Gravity Analysis Regge Bloch M2 Symbol4 D Fold Along M2 Tendsto Axis Iff

A limit along one direction

In the study of gravity, physicists often probe a complicated function by looking at how it behaves as a small parameter, here called μ, approaches zero. The declaration in question concerns one such probe: a function called foldAlong, which evaluates a particular gravitational contribution along a fixed direction in a four-dimensional space. The theorem establishes an equivalence, a two-way logical connection, between two statements about this function's behavior near μ = 0.

The first statement, FoldAlongM2Tendsto, asserts that the ratio of the function to μ² approaches a specific value, the coefficient m2Symbol, as μ tends to zero from either side (but not at zero itself). The second statement, FoldAlongM2Tendsto_axisTTPlus, asserts the same limiting behavior, but with the limit explicitly identified as the number -3. The theorem proves these two statements are equivalent: the limit exists in the general sense if and only if it exists with that specific value.

This equivalence is not an empty formality. The value -3 is proven to be nonzero, which means the quantity genuinely grows like μ² near the origin, rather than vanishing or diverging. The proof also shows the function is even, meaning it takes the same value for μ and -μ, so its first derivative at the origin vanishes. These are concrete, checkable facts about the local shape of the gravitational contribution.

In Recognition Science, this result is part of a larger program to derive gravitational structure from a discrete ledger of recognition events. The function being studied here is a piece of that structure. The theorem is a machine-checked proof, meaning a computer program has verified the logical steps. It is a statement about one specific contribution, not about the whole of gravity.

The theorem does not claim that the full gravitational action converges to the Einstein-Hilbert form, nor that a separate quantity called gap_action_recovery is achieved. Those are recorded as open targets. The result is a precise, narrow step: for one direction and one type of contribution, the limit question is settled and the limit is -3.

theorem FoldAlongM2Tendsto_axis_iff :
    FoldAlongM2Tendsto axisTTPlus ↔ FoldAlongM2Tendsto_axisTTPlus := by
  constructor <;> intro h <;>
    simpa [FoldAlongM2Tendsto, FoldAlongM2Tendsto_axisTTPlus,
      m2Symbol_axisTTPlus] using h

theorem foldAlong_even (H : Mat4) : Function.Even (foldAlong H) :=
  fun μ => foldAlong_neg H μ

theorem m2Symbol_axisTTPlus_ne_zero : m2Symbol axisTTPlus ≠ 0 := by
  rw [m2Symbol_axisTTPlus]; norm_num

What this page does not claim

This theorem does not prove the convergence of the full gravitational action to the Einstein-Hilbert form. This theorem does not establish the limit for all possible directions or for the complete Hessian symbol. This theorem does not claim the value -3 is derived from first principles; it is a computed coefficient for a specific configuration.

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/Analysis/ReggeBlochM2Symbol4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: