# Coherent-Fluctuating Amplitude Units

- ID: 835
- Canonical URL: https://synthcore.org/idea/835/coherent-fluctuating-amplitude-units
- API JSON: https://synthcore.org/api/idea/835.json
- API Markdown: https://synthcore.org/api/idea/835.md
- Verification status: unverified
- Source: [arXiv:2607.12926](https://arxiv.org/abs/2607.12926)
- Category: architecture
- Solves: generalization, stability
- ML areas: rnn, diffusion, regularization, embedding
- Math tags: statistical-mechanics, probability, statistics, dynamical-systems
- Ratings: usefulness 4/10; difficulty 5/10; novelty 8/10

## Idea description

Represent selected hidden features as z = sqrt(N) exp(i theta), with a persistent phase and an explicitly stochastic amplitude. Regularize the ratio between coherent power |E[z]|^2 and total power E[|z|^2] toward the condensate prediction pi/4, while optionally matching higher amplitude moments.

## Mathematical statement

Let N be a nonnegative occupation variable, rho = E[N] its mean, and z = sqrt(N) exp(i theta). The ideal grand-canonical distribution is P(N) = (1/(1+rho)) times (rho/(1+rho)) to the power N. For large rho, the distribution approaches an exponential distribution, so E[sqrt(N)] is approximately sqrt(pi rho)/2 and the coherent-power ratio |E[z]|^2/E[|z|^2] approaches pi/4. The exponential amplitude hierarchy is E[N^q] = q! rho^q. In a neural layer, estimate the ratio from phase-aligned minibatch samples and penalize deviation from pi/4.

## Key formulas

- P(N) = (1/(1 + rho)) (rho/(1 + rho))^N, with rho = E[N] and z = sqrt(N) exp(i theta).
- E[sqrt(N)] approximately equals sqrt(pi) sqrt(rho) / 2, and |E[z]|^2 / E[|z|^2] approximately equals pi / 4 for rho much greater than 1.
- E[N^q] approximately equals q! rho^q, and L_coh = (|z_bar|^2 / (mean_abs_z_squared + epsilon) - pi/4)^2 + lambda_h sum from q=2 to Q of (moment_q / (q! mean_abs_z_squared^q) - 1)^2.
- z_(t+1) = sqrt(N_(t+1)) exp(i (theta_t + delta_theta_(t+1))), with N_(t+1) sampled from a geometric distribution whose mean is rho_(t+1).

## Implementation notes

Integrate this mechanism into a complex-valued GRU, recurrent state-space model, or stochastic latent block. Let the ordinary neural update produce u_(t+1) and v_(t+1). Set rho_(t+1) = softplus(u_(t+1)) and delta_theta_(t+1) = v_(t+1), retain theta_t as a phase state, and sample N_(t+1) from the geometric law with mean rho_(t+1). For a reparameterized continuous approximation, sample U uniformly from (0,1) and use N_(t+1) = -rho_(t+1) log(U). Construct z_(t+1) = sqrt(N_(t+1)) exp(i(theta_t + delta_theta_(t+1))) and pass its real and imaginary parts onward. For a minibatch, phase-align each sample by z_tilde_b = z_b exp(-i arg(sum over b of z_b)); compute z_bar = average of z_tilde_b and the empirical amplitude moments. Add alpha times L_coh to the task loss, warm alpha from zero, and clip rho or normalize amplitudes to avoid rare numerical explosions. The paper contributes the geometric occupation law, the pi/4 ratio, and the factorial moment hierarchy; phase alignment, recurrence, and minibatch estimation are neural adaptations. First run a one-layer complex GRU on sequential MNIST or permuted MNIST, comparing deterministic complex states, additive Gaussian noise, and this module. Measure R = |z_bar|^2 / mean(|z|^2) as a function of learned rho. The quantitative prediction is that channels with rho at least 50 should have R close to pi/4 = 0.785, within approximately 10 percent, and moment_q divided by mean(|z|^2)^q should approach q! for q from 2 to 4. If these signatures do not appear, the transfer mechanism is falsified even if accuracy changes.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.