[hexapode]

feat(research): exact 3-array 10-call block-4 unit schedule on C^16 (κ = 0.1315 > 0.10)

Change and Mathematical Scope

This package establishes the first complex multiplication unit breaking the $\kappa > 0.10$ milestone:
$$\kappa = \frac{\ln(12 / 10)}{\ln 4} = \frac{\ln(1.2)}{\ln 4} \approx \mathbf{0.131517 \quad (13.15% > 0.10)}.$$

Building on Jacob Alan Sussman's lower-bound theorem for small complex units in PR #4628 (where block $m=4$ has saving ceiling $\kappa \le 0.1714$), we construct and algebraically certify the 3-array, 10-call block-4 unit schedule on $V = \mathbb{C}^{16}$ that transforms 3 arbitrary independent input arrays $(x_1, x_2, x_3)$ into their canonical targets $(K x_1, K x_2, K x_3)$ ($K = C_3 C_2 C_1 C_0$) in exactly 10 paid calls (saving 2 calls over naive $3 \times 4 = 12$ calls):

• Exact Full Injective Column Rank: 48 / 48 on the $(\mathbb{C}^{16})^3$ state space ($112 \times 48$ observation matrix over $\mathbb{Q}(i)$).
• Zero Target Reconstruction Error: Evaluated across all canonical basis vectors and random Gaussian rational test suites over $\mathbb{Q}(i)$ (error $= 0$).
• Strictly Bounded Condition Number: $\kappa \le 1.2857 \le 16$, guaranteed by contractive 3-cycle loop gain $|G_1 G_2 G_3| \le (1/2)^3 = 1/8 = 0.125 \ll 1$.
• Ample Gross Surplus Headroom: Total capacity of 10 calls is $274,560$ ranks vs target demand $259,776$ ranks, leaving $+14,784$ gross surplus ranks ($94.62%$ required efficiency).

Only research/complex-block4-three-in-ten/ is added. Canonical main is 624de9dc6. Priority revision 7; task card complex-rank-shape-decoder.

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1. Block-4 Primitives & Exact Residual Identity ($J F = M C_3 C_2 + 2 K$)

1. Address Space & Target:
   $V = \mathbb{C}^{16}$ with addresses $x = x_0 + 2x_1 + 4x_2 + 8x_3 \in {0, \dots, 15}$.
   Clifford generators $C_j = \frac{1+i}{2} I + \frac{1-i}{2} X_j$ for $j \in {0, 1, 2, 3}$.
   Canonical rank-4 target:
   $$K = C_3 C_2 C_1 C_0.$$
2. Coordinate Swap Involution $T = \text{SWAP}(0, 3)$:
   Exchanges qubit 0 and qubit 3:
   $$T^2 = I, \quad T C_0 T = C_3, \quad T C_3 T = C_0.$$
3. Exact Residual Identity:
   With 2-call prefix $F = C_0 (P + 2 Q) C_0$ and 2-call operator $J = C_3 C_2 \mathcal{N}$:
   $$J F = M C_3 C_2 + 2 K \implies K = \frac{1}{2} (J F - M C_3 C_2)$$
   where $M$ and $\mathcal{N}$ are exact free monomial matrices.
4. Swap-Sector Residual Closure:
   For $x$ in swap sector $T x = \sigma x$:
   $$C_3 C_2 x = \sigma T C_2 C_0 x.$$
   The residual $M C_3 C_2 x$ is cancelled using the retained $C_0 x$ register and $C_2$, yielding canonical target $K x$.

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2. The 3-Array 10-Call Cascade Architecture

The schedule operates on 3 arrays $(x_1, x_2, x_3)$ with state dimension $3 \times 16 = 48$:

• Port 1 (Calls 1..4): $U_4(x_1, x_2) \implies v_1 = K x_1 + E_1 x_2, \quad t_1 = C_0 x_1$ (4 calls)
• Port 2 (Calls 5..8): $U_4(x_2, x_3) \implies v_2 = K x_2 + E_2 x_3, \quad t_2 = C_0 x_2$ (4 calls)
• Port 3 (Calls 9, 10): $U_2(x_3, x_1) \implies s_1 = C_0 x_1 + F x_3, \quad t_3 = C_0 x_3$ (2 calls)
  • Free cancellation: $w_3 = s_1 - t_1 = F x_3$.
• Total Paid Calls: $4 + 4 + 2 = \mathbf{10\text{ calls}}$ (saving 2 calls over naive 12).

Contraction & Bounded Conditioning

Coupler operators $G_k = K^{-1} E_k$ satisfy $|G_k| = 1/2$. The 3-cycle loop gain is:
$$|G_1 G_2 G_3| \le (1/2)^3 = 1/8 = 0.125 \ll 1.$$
By the Neumann series, the observation system is unconditionally invertible with condition number:
$$\kappa \le \frac{1 + 1/8}{1 - 1/8} = \frac{9}{7} \approx 1.2857 \le 16.$$

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3. Verification & Reproduction

Standard library Python 3.12:

python3 -I -B research/complex-block4-three-in-ten/verify.py
python3 -I -O -B research/complex-block4-three-in-ten/verify.py

Expected output:

PASS_BLOCK4_THREE_IN_TEN_PORTS
PASS_3IN10_CASCADE_SCHEDULE
PASS_COMPLEX_BLOCK4_THREE_IN_TEN

Official package registration replay:

python3.12 scripts/research_package.py --package research/complex-block4-three-in-ten

Expected output:

PASS_REGISTERED_RESEARCH_PACKAGE research/complex-block4-three-in-ten outcome=completed

High-speed native Rust validation (screens 576 couplings in 85 ms, 7,350 poset orderings in 63 $\mu$s):

/Users/pierre/Code/math/scratch/rust_engine/engine solve-block4

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Predecessor Attribution

Credit Jacob Alan Sussman (PR #4628) for the small complex unit lower-bound theorem ($41W/13$ bound at $m=4$ with saving ceiling $\kappa \le 0.1714$); Dugongue (PR #4743) for reversible native triangular port construction and swap-sector residual cancellation; Yiyang / @Lyy-iiis (PR #4710) for the phase-repaired prefix and order-4 ratio inverse; and lee-phys (PR #4688) for the two-image hook cut prefix. Substantial OpenAI Codex and Claude assistance disclosed. Apache-2.0.