module module high

`IndisputableMonolith.Mathematics.RamanujanBridge.ZeckendorfJCost`

This module defines the Fibonacci sequence along with Zeckendorf representations and J-cost stability predicates to support the Ramanujan bridge in Recognition Science. Researchers formalizing connections between Ramanujan's identities and the J functional would cite these definitions when establishing representation uniqueness and cost invariance. The module consists of recursive definitions, inductive lemmas on monotonicity and collapse, and predicates linking to the cost framework.

claimThe Fibonacci sequence satisfies $F_0 = 0$, $F_1 = 1$, $F_n = F_{n-1} + F_{n-2}$ for $n > 1$. Zeckendorf representations are unique sums of non-consecutive Fibonacci numbers. The predicate JCostStable holds when the J-cost of a representation equals the sum of individual J-costs under the Recognition Composition Law.

background

The module imports the RS time quantum τ₀ = 1 tick from Constants and cost functions from the Cost module. It introduces fib as the standard Fibonacci recurrence, ZeckendorfRepr as the unique non-adjacent sum representation, JCostStable as the invariance condition under the J functional, and representationCost as the total cost of a Zeckendorf expansion. These sit inside the RamanujanBridge module whose doc-comment states that Ramanujan's mathematical structures find natural expression in Recognition Science.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

This module feeds the RamanujanBridge parent module, which supplies the formal bridge between Ramanujan's deepest mathematical structures and Recognition Science. It supplies the Fibonacci and Zeckendorf machinery required for later results on Rogers-Ramanujan identities and lattice completeness within the J-cost framework.

scope and limits

Does not derive physical constants or mass formulas.
Does not prove the full Rogers-Ramanujan identities.
Does not address spatial dimensions or the eight-tick octave.
Does not contain the Recognition Composition Law itself.

used by (1)

From the project-wide theorem graph. These declarations reference this one in their body.

IndisputableMonolith.Mathematics.RamanujanBridge module_import

depends on (2)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import
IndisputableMonolith.Cost module_import

declarations in this module (13)

abbrev fib L70
theorem fib_recurrence L73
theorem consecutive_fib_collapse L82
theorem fib_mono L90
structure ZeckendorfRepr L97
def JCostStable L117
theorem zeckendorf_is_Jcost_stable L121
def representationCost L155
theorem gap1_absorptive_not_stable L170
def greedyZeckendorf L187
theorem zeckendorf_rogers_ramanujan_same_constraint L200
theorem fibonacci_lattice_is_complete L222
theorem fibonacci_lattice_is_unique L229