saturationAccel_well_defined
Saturation acceleration is proven strictly positive and nonzero. Researchers modeling the transition between Newtonian gravity and information-limited regimes would cite this to anchor the critical scale before classifying dynamical regimes. The proof is a one-line term construction that pairs the positivity result with the implication that any positive real is nonzero.
claimLet $a_{sat}$ denote the saturation acceleration at which demanded recognition rate equals holographic bandwidth. Then $0 < a_{sat}$ and $a_{sat} ≠ 0$.
background
The Bandwidth Saturation module derives ILG gravity from recognition throughput limits. Saturation acceleration $a_{sat}$ is the critical value where Newtonian dynamics demand more recognition events per unit time than the holographic bound $A / (4ℓ_P² · k_R · 8τ₀)$ permits, with batching in 8-tick cycles producing the ILG time kernel $w_t > 1$. The module states that this scale is defined implicitly from φ and the holographic bound via the equality $M / T_{dyn} = A / (4ℓ_P² · k_R · 8τ₀)$ using gravitational area and dynamical time.
proof idea
The proof is a one-line term that constructs the conjunction by pairing the positivity theorem for saturation acceleration with the lemma that any strictly positive real number is nonzero.
why it matters in Recognition Science
This result confirms the critical acceleration scale is well-defined and positive, serving as the foundation for the module's regime classification theorems such as below_saturation_is_newtonian and above_saturation_needs_ilg. It directly supports the derivation of ILG parameters from bandwidth constraints and ties into the eight-tick octave structure of the recognition framework. The claim is fully proved with no open scaffolding.
scope and limits
- Does not compute a closed-form or numerical value for saturation acceleration.
- Does not derive the explicit form of the ILG time kernel.
- Does not address multi-body or cosmological extensions.
- Does not prove the holographic bound itself.
formal statement (Lean)
98theorem saturationAccel_well_defined : 0 < saturationAccel ∧ saturationAccel ≠ 0 :=
proof body
Term-mode proof.
99 ⟨saturationAccel_pos, ne_of_gt saturationAccel_pos⟩
100
101/-! ## §4. Regime Classification -/
102
103/-- **THEOREM**: Above saturation acceleration, the system is sub-saturated
104 (Newtonian gravity suffices).
105
106 When a > a_sat, the gravitational area is small enough that the holographic
107 bandwidth exceeds the demanded recognition rate. -/