C  LAYERNORM — layer normalisation, double-precision, single row.
C  Computes the per-row mean and variance, normalises, then applies
C  the learnable affine transform (gamma, beta).
C
C  Reference: J. L. Ba, J. R. Kiros, G. E. Hinton, "Layer
C  Normalization", arXiv:1607.06450 (2016). Standard normalisation
C  layer in transformer architectures.
C
C  Hand-written reference for the Dark Factory's Phase 3 inference
C  kernel ladder, 2026-05-24. Public domain.
C
C  Mathematical definition (layer normalisation, per row of width D):
C      mean  = sum over I of X(I) / D
C      var   = sum over I of (X(I) - mean)^2 / D
C      Y(I)  = (X(I) - mean) / sqrt(var + EPS) * GAMMA(I) + BETA(I)
C  for I in 1..D.
C
C  Shapes:
C    X     is a vector of length D                     (input row)
C    GAMMA is a vector of length D                     (scale)
C    BETA  is a vector of length D                     (shift)
C    Y     is a vector of length D                     (output row)
C    EPS   is a small positive scalar (numerical stability)
C
C  Two passes over X: first for the mean, second for the variance.
C  Then a single normalisation pass writes Y.

      SUBROUTINE LAYERNORM (D, X, GAMMA, BETA, EPS, Y)
C  Inputs:
C    D       — vector length (positive)
C    X       — D-element input row
C    GAMMA   — D-element scale parameters
C    BETA    — D-element shift parameters
C    EPS     — variance-stabilisation epsilon (positive)
C  Output:
C    Y       — D-element normalised, scaled and shifted row
      INTEGER D
      DOUBLE PRECISION X(*), GAMMA(*), BETA(*), Y(*)
      DOUBLE PRECISION EPS
      INTEGER I
      DOUBLE PRECISION SUM, MEAN, DIFF, VAR, INV_STD, DD
      DD = DBLE(D)
C  Mean pass.
      SUM = 0.0D0
      DO 10 I = 1, D
         SUM = SUM + X(I)
   10 CONTINUE
      MEAN = SUM / DD
C  Variance pass.
      SUM = 0.0D0
      DO 20 I = 1, D
         DIFF = X(I) - MEAN
         SUM = SUM + DIFF * DIFF
   20 CONTINUE
      VAR = SUM / DD
      INV_STD = 1.0D0 / SQRT(VAR + EPS)
C  Normalisation, scale, shift pass.
      DO 30 I = 1, D
         Y(I) = (X(I) - MEAN) * INV_STD * GAMMA(I) + BETA(I)
   30 CONTINUE
      RETURN
      END