Almost-Everywhere Near-Cubic Gate Lower Bounds for SYM ◦THR Circuits

We prove near-cubic, almost-everywhere gate lower bounds for depth-two circuits consisting of an arbitrary symmetric output gate over linear threshold gates with unrestricted fan-in and arbitrary real weights (SYM∘THR).

There are constants c, d > 0 and a single language in E^NP that disagrees on more than a d fraction of inputs with every SYM∘THR circuit having at most

c · n³ / ((log n)^(21/4) · (log log n)⁷)

bottom gates, at every sufficiently large input length n. The same language works across all sufficiently large lengths and all eligible circuits. Here E^NP denotes deterministic time 2^O(n) with an NP oracle. More generally, the theorem allows signed integer weights in the top sum, charging the sum of their absolute values, equivalently top-wire multiplicity. With one top connection per bottom gate, this is the ordinary bottom-gate count.

Chen–Tal–Wang (STOC 2026) proved n^(5/2−ε) gate lower bounds for SYM∘THR and THR∘THR. For the symmetric-top class, this paper raises the exponent to 3, up to logarithmic factors, with almost-everywhere constant-error hardness.

The central algorithm reconstructs exact local score distributions so that an arbitrary symmetric lookup can be evaluated without approximating it. Building on Kane–Williams restrictions and Chen–Tal–Wang’s circuit-analysis framework, restrictions leave few live threshold gates. Integer-ratio nested sampling encodes score estimates through short corrections; conditioning on these corrections confines score pairs to small rectangles. Modular histograms then recover the exact joint distributions through a quantitative form of Hajdu–Tijdeman’s discrete-tomography method. The reconstruction window is optimal in its polynomial dependence on the prime scale for arbitrary histograms supported on a known rectangle.

Even-moment tests certify the required bounds for every guessed sampling certificate. Every certificate yields a valid interval for the disagreement probability, and some certificate gives the requested accuracy. Williams’ batch evaluation and Matoušek’s bucketing make these measurements efficient. For fixed accuracy, the certified counter runs in time 2^n / n^(Ω(log log n)) within the near-cubic weight budget.

The transfer to constant-error hardness uses a weighted stability analysis of the classical Garey–Johnson–Stockmeyer MAX-2SAT gadget applied to three-query PCP-of-proximity constraints. This makes single-output and pairwise disagreement counts sufficient. Chen–Lyu–Williams’ constructive refuter framework supplies hardness at every sufficiently large length.

A companion Lean 4 formalization proves the main theorem from seven explicitly stated published results. Those results remain external hypotheses; their required adaptations and the remaining derivation are formally proved and checked by the Lean kernel. The repository includes the paper source and documentation of the formalization boundary: https://github.com/devnag/near-cubic-gates-lean

Version 3 focuses on the arbitrary-symmetric-top constant-error theorem. It retains the main lower-bound and certified-counting statements from version 2 while completing the short-PCP, machine-model, and source-interface arguments and their formalization. The periodic-top results and their separate transfer remain in the archived version 2 and are not included here.

These results do not establish near-cubic gate lower bounds for general THR∘THR circuits with uncharged output weights. Vanishing correlation for arbitrary symmetric tops at a near-cubic gate count remains open.