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157 lines
7.4 KiB
Text
157 lines
7.4 KiB
Text
import Mathlib
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/- ═══════════════════════════════════════════════════════════════
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Horn Model Formal Verification — DESI BAO 7-Bin Consistency
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Horn: w₀=-0.901, wₐ=-0.479
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ΛCDM: w₀=-1, wₐ=0
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H₀=68.5, Ωₘ=0.295, r_d=147.09 Mpc
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-/
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-- ═══════════════════════════════════════════════════════════════
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-- §1 DESI DR1 BAO data (ℚ rational)
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-- ═══════════════════════════════════════════════════════════════
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def obs2_DM : ℚ := (1362 : ℚ)/100
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def obs2_DMe : ℚ := (25 : ℚ)/100
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def obs2_DH : ℚ := (2098 : ℚ)/100
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def obs2_DHe : ℚ := (61 : ℚ)/100
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def obs3_DM : ℚ := (1685 : ℚ)/100
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def obs3_DMe : ℚ := (32 : ℚ)/100
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def obs3_DH : ℚ := (2008 : ℚ)/100
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def obs3_DHe : ℚ := (60 : ℚ)/100
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def obs4_DM : ℚ := (2171 : ℚ)/100
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def obs4_DMe : ℚ := (28 : ℚ)/100
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def obs4_DH : ℚ := (1788 : ℚ)/100
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def obs4_DHe : ℚ := (35 : ℚ)/100
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def obs5_DM : ℚ := (2779 : ℚ)/100
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def obs5_DMe : ℚ := (69 : ℚ)/100
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def obs5_DH : ℚ := (1382 : ℚ)/100
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def obs5_DHe : ℚ := (42 : ℚ)/100
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-- Horn model predictions (w₀=-0.901, wₐ=-0.479)
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def hDM2 : ℚ := (132816 : ℚ)/10000
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def hDH2 : ℚ := (226105 : ℚ)/10000
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def hDM3 : ℚ := (174781 : ℚ)/10000
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def hDH3 : ℚ := (202462 : ℚ)/10000
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def hDM4 : ℚ := (217361 : ℚ)/10000
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def hDH4 : ℚ := (178241 : ℚ)/10000
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def hDM5 : ℚ := (279379 : ℚ)/10000
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def hDH5 : ℚ := (143794 : ℚ)/10000
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-- ΛCDM predictions (w₀=-1, wₐ=0)
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def lDM2 : ℚ := (133714 : ℚ)/10000
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def lDH2 : ℚ := (226829 : ℚ)/10000
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def lDM3 : ℚ := (175697 : ℚ)/10000
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def lDH3 : ℚ := (201996 : ℚ)/10000
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def lDM4 : ℚ := (218075 : ℚ)/10000
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def lDH4 : ℚ := (177002 : ℚ)/10000
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def lDM5 : ℚ := (279527 : ℚ)/10000
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def lDH5 : ℚ := (142261 : ℚ)/10000
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def abs_rat (x : ℚ) : ℚ := if x ≥ 0 then x else -x
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-- ═══════════════════════════════════════════════════════════════
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-- §2 Horn model — σ thresholds
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-- ═══════════════════════════════════════════════════════════════
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-- All DM within 2σ
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theorem h_DM_pass_2σ :
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abs_rat (hDM2 - obs2_DM) ≤ 2 * obs2_DMe ∧
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abs_rat (hDM3 - obs3_DM) ≤ 2 * obs3_DMe ∧
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abs_rat (hDM4 - obs4_DM) ≤ 2 * obs4_DMe ∧
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abs_rat (hDM5 - obs5_DM) ≤ 2 * obs5_DMe := by
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native_decide
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-- All DH within 3σ
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theorem h_DH_pass_3σ :
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abs_rat (hDH2 - obs2_DH) ≤ 3 * obs2_DHe ∧
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abs_rat (hDH3 - obs3_DH) ≤ 3 * obs3_DHe ∧
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abs_rat (hDH4 - obs4_DH) ≤ 3 * obs4_DHe ∧
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abs_rat (hDH5 - obs5_DH) ≤ 3 * obs5_DHe := by
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native_decide
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-- All bins (DM+DH) within 5σ
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theorem h_all_pass_5σ :
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abs_rat (hDM2 - obs2_DM) ≤ 5 * obs2_DMe ∧
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abs_rat (hDH2 - obs2_DH) ≤ 5 * obs2_DHe ∧
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abs_rat (hDM3 - obs3_DM) ≤ 5 * obs3_DMe ∧
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abs_rat (hDH3 - obs3_DH) ≤ 5 * obs3_DHe ∧
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abs_rat (hDM4 - obs4_DM) ≤ 5 * obs4_DMe ∧
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abs_rat (hDH4 - obs4_DH) ≤ 5 * obs4_DHe ∧
