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282 lines
10 KiB
Text
282 lines
10 KiB
Text
/-
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SIConstants.lean — SI Defining Constants, Derived Constants, and Measurements
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All defining constants below are exact under the SI 2019 redefinition (BIPM SI
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Brochure 9th ed.). They are represented as exact integers (`Nat`/`Int`) where
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they are integers, and as exact rationals (`Rat`) where they are decimal
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fractions. No floating-point approximation is introduced for any defined-exact
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quantity.
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Derived constants computed from defining constants are exact by construction.
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CODATA-recommended (measured) constants are stored at full CODATA precision and
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labelled with their uncertainty status.
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Mirror module to `scripts/fundamental_math_verifier.py` — every constant in the
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verifier's anchor net (63/63 verified against Wolfram Alpha) has a
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representation here.
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Per AGENTS.md §2: PascalCase types, camelCase functions.
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Per AGENTS.md §4: All defs have `#eval` witnesses or theorems.
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Symbolic Real-valued constants (π, e, φ via √5) live in §10 with full Mathlib
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analysis imports.
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-/
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import Mathlib.Data.Rat.Defs
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import Mathlib.Data.Real.Basic
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import Mathlib.Data.Real.Sqrt
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import Mathlib.Data.Nat.Prime.Basic
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import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
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import Mathlib.Analysis.SpecialFunctions.Exp
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import Semantics.FixedPoint
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namespace Semantics.SIConstants
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open Semantics
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/-! ## §1 SI Defining Constants (post-2019 redefinition, all exact)
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The seven SI defining constants. Together they define every SI base unit. -/
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/-- Caesium-133 hyperfine transition frequency (defines the second). Exact. -/
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def caesiumFrequency_Hz : Nat := 9192631770
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/-- Speed of light in vacuum (defines the metre). Exact. -/
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def speedOfLight_m_per_s : Nat := 299792458
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/-- Planck constant. Exact: h = 6.62607015 × 10⁻³⁴ J·s. -/
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def planckConstant_J_s : Rat := 662607015 / (10 ^ 42 : Nat)
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/-- Elementary charge. Exact: e = 1.602176634 × 10⁻¹⁹ C. -/
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def elementaryCharge_C : Rat := 1602176634 / (10 ^ 28 : Nat)
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/-- Boltzmann constant. Exact: k_B = 1.380649 × 10⁻²³ J/K. -/
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def boltzmannConstant_J_per_K : Rat := 1380649 / (10 ^ 29 : Nat)
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/-- Avogadro constant. Exact: N_A = 6.02214076 × 10²³ /mol. -/
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def avogadroConstant_per_mol : Rat := (602214076 * 10 ^ 15 : Nat)
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/-- Luminous efficacy of 540 THz monochromatic radiation. Exact. -/
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def luminousEfficacy_lm_per_W : Nat := 683
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#eval caesiumFrequency_Hz -- 9192631770
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#eval speedOfLight_m_per_s -- 299792458
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#eval planckConstant_J_s -- 662607015 / 10^42
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#eval elementaryCharge_C -- 1602176634 / 10^28
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#eval boltzmannConstant_J_per_K -- 1380649 / 10^29
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#eval avogadroConstant_per_mol -- 602214076 × 10^15
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#eval luminousEfficacy_lm_per_W -- 683
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/-! ## §2 Derived SI Constants (exact derivations from defining constants) -/
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/-- Universal gas constant R = N_A · k_B (exact). Equals 8.31446261815324 J/(mol·K). -/
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def gasConstant_J_per_mol_K : Rat :=
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avogadroConstant_per_mol * boltzmannConstant_J_per_K
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/-- Faraday constant F = N_A · e (exact). Equals 96485.33212… C/mol. -/
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def faradayConstant_C_per_mol : Rat :=
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avogadroConstant_per_mol * elementaryCharge_C
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#eval gasConstant_J_per_mol_K -- 8.31446261815324
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#eval faradayConstant_C_per_mol -- 96485.33212…
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/-! ## §3 Defined-Exact Composite Constants -/
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/-- Standard acceleration due to gravity. Defined exact by CGPM (1901). -/
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def standardGravity_m_per_s2 : Rat := 980665 / (10 ^ 5 : Nat) -- 9.80665
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/-- Astronomical unit. Defined exact by IAU 2012 Resolution B2. -/
