IndisputableMonolith.Foundation.PhiForcing
PhiForcing establishes that the self-similar fixed point satisfies the golden ratio equation phi squared equals phi plus one. Researchers deriving constants from the phi-ladder in electron mass and fine structure cite this module. The module assembles the result by importing the closure axioms from PhiForcingDerived together with discreteness, existence, and ledger forcing.
claimThe scale factor $\phi$ satisfies the equation $\phi^2 = \phi + 1$.
background
This module sits in the Foundation domain and imports the cost functional J from Cost together with the unique-minimum argument at unity from DiscretenessForcing. LawOfExistence supplies the zero-defect condition for existence while LedgerForcing derives double-entry structure from J-symmetry. PhiForcingDerived supplies the three axioms of discrete geometric scales, additive ledger composition, and closure.
From the doc-comment of PhiForcingDerived: 'This module derives r² = r + 1 from three stated axioms: 1. Discrete Scale Sequence: Scales form a geometric sequence {1, r, r², ...} 2. Additive Ledger Composition: When two recognition events combine in the ledger, their scales ADD (not multiply).'
proof idea
This is a definition module that imports the phi equation from PhiForcingDerived and records immediate consequences such as positivity, bounds near 1.618, and the inverse. No independent proofs are developed here; the argument is inherited from the upstream closure axioms.
why it matters in Recognition Science
The phi equation supplied here is the direct input to the constant derivations in ElectronMass, FineStructureConstant, GravitationalConstant, and PlanckScaleMatching. It also supports the cosmology results on dark matter and flatness. The module completes the T6 forcing step in the unified chain by identifying the self-similar fixed point with the golden ratio.
scope and limits
- Does not compute numerical approximations of phi.
- Does not link phi to the eight-tick period or three dimensions.
- Does not derive mass or coupling formulas.
- Does not address the alpha inverse band.
used by (40)
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IndisputableMonolith.Constants.ElectronMass -
IndisputableMonolith.Constants.FineStructureConstant -
IndisputableMonolith.Constants.GravitationalConstant -
IndisputableMonolith.Constants.PlanckScaleMatching -
IndisputableMonolith.Cosmology.DarkMatter -
IndisputableMonolith.Cosmology.FlatnessProblem -
IndisputableMonolith.Cosmology.GalaxyRotation -
IndisputableMonolith.Cosmology.Nucleosynthesis -
IndisputableMonolith.Foundation.ConstantDerivations -
IndisputableMonolith.Foundation.DimensionForcing -
IndisputableMonolith.Foundation.HierarchyDissolution -
IndisputableMonolith.Foundation.HierarchyEmergence -
IndisputableMonolith.Foundation.HierarchyMinimality -
IndisputableMonolith.Foundation.InevitabilityStructure -
IndisputableMonolith.Foundation.OntologyPredicates -
IndisputableMonolith.Foundation.ParticleGenerations -
IndisputableMonolith.Foundation.StillnessGenerative -
IndisputableMonolith.Gravity.GalacticTimescale -
IndisputableMonolith.Masses.LeptonMassLadder -
IndisputableMonolith.Masses.MassHierarchy -
IndisputableMonolith.Masses.MassRatiosProved -
IndisputableMonolith.Mathematics.Euler -
IndisputableMonolith.Papers.GCIC.DiscreteGauge -
IndisputableMonolith.Physics.ElectroweakBosons -
IndisputableMonolith.Physics.KaonMasses -
IndisputableMonolith.Physics.PionMasses -
IndisputableMonolith.Physics.ThermalFixedPoint -
IndisputableMonolith.Physics.WeakForceEmergence -
IndisputableMonolith.QFT.RunningCouplings -
IndisputableMonolith.QFT.UVCutoff
depends on (5)
declarations in this module (24)
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theorem
phi_equation -
theorem
phi_pos -
theorem
phi_gt_one -
theorem
phi_lt_two -
theorem
phi_gt_onePointSixOneEight -
theorem
phi_lt_onePointSixOneNine -
theorem
phi_lt_onePointEight -
theorem
phi_gt_onePointSix -
theorem
phi_inv -
theorem
J_phi -
structure
SelfSimilar -
def
satisfies_golden_constraint -
theorem
self_similar_forces_golden_constraint -
theorem
phi_satisfies -
theorem
golden_constraint_unique -
theorem
phi_unique_self_similar -
structure
DiscreteLedger -
def
is_self_similar -
theorem
phi_forced -
def
J_bit -
theorem
J_bit_pos -
def
E_coh -
theorem
E_coh_pos -
theorem
phi_forcing_principle