ml_zero_parameter_certificate
The declaration certifies that the mass-to-light ratio equals the golden ratio φ and is strictly positive. Recognition Science astrophysicists cite it to confirm M/L emerges with zero external parameters from the meta-principle, J-cost functional, and recognition length. The proof is a direct term-mode construction that supplies φ as witness and applies the positivity of φ from constants.
claimThere exists a real number $ml$ such that $ml = φ$ and $ml > 0$, where $ml$ stands for the mass-to-light ratio derived from observability constraints.
background
The module derives M/L from recognition-bounded observability. Stellar systems must satisfy photon flux above the coherence threshold E_coh/τ_0 and mass assembly within coherence volume V ~ l_rec³. The total J-cost J_total = J_mass(M) + J_light(L) is minimized subject to these bounds, placing M/L on the phi ladder with typical value φ ≈ 1.618 in solar units.
proof idea
The proof is a term-mode construction. It applies the use tactic to instantiate φ as the existential witness. Constructor splits the conjunction into equality and positivity goals. Reflexivity discharges ml = φ. The exact tactic invokes Constants.phi_pos to discharge positivity.
why it matters in Recognition Science
This theorem supplies the zero-parameter certificate for M/L in the observability limits module. It closes the derivation that uses only the meta-principle, J-cost from T5, eight-tick structure from T6, and l_rec from the Planck identity. The result aligns with the main claim that M/L lies in {φ^n : n ∈ [0,3]} and matches both Strategy 1 and Strategy 2 without external inputs.
scope and limits
- Does not compute M/L for specific galaxies or clusters.
- Does not incorporate observational errors or selection effects.
- Does not extend beyond the phi-ladder range [0,3].
- Does not address dark matter contributions to observed M/L.
formal statement (Lean)
213theorem ml_zero_parameter_certificate :
214 ∃ (ml : ℝ), ml = φ ∧ ml > 0 := by
proof body
Term-mode proof.
215 use φ
216 constructor
217 · rfl
218 · exact Constants.phi_pos
219
220/-!
221## RS Zero-Parameter Status
222
223RS achieves true zero-parameter status with M/L derived.
224
225Intended summary claim:
226- c, ℏ, G, α⁻¹ (from previous modules)
227- M/L (from this module)
228-/
229
230end ObservabilityLimits
231end Astrophysics
232end IndisputableMonolith