Constants Alpha Genesis Measurement Verdict Exp 048122 Taylor Floor

The Taylor floor

The exponential function exp(x) can be approximated by adding up a few terms of its Taylor series, the polynomial 1 + x + x²/2 + x³/6 and so on. The declaration exp_048122_taylor_floor is a lemma in the Recognition Science library that certifies a lower bound for such a partial sum at a specific input. The input is x = 0.48122, and the lemma proves that the 12-term Taylor polynomial, minus a bound on the error of truncation, is greater than 80902/50000, which equals 1.61804. This number is not arbitrary: it is just above the golden ratio φ ≈ 1.6180339.

The point of the bound is to compare two real numbers without computing them exactly. The library wants to show that the natural logarithm of φ is less than 0.48122. Since the exponential function is the inverse of the logarithm, this is equivalent to showing that exp(0.48122) > φ. The lemma provides a certified lower bound for exp(0.48122) that clears φ by a small margin. The margin is tiny, but it is enough to make the inequality rigorous.

In Recognition Science, this inequality is a step in a larger argument about the fine-structure constant α. The framework derives a first-order value for α from a seed of 44π, and this value is too large. The lemma exp_048122_taylor_floor is a piece of the proof that the first-order value exceeds the CODATA 2022 measured value by more than 0.0007, which is over 30000 standard deviations. The comparison is a theorem in the library, but it is quarantined because it imports the CODATA value from an external anchor.

The lemma itself is a narrow technical result. It does not claim that the first-order Alpha Genesis value is correct, nor does it adjust the CODATA value. It does not prove anything about the true value of α or about any higher-order corrections. The lemma only certifies a numerical lower bound for a Taylor polynomial, which is a small but necessary step in a larger measurement verdict.

private lemma exp_048122_taylor_floor :
    (80902 / 50000 : ℚ) < exp_taylor_12_at_048122 - exp_error_12_at_048122 := by
  norm_num [exp_taylor_12_at_048122, exp_error_12_at_048122, Nat.factorial]

/-- Tight enough upper bound on `log φ` for the measurement verdict. -/
theorem log_phi_lt_048122 : Real.log Constants.phi < (0.48122 : ℝ) := by
  rw [Real.log_lt_iff_lt_exp Constants.phi_pos]
  have hphi_hi : Constants.phi < (1.6180340 : ℝ) :=
    Numerics.W8Bounds.phi_lt_16180340
  exact lt_trans hphi_hi (by linarith [exp_048122_gt])

/-- The first-order Alpha Genesis value is above CODATA by at least `0.0007`.
This is the formal version of Anil's measurement objection, stated coarsely
but with a fully certified margin. -/
theorem alphaInvGenesis_exceeds_CODATA_by_0007 :
    alpha_inv_CODATA + (0.0007 : ℝ) < alphaInvGenesis := by
  rw [alphaInvGenesis_eq_alphaInv]
  unfold Constants.alphaInv
  have hseed_lo : (138.230048 : ℝ) < Constants.alpha_seed := Numerics.alpha_seed_gt
  have hseed_pos : 0 < Constants.alpha_seed := lt_trans (by norm_num) hseed_lo
  have hload_hi : Constants.f_gap / Constants.alpha_seed < (0.0086705 : ℝ) :=
    exponentialLoad_lt_0086705
  have hexp_mono : Real.exp (-0.0086705 : ℝ) < Real.exp (-(Constants.f_gap / Constants.alpha_seed)) := by
    exact Real.exp_lt_exp.mpr (by linarith)
  have hexp_lo : (0.99136694 : ℝ) < Real.exp (-(Constants.f_gap / Constants.alpha_seed)) :=
    lt_trans exp_neg_0086705_gt hexp_mono
  have hmul :
      (138.230048 : ℝ) * (0.99136694 : ℝ) <
        Constants.alpha_seed * Real.exp (-(Constants.f_gap / Constants.alpha_seed)) := by
    have h1 :
        (138.230048 : ℝ) * (0.99136694 : ℝ) <
          Constants.alpha_seed * (0.99136694 : ℝ) :=
      mul_lt_mul_of_pos_right hseed_lo (by norm_num)
    have h2 :
        Constants.alpha_seed * (0.99136694 : ℝ) <
          Constants.alpha_seed * Real.exp (-(Constants.f_gap / Constants.alpha_seed)) :=
      mul_lt_mul_of_pos_left hexp_lo hseed_pos
    exact lt_trans h1 h2
  have htarget :
      alpha_inv_CODATA + (0.0007 : ℝ) <
        (138.230048 : ℝ) * (0.99136694 : ℝ) := by
    norm_num [alpha_inv_CODATA]
  exact lt_trans htarget hmul

What this page does not claim

The lemma does not prove that the first-order Alpha Genesis value is correct. The lemma does not adjust or correct the CODATA 2022 value of the inverse fine-structure constant. The lemma does not establish any bound on the true value of α beyond the first-order comparison.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Constants/AlphaGenesis/MeasurementVerdict.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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