[hexapode]

Summary

This pull request constructs, algebraically certifies, and verifies over $\mathbb{Q}(i)$ the 3-array 9-call ($W=3, s=9$) unit schedule on Block $m=5$ ($V = \mathbb{C}^{32}$), achieving saving exponent:
$$\kappa = 1 - \frac{\ln(9/3)}{\ln 5} = 1 - \frac{\ln 3}{\ln 5} \approx \mathbf{0.31739 \quad (31.74%)}.$$

This breaks through into Block $m=5$ and realizes the open shape explicitly identified by Jacob Alan Sussman in PR #4628 (research/complex-small-m-lower-bounds).

Key Highlights

1. Exact Optimality Against Sussman's Lower Bound (PR Research: lower bounds for small complex units: m = 2 never saves, m = 3 needs 14W/5 calls (Lean, engine game), m = 4 needs 41W/13, and the smallest open cases (no new κ) #4628):

   • Sussman proved that small complex units at $m=5$ must satisfy $s \ge 2.99172 W$.
   • For $W=3$ arrays, the theoretical minimum integer call count is $\lceil 3 \times 2.99172 \rceil = \lceil 8.975 \rceil = \mathbf{9\text{ calls}}$.
   • This schedule attains the exact theoretical minimum call count allowed on $V = \mathbb{C}^{32}$.
   • Naive uncoupled cost: $3 \times 5 = 15$ calls.
   • Schedule cost: 9 calls, saving 6 paid calls (40.0% reduction).

2. Block-5 Primitives & Invariants on $V = \mathbb{C}^{32}$:

   • Canonical target operator: $K = C_4 C_3 C_2 C_1 C_0$ ($32 \times 32$ unitary matrix over $\mathbb{Q}(i)$).
   • Coordinate swap involution: $T = \text{SWAP}(0, 4)$ satisfying $T^2 = I$, $T C_0 T = C_4$, $T C_4 T = C_0$.
   • Phase & permutation generators: $P = S U S$, $Q = S V_{01} S = Z_{\text{all}} V_{01}$.
   • Ratio $R = P^{-1} Q$ has exact order 4 ($R^4 = I_{32}$), yielding the uniform all-width polynomial inverse $(I+2R)(I-2R+4R^2-8R^3) = -15 I_{32}$.
   • Exact residual identity: $J F = M C_4 C_3 C_2 + 2 K$, where $J = C_4 C_3 C_2 \mathcal{N}$ uses 3 paid calls, and $\mathcal{N}, M$ are free monomial matrices.

3. 3-Port Symmetric Architecture on $(\mathbb{C}^{32})^3$:

   • State dimension: $3 \times 32 = 96$ variables over $\mathbb{Q}(i)$.
   • Budget: 3 ports of 3 calls each ($3 \times 3 = 9$ calls):
     • Port 1 on $(x_1, x_2)$: $v_1 = K x_1 + E_1 x_2$, $t_1 = C_0 x_1$
     • Port 2 on $(x_2, x_3)$: $v_2 = K x_2 + E_2 x_3$, $t_2 = C_0 x_2$
     • Port 3 on $(x_3, x_1)$: $v_3 = K x_3 + E_3 x_1$, $t_3 = C_0 x_3$

4. Strictly Contractive Loop Gain & Bounded Condition Number:

   • Intermediate couplers $G_k = K^{-1} E_k = \frac{1}{2} X_k$ satisfy $|G_k| = 1/2$.
   • 3-cycle loop operator: $\mathcal{L}3 = G_1 G_2 G_3 = \frac{1}{8} X{012}$, with $|\mathcal{L}_3| = (1/2)^3 = 0.125 \ll 1$.
   • Cascade condition number: $\kappa(M_{\text{cascade}}) \le \frac{1 + |\mathcal{L}_3|}{1 - |\mathcal{L}_3|} = \frac{9}{7} \approx \mathbf{1.2857 \ll 16}$.

5. Validation:

   • Local registered research package replay passes: PASS_REGISTERED_RESEARCH_PACKAGE research/complex-block5-three-in-nine outcome=completed.
   • Local committed byte check passes: PASS_RESEARCH_COMMIT_BYTES.
   • Exact matrix ranks over $\mathbb{Q}(i)$: $\text{rank}(M_{\text{cascade}}) = 96/96$, $\text{rank}(O_{\text{primary}}) = 96/96$, $\text{rank}(O_{\text{full}}) = 96/96$.
   • Exact zero reconstruction error on all canonical basis vectors, Gaussian rational inputs, and swap sectors.