Gravity Coercive Projection Ilg Alpha Is Alpha Lock

The locked constant

The theorem ilg_alpha_is_alphaLock in the framework's machine-checked library of formal theorems states a simple identity: a parameter called alphaLock, which appears in the Information-Limited Gravity (ILG) model, equals (1 - 1/φ)/2, where φ is the golden ratio. Numerically, this is about 0.190983. The declaration is a definitional equality, proved by unfolding definitions, not by a deep chain of reasoning. It is a fact about how the framework's authors chose to define alphaLock.

The golden ratio φ is the number satisfying φ² = φ + 1, approximately 1.618034. The value (1 - 1/φ)/2 is exactly 1/(2φ²), which is also equal to (3 - √5)/4. This quantity appears in the ILG model's energy functional, which the framework uses to describe gravity as a minimization problem. The broader context in the same file shows that this functional has a unique minimizer, with a coercivity constant c = 49/162, and that the model's net constant K_net = (9/7)² arises from an eight-tick cycle with epsilon = 1/8. These are the framework's internal results, tagged as theorems within its own system.

In Recognition Science, the framework models gravity through a cost or energy function that must be minimized. The ILG model adds a weight operator w that multiplies the density, and the framework proves that this operator is positive, meaning it never decreases the energy. The alphaLock constant is one piece of this model's structure. The declaration that alphaLock equals (1 - 1/φ)/2 is a precise statement about that piece, but it is not a derivation of any physical constant from first principles. It is a definitional choice, a lock placed on a parameter, not a law that forces the parameter's value.

The theorem does not claim that alphaLock is the fine-structure constant α. The fine-structure constant is a measured quantity in physics, approximately 1/137.036, and the framework does not derive it. The name alphaLock might suggest a connection, but the declaration itself only fixes the ILG parameter's value in terms of the golden ratio. It also does not claim that the ILG model is a complete theory of gravity, nor that its energy minimization has been observed in nature. The theorem is a formal statement inside the framework's library, and its meaning is confined to that system.

/-- The ILG alpha exponent: alpha = (1 - 1/phi) / 2 = alphaLock. -/
theorem ilg_alpha_is_alphaLock : alphaLock = (1 - 1/phi) / 2 := rfl

/-- The ILG alpha exponent: alpha = (1 - 1/phi) / 2 = alphaLock. -/
theorem ilg_alpha_is_alphaLock : alphaLock = (1 - 1/phi) / 2 := rfl

theorem c_coercive_value : c_coercive = 49 / 162 := rfl

theorem K_net_value : K_net = 81 / 49 := by
  unfold K_net; norm_num

What this page does not claim

The theorem does not derive the fine-structure constant α, which remains an open target in the framework. The declaration does not claim that alphaLock is a measured physical quantity. The theorem does not assert that the ILG model is a complete or observationally confirmed theory of gravity.

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/Gravity/CoerciveProjection.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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