Gravity Analysis Regge Bloch M2 Tendsto4 D Fold Along M2 Tendsto Of Axis Ttplus

The punctured limit

In the Recognition Science framework's study of gravity, a central object is the ledger, a discrete record of geometric events. The framework's library, a machine-checked collection of formal theorems, analyzes how certain quantities built from this ledger behave as a probe scale, written μ, shrinks toward zero. The declaration FoldAlongM2Tendsto_of_axisTTPlus is a theorem in this library. It establishes a precise limit statement for a specific configuration of the ledger, identified by the name axisTTPlus.

The theorem proves that a quantity called foldAlong, which aggregates geometric data along a chosen direction, divided by the square of the scale μ, approaches a finite value as μ tends to zero. This is a second-order limit: it shows not just that the quantity vanishes, but that it vanishes at a rate proportional to μ². The proof relies on two ingredients: a cosine expansion for the geometric terms, and the fact that a certain deficit kernel, a measure of curvature defect, is exactly zero at μ = 0 for this configuration. The result is one of a pair; a companion theorem establishes the same limit for a configuration called decoyGauge.

The theorem's scope is deliberately narrow. It applies only to the two named configurations, axisTTPlus and decoyGauge, not to a general configuration. The library's documentation states this plainly: the theorem closes the limit for those two polarizations, while the general case remains an open proposition. The theorem does not assert that gravity itself is discrete, nor that the ledger is the correct description of physical spacetime. It is a statement within the framework's formal system, about the behavior of a defined quantity under a specific limit.

What the theorem changes is the status of a question: for these two configurations, the second-order limit is no longer open. It is a proved result, available for further formal work. The general case, for arbitrary configurations, remains a target. The theorem also demonstrates a method: combining a cosine expansion with a vanishing condition at zero to close a limit argument. This is a concrete step in the framework's larger project of deriving gravitational structure from its ledger, but it is a step, not the whole edifice.

theorem FoldAlongM2Tendsto_of_axisTTPlus :
    FoldAlongM2Tendsto axisTTPlus := by
  unfold FoldAlongM2Tendsto
  exact tendsto_foldAlong_div_sq axisTTPlus fun s t => kerAlong_axis_zero s t

theorem kerAlong_axis_zero (s : Fin 24) (t : Fin 10) :
    kerAlong axisTTPlus s t 0 = 0 := by
  rw [kerAlong_zero]
  simpa [classDot] using classDot_slotDeficitKer_axis s t

theorem FoldAlongM2Tendsto_of_decoyGauge :
    FoldAlongM2Tendsto decoyGauge := by
  unfold FoldAlongM2Tendsto
  exact tendsto_foldAlong_div_sq decoyGauge fun s t => kerAlong_gauge_zero s t

What this page does not claim

The theorem does not apply to general configurations, only to axisTTPlus and decoyGauge. The theorem does not assert that physical gravity is discrete or that the ledger is the correct description of spacetime. The theorem does not establish the limit for the general case, which remains an open proposition.

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/ReggeBlochM2Tendsto4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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