Let \(M\) be the minimum in the candidate question. Emdin, Kulikov, Mihajlin, and Slezkin prove that every CNF encoding of parity on \(n\) visible variables, with any number of auxiliary variables, has at least \(3n-9\) clauses. The relation in this problem is even parity on the six visible variables \(x_1,\ldots,x_5,y\). Flipping the polarity of \(y\) carries an even-parity encoding to an odd-parity encoding without changing its clause count or auxiliary-variable count. Their theorem therefore gives \(M\geq 3\cdot6-9=9\).

For the other endpoint, introduce \(z_1,z_2,z_3\) and impose \[ z_1=x_1\oplus x_2,\qquad z_2=z_1\oplus x_3,\qquad z_3=z_2\oplus x_4,\qquad y=z_3\oplus x_5. \] Each binary XOR equivalence uses its four prime implicates, giving 16 clauses. The executable artifact checks the existential projection and all 729 partial assignments of the visible variables. It also checks all 19,683 partial assignments of all nine variables, a stronger propagation condition. Hence \[ 9\leq M\leq16. \] The available evidence leaves clause counts 9 through 15 open. In particular, it does not establish the candidate's proposed minimum of 16.

Theorem 1.1(3) of Emdin, Kulikov, Mihajlin, and Slezkin states that a CNF encoding of \(\operatorname{PAR}_n\), allowing auxiliary variables, has \(m\geq3n-9\) clauses. The paper defines an encoding by existential projection over the auxiliary variables, matching the semantic convention in the candidate.

The target relation satisfies \[ x_1\oplus x_2\oplus x_3\oplus x_4\oplus x_5\oplus y=0. \] Replacing \(y\) by \(\neg y\) turns this predicate into \(\operatorname{PAR}_6\). Literal complementation preserves clauses one for one. Substitution into the published theorem gives \(m\geq3\cdot6-9=9\). This semantic lower bound applies to the candidate's smaller class with at most three auxiliaries and its added propagation requirement.

Gwynne and Kullmann distinguish two propagation criteria. Their arc-consistency condition applies partial assignments to the variables of the represented XOR system and asks unit propagation to derive every forced original literal. Their propagation-completeness condition quantifies over all variables in the CNF, including auxiliaries. Under that terminology, the candidate's stated visible-variable condition is arc consistency. The 16-clause artifact passes both criteria, so the terminology does not affect its upper bound.

The same paper proves that the standard translation of one XOR equation is propagation complete. Its construction breaks a long XOR into binary equations with fresh variables, matching the four-gate chain checked here. Emdin et al. prove the unrestricted clause lower bound \(3n-9\), which yields 9 clauses at \(n=6\). Their limited-auxiliary estimates are asymptotic or width bounds and give no 16-clause lower bound for this fixed parameter choice.

A focused search of these papers and their cited parity-encoding results found no exact clause minimum for six visible variables with at most three auxiliaries, under either propagation convention. This records the search result rather than a novelty theorem. A proof excluding 15 clauses, or an encoding with at most 15 clauses, would still meet the candidate's acceptance condition.

Variables 1 through 6 are \(x_1,x_2,x_3,x_4,x_5,y\), and variables 7 through 9 are \(z_1,z_2,z_3\). For an equation \(c=a\oplus b\), the verifier emits \[ (a\vee b\vee\neg c),\ (a\vee\neg b\vee c),\ (\neg a\vee b\vee c),\ (\neg a\vee\neg b\vee\neg c). \] It applies this template to \((x_1,x_2,z_1)\), \((z_1,x_3,z_2)\), \((z_2,x_4,z_3)\), and \((z_3,x_5,y)\).

The semantic check enumerates all \(2^9\) total assignments. Exactly 32 satisfy the formula, their visible projections are exactly the 32 parity tuples, and every valid visible tuple has one auxiliary extension. For each ternary partial assignment, the program enumerates compatible total models. Unit propagation must find a conflict when that model set is empty and must derive every unassigned variable fixed across the compatible models.

Among the 729 visible partial assignments, 697 are consistent and 32 are inconsistent. There are 192 newly forced visible literals. Among all 19,683 partial assignments to the nine formula variables, 9,315 are consistent, 10,368 are inconsistent, and 21,960 unassigned literals are forced. Every obligation passes. The candidate record's number 2,916 is the total number of assigned visible positions across the 729 ternary rows, so it includes assumptions and inconsistent rows.