Gravity Analysis Regge Exact Flat Hessian Symbol4 D Measured Small Kalpha Symbol

The small-k measurement

The quantity measuredSmallKAlphaSymbolDir is a number recorded in the framework's machine-checked library of formal theorems. It is not a proved equality but a measured value, the result of a finite computer calculation on a discrete model of spacetime. The number is -0.249884, and the declaration measuredSmallKAlphaSymbolDir_near_quarter records that this value lies within 0.01 of -1/4. That is the entire content of the declaration: a closeness statement about a specific computed number.

The context matters. The framework models spacetime as a discrete ledger, a record of events on a grid, and this measurement concerns the second variation of the Regge action, a discrete version of the Einstein-Hilbert action from general relativity. On a flat background, where all curvature vanishes, this second variation reduces to a cross term. The declaration measures this term for a particular polarization mode at a small wave number, and finds it close to the continuum value -1/4. The closeness is the result, not a derivation.

What the declaration does not claim is equally precise. It does not prove that the discrete model converges to the continuum theory. The library explicitly records that the limit statement, the inhabitability of the ledger S_RS, and the recovery of the gap action all remain open. The algebraic table of all coupling coefficients is also absent. The measurement is a single point of evidence, a certificate that a specific computation landed near a target, not a general theorem about the model.

The value -0.249884 differs from -1/4 by about 0.000116, which is well within the declared 0.01 window. The declaration is honest about its own strength: it states a numerical proximity, nothing more. In the framework's vocabulary, this is a MEASURED fact, one tier below a THEOREM. The reader should take it as a concrete data point that supports but does not establish the connection between the discrete model and the continuum action.

/-- Two-point small-`k` intercept for `axisTTPlus`/`symbolDir`:
stand-in `-249884/1000000` for measured `α ≈ -0.249884`. -/
def measuredSmallKAlphaSymbolDir : ℝ := -(249884 / 1000000)

What this page does not claim

It does not claim the discrete model converges to the continuum theory. It does not claim the value -1/4 is derived from the framework's axioms. It does not claim a general algebraic table of coupling coefficients exists.

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/ReggeExactFlatHessianSymbol4D.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

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