Foundation Logic Real Constants Alpha Inv L Bounds

A bounded mirror

The inverse fine-structure constant, written 1/α, is a number physicists use to measure the strength of the electromagnetic force between charged particles. Its measured value is about 137.036, a figure known to extraordinary precision. The declaration alphaInvL_bounds is a formal statement, checked by a machine, that this number lies between 137.030 and 137.039. That is its entire content: a narrow interval, not a derivation of the number itself.

The statement lives inside a machine-checked library of formal theorems, a collection where every proof is verified by a computer. The library works in a special number system called LogicReal, which mirrors the ordinary real numbers. The declaration defines a constant, alphaInvL, as the mirror image of a real-valued constant already present in the framework. The theorem then proves that this mirrored constant falls inside the stated bounds, by transporting a previously established fact about the real-valued constant into the LogicReal setting.

In Recognition Science, this is part of a larger project. The framework models physical structure as arising from a discrete record of recognition events, and it derives constants like the golden ratio and Planck's constant from a forcing chain of theorems. The inverse fine-structure constant appears in this framework as a defined quantity, not as a derived one. The bounds theorem is a consistency check: it shows that the framework's definition, whatever its origin, lands within a window that contains the measured value.

What the declaration does not do is explain why 1/α has the value it does. The framework does not derive the constant from its first principles. The seed of the definition, which involves the number 44π, is an identification, not a derived coupling. The exact value of the inverse fine-structure constant remains an open problem in this framework, as it does in conventional physics. The bounds theorem merely certifies that the framework's chosen number is not wildly wrong.

The practical consequence is modest but real. Anyone working inside the framework can rely on this interval without rechecking the arithmetic. The theorem is a small, verified brick in a larger wall. It does not tell you why the wall stands, only that this particular brick is in place.

/-- Recovered alpha inverse has the verified EM range. -/
theorem alphaInvL_bounds : fromReal (137.030 : ℝ) < alphaInvL ∧
    alphaInvL < fromReal (137.039 : ℝ) := by
  have h := (Verification.EMAlpha.EMAlphaCert.verified_any ⟨⟩).2.2.2
  constructor
  · rw [lt_iff_toReal_lt, toReal_fromReal, toReal_alphaInvL]
    exact h.1
  · rw [lt_iff_toReal_lt, toReal_fromReal, toReal_alphaInvL]
    exact h.2

@[simp] theorem toReal_alphaInvL : toReal alphaInvL = Constants.alphaInv := toReal_fromReal _

What this page does not claim

The declaration does not derive the value of the inverse fine-structure constant from first principles. The declaration does not claim that the framework's definition of the constant is unique or forced. The declaration does not address the physical interpretation of the constant within the framework.

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/Foundation/LogicRealConstants.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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