Gravity Analysis Regge Bloch M2 Symbol4 D

The small-momentum symbol

In the Recognition Science framework's study of gravity, a ledger (a discrete record of events) is used to model spacetime. The module ReggeBlochM2Symbol4D examines one piece of this model: the behavior of a function called blochFold11 as a small momentum parameter μ approaches zero. This function is built from a sum over 24 slot types and 10 triangle types, each weighted by coefficients derived from a 4x4 matrix H that encodes the spacetime configuration.

The central result is about the coefficient of μ² in this expansion, called the m2Symbol. For a specific test configuration called axisTTPlus, the library proves this coefficient is exactly -3, not zero. This is significant because a nonzero coefficient means the function has a genuine quadratic term at small momentum, rather than being flat. The proof is carried out in a machine-checked library of formal theorems, meaning every step is verified by a computer. The library also shows that for a different configuration called decoyGauge, the coefficient is exactly 0, demonstrating that the result depends on the specific spacetime configuration.

The module also establishes a symmetry property: the function foldAlong, which is blochFold11 evaluated along a specific direction in momentum space, is an even function of μ. This means that its first derivative at μ=0 vanishes, a mathematical consequence of the evenness. This evenness is a structural property that holds for any configuration H, not just the specific test cases.

However, the module is explicitly scoped. It only proves results for this (1,1)-orbit contribution, not the full Hessian symbol of the entire gravity model. The status structure within the module records that the full Hessian symbol, the convergence to the Einstein-Hilbert action, and the recovery of the gap action remain open problems. The module also leaves open the precise limit statement FoldAlongM2Tendsto, which would formally connect the coefficient to the limit of the ratio foldAlong H μ / μ² as μ tends to zero. This limit is proven only for the two specific configurations, not for a general H.

What this means in practice is that the framework has a verified, concrete handle on one small piece of its gravity model: for a particular test configuration, the small-momentum behavior is nontrivial and computable. This is a building block, not the whole edifice. The framework's library proves this specific coefficient is -3, and it does so with complete formal rigor, but it does not yet claim to have derived all of gravity from its principles.

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

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

theorem sum_m2SlotCertZ_gauge :
    (∑ s : Fin 24, ∑ t : Fin 10, m2SlotCertZ decoyGaugeCoeffZ s t) =
      (0 : ℤ) := by
  decide

What this page does not claim

The module does not prove the full Hessian symbol for the gravity model. The module does not establish convergence to the Einstein-Hilbert action. The limit statement FoldAlongM2Tendsto is proven only for specific configurations, not for a general H.

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)

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