computation_limits_structure
Recognition Science derives a structural limit on exact computation from the irrationality of the golden ratio φ. Researchers modeling discrete spacetime or fundamental physics bounds would cite this when showing that RS dynamics cannot be captured by finite rational arithmetic. The result sits in the IC-002 chain on computation limits and is invoked by downstream theorems linking irrationality to Church-Turing and complexity structures. The proof is a one-line term that applies the established irrationality of φ directly to the ledger-derived
claimThe golden ratio satisfies Irrational($φ$), which defines the computation limits structure arising from the ledger in Recognition Science.
background
The module IC-002 treats fundamental computation limits as emerging from temporal discreteness (minimum time quantum τ₀), irrational constants, and thermodynamic erasure costs. The sibling definition computation_limits_from_ledger is the proposition Irrational(φ), presented as the core structural constraint: exact representation of RS constants requires transcendental arithmetic. Upstream, phi_irrational proves φ equals Real.goldenRatio and is therefore irrational because √5 is irrational (5 prime, not a square). The local setting quotes the module: “φ is irrational, so φ-based states cannot be exactly represented with finite rational arithmetic.”
proof idea
The proof is a term-mode one-liner that applies the theorem phi_irrational to discharge the definition computation_limits_from_ledger. No tactics are used; the identifier phi_irrational is substituted directly for the target Prop.
why it matters in Recognition Science
This fills the IC-002.5 slot and carries the irrationality constraint into has_computation_limits_structure (Church-Turing physics) and physics_complexity_structure (IC-005.12). It underscores how the self-similar fixed point φ (T6) prevents exact finite representations, consistent with the eight-tick octave and D=3. The result closes the gap between the ledger definition and the higher-level physics-complexity theorems.
scope and limits
- Does not derive numerical bounds on maximum bit rates or clock frequencies.
- Does not incorporate Landauer energy or Bremermann E×t limits.
- Does not address concrete simulation algorithms or precision requirements.
- Does not prove that transcendental arithmetic is practically unattainable.
formal statement (Lean)
76theorem computation_limits_structure : computation_limits_from_ledger := phi_irrational
proof body
Term-mode proof.
77
78/-- **THEOREM IC-002.5**: No rational approximation equals φ exactly. -/