verified 10 stabilizer terms reproduce |N>^{ot 5}, checked symbolically
Implied per-copy exponent γ ≤ 0.4192 against a published 0.4206, an improvement.
6 runs; 6 CPU-hours; 5 h wall-clock; on Apple silicon laptop, one core per solve; one confirming solve on a Perlmutter CPU node; claude-fable-5-1, designed and ran the search and the exact refit.
Two-copy census: among the 360 two-qutrit stabilizer states, the three-term decompositions of |N⟩⊗2 span 13 distinct three-spaces, three of which contain four stabilizer rays rather than three, and the plane through |N⟩ itself contains four rays. The sparsest representation of |N⟩⊗5 as a combination of products (ray of such a space) x (ray of the same space) x (ray of the plane) has exactly 10 terms on each of the three four-ray spaces (HiGHS, optimal; 64 words per cell), against 12 by tensoring 4 x 3. The ten terms are products of two two-qutrit and one one-qutrit stabilizer state; the coefficients were solved exactly over Q(omega_3) after factoring the term normalizations √(3^k) and 36 √ 6 out, and are of the forms (+-√ 6 +- 3 √ 2 I)/16 and (-√ 2 - √ 6 I)/16. Implied exponent log_3(10)/5 = 0.4192, below log_3(4)/3 = 0.4206 from χ(N^3) ≤ 4. Barren routes recorded: the record span at three copies is a basis dictionary (3+3 words give exactly 16); the same construction gives nothing for H3 (exactly 8 words at four copies over every four-ray span), T3 (its two-copy decomposition is unique with a basis span), or S (no span with extra rays through S, S^2, or the three-copy record). Word ranks at six copies capped at 15 were undecided after 900 s locally and are running longer elsewhere.
Vincent Russo · this repository (stabrank-research, round 28 letter lifts) · sparsest word representation of |N⟩⊗5 over (four-ray stabilizer-spanned three-space through |N⟩⊗2) x (same) x (four-ray plane through |N⟩), HiGHS MILP, coefficients solved exactly over Q(omega_3)
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"(-sqrt(6) + 3*sqrt(2)*I)/16",
"(3*sqrt(2) + sqrt(6)*I)/16",
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},
"provenance": {
"author": "Vincent Russo",
"reference": "this repository (stabrank-research, round 28 letter lifts)",
"method": "sparsest word representation of |N>^5 over (four-ray stabilizer-spanned three-space through |N>^2) x (same) x (four-ray plane through |N>), HiGHS MILP, coefficients solved exactly over Q(omega_3)",
"date": "2026-10-16",
"github": [
"vprusso"
],
"compute": {
"cpu_hours": 6,
"wall_clock_hours": 5,
"runs": 6,
"hardware": "Apple silicon laptop, one core per solve; one confirming solve on a Perlmutter CPU node",
"llm": [
{
"model": "claude-fable-5-1",
"role": "designed and ran the search and the exact refit"
}
]
}
},
"notes": "Two-copy census: among the 360 two-qutrit stabilizer states, the three-term decompositions of |N>^2 span 13 distinct three-spaces, three of which contain four stabilizer rays rather than three, and the plane through |N> itself contains four rays. The sparsest representation of |N>^5 as a combination of products (ray of such a space) x (ray of the same space) x (ray of the plane) has exactly 10 terms on each of the three four-ray spaces (HiGHS, optimal; 64 words per cell), against 12 by tensoring 4 x 3. The ten terms are products of two two-qutrit and one one-qutrit stabilizer state; the coefficients were solved exactly over Q(omega_3) after factoring the term normalizations sqrt(3^k) and 36 sqrt 6 out, and are of the forms (+-sqrt 6 +- 3 sqrt 2 I)/16 and (-sqrt 2 - sqrt 6 I)/16. Implied exponent log_3(10)/5 = 0.4192, below log_3(4)/3 = 0.4206 from chi(N^3) <= 4. Barren routes recorded: the record span at three copies is a basis dictionary (3+3 words give exactly 16); the same construction gives nothing for H3 (exactly 8 words at four copies over every four-ray span), T3 (its two-copy decomposition is unique with a basis span), or S (no span with extra rays through S, S^2, or the three-copy record). Word ranks at six copies capped at 15 were undecided after 900 s locally and are running longer elsewhere."
}