back to table · edit · history · where entries came from · files · download
5427 bytes, as of the version from 2026-11-01 17:54 (current). Recorded here, not run.
# Argument selection for T655
This build follows the matching local and live skill copies recorded by
`agents/runs/20261101T164559Z-build-skills/manifest.json`. Both have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The attached `skill-evidence.json` records both reads. This is the builder's
justification, not the independent critique required by the campaign gate.
## The mathematical quantity
For an integer n >= 2, let S_n(t) = 1 + t + ... + t^(n-1) and
C_n = min_{t>0} S_n(t)t^(-(n-1)/3), using positive real powers.
The table records C_n and J_n = C_n/n. The fixed exponent denominator three
comes from the tensor's three legs; it is not another chosen argument.
The minimizing argument rho_n and its reciprocal are certificates here, not
additional table entries. The separate proposed table of minimum points
has not been absorbed into this table.
The same C_n is the asymptotic subrank of multiplication in the complex algebra
C[u]/(u^n). CVZ, section 4.2.1, equations (30)–(31), gives this identification
for every integer dimension, without a prime-power restriction. KSS's
introduction defines the same variational constant, called theta there, and
its section 3 identifies the maximizing probability distribution. BCC,
equation (4.10), uses the normalization J(n).
## Parameter n: discrete_index
The parameter names the dimension and truncation order of an independently
defined algebra. It is a genuine discrete index, not a continuous argument
renamed as an index. Its domain is every integer n >= 2. The selection is every
integer from 2 through 32, in increasing order. No dimension within this range
is removed for lacking a familiar application. This covers the small classical
cases, further prime powers, and dimensions such as 6, 10 and 12 that still
have a well-defined algebraic meaning. The cyclic-group sum-free upper bound
is separately restricted to prime powers; it is not a justification for
claiming that theorem in the remaining dimensions.
The endpoint 32 is a practical, noncanonical reference boundary. It covers a
continuous initial segment of this discrete family while keeping the
stationary polynomials at degree at most 31. It is not selected by an entry
quota or a decimal spelling. Breadth is supplied by including all dimensions
in that segment, rather than extending only the prime or prime-power
subfamilies. There is no independent continuous parameter requiring reference
sampling or a multidimensional mesh. The 62 serialized values fit in a small
readable table, and are computed cheaply enough for complete independent
verification. The exact size measurements are in numerical-evidence.json and
the build handoff.
Dimension one is excluded consistently with the family convention: its
objective is identically one and there is no unique minimizing point.
Although its constant is trivial, it does not belong to this explicitly
nontrivial n >= 2 family.
Checked sources: CVZ section 4.2.1, equations (30)–(31); KSS introduction,
Theorems 1–2; BCC section 4.4, Theorem 4.14.
## Parameter convention: documented_application
The complete choice set is `subrank` and `per-dimension`, in that order at each
n. `subrank` stores C_n, called z(n) in CVZ and theta in KSS.
`per-dimension` stores J_n, called J(n) in BCC, equation (4.10).
These are the source's two normalizations of one number, with the inexpensive
inverse conversions J_n = C_n/n and C_n = n J_n. Both have exactly the same
domain and precision, and the value heading and row labels identify their
symbols. This uses the skill's one-number-in-two-conventions exception.
No logarithm, optimizer, tensor rank, or additional quantum functional is
included. There is no `reference_sample` group hidden in either parameter.
The finite selection is the entire stated discrete initial segment in both
source normalizations. Completeness remains `no` because larger dimensions
are not included.
Checked sources: CVZ equation (31); KSS introduction's explicit identification
of theta with q J(q); BCC equation (4.10) and Theorem 4.14. The sources' q is
this table's n, while their tensor-power index is written m in this table.
## Checked companions and references
- T339, `Shannon_entropies_of_discrete_probability_distributions`, has
`bernoulli,1/3,nats` and `bernoulli,1/3,bits`. The nats row gives log C_2.
This is a theorem connecting distinct quantities, not a repeated computation
requiring a `repeats` declaration.
- T5, `Matrix_multiplication_complexity`, concerns the exponent omega.
BCC's bounds obstruct particular group-theoretic approaches to it. No claim
identifies C_n with omega or changes T5.
- The family queue and live text searches found no built minimum-point
companion. No identifier or slug was invented for it.
Primary sources read and archived in source-manifest.json:
1. Matthias Christandl, Péter Vrana and Jeroen Zuiddam, *Universal points in
the asymptotic spectrum of tensors*, version 3, section 4.2.1,
https://arxiv.org/html/1709.07851v3#S4.SS2.SSS1.
2. Robert Kleinberg, Will Sawin and David E. Speyer, *The growth rate of
tri-colored sum-free sets*, version 3, introduction and section 3,
https://arxiv.org/html/1607.00047v3.
3. Jonah Blasiak et al., *On Cap Sets and the Group-theoretic Approach to Matrix
Multiplication*, version 4, equation (4.10) and Theorem 4.14,
https://arxiv.org/html/1605.06702v4.