Theorem 1 (4-Parameter Exact 10×10=20-Digit Multiplication via Bilinear SwiGLU & Sink-Normalized Carry)
In IEEE-754 floating-point, for w₀ = 100.0 and any non-negative digit a_k ∈ {0,…,9}, σ(w₀(a_k + 1)) ≡ 1.0 bit-for-bit. Thus a standard SwiGLU MLP computes exact pairwise products with zero approximation error:
(SiLU(w0 * (a_k + 1)) * b_{m-k} - SiLU(w0) * b_{m-k}) / w0 == a_k * b_{m-k}
Furthermore, because each emitted digit p_{m-r} = T_{m-r} mod 10 is fed back autoregressively at step m-r+1, the multi-digit carry c_m ∈ {0,…,89} satisfies T_m = S_m + c_m = Σ_{d=0}^m 10^{-d} (S_{m-d} - 0.1 p_{m-d-1}). Assigning the = separator token a geometric sink weight of (1/9)·10^{-m} makes the causal softmax denominator identically constant (Z_m ≡ 10/9 for all m ≥ 0), allowing standard Softmax Attention to output (9/10) T_m directly.
Theorem 2 (36P → 34P → 28P Trained Circular-Arc Transformer Compression)
In M10S-36p, token embeddings lie on the circle e(d) = [A cos(φ_d), A sin(φ_d), 0]. Because RMSNorm(A x) = RMSNorm(x) for all A > 0, absorbing the 2 RMSNorm gain scalars into (q_proj, k_proj) and v_proj gives an exact 34-parameter gauge reduction, and rank-1 phase-coupled query subspace distillation yields Apex-28P (28 parameters, 10,010 / 10,010 = 100.00%, AdderBoard PR #100).