TauLib.BookIV.Calibration.DimensionlessNearMatch
TauLib.BookIV.Calibration.DimensionlessNearMatch
Maps τ-derived dimensionless couplings to physical constants with range proofs.
Registry Cross-References
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[IV.D28] Weinberg Near-Match —
WeinbergNearMatch,weinberg_range -
[IV.D29] Strong Near-Match —
StrongNearMatch,strong_range -
[IV.P05] All Near-Matches in Range —
all_near_matches_in_range
Mathematical Content
Dimensionless Near-Matches
The τ-framework produces dimensionless couplings from ι_τ alone. Three such couplings can be compared to measured physical constants:
τ-coupling Formula τ-value SI value Deviation
α_EM (8/15)·ι_τ⁴ ~1/137.9 1/137.036 0.6%
sin²θ_W ι_τ(1−ι_τ) ~0.2249 0.23121 2.7%
α_s(M_Z) 2·ι_τ³/(1−ι_τ) ~0.1208 0.1180 2.4%
All three are CONJECTURAL: the numerical near-matches are observed but the exact mechanism connecting ι_τ formulas to measured values requires the full calibration cascade (Parts III–X).
Ground Truth Sources
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Book IV Part II ch10 (Sector Couplings), ch11 (Fine-Structure α)
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CODATA 2022 for experimental values
Tau.BookIV.Calibration.WeinbergNearMatch
source structure Tau.BookIV.Calibration.WeinbergNearMatch :Type
[IV.D28] Weinberg angle near-match: κ(A,D) = ι_τ(1−ι_τ) vs sin²θ_W.
τ-derived: κ(A,D) ≈ 0.2249 Experimental: sin²θ_W = 0.23121(4) (on-shell, CODATA 2022) Deviation: ~2.7%
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tau_coupling : Sectors.CouplingFormula τ-derived coupling formula.
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si_value : SIConstant SI reference value.
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scope : String Scope: established (near-match with range proof).
Instances For
Tau.BookIV.Calibration.weinberg_range
source theorem Tau.BookIV.Calibration.weinberg_range :Sectors.kappa_AD.numer * 1000 > 224 * Sectors.kappa_AD.denom ∧ Sectors.kappa_AD.numer * 1000 < 226 * Sectors.kappa_AD.denom
κ(A,D) is in the range (0.224, 0.226).
Tau.BookIV.Calibration.weinberg_undershoots
source theorem Tau.BookIV.Calibration.weinberg_undershoots :Sectors.kappa_AD.numer * si_weinberg_sin2.denom < si_weinberg_sin2.numer * Sectors.kappa_AD.denom
κ(A,D) < sin²θ_W: the τ-approximation undershoots the experimental value.
Tau.BookIV.Calibration.weinberg_deviation_lt_3pct
source theorem Tau.BookIV.Calibration.weinberg_deviation_lt_3pct :(si_weinberg_sin2.numer * Sectors.kappa_AD.denom - Sectors.kappa_AD.numer * si_weinberg_sin2.denom) * 100 < 3 * si_weinberg_sin2.numer * Sectors.kappa_AD.denom
The Weinberg deviation is less than 3%. (sin²θ_W − κ(A,D)) × 100 < 3 × sin²θ_W Cross-multiplied on Nat pairs.
Tau.BookIV.Calibration.StrongNearMatch
source structure Tau.BookIV.Calibration.StrongNearMatch :Type
[IV.D29] Strong coupling near-match: 2·κ(C,C) vs α_s(M_Z).
τ-derived: 2·κ(C) = 2·ι_τ³/(1−ι_τ) ≈ 0.1208 Experimental: α_s(M_Z) = 0.1180(9) Deviation: ~2.4%
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tau_coupling : Sectors.CouplingFormula τ-derived coupling formula (doubled).
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factor : ℕ Multiplication factor.
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si_value : SIConstant SI reference value.
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scope : String Scope: established (near-match with range proof).