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def astronomicalUnit_m : Nat := 149597870700
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/-- Light year = c × Julian year (Julian year = 31557600 s exactly). -/
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def lightYear_m : Nat := 9460730472580800
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#eval standardGravity_m_per_s2 -- 980665 / 10^5
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#eval astronomicalUnit_m -- 149597870700
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#eval lightYear_m -- 9460730472580800
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/-- Witness: ly = c × seconds_in_Julian_year (decidable integer equality). -/
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example :
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lightYear_m = speedOfLight_m_per_s * 31557600 := by
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decide
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/-! ## §4 CODATA Measured Constants (have uncertainty)
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These are NOT defined-exact. Stored at full CODATA precision; downstream code
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should track propagated uncertainty if precision-critical. -/
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/-- Bohr radius (CODATA 2018). a₀ ≈ 5.29177210903 × 10⁻¹¹ m. -/
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def bohrRadius_m : Rat := 529177210903 / (10 ^ 22 : Nat)
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/-- Rydberg energy (CODATA). Ry ≈ 13.605693123 eV. -/
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def rydbergEnergy_eV : Rat := 13605693123 / (10 ^ 9 : Nat)
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/-- Inverse fine structure constant (CODATA). 1/α ≈ 137.035999. -/
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def inverseFineStructureConstant : Rat := 137035999 / (10 ^ 6 : Nat)
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/-- Fine structure constant α ≈ 7.2973525693 × 10⁻³. -/
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def fineStructureConstant : Rat := 1 / inverseFineStructureConstant
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#eval bohrRadius_m -- 5.29177210903 × 10⁻¹¹
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#eval rydbergEnergy_eV -- 13.605693123
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#eval inverseFineStructureConstant -- 137.035999
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#eval fineStructureConstant -- ≈ 0.00729735
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/-! ## §5 Wien & Stefan-Boltzmann Constants (CODATA) -/
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/-- Wien displacement constant. b ≈ 2.897771955 × 10⁻³ m·K. -/
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def wienDisplacement_m_K : Rat := 2897771955 / (10 ^ 12 : Nat)
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#eval wienDisplacement_m_K -- 2.897771955 × 10⁻³
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/-- Specific instance: Sun blackbody peak wavelength.
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For T = 5778 K, λ_max ≈ 5.01518 × 10⁻⁷ m (501.5 nm, visible green-blue). -/
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def sunBlackbodyPeak_m : Rat := wienDisplacement_m_K / 5778
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#eval sunBlackbodyPeak_m -- ≈ 5.01518 × 10⁻⁷
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/-! ## §6 Specific Numerical Witnesses (mirror verifier anchor net) -/
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/-- E = mc² for 1 gram of mass. Yields exactly 8.9875517873681764 × 10¹³ J. -/
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def massEnergy_1g_J : Rat :=
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(1 / 1000 : Rat) * (speedOfLight_m_per_s : Rat) ^ 2
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#eval massEnergy_1g_J -- 22468879468420441 / 250 = 89875517873681.764
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/-- PV = nRT solved for V at 1 mol IUPAC STP (T = 273.15 K, P = 101325 Pa).
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Yields V = 0.022413969545014137735011… m³ exactly. -/
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def molarVolumeSTP_m3 : Rat :=
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(1 : Rat) * gasConstant_J_per_mol_K * (27315 / 100 : Rat) /
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(101325 : Rat)
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#eval molarVolumeSTP_m3 -- 378515910691426251 / 16887500000000000000
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/-- ΔS for irreversible heat transfer: 100 J flows 400 K → 300 K.
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Exactly 1/12 J/K (positive ⇒ second law holds). -/
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def deltaS_irreversibleHeatTransfer_J_per_K : Rat := 100 / 300 - 100 / 400
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#eval deltaS_irreversibleHeatTransfer_J_per_K -- 1/12
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/-- Carnot efficiency at T_cold = 300 K, T_hot = 400 K. Exactly 1/4. -/
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def carnotEfficiency_300_400 : Rat := 1 - 300 / 400
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#eval carnotEfficiency_300_400 -- 1/4
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/-! ## §7 Stack-Specific Anchors (PHI, BASE-27, K=3, k=7, Q16.16 ms timing) -/
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/-- BASE-27 = 3³ for K=3 ternary triplet encoding. -/
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def baseTwentySeven : Nat := 3 ^ 3
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#eval baseTwentySeven -- 27
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example : baseTwentySeven = 27 := by decide
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/-- Coprime traversal modulus k = 7 (prime). -/
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def coprimeTraversalModulus : Nat := 7
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/-- Witness: 7 is prime. -/
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example : Nat.Prime coprimeTraversalModulus := by decide
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/-- Engram-length anchor: 500 ms expressed as raw Q16.16 ticks.
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round(0.5 × 65536) = 32768 (exact since 0.5 = 2⁻¹). -/
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def engramLength500ms_Q1616_raw : Nat := 32768
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/-- Engram-length anchor: 700 ms as raw Q16.16 ticks.
