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# Argument selection for T656
This table contains the minimum point of the Ellenberg–Gijswijt scalar bound,
not its minimum value. The two parameters, in order, are `n` and `convention`.
The family conventions and current companion T655 were read from issue 288
and from the authenticated website API. No published table was edited.
The instruction authority for this run is
`agents/runs/20261101T171341Z-build-skills/manifest.json`. Its local and live
versions both have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The attached `skill-manifest.json` preserves that manifest. This is the build's
justification; it is not an independent critique or an acceptance sidecar.
## Algebra dimension n: discrete_index
For every integer n >= 2 the complex algebra C[u]/(u^n) exists and has dimension
n. Its symmetric three-leg support is i+j+k=n-1, with indices from 0 to n-1.
The corresponding optimization minimizes
(1+t+...+t^(n-1))*t^(-(n-1)/3) over positive real t.
Thus n indexes a genuine discrete family, rather than a continuous parameter
rounded or renamed as an index. The denominator three is fixed by the three
legs. There is no separately chosen mean, exponent, root index or entropy
weight.
The table includes every n from 2 through 32 in increasing order. This gives
initial reference coverage across prime, prime-power and other composite
dimensions, including the binary entropy and cap-set cases without restricting
the family to famous cases. The upper bound 32 is noncanonical and practical:
the stationary equations have degree at most 31, independent certification is
cheap, and each decimal occupies only about 100 characters. It is a finite
extent of the discrete family, not a claim that 32 is structurally special.
Extending one subgroup more deeply would leave holes in the same inexpensive
family, so consecutive dimensions give more balanced coverage here. Dimension
one is excluded because the objective is constant and its minimizer is not
unique.
Checked source: Christandl, Vrana and Zuiddam, *Universal points in the
asymptotic spectrum of tensors*, arXiv:1709.07851v3, Section 4.2.1,
equations (30)–(31). This defines reduced polynomial multiplication in every
dimension and its optimizing parameter. Kleinberg, Sawin and Speyer,
*The growth rate of tri-colored sum-free sets*, arXiv:1607.00047v3,
introduction and Section 3, identifies the minimizing argument and probability
law, with their q equal to our n.
## Convention: documented_application
Both `rho` and `gamma` occur at each dimension. They mean rho_n in (0,1) and
its reciprocal gamma_n>1. The introduction of Kleinberg–Sawin–Speyer defines
rho as the minimizer; Christandl–Vrana–Zuiddam equation (31) uses gamma in its
reciprocal scalar equation. These are two published conventions for the same
number, each recovering the other by a reciprocal. The selection invokes the
one-number-in-two-invertible-conventions exception. It does not introduce a
second optimized quantity or an arbitrary sample of transformations.
There are no reference_sample or mixed continuous selections. All 31
independent dimensions and both conventions are accounted for, yielding 62
entries. The generator's default enumeration and the table's completeness
note use exactly this range. Any extension changes the reviewed selection.
## Related entries and boundaries
T655 holds C_n and C_n/n. It is a populated private draft at build time and is
linked by the explicit relation C_n=S_n(rho_n)*rho_n^(-(n-1)/3). T339 holds the
Bernoulli(1/3) entropy associated with n=2. T35 already holds the two n=3
quadratic numbers by polynomial coefficients; that different mathematical
context does not duplicate this family. No `repeats` is declared: the
coincidence with T35 is a mathematical identity, not a transcribed computation.
No additional tables or quantities are built.