IndisputableMonolith.Gravity.ZeroParameterGravity
The module derives the Einstein gravitational coupling as exactly κ = 8φ⁵ from the J-cost ledger and forced D = 3. Researchers modeling emergent gravity or zero-parameter unification cite it to link the phi-ladder to Newtonian limits. It assembles targeted lemmas from Cost and DimensionForcing imports into closed-form expressions, positivity results, and ledger-to-gravity implications.
claimThe Einstein gravitational coupling satisfies \( \kappa = 8 \phi^5 \), where \( \phi \) is the self-similar fixed point forced by the Recognition Composition Law.
background
The module operates in the Gravity domain and imports the RS time quantum τ₀ = 1 tick, the Cost framework for J-cost and defectDist, LawOfExistence (x exists ⟺ defect(x) = 0), and DimensionForcing (D = 3 is forced by the RS framework). These supply the phi-ladder mass formula and eight-tick octave used to construct gravitational potential and coupling without free parameters.
Upstream results establish the ledger as the sole source: Constants fixes τ₀, while DimensionForcing supplies the topological and self-similar arguments that pin spatial dimension at three. The module then defines kappa_rs directly from the cost function and derives its properties.
proof idea
The module structures its content as a chain of definitions and lemmas rather than a single proof. It begins with kappa_rs from the J-cost, proves the closed form kappa_rs_closed_form via algebraic reduction on the phi-ladder, then establishes kappa_pos, kappa_ne_zero, and equivalence_principle_automatic. Further lemmas show gravity_from_ledger implies the eight-tick structure and kappa bounds.
why it matters in Recognition Science
This module supplies the derived κ = 8φ⁵ that SpacetimeEmergence imports to force the full Lorentzian metric, causal structure, and arrow of time. It directly realizes the DOC_COMMENT claim that the RS prediction for the Einstein coupling is derived rather than assumed, connecting T8 dimension forcing and the Recognition Composition Law to classical gravity.
scope and limits
- Does not derive the Einstein field equations.
- Does not include quantum corrections or loop effects.
- Does not treat cosmological solutions or black holes.
- Does not perform numerical fits beyond the stated alpha band.
used by (1)
depends on (4)
declarations in this module (12)
-
def
kappa_rs -
theorem
kappa_pos -
theorem
kappa_rs_closed_form -
theorem
kappa_ne_zero -
theorem
equivalence_principle_automatic -
def
gravitational_potential -
theorem
potential_negative -
theorem
gravity_from_ledger -
theorem
gravity_from_ledger_implies_eight_tick -
theorem
gravity_from_ledger_implies_kappa_pos -
theorem
gravity_from_ledger_implies_kappa_ne_zero -
theorem
kappa_bounds