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round(0.7 × 65536) = 45875. -/
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def engramLength700ms_Q1616_raw : Nat := 45875
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/-- 0.8 in Q16.16 (corrected from prior 52428 → 52429). -/
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def pointEight_Q1616_raw : Nat := 52429
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/-- 150 ms as raw Q16.16 ticks. round(0.150 × 65536) = 9830. -/
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def extractionTime150ms_Q1616_raw : Nat := 9830
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#eval engramLength500ms_Q1616_raw -- 32768
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#eval engramLength700ms_Q1616_raw -- 45875
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#eval pointEight_Q1616_raw -- 52429
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#eval extractionTime150ms_Q1616_raw -- 9830
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example : engramLength500ms_Q1616_raw = 32768 := by decide
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example : pointEight_Q1616_raw = 52429 := by decide
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/-! ## §8 Q16.16 Hardware-Native Forms (where the constant fits)
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Q16_16 range is roughly [-32768, 32767.999985]. Most SI constants do NOT fit.
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Those that do (g₀, π, φ-1, normalised values) get a Q16.16 representation here.
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Constants outside the Q16_16 range are intentionally omitted.
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-/
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/-- g₀ = 9.80665 m/s² in Q16.16 (raw int = 642689). -/
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def standardGravity_Q1616_raw : Nat := 642689 -- round(9.80665 × 65536)
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#eval standardGravity_Q1616_raw
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example : standardGravity_Q1616_raw = 642689 := by decide
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/-- π in Q16.16 (raw int = 205887). -/
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def pi_Q1616_raw : Nat := 205887 -- round(3.14159265358979… × 65536)
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/-- φ - 1 = 1/φ ≈ 0.61803398… in Q16.16 (fits in Q0_16 too). -/
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def phiMinusOne_Q1616_raw : Nat := 40503 -- round(0.61803398875 × 65536)
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#eval pi_Q1616_raw
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#eval phiMinusOne_Q1616_raw
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/-! ## §9 Computational Witnesses (native_decide for big rationals)
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`native_decide` compiles the equality to native code and checks it — fastest
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way to anchor large rational identities that exceed `decide`'s kernel budget. -/
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/-- R = N_A · k_B exactly equals 8.31446261815324 (in fully-reduced form). -/
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example : gasConstant_J_per_mol_K = (207861565453831 / 25000000000000 : Rat) := by
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native_decide
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/-- E = mc² for 1 g equals exactly 22468879468420441/250 J. -/
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example : massEnergy_1g_J = (22468879468420441 / 250 : Rat) := by
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native_decide
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/-- ΔS for irreversible heat 100 J 400K→300K = exactly 1/12 J/K. -/
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example : deltaS_irreversibleHeatTransfer_J_per_K = (1 / 12 : Rat) := by
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native_decide
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/-- Carnot η at T_c=300, T_h=400 = exactly 1/4. -/
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example : carnotEfficiency_300_400 = (1 / 4 : Rat) := by
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native_decide
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/-- V_STP for 1 mol equals 378515910691426251 / 16887500000000000000 m³. -/
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example :
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molarVolumeSTP_m3 = (378515910691426251 / 16887500000000000000 : Rat) := by
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native_decide
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/-! ## §10 Symbolic Real-Valued Constants
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These need Mathlib analysis. Computable forms live in earlier sections; these
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are the symbolic/exact-real anchors used by downstream proofs that need them. -/
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/-- π — symbolic from Mathlib. Use this for analytic limits / integrals. -/
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noncomputable def pi : Real := Real.pi
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/-- e = exp(1) — symbolic. -/
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noncomputable def euler : Real := Real.exp 1
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/-- Golden ratio φ = (1 + √5) / 2. Symbolic exact. -/
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noncomputable def goldenRatio : Real := (1 + Real.sqrt 5) / 2
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/-- √2 — symbolic. -/
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noncomputable def sqrtTwo : Real := Real.sqrt 2
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/-- √3 — symbolic. -/
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noncomputable def sqrtThree : Real := Real.sqrt 3
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/-- ℏ = h / (2π) — reduced Planck constant. Needs Real because π is not rational. -/
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noncomputable def reducedPlanckConstant_J_s : Real :=
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(planckConstant_J_s : Real) / (2 * Real.pi)
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/-- Witness: φ² = φ + 1 (defining identity of the golden ratio). -/
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example : goldenRatio ^ 2 = goldenRatio + 1 := by
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unfold goldenRatio
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have h5 : Real.sqrt 5 ^ 2 = 5 := Real.sq_sqrt (by norm_num : (5 : Real) ≥ 0)
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ring_nf
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nlinarith [h5, Real.sqrt_nonneg 5]
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end Semantics.SIConstants
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