Resolution tax τ_4 in its smallest open form

Result

The resolution tax τ_4 satisfies:

τ_4 = MC(K_4) − MC(J_5) ∈ {1, 2}

This range follows from:

1. MC(K_4) = 9 (exact, reference MF-181).
2. MC(J_5) ∈ [7, 8].

Deciding whether MC(J_5) = 7 for the 16-input, 5-output map remains open, defining a solver wall in the 13-wire, 7-gate synthesis class.

Setting and definitions

Circuits are evaluated in the XOR-free multiplicative complexity model GF(2) XAG, where linear XOR operations cost zero and nonlinear AND gates cost 1.

• MC(f) is the multiplicative complexity of a multi-output Boolean function f over GF(2).
• K_4 and J_5 are multi-output Boolean function families; J_5 maps 16 input bits to 5 output bits.
• The resolution tax is τ_4 = MC(K_4) − MC(J_5).
• deg(f) is the algebraic degree of f. For the outputs of J_5, deg(J_5 outputs) ≤ 8.
• Target parameter p denotes the candidate number of multiplicative gates in the synthesis search (p = 7 for J_5).

Method

Bounds on MC(J_5) and τ_4 were tested using automated synthesis across several solver gauges:

• Monolithic synthesis via tile_synth.
• Gate-0 selector cubing.
• Shadow pins.
• RREF factor-pair gauge encodings.
• Impurity cube formulations (calibrated on J_4 at p = 5, yielding no speedup).
• Row restriction encodings (which increased runtime on UNSAT instances).

Equivalent formulations were also evaluated:

• Scalar TOP_4 at p = 5.
• F_5 = 7 exact + E1.
• Sum-of-dots columns at p = 5.
• Binary fork columns at p = 5.

All runs timed out between 3,600 and 7,200 seconds per cell across both tested gauges. Algebraic degree bounds cannot resolve the instance: deg(J_5 outputs) ≤ 8 = p + 1 at p = 7, leaving the missing gate undetectable by degree-based lower bounds.

Discussion

The resolution tax τ_4 remains open due to an explicit solver wall. Synthesizing J_5 at p = 7 requires searching a 13-wire, 7-gate space for a 16-input, 5-output system.

Neither direct SAT synthesis nor algebraic degree bounding determines whether MC(J_5) is 7 or 8. Degree arguments cannot force a lower bound of 8 because the maximum output degree of 8 does not exceed 2^(p - 4) or the p + 1 gate threshold at p = 7. Pruning techniques, including impurity cubing and row restriction, failed to deliver tractability or an UNSAT certificate. The problem status remains OPEN.

Register references

• Register entry: ML-089 (2026-09-03)
• Cross-reference: MF-181 (for MC(K_4) = 9)
• Receipt artifacts: ZKGOLF-COUNCIL4-DIGEST-2026-09-02.md, RECORD-RREF-SYNTH.log, RECORD-COUNCIL4-FIVECENT.log

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