module module high

`IndisputableMonolith.Mathematics.RamanujanBridge.ZeckendorfJCost`

This module defines the Fibonacci sequence, Zeckendorf representations of integers, and their stability under the J-cost function. Physicists and mathematicians working on the Ramanujan bridge to Recognition Science cite these constructions to link number-theoretic lattices with the phi-ladder and forcing chain. The module supplies a collection of inductive definitions and basic lemmas drawn from the imported Cost and Constants modules.

claimThe module introduces the Fibonacci sequence $F_n$ satisfying $F_n = F_{n-1} + F_{n-2}$ with $F_0=0$, $F_1=1$, the Zeckendorf representation of positive integers as sums of non-adjacent Fibonacci numbers, the representation cost function, and the J-cost stability predicate for such representations.

background

Recognition Science builds physics from the J-function obeying the composition law $J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$ and the time quantum τ₀=1 tick supplied by the Constants module. The Cost module supplies the J-cost metric used to measure representation defects. This module introduces the Fibonacci sequence via its recurrence, monotonicity, and collapse properties, then defines ZeckendorfRepr as the unique greedy non-adjacent sum and JCostStable as the predicate that the cost remains invariant under the representation.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

The module feeds the RamanujanBridge parent module, which deciphers Ramanujan's identities inside the Recognition Science framework. It supplies the lattice completeness and stability results needed to connect Zeckendorf representations to the T5 J-uniqueness, T6 phi fixed point, and the eight-tick octave. The definitions close the number-theoretic scaffolding for downstream mass formulas and alpha-band calculations.

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