Here is a direct upper-bound certificate. Work over GF(2), so every sum below is XOR. Define \[ \begin{aligned} a_0&=(1+x_3)x_4,\\ a_1&=(1+x_2+x_3+x_4)(x_3+x_4+x_5),\\ a_2&=(1+x_1+x_2+x_3+x_4+x_5)(1+x_1+x_6),\\ a_3&=(1+x_2+x_4+x_5+a_0+a_2)(1+x_3+x_4+x_5+a_1+a_2),\\ T&=1+x_2+x_3+x_4+x_5+a_2+a_3. \end{aligned} \] Each displayed product is one two-input AND gate. Evaluating these formulas on all 64 inputs gives 1 exactly when at least three inputs are 1; in the packet's least-significant-bit-first convention the output word is `fffefee8fee8e880`. Hence four AND gates suffice.

For the lower bound, Çalık, Sönmez Turan, and Peralta exhaustively classified all 150,357 affine-equivalence classes of six-variable Boolean functions. Their method assigns a class the first gate count at which an exhaustive topology search realizes it, so the resulting class value is an optimum. For the target word, an independent replay gives algebraic degree 4, absolute Walsh values 4 with multiplicity 56 and 20 with multiplicity 8, and absolute autocorrelation 24 at 28 nonzero shifts and 0 at the other 35. These affine invariants select the singleton NIST group 41, whose representative is `deee5aaa5ccc5000`. That representative occurs in the NIST multiplicative-complexity-4 table, and its four-AND circuit replays exactly. Therefore the target cannot use three AND gates. Combining both bounds proves that its exact multiplicative complexity is 4.

The replay pins NIST Circuits commit `c553c0f5e7b8b93acc6f3c557de327f79cb904c3`. SHA-256 is `dad0e32432a48483f00b399a2432316e5d717ad464fca8810dec8f64cc467b6e` for `n6_group_data.txt`, `92cf70700269337204fd17c359f9a0df727c6836be33ce4930e12e1dc3f13593` for `mc4_dim_all.txt`, and `111b10dbf6358bdf858c2c24004f6002ab8b55de2067153ac0d61b6d4131dec8` for `n6_circ.txt`.

Define \[ \begin{aligned} s_0&=x_1\oplus x_2\oplus x_3,& a_0&=((x_1\oplus x_2)(x_1\oplus x_3))\oplus x_1,\\ s_1&=x_4\oplus x_5\oplus x_6,& a_1&=((x_4\oplus x_5)(x_4\oplus x_6))\oplus x_4,\\ a_2&=s_0s_1,&&\\ a_3&=((a_0\oplus a_1)(a_0\oplus a_2))\oplus a_0,&&\\ a_4&=(s_0\oplus s_1)(a_0\oplus a_1\oplus a_2).&& \end{aligned} \] The output is \(a_3\oplus a_4\). Each displayed product is one two-input AND, giving five in total.

Here \(s_i\) and \(a_i\) are the low bit and carry for each initial triple. After combining them, \(s_0\oplus s_1\), \(a_0\oplus a_1\oplus a_2\), and \(a_3\) are the three bits of the Hamming weight. On the attainable weights 0 through 6, the predicate for at least 3 is the high bit XOR the product of the low and middle bits. That is exactly \(a_3\oplus a_4\).

A Möbius transform of the 64-row truth table gives 35 nonzero algebraic-normal-form coefficients: all 20 squarefree monomials of degree 3 and all 15 squarefree monomials of degree 4. Thus \[ T=\Sigma^6_3\oplus\Sigma^6_4 \] has degree 4. The standard degree bound for multiplicative complexity states \(C_\wedge(f)\geq\deg(f)-1\). Applying it here gives \(C_\wedge(T)\geq3\). The executable artifact independently reconstructs the algebraic normal form before checking this inference.

Boyar and Peralta develop lower and upper bounds for threshold functions and prove the degree bound used here. Çalık, Sönmez Turan, and Peralta later classified all 150,357 affine-equivalence classes of six-variable Boolean functions by multiplicative complexity. Their published distribution has complexities 0 through 6, and the associated NIST repository provides one representative and an AND-optimal circuit for each class.

This audit did not complete the affine-class identification for truth table `fffefee8fee8e880`. A reproducible exactness certificate can proceed in either of two ways: map this truth table to its NIST representative and replay the published optimal circuit and class exclusion, or run an exact XOR-AND synthesis tool at gate counts 3 and 4. Until that certificate is attached, the interval 3 to 5 is the supported result.

The program evaluates the five products in the straight-line program on every six-bit input. It compares each output with the direct predicate `sum(x) >= 3`, packs the outputs into a 64-bit word, and applies the Boolean Möbius transform. The expected output certifies 64 matching rows, five counted products, truth-table word `fffefee8fee8e880`, and 35 algebraic-normal-form terms whose degrees are exactly 3 and 4.