Instances For
Tau.BookIV.Calibration.strong_range
source theorem Tau.BookIV.Calibration.strong_range :2 * Sectors.kappa_CC.numer * 1000 > 119 * Sectors.kappa_CC.denom ∧ 2 * Sectors.kappa_CC.numer * 1000 < 122 * Sectors.kappa_CC.denom
2·κ(C) is in the range (0.119, 0.122), bracketing α_s ≈ 0.1180.
Tau.BookIV.Calibration.strong_overshoots
source theorem Tau.BookIV.Calibration.strong_overshoots :2 * Sectors.kappa_CC.numer * si_strong_coupling.denom > si_strong_coupling.numer * Sectors.kappa_CC.denom
2·κ(C) > α_s: the τ-approximation overshoots the experimental value.
Tau.BookIV.Calibration.strong_deviation_lt_3pct
source theorem Tau.BookIV.Calibration.strong_deviation_lt_3pct :(2 * Sectors.kappa_CC.numer * si_strong_coupling.denom - si_strong_coupling.numer * Sectors.kappa_CC.denom) * 100 < 3 * si_strong_coupling.numer * Sectors.kappa_CC.denom
The strong coupling deviation is less than 3%. (2κ(C) − α_s) × 100 < 3 × α_s Cross-multiplied on Nat pairs.
Tau.BookIV.Calibration.alpha_spectral_overshoots
source theorem Tau.BookIV.Calibration.alpha_spectral_overshoots :Sectors.alpha_spectral_denom * si_alpha_inverse.denom > si_alpha_inverse.numer * Sectors.alpha_spectral_numer
The spectral 1/α overshoots the experimental value: α_spectral < α_exp (i.e. 1/α_spectral > 1/α_exp). Spectral: 1/α ≈ 137.9, Experimental: 1/α ≈ 137.036.
Tau.BookIV.Calibration.all_near_matches_in_range
source theorem Tau.BookIV.Calibration.all_near_matches_in_range :(Sectors.alpha_spectral_denom > 137 * Sectors.alpha_spectral_numer ∧ Sectors.alpha_spectral_denom < 139 * Sectors.alpha_spectral_numer) ∧ (Sectors.kappa_AD.numer * 1000 > 224 * Sectors.kappa_AD.denom ∧ Sectors.kappa_AD.numer * 1000 < 226 * Sectors.kappa_AD.denom) ∧ 2 * Sectors.kappa_CC.numer * 1000 > 119 * Sectors.kappa_CC.denom ∧ 2 * Sectors.kappa_CC.numer * 1000 < 122 * Sectors.kappa_CC.denom
[IV.P05] All three dimensionless near-matches are within their respective range brackets. This is a structural observation, NOT a proof of correctness — all three are CONJECTURAL.
Tau.BookIV.Calibration.NearMatchEntry
source structure Tau.BookIV.Calibration.NearMatchEntry :Type
Near-match entry for the summary table.
- name : String
- tau_numer : ℕ
- tau_denom : ℕ
- si_numer : ℕ
- si_denom : ℕ
- overshoots : Bool Instances For
Tau.BookIV.Calibration.instReprNearMatchEntry.repr
source def Tau.BookIV.Calibration.instReprNearMatchEntry.repr :NearMatchEntry → ℕ → Std.Format
Equations
- One or more equations did not get rendered due to their size. Instances For
Tau.BookIV.Calibration.instReprNearMatchEntry
source instance Tau.BookIV.Calibration.instReprNearMatchEntry :Repr NearMatchEntry
Equations
- Tau.BookIV.Calibration.instReprNearMatchEntry = { reprPrec := Tau.BookIV.Calibration.instReprNearMatchEntry.repr }
Tau.BookIV.Calibration.near_match_table
source def Tau.BookIV.Calibration.near_match_table :List NearMatchEntry
The three dimensionless near-match entries. Equations
- One or more equations did not get rendered due to their size. Instances For
Tau.BookIV.Calibration.near_match_count
source theorem Tau.BookIV.Calibration.near_match_count :near_match_table.length = 3
Three near-match entries.