2 dsimp [a] field_simp [hloPos.ne'] nlinarith [hcenterW] have hdecomp : S / (lo : ℝ) = 1 + (S - (lo : ℝ)) / (lo : ℝ) := by field_simp [hloPos.ne'] ring rw [hdecomp] nlinarith have hNSratio : (5496 / 463 : ℝ) < N / S := by apply (lt_div_iff₀ hS).2 have hlowerMul := (le_div_iff₀ hloPos).mp hratioLower have hSupper := (div_lt_iff₀ hloPos).mp hSratio nlinarith have hClower : (1 / 2 : ℝ) ≤ C := hcell.one_half_le_C have hloOne : (1 : ℝ) ≤ lo := by exact_mod_cast hTube.lo_pos have hlowerMul := (le_div_iff₀ hloPos).mp hratioLower have hNgtEight : (8 : ℝ) < N := by nlinarith have hlogNpos : 0 < Real.log N := Real.log_pos (by nlinarith) have hNsqrt : 0 ≤ N * Real.log N := mul_nonneg hNnonneg hlogNpos.le have hWsq : W ^ 2 = C ^ 2 * (N * Real.log N) := by dsimp [W, N, ndA5TubeWidth] rw [mul_pow, Real.sq_sqrt hNsqrt] have hRsq : ((70 / 71 : ℝ) ^ 2) * W ^ 2 ≤ R ^ 2 := by simpa only [mul_pow] using (sq_le_sq₀ (mul_nonneg (by norm_num) hWpos.le) hRpos.le).2 hRW have hexponent : (1 / 2 : ℝ) * Real.log N ≤ R ^ 2 / (4 * S) := by have hC2 : (1 / 4 : ℝ) ≤ C ^ 2 := by nlinarith [sq_nonneg (C - 1 / 2)] have hNS := (lt_div_iff₀ hS).mp hNSratio rw [hWsq] at hRsq apply (le_div_iff₀ (by positivity : (0 : ℝ) < 4 * S)).2 nlinarith have hlogNGtTwo : (2 : ℝ) < Real.log N := by have hlogMono := Real.strictMonoOn_log (show (0 : ℝ) < 8 by norm_num) hNpos hNgtEight have hlogEight : (2 : ℝ) < Real.log 8 := by rw [show (8 : ℝ) = 2 ^ 3 by norm_num, Real.log_pow] have h := Real.log_two_gt_d9 norm_num at h ⊢ nlinarith exact hlogEight.trans hlogMono have hSleRsq : S ≤ R ^ 2 := by have honeLog : (1 : ℝ) < (1 / 2 : ℝ) * Real.log N := by linarith only [hlogNGtTwo] have hone : (1 : ℝ) < R ^ 2 / (4 * S) := honeLog.trans_le hexponent have hfourS : 4 * S < R ^ 2 := by simpa only [one_mul] using (lt_div_iff₀ (by positivity : (0 : ℝ) < 4 * S)).mp hone have hSfour : S ≤ 4 * S := by linarith only [hS] exact hSfour.trans hfourS.le have hsqrtSpos : 0 < Real.sqrt S := Real.sqrt_pos.2 hS have hsqrtNpos : 0 < Real.sqrt N := Real.sqrt_pos.2 hNpos have hsqrtSleR : Real.sqrt S ≤ R := by rw [Real.sqrt_le_iff] exact ⟨hRpos.le, hSleRsq⟩ have hpeakBasic : ndA6Gaussian S 0 ≤ 1 / Real.sqrt S := by rw [ndA6Gaussian_zero] have hfourPi : (1 : ℝ) ≤ 4 * Real.pi := by nlinarith [Real.pi_gt_three] have hsqrtLe : Real.sqrt S ≤ Real.sqrt (4 * Real.pi * S) := by apply Real.sqrt_le_sqrt nlinarith simpa only [inv_eq_one_div] using one_div_le_one_div_of_le hsqrtSpos hsqrtLe have hNleS : N ≤ 49 * S := by have hupperMul := (div_le_iff₀ hloPos).mp hratioUpper nlinarith have hsqrtNle : Real.sqrt N ≤ 7 * Real.sqrt S := by rw [Real.sqrt_le_iff] constructor · positivity rw [mul_pow, Real.sq_sqrt hS.le] norm_num exact hNleS have hpeak : ndA6Gaussian S 0 ≤ 7 / Real.sqrt N := by calc ndA6Gaussian S 0 ≤ 1 / Real.sqrt S := hpeakBasic _ ≤ 7 / Real.sqrt N := by apply (div_le_div_iff₀ hsqrtSpos hsqrtNpos).2 simpa [mul_comm] using hsqrtNle have hexpTail : Real.exp (-(R ^ 2 / (4 * S))) ≤ 1 / Real.sqrt N := by have hhalf : Real.log N / 2 ≤ R ^ 2 / (4 * S) := by simpa [div_eq_mul_inv, mul_comm] using hexponent have hmono : Real.exp (-(R ^ 2 / (4 * S))) ≤ Real.exp (-(Real.log N / 2)) := Real.exp_le_exp.mpr (neg_le_neg hhalf) calc Real.exp (-(R ^ 2 / (4 * S))) ≤ Real.exp (-(Real.log N / 2)) := hmono _ = 1 / Real.sqrt N := by rw [← Real.log_sqrt hNnonneg, Real.exp_neg, Real.exp_log hsqrtNpos] simp only [one_div] have hgaussR : ndA6Gaussian S R ≤ (1 / Real.sqrt S) * (1 / Real.sqrt N) := by have hprefactor : (Real.sqrt (4 * Real.pi * S))⁻¹ ≤ 1 / Real.sqrt S := by simpa only [ndA6Gaussian_zero] using hpeakBasic unfold ndA6Gaussian exact mul_le_mul hprefactor hexpTail (Real.exp_pos _).le (one_div_nonneg.mpr hsqrtSpos.le) have hinvDelta : 1 / ndA5PhaseDelta < (63 / 25 : ℝ) := by have hscaled : W / ndA5PhaseDelta < (W + 1) / ndA5PhaseDelta := by apply (div_lt_div_iff₀ ndA5PhaseDelta_mem_Ioo.1 ndA5PhaseDelta_mem_Ioo.1).2 nlinarith have hmul : (1 / ndA5PhaseDelta) * W < (63 / 25 : ℝ) * W := by calc (1 / ndA5PhaseDelta) * W = W / ndA5PhaseDelta := by ring _ < (W + 1) / ndA5PhaseDelta := hscaled _ ≤ (63 / 25 : ℝ) * W := hDle exact lt_of_mul_lt_mul_right hmul hWpos.le have hpeakPart : (2 + 1 / ndA5PhaseDelta) * ndA6Gaussian S 0 ≤ (791 / 25 : ℝ) / Real.sqrt N := by have hcoefPeak : 2 + 1 / ndA5PhaseDelta ≤ (113 / 25 : ℝ) := by linarith calc (2 + 1 / ndA5PhaseDelta) * ndA6Gaussian S 0 ≤ (113 / 25 : ℝ) * (7 / Real.sqrt N) := mul_le_mul hcoefPeak hpeak (ndA6Gaussian_nonneg S 0) (by positivity) _ = (791 / 25 : ℝ) / Real.sqrt N := by ring have htailPart : (4 * S / (ndA5PhaseDelta * R)) * ndA6Gaussian S R ≤ (252 / 25 : ℝ) / Real.sqrt N := by have hcoefTail : 4 * (1 / ndA5PhaseDelta) ≤ (252 / 25 : ℝ) := by linarith have hsqrtRatio : Real.sqrt S / R ≤ 1 := by exact (div_le_one hRpos).2 hsqrtSleR calc (4 * S / (ndA5PhaseDelta * R)) * ndA6Gaussian S R ≤ (4 * (1 / ndA5PhaseDelta)) * (Real.sqrt S / R) * (1 / Real.sqrt N) := by have hcoefNonneg : 0 ≤ 4 * S / (ndA5PhaseDelta * R) := div_nonneg (mul_nonneg (by norm_num : (0 : ℝ) ≤ 4) hS.le) (mul_nonneg ndA5PhaseDelta_mem_Ioo.1.le hRpos.le) have hmul : (4 * S / (ndA5PhaseDelta * R)) * ndA6Gaussian S R ≤ (4 * S / (ndA5PhaseDelta * R)) * ((1 / Real.sqrt S) * (1 / Real.sqrt N)) := mul_le_mul_of_nonneg_left hgaussR hcoefNonneg calc (4 * S / (ndA5PhaseDelta * R)) * ndA6Gaussian S R ≤ (4 * S / (ndA5PhaseDelta * R)) * ((1 / Real.sqrt S) * (1 / Real.sqrt N)) := hmul _ = (4 * (1 / ndA5PhaseDelta)) * (Real.sqrt S / R) * (1 / Real.sqrt N) := by field_simp [ndA5PhaseDelta_mem_Ioo.1.ne', hRpos.ne', hsqrtSpos.ne'] rw [Real.sq_sqrt hS.le] _ ≤ (252 / 25 : ℝ) * 1 * (1 / Real.sqrt N) := mul_le_mul_of_nonneg_right (mul_le_mul hcoefTail hsqrtRatio (by positivity) (by positivity)) (one_div_nonneg.mpr hsqrtNpos.le) _ = (252 / 25 : ℝ) / Real.sqrt N := by ring change (2 + 1 / ndA5PhaseDelta) * ndA6Gaussian S 0 + (4 * S / (ndA5PhaseDelta * R)) * ndA6Gaussian S R ≤ 42 / Real.sqrt N calc _ ≤ (791 / 25 : ℝ) / Real.sqrt N + (252 / 25 : ℝ) / Real.sqrt N := add_le_add hpeakPart htailPart _ ≤ 42 / Real.sqrt N := by have hinvSqrt : 0 ≤ 1 / Real.sqrt N := one_div_nonneg.mpr hsqrtNpos.le calc (791 / 25 : ℝ) / Real.sqrt N + (252 / 25 : ℝ) / Real.sqrt N = (1043 / 25 : ℝ) * (1 / Real.sqrt N) := by ring _ ≤ 42 * (1 / Real.sqrt N) := mul_le_mul_of_nonneg_right (by norm_num) hinvSqrt _ = 42 / Real.sqrt N := by ring end end ND end Erdos1135