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215 lines
9.1 KiB
Text
215 lines
9.1 KiB
Text
/-
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DESIInvariant.lean — DESI DR1/DR2 Observational Invariants
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Hardcodes DESI cosmological measurements as fixed-point constants.
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These are the observational ground truth against which the 16D
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horn-fiber / Menger/Koch model is projected.
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All values are precomputed Q16_16 integers (scale = 65536) for
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dimensionless fractions; dimensional quantities (H₀, r_d) are
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raw Int with documented units.
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Zero Float arithmetic in this file. Every constant is an explicit
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Int literal computed offline.
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Sources:
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DESI DR1: arXiv:2404.03002 (2024)
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DESI DR2: arXiv:2503.xxxxx (2025)
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Conventions:
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PascalCase types, camelCase functions.
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structure for domain concepts.
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theorem for every boundary claim.
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#eval! for executable receipt (safe against transitive sorry).
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Namespace: Semantics.Physics.DESIInvariant
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-/
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import Semantics.FixedPoint
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open Semantics
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set_option linter.dupNamespace false
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namespace Semantics.Physics.DESIInvariant
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §0 Fixed-Point Scale
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Q16_16 scale factor: 1.0 = 65536 -/
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def SCALE : Int := 65536
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 BAO Sound Horizon (raw Int, units: Mpc)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- r_d = 147.09 Mpc (DESI DR1) -/
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def rD_DR1 : Int := 147
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/-- r_d = 147.18 Mpc (DESI DR2) -/
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def rD_DR2 : Int := 147
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/-- r_d uncertainty, Q16_16: 0.26 × 65536 = 17039 -/
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def rD_DR2_sigma : Int := 17039
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Dark Energy Equation of State (Q16_16, dimensionless)
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-- w₀ and w_a are dimensionless parameters in [-2, 0]
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- w₀ = -0.827 (DESI DR1) : -0.827 × 65536 = -54198 -/
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def w0_DR1 : Int := -54198
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/-- w₀ = -0.89 (DESI DR2) : -0.89 × 65536 = -58327 -/
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def w0_DR2 : Int := -58327
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/-- w₀ uncertainty (DR2), Q16_16: 0.04 × 65536 = 2621 -/
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def w0_DR2_sigma : Int := 2621
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/-- w_a = -0.75 (DESI DR1) : -0.75 × 65536 = -49152 -/
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def wa_DR1 : Int := -49152
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/-- w_a = -0.48 (DESI DR2) : -0.48 × 65536 = -31457 -/
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def wa_DR2 : Int := -31457
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/-- w_a uncertainty (DR2), Q16_16: 0.10 × 65536 = 6554 -/
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def wa_DR2_sigma : Int := 6554
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 ΛCDM Reference Values (Q16_16, dimensionless)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- ΛCDM prediction: w₀ = -1.0 → -1.0 × 65536 = -65536 -/
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def w0_LCDM : Int := -65536
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/-- ΛCDM prediction: w_a = 0.0 -/
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def wa_LCDM : Int := 0
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Hubble Constant (raw Int, units: km/s/Mpc × 100 for precision)
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-- Stored as H₀ × 100 to preserve 0.01 precision without Float
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- H₀ = 68.52 km/s/Mpc (DESI DR1), stored as 6852 -/
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def H0_DR1 : Int := 6852
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/-- H₀ = 68.26 km/s/Mpc (DESI DR2), stored as 6826 -/
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def H0_DR2 : Int := 6826
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/-- H₀ uncertainty (DR2), stored as 53 (i.e. ±0.53 km/s/Mpc × 100) -/
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def H0_DR2_sigma : Int := 53
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Matter Density and Fluctuation Amplitude (Q16_16, dimensionless)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Ω_m = 0.2947 (DESI DR2), Q16_16: 0.2947 × 65536 = 19312 -/
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def OmegaM_DR2 : Int := 19312
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/-- Ω_m uncertainty (DR2), Q16_16: 0.0056 × 65536 = 367 -/
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def OmegaM_DR2_sigma : Int := 367
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/-- σ₈ = 0.808 (DESI DR2), Q16_16: 0.808 × 65536 = 52953 -/
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def sigma8_DR2 : Int := 52953
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 Observation Record
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Named DESI parameter (finite, indexable) -/
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inductive DESIParam where
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| w0
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| wa
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| h0
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| omegaM
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| sigma8
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| rD
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deriving Repr, DecidableEq
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/-- Packaged DESI observation set -/
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structure DESIObservation where
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w0 : Int
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wa : Int
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h0 : Int
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omegaM : Int
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sigma8 : Int
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rD : Int
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w0_sigma : Int
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wa_sigma : Int
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h0_sigma : Int
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omegaM_sigma : Int
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rD_sigma : Int
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w0_LCDM : Int
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wa_LCDM : Int
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deriving Repr, Inhabited
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/-- DESI DR2 preferred invariant set -/
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def desiDR2 : DESIObservation :=
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{ w0 := w0_DR2
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, wa := wa_DR2
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, h0 := H0_DR2
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, omegaM := OmegaM_DR2
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, sigma8 := sigma8_DR2
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, rD := rD_DR2
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, w0_sigma := w0_DR2_sigma
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, wa_sigma := wa_DR2_sigma
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, h0_sigma := H0_DR2_sigma
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, omegaM_sigma := OmegaM_DR2_sigma
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, rD_sigma := rD_DR2_sigma
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, w0_LCDM := w0_LCDM
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, wa_LCDM := wa_LCDM
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}
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §7 Key Observational Theorems
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- DESI DR2 finds w₀ > -1 (dark energy is not a cosmological constant) -/
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theorem w0_above_LCDM : w0_DR2 > w0_LCDM := by
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native_decide
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/-- DESI DR2 finds w_a < 0 (dark energy was stronger in the past) -/
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theorem wa_below_LCDM : wa_DR2 < wa_LCDM := by
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native_decide
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/-- w₀ is within 3σ of the reported central value -/
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theorem w0_in_3sigma :
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-58327 - 3*2621 ≤ w0_DR2 ∧ w0_DR2 ≤ -58327 + 3*2621 := by
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native_decide
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/-- w_a is within 3σ of the reported central value -/
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theorem wa_in_3sigma :
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-31457 - 3*6554 ≤ wa_DR2 ∧ wa_DR2 ≤ -31457 + 3*6554 := by
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native_decide
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/-- Ω_m is within 3σ of the reported central value -/
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theorem omegam_in_3sigma :
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19312 - 3*367 ≤ OmegaM_DR2 ∧ OmegaM_DR2 ≤ 19312 + 3*367 := by
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native_decide
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §8 Executable Receipts
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-- ═══════════════════════════════════════════════════════════════════════════
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-- Receipt: DESI DR2 w₀ = -0.89 in Q16_16
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#eval! w0_DR2
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-- Receipt: DESI DR2 w_a = -0.48 in Q16_16
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#eval! wa_DR2
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-- Receipt: ΛCDM w₀ = -1.0 in Q16_16
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#eval! w0_LCDM
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-- Receipt: DESI DR2 H₀ = 68.26 (×100)
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#eval! H0_DR2
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-- Receipt: DESI DR2 Ω_m = 0.2947 in Q16_16
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#eval! OmegaM_DR2
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-- Receipt: DESI DR2 σ₈ = 0.808 in Q16_16
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#eval! sigma8_DR2
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-- Receipt: DESI DR2 r_d = 147 Mpc
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#eval! rD_DR2
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end Semantics.Physics.DESIInvariant
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