IndisputableMonolith.Meta.LedgerUniqueness
The Meta.LedgerUniqueness module defines the golden ratio φ and collects lemmas establishing its uniqueness as fixed point together with linking-dimension and cycle-minimality results that rely on Gray-code constructions. Researchers deriving the phi-ladder and spatial dimension D=3 in Recognition Science would cite it. The arguments proceed by algebraic verification of the fixed-point equation combined with combinatorial properties of the binary-reflected Gray code on hypercubes.
claim$φ = (1 + √5)/2$, the unique positive solution to the fixed-point equation $J(x) = x$ with $J(x) = (x + x^{-1})/2 - 1$.
background
The module sits in the meta layer and imports the RS time quantum τ₀ = 1 tick together with the binary-reflected Gray code construction. The latter supplies the recursive definition BRGC(0) = [0] that generates Hamiltonian cycles on the d-dimensional hypercube Q_d. It introduces φ as the self-similar fixed point forced by the unified forcing chain (T5–T6) and pairs it with linking-number and eight-tick cycle definitions.
proof idea
This is a definition module containing multiple lemmas; algebraic identities verify the fixed-point property of φ while Gray-code recursion establishes the minimality of the eight-tick cycle and uniqueness of the linking dimension.
why it matters in Recognition Science
The module supplies the phi definition and uniqueness lemmas that support downstream results on linking-dimension uniqueness, cube uniqueness, and the eight-tick minimal cycle, thereby feeding the derivation of D = 3 and the phi-ladder mass formula in the Recognition framework.
scope and limits
- Does not compute numerical values or approximations of φ.
- Does not address physical units or experimental tests of the uniqueness claims.
- Does not generalize the Gray-code arguments beyond the eight-tick octave.
- Does not contain the full forcing-chain derivation of T0–T8.
depends on (2)
declarations in this module (17)
-
def
phi -
theorem
phi_satisfies_fixed_point -
theorem
phi_unique_fixed_point -
theorem
cost_fixed_point_is_phi -
def
linkingNumber -
def
H_LinkingDimensionUniqueness -
theorem
Q3_unique_linking_dimension -
theorem
cube_uniqueness -
def
grayCodeCycleLength -
theorem
eight_tick_minimal -
theorem
no_shorter_cycle -
theorem
eight_tick_is_minimal -
structure
RSLedger -
structure
DiscreteConservativeSystem -
theorem
ledger_structure_unique -
theorem
complete_ledger_uniqueness -
theorem
rs_ledger_is_unique