p_steepness
The definition supplies the spatial profile steepness p in Recognition Science gravity models. Galaxy dynamicists cite it when constructing density profiles or fitting rotation curves. It arises by direct substitution of the locked fine-structure constant into the expression 1 minus alphaLock divided by 4.
claimThe spatial profile steepness is defined by $p = 1 - (1 - 1/φ)/8$, where the locked fine-structure constant satisfies $α_{lock} = (1 - 1/φ)/2$.
background
The GravityParameters module derives seven phenomenological parameters for galactic gravity from Recognition Science. The steepness p carries an RS basis and is expressed using the locked fine-structure constant. Upstream, alphaLock is defined as the canonical value (1 − 1/φ)/2 that bridges the fine-structure constant to the golden ratio φ.
proof idea
It is a direct definition that substitutes the expression for alphaLock. No lemmas or tactics are applied beyond the substitution itself.
why it matters in Recognition Science
The definition feeds the equality theorem p_steepness_eq and the positivity theorem p_steepness_pos. It occupies the HAS RS BASIS slot for p in the seven-parameter gravity table, delivering a 0.2% match to the fitted value 0.95 ± 0.02. The construction ties to the phi-derived alphaLock and supports the broader parameter set linked to the phi-ladder.
scope and limits
- Does not prove that p lies strictly between zero and one.
- Does not derive the steepness formula from the J-cost or Recognition Composition Law.
- Does not address connections to the eight-tick octave or spatial dimension D = 3.
formal statement (Lean)
117def p_steepness : ℝ := 1 - alphaLock / 4
proof body
Definition body.
118