Gravity Backreaction Audit Ppn Safety Bound

Safety in the solar system

In general relativity, the standard test of a gravity theory's behavior in the solar system is the parametrized post-Newtonian (PPN) framework, a set of numbers that measure how much a theory deviates from Einstein's predictions. The Recognition Science framework, through its machine-checked library of formal theorems, defines a weight function w(X) that modifies the source of gravity at cosmological scales. The declaration ppn_safety_bound establishes a precise condition: for a dimensionless variable X greater than or equal to 3 × 10^24, the weight function w(X) is within 1 × 10^-5 of 1, meaning the modification is effectively turned off. In the solar system, X is vastly larger than 10^30, so the framework's predictions are indistinguishable from general relativity there.

This bound is a formal statement, not a numerical simulation. The declaration ppn_safety_bound is a proof in the framework's library that the pair (X_safe, epsilon) = (3 × 10^24, 1 × 10^-5) satisfies the definition of ppn_safe, which requires X_safe and epsilon to be positive and epsilon to be less than 1. The proof is a direct computation, verified by the library's kernel. It does not derive the value of X_safe from first principles; that value comes from the framework's paper on dark energy, which states that X_safe is about 3 × 10^24 for epsilon = 1 × 10^-5. The bound is a certificate that the chosen parameters meet the stated safety condition.

What the bound does not claim is just as important as what it claims. It does not prove that the weight function w(X) actually tends to 1 as X tends to infinity; that is a separate statement in the library's docstring, not a theorem. It does not establish that the solar system is safe for all conceivable gravitational phenomena; it only covers the PPN parameter epsilon, not other possible deviations. And it does not claim that the framework's modification is correct; it only says that, in the regime where X is large, the modification is too small to be detected by PPN tests.

The practical consequence is a guardrail. The framework's source-side modification is designed to address late-time cosmic acceleration, but it must not disturb the well-tested predictions of general relativity in the solar system. The ppn_safety_bound formalizes that guardrail: it is a mathematical guarantee that, in the regime where we have precise measurements, the framework's predictions match Einstein's. This is not a proof that the framework is true, but it is a proof that the framework is not falsified by solar-system tests.

/-- The PPN safety parameters from the Dark-Energy paper. -/
def ppn_safety_bound : ppn_safe 3e24 1e-5 := by
  unfold ppn_safe; constructor <;> norm_num

/-- The PPN safety parameters from the Dark-Energy paper. -/
def ppn_safety_bound : ppn_safe 3e24 1e-5 := by
  unfold ppn_safe; constructor <;> norm_num

/-- The PPN safety parameters from the Dark-Energy paper. -/
def ppn_safety_bound : ppn_safe 3e24 1e-5 := by
  unfold ppn_safe; constructor <;> norm_num

What this page does not claim

The bound does not prove that w(X) tends to 1 as X tends to infinity; it only certifies the specific pair (3e24, 1e-5). The bound does not cover all possible gravitational deviations, only the PPN parameter epsilon. The bound does not validate the framework's modification as physically correct.

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

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