IndisputableMonolith.Foundation.PhiForcingDerived
The module defines geometric scale sequences with positive ratio r and derives their J-cost composition laws. It establishes commutativity, associativity, and closure results that force the golden equation with closed ratios equal to phi. Researchers tracing discrete ledgers to self-similar fixed points cite these algebraic derivations. The content proceeds via direct identities on the imported cost structure.
claimA geometric scale sequence is a discrete structure scaled by fixed ratio $r > 0$. Ledger composition is commutative and associative. The J-cost obeys the decomposition $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$. Closure conditions force the ratio to satisfy the golden equation.
background
This module sits in the Foundation domain and imports the RS time quantum τ₀ = 1 tick together with the cost structure. It introduces geometric scale sequences as discrete ledgers with constant positive ratio r. Ledger composition is defined to be commutative and associative, while the J-cost function supplies the additive decomposition for independent scales. These objects supply the minimal algebraic data for subsequent closure arguments.
proof idea
The module opens with definitions of the geometric scale sequence and ledger composition. It verifies commutativity and associativity by direct algebraic check. Later lemmas apply the J decomposition identity to show that closure forces the golden equation and that closed ratios equal phi.
why it matters in Recognition Science
These definitions and lemmas feed the Phi Forcing module, which proves φ is forced by self-similarity in a discrete ledger with J-cost. They also supply the discrete geometric ledger required by Hierarchy Minimality for the B1 closure step with scale 0 + scale 1 = scale 2. The Stillness Generative module draws on the ground-state properties established here. The material advances the chain from J-uniqueness to the self-similar fixed point.
scope and limits
- Does not prove uniqueness of the golden-ratio fixed point.
- Does not address the eight-tick octave or spatial dimensions.
- Does not derive physical constants such as alpha or G.
- Does not treat nonzero defects or continuous extensions.
used by (3)
depends on (2)
declarations in this module (14)
-
structure
GeometricScaleSequence -
def
ledgerCompose -
lemma
ledgerCompose_comm -
lemma
ledgerCompose_assoc -
theorem
closure_forces_golden_equation -
theorem
closed_ratio_is_phi -
def
J -
theorem
J_composition_decomposition -
theorem
J_additive_for_independent -
theorem
J_cost_motivates_additive_composition -
structure
of -
theorem
phi_forcing_complete -
def
LedgerComplete -
theorem
minimal_closure_sufficient