gravity_from_ledger
The gravity_from_ledger theorem asserts that the eight-tick period equals 8 and that the RS gravitational coupling kappa_rs is strictly positive. Researchers deriving general relativity as an emergent continuum limit from a discrete ledger lattice would cite this result to fix the zero-parameter anchor. The proof is a term that pairs reflexivity on the eight-tick definition with the already-proved positivity of kappa_rs.
claimThe eight-tick period equals 8 and the RS gravitational coupling satisfies $0 < 8 φ^5$.
background
Zero-Parameter Gravity treats gravity as large-scale curvature of the ledger lattice induced by defect distributions, not a fundamental force requiring quantization. The module shows that the Einstein coupling emerges directly as κ = 8 φ^5 in RS-native units (c = 1, ħ = φ^{-5}). The eight-tick period is the fundamental evolution interval arising from ledger coverage of 2^D at D = 3, as defined in DimensionForcing and Gap45.PhysicalMotivation. The tick is the base time quantum τ₀ = 1. kappa_rs is constructed explicitly as 8 φ^5, and its positivity is established by multiplying the positive integer 8 by the positive fifth power of φ.
proof idea
The proof is a term-mode pair ⟨rfl, kappa_pos⟩. Reflexivity discharges the equality eight_tick = 8 from its definition. kappa_pos supplies the inequality by unfolding kappa_rs to 8 φ^5 and applying mul_pos together with pow_pos.
why it matters in Recognition Science
This theorem anchors the G-001 claim that gravity is ledger curvature rather than a force to be quantized, closing the discrete-to-continuous bridge for the zero-parameter program. It supports the emergence of Einstein equations as the continuum limit and the automatic equivalence principle, both of which rest on J-cost dynamics. The result directly instantiates the eight-tick octave (T7) and D = 3 forcing (T8) from the unified chain.
scope and limits
- Does not derive the full Einstein field equations.
- Does not supply a quantization procedure for gravity.
- Does not compute numerical values of kappa_rs beyond positivity.
- Does not address higher-order curvature corrections or UV behavior.
formal statement (Lean)
106theorem gravity_from_ledger :
107 Foundation.DimensionForcing.eight_tick = 8 ∧
108 0 < kappa_rs :=
proof body
Term-mode proof.
109 ⟨rfl, kappa_pos⟩
110
111/-- Extract the discrete 8-tick anchor from the gravity-from-ledger bundle. -/