gate_phi
The gate_phi definition records the φ-forcing necessity gate as a proven constraint on self-similar discrete ledgers. Researchers checking alternative zero-parameter frameworks cite it to test whether a candidate ledger satisfies the scaling fixed point. The definition is a direct record whose supporting argument is imported from the PhiForcing module.
claimThe φ-forcing necessity gate requires that any self-similar discrete ledger must adopt the unique scaling ratio φ satisfying φ = 1 + 1/φ.
background
The InevitabilityStructure module lists six necessity gates that any alternative framework must violate. A NecessityGate is a record holding a name, a proven flag, and the physical meaning of violation. Gate 5 encodes the self-similarity requirement that selects φ. It depends on the Gate structure from CircuitLedger, which guarantees DAG locality with at most two parents, and on the Dimension abbrev that tracks spatial requirements.
proof idea
One-line definition that constructs the NecessityGate record. It supplies the string name, sets the proven flag to true, and records the violation consequence for a non-φ ratio. The actual derivation of the fixed point is located in the imported PhiForcing module.
why it matters in Recognition Science
The definition populates the all_gates list that states the full inevitability theorem. It fills the self-similarity step (T6) in the forcing chain, where the fixed point of the recurrence is forced once discreteness and the Recognition Composition Law are in place. Any ledger theory that is discrete and self-similar but uses a different ratio must either add parameters or fail to derive observables.
scope and limits
- Does not derive the numerical value of φ.
- Does not apply to continuous or non-discrete ledgers.
- Does not constrain the J-cost function directly.
- Does not prove that every physical system must be self-similar.
formal statement (Lean)
101def gate_phi : NecessityGate := {
proof body
Definition body.
102 name := "φ Forcing"
103 proven := true -- Proven in PhiForcing.lean
104 violation_meaning := "Self-similar discrete ledger with ratio ≠ φ"
105}
106
107/-- Gate 6: Dimension Forcing -/