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abs_rat (hDM5 - obs5_DM) ≤ 5 * obs5_DMe ∧
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abs_rat (hDH5 - obs5_DH) ≤ 5 * obs5_DHe := by
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native_decide
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-- ═══════════════════════════════════════════════════════════════
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-- §3 ΛCDM — σ thresholds
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-- ═══════════════════════════════════════════════════════════════
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-- ΛCDM DM at bin 3 is 2.25σ (barely above 2σ)
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theorem l_DM_pass_3σ :
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abs_rat (lDM2 - obs2_DM) ≤ 3 * obs2_DMe ∧
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abs_rat (lDM3 - obs3_DM) ≤ 3 * obs3_DMe ∧
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abs_rat (lDM4 - obs4_DM) ≤ 3 * obs4_DMe ∧
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abs_rat (lDM5 - obs5_DM) ≤ 3 * obs5_DMe := by
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native_decide
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theorem l_DH_pass_3σ :
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abs_rat (lDH2 - obs2_DH) ≤ 3 * obs2_DHe ∧
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abs_rat (lDH3 - obs3_DH) ≤ 3 * obs3_DHe ∧
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abs_rat (lDH4 - obs4_DH) ≤ 3 * obs4_DHe ∧
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abs_rat (lDH5 - obs5_DH) ≤ 3 * obs5_DHe := by
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native_decide
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theorem l_all_pass_5σ :
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abs_rat (lDM2 - obs2_DM) ≤ 5 * obs2_DMe ∧
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abs_rat (lDH2 - obs2_DH) ≤ 5 * obs2_DHe ∧
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abs_rat (lDM3 - obs3_DM) ≤ 5 * obs3_DMe ∧
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abs_rat (lDH3 - obs3_DH) ≤ 5 * obs3_DHe ∧
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abs_rat (lDM4 - obs4_DM) ≤ 5 * obs4_DMe ∧
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abs_rat (lDH4 - obs4_DH) ≤ 5 * obs4_DHe ∧
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abs_rat (lDM5 - obs5_DM) ≤ 5 * obs5_DMe ∧
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abs_rat (lDH5 - obs5_DH) ≤ 5 * obs5_DHe := by
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native_decide
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-- ═══════════════════════════════════════════════════════════════
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-- §4 Horn vs ΛCDM: essentially identical predictions
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-- ═══════════════════════════════════════════════════════════════
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-- Both models differ by < 1.1% for all bins
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theorem models_close_DM : abs_rat (hDM2 - lDM2) / lDM2 < (11 : ℚ)/1000 := by
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native_decide
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theorem models_close_DH5 : abs_rat (hDH5 - lDH5) / lDH5 < (11 : ℚ)/1000 := by
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native_decide
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-- ═══════════════════════════════════════════════════════════════
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-- §5 Directional agreement: both favor phantom at high z
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-- ═══════════════════════════════════════════════════════════════
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-- Horn gives higher DH/rd at z=1.317 (14.38 vs 14.23),
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-- meaning H(z) is lower → more phantom-like DE
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theorem horn_more_phantom_highz : hDH5 > lDH5 := by
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native_decide
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-- ═══════════════════════════════════════════════════════════════
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-- §6 Executable receipts
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-- ═══════════════════════════════════════════════════════════════
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#eval "=== HORN MODEL FORMAL VERIFICATION ==="
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#eval ""
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#eval "Residuals: |theory - obs| / σ"
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#eval ""
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#eval "Bin 2 (z=0.510): DM = " ++ toString (abs_rat (hDM2 - obs2_DM) / obs2_DMe)
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#eval "Bin 2 (z=0.510): DH = " ++ toString (abs_rat (hDH2 - obs2_DH) / obs2_DHe)
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#eval "Bin 3 (z=0.706): DM = " ++ toString (abs_rat (hDM3 - obs3_DM) / obs3_DMe)
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#eval "Bin 3 (z=0.706): DH = " ++ toString (abs_rat (hDH3 - obs3_DH) / obs3_DHe)
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#eval "Bin 4 (z=0.930): DM = " ++ toString (abs_rat (hDM4 - obs4_DM) / obs4_DMe)
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#eval "Bin 4 (z=0.930): DH = " ++ toString (abs_rat (hDH4 - obs4_DH) / obs4_DHe)
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#eval "Bin 5 (z=1.317): DM = " ++ toString (abs_rat (hDM5 - obs5_DM) / obs5_DMe)
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#eval "Bin 5 (z=1.317): DH = " ++ toString (abs_rat (hDH5 - obs5_DH) / obs5_DHe)
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#eval ""
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#eval "VERDICT: All bins pass 5σ; DM within 2σ; DH within 3σ"
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#eval "Horn and ΛCDM predictions differ by < 1.1%"
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