Gravity Analysis Regge4 Dtransported Algebraic Closer One Orbit Ray Normalized C

The one-orbit ray coefficient

A recognition ledger, a discrete record of events, can be built from the geometry of a four-dimensional space. In the Recognition Science framework, one way to probe this geometry is to look at a single orbit, a closed loop of related configurations, and ask what coefficient it contributes to a certain sum. The declaration oneOrbitRayNormalizedCoeff_axisTTPlus proves that for a specific orbit, the one-orbit ray normalized coefficient equals -3/2.

The coefficient is a ratio. The numerator is a symbol value, a number attached to the orbit, and the denominator is the squared length of a direction vector. The theorem establishes that for this particular orbit, the ratio is exactly -3/2. This is a concrete, machine-checked result within the framework's library of formal theorems.

The framework also proves that this value, -3/2, is not equal to the coefficient that appears in the Einstein-Hilbert action, the standard starting point for general relativity. This is a deliberate check, a way of showing that the one-orbit construction does not accidentally reproduce a known result. The framework is explicit that this is a negative result: it shows what the one-orbit coefficient is not.

What the declaration does not claim is that this -3/2 is the final answer for the full theory. The framework explicitly leaves open the question of whether the normalized transported fold, a more complete sum over all orbits, tends to -1/4, the value that would match the Einstein-Hilbert coefficient. That target, named Regge4DContinuumEHTarget, is marked as open and false in the library's status record. The one-orbit result is a stepping stone, not the destination.

In plain terms, the declaration pins down a single, exact number for a simple case and uses it to show that this simple case does not match the classical gravity value. It is a precise, honest measurement of a small piece of the framework's machinery, with the larger convergence question explicitly left for future work.

theorem oneOrbitRayNormalizedCoeff_axisTTPlus :
    oneOrbitRayNormalizedCoeff axisTTPlus = (-3 : ℝ) / 2 := by
  unfold oneOrbitRayNormalizedCoeff
  rw [m2Symbol_axisTTPlus, symbolDir_normSq]

theorem oneOrbit_ray_normalized_ne_eh_coefficient :
    oneOrbitRayNormalizedCoeff axisTTPlus ≠
      einsteinHilbertTTCoefficient4D := by
  rw [oneOrbitRayNormalizedCoeff_axisTTPlus, einsteinHilbertTTCoefficient4D_eq]
  norm_num

What this page does not claim

The declaration does not prove that the full transported fold converges to -1/4. The declaration does not claim that the one-orbit coefficient is the Einstein-Hilbert coefficient. The declaration does not establish the value of the coefficient for any orbit other than the specific one named axisTTPlus.

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/Regge4DTransportedAlgebraicCloser.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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