Clag_is_coherence_quantum
The theorem identifies the ILG lag constant as exactly phi to the minus five in RS-native units, fixing the coherence energy quantum per recognition event. Bandwidth saturation models of gravity cite this when deriving the ILG time kernel from holographic throughput limits. The proof is a direct reflexivity reduction to the upstream definition of Clag.
claimThe ILG lag constant satisfies $C_{lag} = phi^{-5}$, where $phi$ is the golden ratio and the constant sets the energy cost per recognition event in native units.
background
The BandwidthSaturation module shows how ILG gravity arises when Newtonian demands exceed the holographic bound on recognition events per unit time. Systems then batch updates over the eight-tick cycle, producing the ILG time kernel $w_t > 1$ that restores consistency. Clag is introduced as the fixed energy quantum that scales the kernel amplification for excess events. Upstream, the definition appears in Constants.ILG as the noncomputable real $1/(phi^5)$, and the module documentation states that $C_{lag}$ and the exponent alpha are bandwidth-determined. Related upstream structures include the BIT kernel families and the ledger factorization that calibrate the J-cost function.
proof idea
The proof is a one-line wrapper that applies reflexivity to the definition of Clag in the ILG constants module.
why it matters in Recognition Science
This anchors the ILG parameters inside the bandwidth saturation derivation, where $phi^{-5}$ supplies the coherence scale that links the holographic bound to the critical acceleration $a_{sat}$. It fills the parameter identification step listed in the module documentation and connects directly to the Recognition Science forcing chain through the phi-ladder and the eight-tick octave. The result supports downstream claims that the ILG kernel compensates exactly when dynamical time exceeds the recognition bandwidth.
scope and limits
- Does not derive the numerical value of phi or the holographic bound from more primitive axioms.
- Does not address conversion to SI units or empirical calibration of alpha.
- Does not prove positivity or uniqueness of the saturation acceleration.
- Does not extend the identification to non-bandwidth-limited regimes.
formal statement (Lean)
181theorem Clag_is_coherence_quantum :
182 Clag = 1 / phi ^ (5 : ℕ) := rfl
proof body
Term-mode proof.
183
184/-- **THEOREM**: The ILG α = (1−1/φ)/2 determines the power-law index of
185 the bandwidth kernel's scaling with dynamical time.
186
187 When T_dyn ≫ τ₀, the demanded rate scales as 1/T_dyn while the
188 bandwidth is fixed, so the kernel scales as T_dyn^α where
189 α = (1−1/φ)/2 is the φ-determined exponent. -/