discrete_fibonacci_from_minimality
Natural numbers a and b at least 1 with max(a, b) equal to 1 must both equal 1. This pins the coefficients in the minimal additive recurrence for discrete scales under the zero-parameter condition of Recognition Science. Derivations of the Fibonacci recurrence from posting extensivity cite this to select the unique minimal pair. The argument applies antisymmetry of the natural order after the max bound supplies the reverse inequalities.
claimLet $a, b$ be positive integers satisfying $1 ≤ a$, $1 ≤ b$, and $max(a, b) = 1$. Then $a = 1$ and $b = 1$.
background
This theorem belongs to the PostingExtensivity module, which derives additive scale composition from the Recognition Composition Law without assuming linearity. The RCL states $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$ where $J(x) = (x + x^{-1})/2 - 1$. Posting potentials are defined as $Π(x) = J(x) + 1$, yielding multiplicative composition that forces additive recurrences $ℓ_{k+2} = α ℓ_{k+1} + β ℓ_k$ for geometric scales $σ^k$ (with scale defined as $phi^k$ in LargeScaleStructureFromRS). In this setting α and β are non-negative integers counting sub-events in a countable carrier. The zero-parameter condition forbids extra complexity, forcing $max(α, β) = 1$. The proof relies on the antisymmetry lemma le_antisymm from ArithmeticFromLogic.
proof idea
The proof is a direct term-mode construction of the conjunction. It applies the antisymmetry theorem le_antisymm from ArithmeticFromLogic separately to each component, using the max bound together with the lower bounds to obtain the reverse inequalities via omega.
why it matters in Recognition Science
This result completes the discrete coefficient selection step in the end-to-end bridge from RCL through posting d'Alembert and additive closure to the golden equation forcing phi, as described in the module's closing comment on Proposition 4.3. It ensures the recurrence is the Fibonacci one under minimal descriptional complexity, feeding into the eight-tick octave and D = 3 derivations. The theorem closes the coefficient minimality gap without hidden hypotheses.
scope and limits
- Does not prove the full Fibonacci recurrence, only the coefficient values under the max bound.
- Does not apply to non-natural or real-valued coefficients.
- Does not incorporate the J-cost function or RCL directly.
- Does not derive the geometric scale sequence or posting potentials.
formal statement (Lean)
160theorem discrete_fibonacci_from_minimality
161 (a b : ℕ) (ha : 1 ≤ a) (hb : 1 ≤ b) (hmin : max a b = 1) :
162 a = 1 ∧ b = 1 := by
proof body
Term-mode proof.
163 constructor
164 · exact Nat.le_antisymm (by omega) ha
165 · exact Nat.le_antisymm (by omega) hb
166
167/-! ## Complete Posting-Extensivity Bridge -/
168
169/-- **End-to-end theorem**: From the forced RCL combiner structure
170(specifically, the d'Alembert identity on posting potentials),
171a geometric scale sequence closed under additive posting, with
172discrete minimal coefficients, forces φ.
173
174This chains the entire derivation:
175 RCL → posting d'Alembert → additive closure → golden equation → φ -/