import Erdos1135.ND.Band.A5ExactAffineCorrection import Erdos1135.ND.Band.A5BandNormalizer import Erdos1135.ND.Band.A5StrictBandLevels import Erdos1135.Tao.Syracuse.AffineReciprocal /-! # A5 Exact and Nominal Affine Positions This leaf freezes the sign and endpoint behavior of the once-only affine correction. For an exact affine source, the physical band position is the nominal lattice position minus `delta = -log (1 - F / M)`. The physical band is half-open, whereas the nominal profile deliberately uses strict endpoints. The mismatch lemmas below therefore retain the sole zero-length lower-endpoint case instead of hiding it in a generic boundary estimate. -/ namespace Erdos1135 namespace ND noncomputable section /-- Exact logarithmic position of a positive source relative to a band lower endpoint. -/ noncomputable def ndA5ExactAffinePosition (z : ℝ) (N : ℕ) : ℝ := Real.log ((N : ℝ) / z) /-- Nominal affine position obtained by dropping the offset `F` from the exact denominator `M - F`. This is definitionally the position expression used by `ndA5NominalStrictBandLevel`. -/ noncomputable def ndA5NominalAffinePosition (z M : ℝ) (q L : ℕ) : ℝ := ((L : ℝ) - (ndA5PhysicalPhase z M + (q : ℝ) * logTwoThree)) * Real.log 2 /-- Positive-direction correction from nominal to exact affine position. -/ noncomputable def ndA5AffinePositionCorrection (as : List ℕ+) (M : ℝ) : ℝ := -Real.log (1 - (Tao.taoOffsetList as : ℝ) / M) /-- Strict nominal membership used by the frozen phase profile. -/ def ndA5StrictPosition (beta u : ℝ) : Prop := 0 < u ∧ u < beta /-- Half-open physical membership used by the actual A5 bands. -/ def ndA5HalfOpenPosition (beta u : ℝ) : Prop := 0 ≤ u ∧ u < beta /-- The nominal position predicate is exactly the already checked strict signed-level predicate after specializing its physical phase. -/ theorem ndA5NominalStrictBandLevel_iff_nominalPosition (beta z M : ℝ) (q L : ℕ) : ndA5NominalStrictBandLevel beta (ndA5PhysicalPhase z M) q (L : ℤ) ↔ ndA5StrictPosition beta (ndA5NominalAffinePosition z M q L) := by simp only [ndA5NominalStrictBandLevel, ndA5StrictPosition, ndA5NominalAffinePosition, Int.cast_natCast] /-- A logarithmic position is in the physical half-open interval exactly when its source lies between the corresponding multiplicative endpoints. -/ theorem ndA5HalfOpenPosition_exact_iff {beta z : ℝ} {N : ℕ} (hz : 0 < z) (hN : 0 < N) : ndA5HalfOpenPosition beta (ndA5ExactAffinePosition z N) ↔ z ≤ (N : ℝ) ∧ (N : ℝ) < z * Real.exp beta := by have hNR : (0 : ℝ) < N := by exact_mod_cast hN have hdiv : 0 < (N : ℝ) / z := div_pos hNR hz unfold ndA5HalfOpenPosition ndA5ExactAffinePosition constructor · rintro ⟨hlower, hupper⟩ have hone : (1 : ℝ) ≤ (N : ℝ) / z := (Real.log_nonneg_iff hdiv).mp hlower have hupp : (N : ℝ) / z < Real.exp beta := (Real.log_lt_iff_lt_exp hdiv).mp hupper constructor · have := (le_div_iff₀ hz).mp hone simpa only [one_mul] using