nx_apex_gcc_bench.nx

nx_apex_gcc_bench.nx -- APEX RUNG 1: FAIR head-to-head vs gcc -O3. EXTENDS runtime/nx_eqsat_vs_gcc_battery_test.nx (global rule #15: reuse, don't reinvent). Same 8-kernel fixed battery, same FAIR deduped op-count metric, but ALSO emits the EQUIVALENCE-CERTIFICATE axis the rung-1 brief requires: prov ON, every merge cited through the single nx_eqsat_union_cited chokepoint; after saturation we verify every logged rule_id is in the proven-sound allow-list {1..10} and prov_overflow==0. A RULE_NONE (0) or overflow POISONS the cert. TWO AXES, reported separately: AXIS A (op-count): deduped arithmetic-node count of the extracted min-cost form, vs the arithmetic ops in gcc -O3 -S of the SAME computation (counted by the FROZEN LEA rule, see bench .c). AXIS B (verified-equivalence): membership-proof certificate -- gcc has none. METRIC NOTE: op-count is ONE-PER-ARITHMETIC-INSTRUCTION on both sides; MUL is NOT double-charged. The internal class_cost (mul=3, div=30) is the EXTRACTION objective only and is reported for context, never as the head-to-head number. Output per kernel: "<id> <emit_total> <arith_ops> <cost> <root_op> <sat> <cert>" cert = 1 PROVEN-EQUIVALENT (all merges cited 1..10, no overflow), 0 POISONED.