Gravity Running G Grav Casimir Ratio Negligible

The ratio's meaning

The Casimir force is a real, measurable attraction between two uncharged metal plates placed very close together in a vacuum. It arises from quantum fluctuations of the electromagnetic field, which exert a pressure on the plates. The force becomes significant when the gap between the plates is on the order of nanometers, and it is a standard result in quantum field theory.

Within the Recognition Science framework, a theorem named grav_casimir_ratio_negligible establishes a comparison at such scales. It states that the gravitational pressure between two plates, even when the framework's predicted enhancement of gravity at nanometer scales is included, is less than 1e-10 (one ten-billionth) of a pascal. The theorem uses a specific set of inputs: a gravitational constant of 6.674e-11, a density of 1e4, a time of 1e-6, and an enhancement factor of 32. This is the framework's own model of gravity's strength at a 20-nanometer separation, where it predicts a 32-fold increase over its macroscopic value.

The result is a formal, machine-checked proof that this enhanced gravitational pressure is utterly negligible compared to the Casimir pressure. A second theorem, grav_dominated_by_casimir_on_nano, builds on this to state that for any plate separation where the Casimir pressure exceeds 1e7 pascals, the gravitational pressure is less than that Casimir pressure divided by 1e17. In plain terms, at the nanoscale, the Casimir force completely swamps gravity by many orders of magnitude, even under the framework's own assumptions about gravity strengthening.

What the theorem does not claim is that gravity is absent or that the Casimir force is the only force at play. It is a numerical comparison under a specific model, not a statement about the fundamental nature of either force. It also does not claim that the framework's predicted 32-fold enhancement of gravity is physically real; that is a separate hypothesis. The theorem only shows that, given that hypothesis, the resulting gravitational effect is still far too small to matter in a typical nanoscale experiment.

The practical consequence is a clear separation of scales. For anyone designing or interpreting experiments with plates separated by nanometers, the framework's own mathematics says that gravity can be safely ignored as a background effect. The Casimir force is the dominant player, and any anomaly in such an experiment would need to be explained by something other than the framework's running gravitational constant.

/-- The gravitational contribution is negligibly small vs Casimir. -/
theorem grav_casimir_ratio_negligible :
    gravitational_pressure 6.674e-11 1e4 1e-6 32 < 1e-10 := by
  unfold gravitational_pressure; norm_num

/-- Parameterized Casimir-dominance theorem: once an ideal plate configuration
has a pressure magnitude above `1e7`, the running-G gravitational pressure
example is smaller than `|P_Casimir| / 1e17`.  The legacy numeric inequality
above supplies the gravitational side; `CasimirEffectCertV2` supplies the
canonical pressure object. -/
theorem grav_dominated_by_casimir_on_nano
    (r : PlateSeparation) (hfloor : (1e7 : ℝ) < |QFT.CasimirPlateModes.idealPressure r|) :
    gravitational_pressure 6.674e-11 1e4 1e-6 32 <
      |QFT.CasimirPlateModes.idealPressure r| / 1e17 := by
  have hgrav := grav_casimir_ratio_negligible
  have hratio : (1e-10 : ℝ) < |QFT.CasimirPlateModes.idealPressure r| / 1e17 := by
    nlinarith
  linarith

What this page does not claim

The theorem does not claim that gravity is absent at nanometer scales. The theorem does not claim that the 32-fold enhancement of gravity is a physically measured effect. The theorem does not claim that the Casimir force is the only force acting between the plates.

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

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