this · have := (div_lt_iff₀ hz).mp hupp simpa only [mul_comm] using this · rintro ⟨hlower, hupper⟩ constructor · apply (Real.log_nonneg_iff hdiv).mpr exact (le_div_iff₀ hz).mpr (by simpa only [one_mul] using hlower) · apply (Real.log_lt_iff_lt_exp hdiv).mpr exact (div_lt_iff₀ hz).mpr (by simpa only [mul_comm] using hupper) /-- Actual A5 band membership, with oddness supplied by the source atom, is exactly half-open membership of the exact logarithmic position. -/ theorem mem_ndA5OddBand_iff_halfOpen_exactPosition {B N j : ℕ} {branch : Tao.TaoSection5SourceBranch} (hB : 1 ≤ B) (hodd : N % 2 = 1) : N ∈ ndA5OddBand B branch j ↔ ndA5HalfOpenPosition (ndA5BandBeta B branch) (ndA5ExactAffinePosition (ndA5BandLower B branch j) N) := by have hBpos : (0 : ℝ) < B := by exact_mod_cast hB have hYpos : 0 < Tao.taoSection5SourceY B branch := by rw [Tao.taoSection5SourceY_eq_branch_rpow] exact Real.rpow_pos_of_pos hBpos _ have hz : 0 < ndA5BandLower B branch j := by unfold ndA5BandLower exact mul_pos hYpos (Real.exp_pos _) have hN : 0 < N := by omega rw [mem_ndA5OddBand_exp_iff] constructor · rintro ⟨hlower, hupper, _⟩ exact (ndA5HalfOpenPosition_exact_iff hz hN).2 ⟨hlower, hupper⟩ · intro hposition rcases (ndA5HalfOpenPosition_exact_iff hz hN).1 hposition with ⟨hlower, hupper⟩ exact ⟨hlower, hupper, hodd⟩ /-- A zero-length valuation tuple has zero affine offset. -/ theorem ndA5_taoOffsetList_eq_zero_of_length_eq_zero (as : List ℕ+) (hzero : as.length = 0) : Tao.taoOffsetList as = 0 := by have has : as = [] := List.length_eq_zero_iff.mp hzero subst as rfl /-- Every nonempty valuation tuple has strictly positive affine offset. -/ theorem ndA5_taoOffsetList_pos_of_length_pos (as : List ℕ+) (hpos : 0 < as.length) : 0 < Tao.taoOffsetList as := by have hne : as ≠ [] := List.length_pos_iff_ne_nil.mp hpos obtain ⟨a, rest, rfl⟩ := List.exists_cons_of_ne_nil hne unfold Tao.taoOffsetList exact add_pos_of_pos_of_nonneg (div_pos (pow_pos (by norm_num) _) (pow_pos (by norm_num) _)) (Tao.taoOffsetList_nonneg rest) /-- Exact sign identity between physical and nominal affine positions. The plus sign in the intermediate formula is load-bearing: `uExact = uNominal + log (1 - F/M)`, hence the named nonnegative correction is subtracted. -/ theorem ndA5_exactAffinePosition_eq_nominal_sub_correction {z : ℝ} {q M : ℕ} {as : List ℕ+} {N : Tao.TaoOddNat} (hz : 0 < z) (hlen : as.length = q) (hAff : Tao.taoAffList as (N.1 : ℚ) = (M : ℚ)) : ndA5ExactAffinePosition z N.1 = ndA5NominalAffinePosition z M q (Tao.taoTupleWeight as) - ndA5AffinePositionCorrection as M := by have hNpos : (0 : ℝ) < N.1 := by exact_mod_cast Odd.pos N.2 have hdenPos : 0 < (M : ℝ) - (Tao.taoOffsetList as : ℝ) := Tao.taoAffList_denominator_pos hlen hAff have hoffset0 : 0 ≤ (Tao.taoOffsetList as : ℝ) := Rat.cast_nonneg.mpr (Tao.taoOffsetList_nonneg as) have hMpos : (0 : ℝ) < M := by linarith have hclosedQ : (M : ℚ) = (3 : ℚ) ^ as.length / (2 : ℚ) ^ Tao.taoTupleWeight as * (N.1 : ℚ) + Tao.taoOffsetList as := hAff.symm.trans (Tao.taoAffList_closed as (N.1 : ℚ)) have hclosedR := congrArg (fun x : ℚ => (x : ℝ)) hclosedQ norm_num only [Rat.cast_add, Rat.cast_mul, Rat.cast_div, Rat.cast_pow, Rat.cast_natCast, Rat.cast_ofNat] at hclosedR have hdenEq : (M : ℝ) - (Tao.taoOffsetList as : ℝ) = (3 : ℝ) ^ q / (2 : ℝ) ^ Tao.taoTupleWeight as * (N.1 : ℝ) := by rw [hlen] at hclosedR linarith have hratioEq : 1 - (Tao.taoOffsetList as : ℝ) / (M : ℝ) = ((M : ℝ) - (Tao.taoOffsetList as : ℝ)) / (M : ℝ) := by field_simp [hMpos.ne'] have hlogExact : Real.log ((N.1 : ℝ) / z) = Real.log (N.1 : ℝ) - Real.log z := by rw [Real.log_div hNpos.ne' hz.ne'] have hlogPhase : Real.log (z / (M : ℝ)) = Real.log z - Real.log (M : ℝ) := by rw [Real.log_div hz.ne' hMpos.ne'] have hlogCorrection : Real.log (1 - (Tao.taoOffsetList as : ℝ) / (M : ℝ)) = Real.log ((M : ℝ) - (Tao.taoOffsetList as : ℝ)) - Real.log (M : ℝ) := by rw [hratioEq, Real.log_div hdenPos.ne' hMpos.ne'] have hlogDenominator : Real.log ((M : ℝ) - (Tao.taoOffsetList as : ℝ)) = (q : ℝ) * Real.log 3 - (Tao.taoTupleWeight as : ℝ) * Real.log 2 + Real.log (N.1 : ℝ) := by rw [hdenEq, Real.log_mul (div_ne_zero (pow_ne_zero _ (by norm_num)) (pow_ne_zero _ (by norm_num))) hNpos.ne', Real.log_div (pow_ne_zero _ (by norm_num)) (pow_ne_zero _ (by norm_num)), Real.log_pow, Real.log_pow] have hlogTwo : Real.log (2 : ℝ) ≠ 0 := (Real.log_pos (by norm_num)).ne' unfold ndA5ExactAffinePosition ndA5NominalAffinePosition ndA5AffinePositionCorrection ndA5PhysicalPhase logTwoThree rw [hlogExact, hlogPhase, hlogCorrection, hlogDenominator] field_simp [hlogTwo] ring /-- Equivalent plus-log form of the exact position identity. This exposes the sign directly at the public interface used by strip consumers. -/ theorem ndA5_exactAffinePosition_eq_nominal_add_log_ratio {z : ℝ} {q M : ℕ} {as : List ℕ+} {N : Tao.TaoOddNat} (hz : 0 < z) (hlen : as.length = q) (hAff : Tao.taoAffList as (N.1 : ℚ) = (M : ℚ)) : ndA5ExactAffinePosition z N.1 = ndA5NominalAffinePosition z M q (Tao.taoTupleWeight as) + Real.log (1 - (Tao.taoOffsetList as : ℝ) / (M : ℝ)) := by rw [ndA5_exactAffinePosition_eq_nominal_sub_correction hz hlen hAff] unfold ndA5AffinePositionCorrection ring /-- The correction vanishes on the zero-length tuple route. -/ @[simp] theorem ndA5AffinePositionCorrection_eq_zero_of_length_eq_zero (as : List ℕ+) (M : ℝ) (hzero : as.length = 0) : ndA5AffinePositionCorrection as M = 0 := by rw [ndA5AffinePositionCorrection, ndA5_taoOffsetList_eq_zero_of_length_eq_zero as hzero] norm_num /-- A tuple whose declared length is zero has no affine correction. -/ theorem ndA5AffinePositionCorrection_eq_zero_of_length_eq {q : ℕ} (as : List ℕ+) (M : ℝ) (hlen : as.length = q) (hq : q = 0) : ndA5AffinePositionCorrection as M = 0 := by apply ndA5AffinePositionCorrection_eq_zero_of_length_eq_zero omega /-- Every exact affine source has a nonnegative position correction. -/ theorem ndA5AffinePositionCorrection_nonneg {q M : ℕ} {as : List ℕ+} {N : Tao.TaoOddNat} (hlen : as.length = q) (hAff : Tao.taoAffList as (N.1 : ℚ) = (M : ℚ)) : 0 ≤ ndA5AffinePositionCorrection as M := by have hdenPos : 0 < (M : ℝ) - (Tao.taoOffsetList as : ℝ) := Tao.taoAffList_denominator_pos hlen hAff have hoffset0 : 0 ≤ (Tao.taoOffsetList as : ℝ) := Rat.cast_nonneg.mpr (Tao.taoOffsetList_nonneg as) have hMpos : (0 : ℝ) < M := by linarith have hratio0 : 0 ≤ (Tao.taoOffsetList as : ℝ) / (M : ℝ) := div_nonneg hoffset0 hMpos.le have hratioLt : (Tao.taoOffsetList as : ℝ) / (M : ℝ) < 1 := (div_lt_one hMpos).2 (by linarith) unfold ndA5AffinePositionCorrection have hlog := Real.log_nonpos (by linarith : 0 ≤ 1 - (Tao.taoOffsetList as : ℝ) / (M : ℝ)) (by linarith) linarith /-- A positive-length exact affine source has a strictly positive correction. -/ theorem ndA5AffinePositionCorrection_pos {q M : ℕ} {as : List ℕ+} {N : Tao.TaoOddNat} (hq : 0 < q) (hlen : as.length = q) (hAff : Tao.taoAffList as (N.1 : ℚ) = (M : ℚ)) : 0 < ndA5AffinePositionCorrection as M := by have hdenPos : 0 < (M : ℝ) - (Tao.taoOffsetList as : ℝ) := Tao.taoAffList_denominator_pos hlen hAff have hoffsetQ : 0 < Tao.taoOffsetList as := ndA5_taoOffsetList_pos_of_length_pos as (by omega) have hoffset : 0 < (Tao.taoOffsetList as : ℝ) := Rat.cast_pos.mpr hoffsetQ have hMpos : (0 : ℝ) < M := by linarith have hratioPos : 0 < (Tao.taoOffsetList as : ℝ) / (M : ℝ) := div_pos hoffset hMpos have hratioLt : (Tao.taoOffsetList as : ℝ) / (M : ℝ) < 1 := (div_lt_one hMpos).2 (by linarith) unfold ndA5AffinePositionCorrection exact neg_pos.mpr (Real.log_neg (by linarith) (by linarith)) /-- The source-facing scalar packet, rewritten in the named position unit. The positive lower bound uses the checked local `t2.nu_pos`, so the integral phase endpoint is decided on the strict side required by frozen v10. -/ theorem ndA5AffinePositionCorrection_ePrime_pos_le {B n M : ℕ} {C : ℝ} {E : Set ℕ} {as : List ℕ+} (facts : Tao.TaoSection5AffineSourceScaleFacts B) (hlogB : (300000 : ℝ) ≤ Real.log B) (hM : M ∈ Tao.taoSection5EPrime B E) (ht2 : NDA5T2ScaleFacts B n C) (hlen : as.length = n - Tao.taoSection5M0 B) : 0 < ndA5AffinePositionCorrection as M ∧ ndA5AffinePositionCorrection as M ≤ (B : ℝ) ^ (-(9 / 10 : ℝ)) := by let r : ℝ := (Tao.taoOffsetList as : ℝ) / (M : ℝ) have hpacket := ndA5_exactAffineCorrection_ePrime_packet facts hlogB hM ht2 hlen have hasLengthPos : 0 < as.length := by rw [hlen] exact ht2.nu_pos have hoffsetQ : 0 < Tao.taoOffsetList as := ndA5_taoOffsetList_pos_of_length_pos as hasLengthPos have hoffset : 0 < (Tao.taoOffsetList as : ℝ) := Rat.cast_pos.mpr hoffsetQ have hMposNat : 0 < M := Odd.pos (Tao.mem_taoSection5EPrime_iff.mp hM).2.2.1 have hMpos : (0 : ℝ) < M := by exact_mod_cast hMposNat have hrpos : 0 < r := by dsimp [r] exact div_pos