Constants Fine Structure Constant Alpha Lock Numerical Bounds

The bounds and their limits

The fine-structure constant α, approximately 1/137, governs the strength of electromagnetic interactions between charged particles. In the Recognition Science framework, a different number appears under a similar name: alphaLock, defined as (1 − 1/φ)/2, where φ is the golden ratio. This number is approximately 0.19. The framework's machine-checked library of formal theorems proves that alphaLock lies strictly between 0.18 and 0.21, a coarse bound derived from the golden ratio's position between 1.61 and 1.62.

The theorem named alphaLock_numerical_bounds establishes exactly this interval: 0.18 < alphaLock < 0.21. It also proves the number is positive and less than one, placing it in the open unit interval. These are structural facts about how the framework defines this particular quantity, not measurements of any physical constant. The bound follows from the definition and from elementary properties of the golden ratio.

In Recognition Science, alphaLock plays a role as the information-limited-gravity kernel exponent. The framework models certain recognition processes using this exponent, and the theorem confirms the number stays in a modest range. This matters because it shows the framework's internal quantities are well-behaved and bounded, rather than growing without limit or collapsing to zero.

The name is historical and misleading. The framework's own documentation explicitly retracts any earlier claim that alphaLock resolves the question of what determines the electromagnetic fine-structure constant. No conversion between alphaLock and α exists in the repository. The exact value of α itself remains an open target within the framework, treated as a free boundary datum, not something the framework derives.

What a reader should take away: the theorem gives tight numerical bounds for a framework-internal number, and those bounds have no bearing on the measured fine-structure constant. The honest position is that α's value is not determined by this result. The framework's own records state this plainly, and the theorem's modest scope reflects that.

/-- α_lock is between 0.18 and 0.21 (coarse bound from φ ∈ (1.61, 1.62)). -/
theorem alphaLock_numerical_bounds :
    (0.18 : ℝ) < alphaLock ∧ alphaLock < (0.21 : ℝ) := by
  unfold alphaLock
  have h_phi := phi_gt_onePointSixOne
  have h_phi' := phi_lt_onePointSixTwo
  constructor
  · have h_inv : 1 / phi < 1 / 1.61 := by
      rw [div_lt_div_iff_of_pos_left (by norm_num) phi_pos (by norm_num)]
      exact h_phi
    linarith
  · have h_inv : 1 / 1.62 < 1 / phi := by
      rw [div_lt_div_iff_of_pos_left (by norm_num) (by norm_num) phi_pos]
      exact h_phi'
    linarith

/-- α_lock structure: `alphaLock = (1 − 1/φ)/2` with unit-interval bounds.
    This is a φ-structural fact about the ILG kernel exponent. It does NOT
    determine the EM fine-structure constant; see the module header. -/
theorem alphaLock_structure :
    0 < alphaLock ∧ alphaLock < 1 ∧
    alphaLock = (1 - 1 / phi) / 2 :=
  ⟨alphaLock_pos, alphaLock_lt_one, rfl⟩

@[deprecated alphaLock_structure (since := "2026-07-06")]
alias fine_structure_derived := alphaLock_structure

/-- α_lock lies in the open unit interval. -/
theorem alphaLock_in_unit_interval : 0 < alphaLock ∧ alphaLock < 1 :=
  ⟨alphaLock_pos, alphaLock_lt_one⟩

/-- α_lock structure: `alphaLock = (1 − 1/φ)/2` with unit-interval bounds.
    This is a φ-structural fact about the ILG kernel exponent. It does NOT
    determine the EM fine-structure constant; see the module header. -/
theorem alphaLock_structure :
    0 < alphaLock ∧ alphaLock < 1 ∧
    alphaLock = (1 - 1 / phi) / 2 :=
  ⟨alphaLock_pos, alphaLock_lt_one, rfl⟩

@[deprecated alphaLock_structure (since := "2026-07-06")]
alias fine_structure_derived := alphaLock_structure

What this page does not claim

The theorem does not determine the electromagnetic fine-structure constant α. The bounds do not connect alphaLock to any measured physical quantity. The framework does not provide a ledger-to-lab conversion between alphaLock and α.

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/FineStructureConstant.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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