LocalityCert
LocalityCert packages the uniform ratio and geometric progression properties that any zero-parameter ledger must obey. Researchers closing the T5 to T6 bridge in Recognition Science cite it when establishing that level-(k+2) composites depend only on adjacent levels. The structure is a direct packaging of the no_external_scale axiom into constant-ratio and geometric-sequence forms with no internal proof steps.
claimA zero-parameter ledger with levels function $L : ℕ → ℝ$ (strictly positive) satisfies $L(n)/L(n+1) = L(0)/L(1)$ for every $n$, and there exists $r > 0$ such that $L(k+1) = r · L(k)$ for every $k$.
background
The Locality from Ledger module addresses Gap 1b by deriving LocalBinaryRecurrence from ledger axioms that admit no external scales. A ZeroParameterLedger is defined by a levels map ℕ → ℝ that is everywhere positive and obeys the no_external_scale axiom: every adjacent ratio equals the base ratio levels(0)/levels(1) independently of any positive scaling factor λ. This forces all composition rules to be expressible solely in terms of dimensionless ratios of adjacent levels.
proof idea
The declaration is a structure definition that directly assembles the uniform ratio field (restating no_external_scale) and the geometric progression field. No lemmas or tactics are invoked inside the definition; it functions as a packaging of the two required properties for the downstream existence witness.
why it matters in Recognition Science
LocalityCert is instantiated by locality_cert_exists to prove Nonempty LocalityCert, which in turn supports locality_from_ledger. This step closes the T5→T6 bridge by guaranteeing that composites at level k+2 depend only on levels k+1 and k, before the phi fixed point and eight-tick octave are introduced. It directly implements the module's claim that zero parameters force ratio-only, binary composition.
scope and limits
- Does not prove existence of any ZeroParameterLedger.
- Does not derive the binary recurrence theorem locality_from_ledger.
- Does not incorporate J-cost, defectDist, or the phi-ladder.
- Does not address non-geometric or higher-order composition rules.
formal statement (Lean)
97structure LocalityCert where
98 uniform : ∀ (L : ZeroParameterLedger), ∀ n,
99 L.levels n / L.levels (n + 1) = L.levels 0 / L.levels 1
100 geometric : ∀ (L : ZeroParameterLedger),
101 ∃ r : ℝ, r > 0 ∧ ∀ k, L.levels (k + 1) = r * L.levels k
102