hoffset hMpos have hrlt : r < 1 := by have := hpacket.2.2.1 simpa only [r] using (sub_pos.mp this) unfold ndA5AffinePositionCorrection constructor · exact neg_pos.mpr (Real.log_neg (by linarith) (by linarith)) · simpa only [r] using hpacket.2.2.2.2 /-- With strict membership on both sides, a nominal-only point lies in the closed-at-correction lower mismatch strip. -/ theorem ndA5_strict_nominal_not_strict_exact_mem_lowerStrip {beta uExact uNominal delta : ℝ} (heq : uExact = uNominal - delta) (hdelta : 0 ≤ delta) (hnominal : ndA5StrictPosition beta uNominal) (hexact : ¬ ndA5StrictPosition beta uExact) : uNominal ∈ Set.Ioc (0 : ℝ) delta := by rcases hnominal with ⟨hnominal0, hnominalBeta⟩ constructor · exact hnominal0 · by_contra hnot have hdeltaLt : delta < uNominal := lt_of_not_ge hnot apply hexact constructor <;> unfold ndA5StrictPosition at * <;> linarith /-- With strict membership on both sides, an exact-only point lies in the half-open upper mismatch strip. -/ theorem ndA5_strict_exact_not_strict_nominal_mem_upperStrip {beta uExact uNominal delta : ℝ} (heq : uExact = uNominal - delta) (hdelta : 0 ≤ delta) (hexact : ndA5StrictPosition beta uExact) (hnominal : ¬ ndA5StrictPosition beta uNominal) : uNominal ∈ Set.Ico beta (beta + delta) := by rcases hexact with ⟨hexact0, hexactBeta⟩ constructor · by_contra hnot apply hnominal constructor <;> unfold ndA5StrictPosition at * <;> linarith · linarith /-- A nominal strict point excluded from the actual half-open band lies in the genuinely open lower strip. Equality `uNominal = delta` gives `uExact = 0`, which is physically included and is therefore absent here. -/ theorem ndA5_strict_nominal_not_halfOpen_exact_mem_lowerStrip {beta uExact uNominal delta : ℝ} (heq : uExact = uNominal - delta) (hdelta : 0 ≤ delta) (hnominal : ndA5StrictPosition beta uNominal) (hexact : ¬ ndA5HalfOpenPosition beta uExact) : uNominal ∈ Set.Ioo (0 : ℝ) delta := by rcases hnominal with ⟨hnominal0, hnominalBeta⟩ constructor · exact hnominal0 · by_contra hnot have hdeltaLe : delta ≤ uNominal := le_of_not_gt hnot apply hexact constructor <;> unfold ndA5HalfOpenPosition at * <;> linarith /-- Exact half-open membership but failed nominal strict membership is either the sole zero-correction lower endpoint or lies in the upper mismatch strip. -/ theorem ndA5_halfOpen_exact_not_strict_nominal_boundary_or_upperStrip {beta uExact uNominal delta : ℝ} (heq : uExact = uNominal - delta) (hdelta : 0 ≤ delta) (hexact : ndA5HalfOpenPosition beta uExact) (hnominal : ¬ ndA5StrictPosition beta uNominal) : (uNominal = 0 ∧ uExact = 0 ∧ delta = 0) ∨ uNominal ∈ Set.Ico beta (beta + delta) := by rcases hexact with ⟨hexact0, hexactBeta⟩ have hnominal0 : 0 ≤ uNominal := by linarith by_cases hzero : uNominal = 0 · left have hdeltaZero : delta = 0 := by linarith exact ⟨hzero, by linarith, hdeltaZero⟩ · right have hnominalPos : 0 < uNominal := lt_of_le_of_ne hnominal0 (Ne.symm hzero) constructor · by_contra hnot apply hnominal constructor <;> unfold ndA5StrictPosition at * <;> linarith · linarith /-- With a genuinely positive affine correction, the zero endpoint branch is impossible, so every physical-only point lies in the upper mismatch strip. -/ theorem ndA5_halfOpen_exact_not_strict_nominal_mem_upperStrip_of_delta_pos {beta uExact uNominal delta : ℝ} (heq : uExact = uNominal - delta) (hdelta : 0 < delta) (hexact : ndA5HalfOpenPosition beta uExact) (hnominal : ¬ ndA5StrictPosition beta uNominal) : uNominal ∈ Set.Ico beta (beta + delta) := by rcases ndA5_halfOpen_exact_not_strict_nominal_boundary_or_upperStrip heq hdelta.le hexact hnominal with hboundary | hupper · exfalso linarith [hboundary.2.2] · exact hupper /-- Endpoint-shaped once-only strip packet for an exact affine source in an actual A5 band. This is the direct `ℓ7` interface: actual membership is read from the half-open finite carrier, nominal membership is strict, and every mismatch is confined to a strip of width at most `B^-0.9`. -/ theorem ndA5_affineBandMismatch_mem_strips_ePrime {B n M j : ℕ} {C : ℝ} {E : Set ℕ} {branch : Tao.TaoSection5SourceBranch} {as : List ℕ+} {N : Tao.TaoOddNat} (facts : Tao.TaoSection5AffineSourceScaleFacts B) (hlogB : (300000 : ℝ) ≤ Real.log B) (hM : M ∈ Tao.taoSection5EPrime B E) (ht2 : NDA5T2ScaleFacts B n C) (hlen : as.length = n - Tao.taoSection5M0 B) (hAff : Tao.taoAffList as (N.1 : ℚ) = (M : ℚ)) : let beta := ndA5BandBeta B branch let z := ndA5BandLower B branch j let uNominal := ndA5NominalAffinePosition z M (n - Tao.taoSection5M0 B) (Tao.taoTupleWeight as) let delta := ndA5AffinePositionCorrection as M 0 < delta ∧ delta ≤ (B : ℝ) ^ (-(9 / 10 : ℝ)) ∧ (ndA5StrictPosition beta uNominal ∧ N.1 ∉ ndA5OddBand B branch j → uNominal ∈ Set.Ioo (0 : ℝ) delta) ∧ (N.1 ∈ ndA5OddBand B branch j ∧ ¬ ndA5StrictPosition beta uNominal → uNominal ∈ Set.Ico beta (beta + delta)) := by dsimp only have hB := facts.schedule.one_le_B have hz : 0 < ndA5BandLower B branch j := by have hBpos : (0 : ℝ) < B := by exact_mod_cast hB have hYpos : 0 < Tao.taoSection5SourceY B branch := by rw [Tao.taoSection5SourceY_eq_branch_rpow] exact Real.rpow_pos_of_pos hBpos _ unfold ndA5BandLower exact mul_pos hYpos (Real.exp_pos _) have heq := ndA5_exactAffinePosition_eq_nominal_sub_correction hz hlen hAff have hdelta := ndA5AffinePositionCorrection_ePrime_pos_le facts hlogB hM ht2 hlen have hcarrier := mem_ndA5OddBand_iff_halfOpen_exactPosition (branch := branch) (j := j) hB (Nat.odd_iff.mp N.2) refine ⟨hdelta.1, hdelta.2, ?_, ?_⟩ · rintro ⟨hnominal, hnotMem⟩ apply ndA5_strict_nominal_not_halfOpen_exact_mem_lowerStrip heq hdelta.1.le hnominal intro hhalfOpen exact hnotMem (hcarrier.2 hhalfOpen) · rintro ⟨hmem, hnotNominal⟩ apply ndA5_halfOpen_exact_not_strict_nominal_mem_upperStrip_of_delta_pos heq hdelta.1 (hcarrier.1 hmem) hnotNominal end end ND end Erdos1135