Gravity Running Gderivation

Gravity running G derivation

Gravity running G derivation is the Recognition Science result that the effective gravitational constant G_eff (the strength of gravity as actually measured at a given scale) is not constant. It changes with distance, and the way it changes is forced. The module RunningGDerivation.lean proves that the running exponent, the number that controls how fast gravity strengthens toward small scales, is uniquely determined by the ratio of the recognition lag to the self-similarity scaling factor. The lag is the constant C_lag = phi^-5, where phi is the golden ratio. The theorem beta_running_derived proves the exponent equals -(phi - 1) / phi^5.

The derivation starts from voxel density scaling. The effective number of recognition voxels N(r) as a function of radius is defined as r raised to the running exponent beta_running. At nanometer scales, the effective voxel density is proportional to r^beta, where beta is the strain induced by the phi^-5 lag. The effective G is proportional to the local resolution, meaning that where recognition resolves more finely, gravity appears stronger. As r approaches zero, the exponent is negative, so the density and therefore G_eff increase. This is the nanoscale strengthening scaling, established in the theorem running_g_scaling.

The consequence is that gravity running G derivation is not a fitted model with adjustable parameters. The exponent is derived, not chosen. The framework forces the specific value from the recognition cost structure, and the strengthening behavior follows as a theorem. This places the running of G inside the same forced chain that produces the golden ratio and the eight-tick cycle, rather than as an empirical anomaly requiring separate explanation.

THEOREM beta_running_derived · IndisputableMonolith/Gravity/RunningGDerivation.lean

MODEL voxel_density_scaling · IndisputableMonolith/Gravity/RunningGDerivation.lean

THEOREM running_g_scaling · IndisputableMonolith/Gravity/RunningGDerivation.lean

What this page does not claim

This page does not claim that the running G derivation is an empirical measurement of gravity at nanometer scales. This page does not claim that the physical recognition-to-linking bridge is established; that bridge remains OPEN. This page does not claim that the running exponent applies beyond the nanometer scale regime described.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/RunningGDerivation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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