χ_R(∣N⟩^⊗4) ≥ 7

{
  "schema_version": "0.1",
  "orbit": "N",
  "m": 4,
  "direction": "lower",
  "rank": 7,
  "certificate": {
    "script": "verify_challenge/cert_n_m4_rank6_attested.py",
    "expect": "CERTIFIED chi(N^4) >= 7",
    "budget_s": 3600,
    "attested": {
      "batches": "research/n4_rank6/batch_manifest.json",
      "recomputed": 2,
      "compute_hours": 121.5,
      "hardware": "RunPod CPU pod, 16 vCPU AMD EPYC 4564P on a shared host, fifteen batches in parallel at nice 19 (gcc-10 kernels, Python 3.12, numpy 2.3.5), 2026-09-25 21:04 to 2026-09-26 05:40 UTC for the 1,057-batch loop; the 23 stage B6 batches whose 24 items the first aggregate left undecided (the fresh-term filter of PR #153) rerun with --force on the same pod 06:12 to 06:26 UTC; the aggregate's two seeded re-runs (batches 168 and 799, seed 20260925, 255 s and 283 s) on the same pod at 06:26 to 06:46 UTC; the --no-native replays of batches 53 and 764 (05:50 to 09:22 UTC, deterministic hashes equal to the stored ones) under research/n4_rank6/results/nonative/",
      "note": "The exclusion of every rank-6 decomposition of |N>^4 lives in the stored batch outputs under research/n4_rank6/results/ (54 stage A6 batches over the 37,201,212 full 6-covers of |N>^2 with distinct independent base states, 807 stage B6 batches over the 259,655 G_2 orbit representatives of the dependent 6-sets, 166 stage C6 batches over the 321,553 representatives of the repeated 6-multisets, 27 stage beta' batches over the 54,488 representatives of the full 5-multisets with the invisible term on each of the 16 flats missing (2, 2), 3 stage gamma batches over the 505 representatives of the full 4-multisets with each of the 136 flat multisets), every base matched at x_0 = (2, 2) along qutrits 1, 2, every candidate re-decided mod 2013265921 and numerically, 0 hits, 0 refused, 0 undecided. The certificate checks the partition against the enumerator, re-runs the 6-cover kernel over every pivot pair, re-enumerates the orbit representatives, verifies the hashes and geometry of every output, re-decides every stored hit, fails on any refusal or undecided run, and re-runs two batches from scratch; completeness rests on the stored outputs and on research/n4_rank6/batch.py, which regenerates any batch (batch.py K)."
    }
  },
  "provenance": {
    "author": "this repository",
    "reference": "stabrank research/n4_rank6",
    "method": "two-qutrit base slice at (2, 2): every rank-6 decomposition of |N>^4 has four, five or six terms visible at that point (Fact B of PR #86), so its base is a full 4-, 5- or 6-multiset of |N>^2 and the invisible terms lie on the 16 flats missing (2, 2); every base is enumerated up to the unitary symmetry of |N>^2 and completed slice by slice under the two-qutrit slice structure lemma, modulo 65521 with exact re-decision",
    "date": "2026-09-26",
    "github": [
      "vprusso"
    ],
    "compute": {
      "cpu_hours": 121.5,
      "wall_clock_hours": 9.7,
      "runs": 1,
      "hardware": "RunPod CPU pod, 16 vCPU AMD EPYC 4564P on a shared host, fifteen batches in parallel at nice 19 (gcc-10 kernels, Python 3.12, numpy 2.3.5), 2026-09-25 21:04 to 2026-09-26 05:40 UTC for the 1,057-batch loop; the 23 stage B6 batches whose 24 items the first aggregate left undecided (the fresh-term filter of PR #153) rerun with --force on the same pod 06:12 to 06:26 UTC; the aggregate's two seeded re-runs (batches 168 and 799, seed 20260925, 255 s and 283 s) on the same pod at 06:26 to 06:46 UTC; the --no-native replays of batches 53 and 764 (05:50 to 09:22 UTC, deterministic hashes equal to the stored ones) under research/n4_rank6/results/nonative/"
    }
  },
  "notes": "With the Lean rank-7 witness bounds/N-m4-upper-7.json this settles chi(N^4) = 7. The argument (docs/notes/n4_rank6_exclusion.md) depends on: (1) Fact B of PR #86 (not re-run): no two-qutrit slice point of a rank-5 or rank-6 decomposition of |N>^4 has exactly three nonzero terms, and chi(N^2) = 3 leaves at least three, so at least four terms are visible at every point. (2) The orbit lemma: a unitary symmetry U of |N>^2 on qutrits 3, 4 commutes with the slicing along qutrits 1, 2 and carries a decomposition with base S at (2, 2) to one with base U S and the same invisible flats, so one base per G_2 orbit suffices; the lists hold the least element of each orbit and the aggregate recomputes them. (3) The completeness of the lists: A6 by the pivot and partner reductions of the census (the H^6 argument), B6 by the kappa reduction of the design note (an independent full k-cover plus states of its span or parallel modulo it), C6 by the merged-coefficient argument with cancelling blocks, k5 and k4 from the attested rank-5 lists. (4) The two-qutrit slice structure lemma of PR #86; the matchers allow every flat through (2, 2) for every visible term and assume nothing beyond the base point and the invisible flats. (5) Exactness: amplitudes in Q(omega_3), both primes 1 mod 3, every modular kernel a superset, every candidate and hit re-decided mod 2013265921 and numerically. (6) The fresh-term filter of stage B6 lists the reference's first-slice solutions exactly (design note, section 3.1; PR #153 fixed the vanishing-coefficient case that left 24 items undecided in the first aggregate), and a base it cannot take falls back to the reference (none did). (7) Controls (results/control_*.json): planted instances for every stage, flat and flat multiset, the orbit lemma on planted decompositions, the rank-4 decompositions of |N>^3 along a qutrit pair, the lists and census by hash, the rank-7 witness at its bases. Attested because no certificate budget re-runs the pod hours."
}