Gravity Analysis Regge Bloch M2 Tendsto4 D Fold Along M2 Tendsto Decoy Gauge Hol

The decoy gauge limit

In Recognition Science, a framework that models physical structure from discrete recognition events, a central question is how a discrete ledger of events behaves as the scale shrinks toward zero. The declaration FoldAlongM2Tendsto_decoyGauge_holds is a formal statement in the framework's machine-checked library. It establishes that a particular quantity, foldAlong, which measures how the ledger's structure folds along a direction, has a finite limit when divided by the square of the scale parameter, for the configuration called decoyGauge.

To understand what this means, consider a discrete record of events, the ledger. As the scale parameter μ approaches zero, the raw folding quantity may blow up or oscillate. The declaration proves that, for the decoy gauge configuration, the ratio foldAlong(μ)/μ² converges to a finite number, the m2Symbol. This is a precise statement about the second-order behavior of the folding at zero scale, a kind of curvature or acceleration of the ledger structure.

The proof relies on two ingredients. First, a proved cosine two-jet statement shows that the deficit kernel, a measure of curvature, vanishes at zero momentum for the decoy gauge. Second, the declaration tendsto_foldAlong_div_sq shows that if this kernel vanishes at zero for all slots, then the folding ratio has the stated limit. The declaration combines these to close the punctured limit along the symbol direction for the decoy gauge's two polarizations.

What the declaration does not claim is broader. It does not establish the limit for a general configuration H; that remains a named proposition, not a proved statement. It only covers the decoy gauge and the axis TT plus configuration, not all possible configurations. The physical interpretation of this limit, such as what it means for gravity in the framework, is not part of the declaration's statement.

The consequence is that the framework now has a rigorous handle on the small-scale behavior of one specific configuration, a step toward understanding how discrete recognition events give rise to continuous geometry. The limit being finite and well-defined is a necessary condition for the decoy gauge to be a viable physical configuration in the framework's account of gravity.

theorem FoldAlongM2Tendsto_decoyGauge_holds :
    FoldAlongM2Tendsto_decoyGauge :=
  (FoldAlongM2Tendsto_gauge_iff).mp FoldAlongM2Tendsto_of_decoyGauge

theorem kerAlong_gauge_zero (s : Fin 24) (t : Fin 10) :
    kerAlong decoyGauge s t 0 = 0 := by
  rw [kerAlong_zero]
  simpa [classDot] using classDot_slotDeficitKer_gauge s t

What this page does not claim

The declaration does not prove the limit for all configurations, only for the decoy gauge and axis TT plus. The declaration does not provide a physical interpretation of the limit or its significance for gravity. The declaration does not establish that the decoy gauge is physically realized, only that its small-scale limit is well-defined.

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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