M10 extension: classical counterexamples for every noninteger alpha>1.
This supplements, and does not replace, the rational alpha=5/2 certificate.

Fix a noninteger alpha>1, beta=alpha-1>0, m=floor(beta), k=m+2, n=k+1.
Let T be the n-vertex path adjacency matrix and A(e)=I+eT, 0<e<1/2.
Its binomial series converges absolutely in operator norm. Because the graph
distance from1 to n is k, (T^j)_(1n)=0 for j<k, (T^k)_(1n)=1, and the next
nonzero term can only have j=k+2. Therefore
 (A(e)^beta)_(1n)=binom(beta,k)e^k+O_alpha(e^(k+2)).
There is exactly one negative factor in beta(beta-1)...(beta-k+1): the final
factor beta-(m+1). Thus this entry is strictly negative for all sufficiently
small positive e. This is the inherited noninteger matrix-power mechanism,
not a claim to a new matrix-analysis theorem (compare1303.4701v2).

Put G=A(e)/n and H(t)=G+t(E_1n+E_n1). Both are positive definite for small t.
For f(t)=Tr H(t)^alpha, differentiability of positive matrix functional
calculus gives f'(0)=2alpha(G^(alpha-1))_(1n)<0. Hence f(t)<f(0) for all
sufficiently small positive t. Trace G=Trace H=1. As1/(1-alpha)<0, the
Rényi entropy of H(t) exceeds that of G.

Explicit realizable state, dimensions A=n, B=2n, C=2:
* For each path edge i--i+1, give a distinct B label, C=0, and probabilities
  e/n at each of its two A endpoints. There are n-1 such B labels.
* For one further common B label, put probability t at(A=1,C=0) and t at
  (A=n,C=1).
* For each A=i, give a distinct remaining B label, C=0, with probability
  (1-deg_T(i)e)/n - t 1_{i in {1,n}}.
All probabilities are nonnegative if0<t<(1-e)/n and e<1/2. Their sum is1.
For the bipartition A:(BC), the reflected Gram matrix is G: edge labels
supply e/n off-diagonals and all diagonals are1/n. After ordinary partial
trace of C, the special common B label supplies the additional t endpoint
off-diagonal, and no other entries change. Its reflected matrix is H(t).
Thus this density matrix supplies the required strict reverse inequality.
The canonical purification has additional zero eigenvalues, irrelevant to
positive alpha powers. No arbitrary-Gram relaxation is used.

Scope: for every noninteger alpha>1 there exists a finite-dimensional
classical counterexample, with A dimension floor(alpha)+2. The integer
orders are excluded because the binomial sign argument vanishes there.
For the exact M10 universal question, v2 already gives an explicit rational
state at alpha=5/2 and a certified positive gap. This theorem strengthens
the exponent range but does not establish global novelty, external review,
or a quantitative gap uniform as alpha approaches an integer.

Constructive strengthening of M10-all-noninteger-theorem-v1.
For every noninteger alpha>1 set beta=alpha-1, k=floor(beta)+2,
n=k+1, e=2^(-k-3), c=binom(beta,k)<0. Then the same path construction works
with explicit finite parameters, rather than an unspecified small e.

For j>=k, the recurrence for binomial coefficients has absolute ratio
|beta-j|/(j+1)<1, so |binom(beta,j)|<=|c|. Path bipartiteness removes j=k+1,
and ||T||<=2. Thus the absolute remainder after c e^k is at most
 |c| (2e)^(k+2)/(1-2e).
Its ratio to |c|e^k is 2^(k+2)e^2/(1-2e)<1/2 for every k>=2.
It follows that (A(e)^beta)_(1n)<=-|c|e^k/2.

Let d0=n^(-ceil(beta))|c|e^k, L0=8n ceil(alpha)^2 and choose
 t=min{(1-2e)/(2n), d0/(2L0)}.
Then for f(t)=Tr(G+tB)^alpha, B=E_1n+E_n1, one has f'(0)<=-d0.
On0<=u<=t, the matrix spectrum lies between1/(4n) and1. The Hessian formula
for the spectral trace function gives
 f''(u)<=2alpha(alpha-1) max_{x in[1/(4n),1]}x^(alpha-2)<=L0.
Indeed in an eigenbasis this Hessian is the sum of alpha times the divided
differences of x^(alpha-1), multiplied by |B_ij|^2; the scalar mean-value
bound and ||B||_F^2=2 give the displayed inequality. Therefore
 Tr G^alpha-Tr H(t)^alpha >=d0 t-L0 t^2/2 >=3d0t/4>0.
All state probabilities are nonnegative because t<=(1-2e)/(2n)<(1-e)/n.
For rational noninteger alpha, e,c,d0,L0,t and every state probability are
rational, although the matrix powers need not be rational. The analytic
inequalities certify the sign for every such alpha and also for real alpha.
Since Tr G^alpha<=1 for alpha>1, the reverse entropy gap is at least
3d0t/[4(alpha-1)] by log(x/y)>=(x-y)/x. Constants are deliberately conservative.

This proof credits the known sign failure of noninteger matrix powers; the
additional step is its explicit realization by a state and ordinary traceout.
The separate much-larger alpha=5/2 gap in M10-v2 remains the practical witness.
