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# The minimum, its certificate, and its growth interpretation
Let H(t)=sum a_j t^j be a nonconstant augmentation Hilbert polynomial of degree
d. Jennings' factors imply a_0=a_d=1, a_j>=0 and a_j=a_(d-j).
Put P(t)=3tH'(t)-dH(t). Its nonzero coefficients (3j-d)a_j change sign once,
from negative to positive. P(0)<0 and P(1)=dH(1)/2>0. Descartes' rule
therefore gives exactly one positive zero t_*, and it lies in (0,1).
The derivative of H(t)t^(-d/3) has the sign of P(t), so this zero is its
global minimum on (0,1]. The value is strictly between 1 and |G|.
Alternatively, write s=log t and q_j=a_j exp(js)/H(exp(s)). The second
derivative of log H(exp(s)) is Var_q(j)>0. At the minimum the mean is d/3.
The entropy identity is
log C = -sum_{a_j>0} q_j log(q_j/a_j).
This identity concerns coefficients with multiplicity a_j; dropping their
multiplicity would give a different entropy. The build checks this identity
numerically for all 25 groups using a separate mpmath computation.
For the direct power G^k the augmentation Hilbert polynomial is H(t)^k,
by the tensor product filtration. Put
L_G(k) = sum_{j=0}^{floor(kd/3)} [t^j] H(t)^k.
For every 0<t<=1, L_G(k) <= H(t)^k t^(-kd/3), hence L_G(k)<=C^k.
There is also a matching exponential lower bound. Tilt the coefficient
weights to q_j at t_*. Choose integer counts m_j with sum m_j=k and
m_j=k q_j+O(1), adjusting only O(1) counts to force sum j m_j<=floor(kd/3).
This is possible for sufficiently large k because q_0,q_d>0, the total
rounding discrepancy is bounded, and transferring a bounded number of
counts from degree d to degree 0 reduces that discrepancy. The multinomial
term k!/prod m_j! times prod a_j^(m_j) is part of L_G(k). Stirling's formula
gives its k-th root tending to exp(-sum q_j log(q_j/a_j))=C.
Thus lim L_G(k)^(1/k)=C. This derivation specializes the Chernoff method;
it does not identify C with tensor rank or with the matrix multiplication
exponent. Finite-k numerical checks verify the upper bound, not the limit
proof itself.
The high-degree ideal X=I^(floor(kd/3)+1) in F_p[G^k] satisfies X^3=0,
because its triple product has degree greater than kd. Its codimension is
L_G(k). This is the origin of the fixed one-third threshold.
For certified numerical computation, exact integer Horner evaluation at
dyadic rationals bisects P's unique zero. With B working bits, the root
bracket has width at most 2^(-B). The generator forms an outward-rounded
Arb ball containing this entire bracket, evaluates H on it, and multiplies
by exp(-(d/3) log t). Every operation propagates enclosure error. No
approximate optimizer is wrapped in a ball and called a proof.
The 100-digit run uses B=413 (80 extra working bits). A second run at
140 requested digits and an independent 180-decimal-digit mpmath computation
check the enclosures. The attached evidence gives the measured accuracy,
exact root brackets, serialized values and finite-power count checks.
Known-answer controls use H=1+t, with t_*=1/2 and C^3=27/4, and H=(1+t)^3,
with C=27/4. Controls are not stored as extra rows.