incoherent_effective_source
The definition computes the net source strength for an ensemble of N particles with random phases as the square root of N times the individual amplitude, following the central limit theorem for vector summation. Researchers modeling gravitational effects from coherent matter, such as in superconductors, would reference this when quantifying the baseline incoherent contribution. It is realized as a direct arithmetic expression without additional lemmas.
claimLet $E$ be an ensemble with $N$ particles each of amplitude $a > 0$. The incoherent effective source magnitude is given by $a /sqrt{N}$.
background
The Ensemble structure packages the particle count $N$ (with a positivity witness), the individual amplitude $a$ (with its positivity witness), and records that the net source depends on phase relations among the contributions. In the Gravity.CoherenceGain module the focus lies on how phase coherence modifies the effective gravitational source strength relative to the incoherent baseline supplied by random phases. The upstream Ensemble definition supplies the data fields $N$ and individual_amplitude used in the expression.
proof idea
The definition is a one-line expression that multiplies the square root of the particle count by the single-particle amplitude, directly implementing the random-walk magnitude for uncorrelated phases.
why it matters in Recognition Science
This definition supplies the denominator for the coherence_gain ratio, which equals the square root of $N$ and appears in the CoherenceGainCert structure that certifies the coherence gate for superconductors. It anchors the comparison between incoherent and coherent regimes in the Recognition Science treatment of gravity, where phase coherence in matter amplifies the effective source by √N. It closes the classical random-walk baseline needed before invoking further scalings in the framework.
scope and limits
- Does not assume any specific phase distribution beyond uncorrelated randomness.
- Does not incorporate relativistic or quantum-field corrections to the source.
- Does not constrain $N$ beyond the positivity already required by Ensemble.
- Does not model interactions between multiple ensembles.
formal statement (Lean)
47def incoherent_effective_source (ens : Ensemble) : ℝ :=
proof body
Definition body.
48 Real.sqrt ens.N * ens.individual_amplitude
49