round161-support-counterexample-closure-frozen

Immutable full-result SHA256: c0d22e76dbce0a0272c6b9dd6afb0bd83045f822ee3732869e995f739f65d212

STATEMENT
Let e=2^-16, a=1-e, p=4/5, A=diag(a,e), B=HAH, and rho=pA direct-sum (1-p)B, sigma=(1-p)B direct-sum pA on C^4. Consider precisely the stated two-pass majority-label policy, with optimal classical likelihood decision. Put d=log(a/e), u=log(2a)/d, H(u)=-u log u-(1-u)log(1-u), and E=log(5/4)+log 2-H(u). For every integer n>=1 its error satisfies exp(-d-nE)/(4(n+1)^2) <= p_e(policy,n) <= (n+1)exp(-nE)/2. Thus its exact exponent is E=log(5/4)+C_cl((a,e),(1/2,1/2)). These faithful noncommuting states satisfy C_M<log(5/4)+454/1000 whereas E>log(5/4)+457/1000. Consequently both optimal lower exponents, LOCC and SEP, exceed C_M by more than 3/1000. This is a complete counterexample to the universal equality Eq.(4), not a determination of the optimal class exponents.

PROOF
All logarithms are natural. The proof is analytic; the attached independently written rational checks are supplementary, and are not asserted to have been executed during this final turn.

1. Arbitrary-POVM bound. For density matrices X,Y>0, alpha in [1/2,1), beta=(1-alpha)/alpha, and a POVM {M_i}, set x_i=Tr(XM_i), y_i=Tr(YM_i), omitting zero effects. Scalar minimization, with minimizer t_i=(x_i/y_i)^alpha, gives
sum_i x_i^alpha y_i^(1-alpha)=inf_{t_i>0}[alpha sum_i x_i t_i^-beta+(1-alpha)sum_i y_i t_i].
Set G=sum_i t_i M_i>0 and v=Tr(YG)>0. Operator Jensen for the operator-convex function z^-beta, valid for 0<=beta<=1, gives sum_i t_i^-beta M_i>=G^-beta. The positive matrix G^(1/2)YG^(1/2) is at most its trace v times I; conjugating gives Y<=vG^-1. Loewner-Heinz, in the same range of beta, gives Y^beta<=v^beta G^-beta. Hence every expression minimized above is at least alpha c v^-beta+(1-alpha)v, where c=Tr(XY^beta). Its minimum is c^alpha, attained at v=c^alpha. Therefore
sum_i x_i^alpha y_i^(1-alpha)>=(Tr(XY^beta))^alpha. (A)
The operator properties invoked are the usual inverse Jensen inequality and fractional-power convexity/monotonicity, with their valid ranges explicitly respected. No interchange of minimizations or optimization over operators is needed.

2. Uniform measured bound. Refine every single-copy POVM by the two orthogonal direct-sum blocks. Scalar Holder implies that this refinement cannot increase the Chernoff coefficient. Within each block the restricted effects sum to the block identity. Since H interchanges A and B,
Tr(AB^beta)=Tr(BA^beta)=(a^beta+e^beta)/2.
For any interior Chernoff parameter, take alpha=max(s,1-s), interchanging X,Y in (A) as necessary. Every coefficient is at least
(4/5) cosh((2alpha-1)log 2) [(a^beta+e^beta)/2]^alpha.
Since a^beta+e^beta>=a^beta(1+e^beta), its logarithm is at least log(4/5)+h(alpha)+(1-alpha)log a, where
h(alpha)=log cosh((2alpha-1)L)-alpha L+alpha log(1+exp(-16L(1-alpha)/alpha)), L=log 2.
At alpha=3/4, writing x=2^-16/3,
h=log 3-(9/4)L+(3/4)log(1+x),
h'=-L/3+log(1+x)+(64L/3)x/(1+x).
The rational bounds .6931<L<.6932, log 3>1.0986, and .0248<x<.0249 imply h(3/4)>-.443 and 0<h'(3/4)<.154, using x-x^2/2<=log(1+x)<=x. These logarithm bounds can be verified directly from log z=2 sum_{j>=0} w^(2j+1)/(2j+1), w=(z-1)/(z+1), bounding the positive remainder after j=m-1 by 2w^(2m+1)/((2m+1)(1-w^2)); the x bounds follow by cubing.
The last term of h is a convex perspective of log(1+exp z). The first term has second derivative 4L^2 sech^2((2alpha-1)L)>6/5 on [1/2,1], since sech^2 L=16/25 and (64/25)(.6931)^2>6/5. Thus strong convexity and completing the square give
h(alpha)>-443/1000-(77/500)^2/(12/5).
Furthermore log a>=-1/65535 and 1-alpha<=1/2. Consequently
h(alpha)+(1-alpha)log a > -delta,
delta=443/1000+(77/500)^2/(12/5)+1/131070 <454/1000.
This is a uniform strict margin. Endpoint coefficients equal one. Taking the infimum over all POVMs and parameters proves C_M<log(5/4)+454/1000.

3. Exact transcript laws and admissibility. Extraction of block j is the instrument X -> V_j X V_j*, retaining the conditional qubit. Its flag probabilities are (p,1-p) under rho and (1-p,p) under sigma. Conditional states are respectively (A,B) and (B,A). All flags are communicated before any retained qubit is measured. For k>=n/2 choose the rho eigenbases; for k<n/2 choose the sigma eigenbases. Measuring either A or B in the other's eigenbasis gives the uniform law Q=(1/2,1/2), whereas its own eigenbasis gives P=(a,e). Thus conditional on any flag string the final outcomes are independent, with laws (P^n,Q^n) in the first branch and (Q^n,P^n) in the second. For a flag string with k zeros its weights are a_k=p^k(1-p)^(n-k) and b_k=(1-p)^k p^(n-k). The exact error is one half the sum over all transcripts of the minimum of these weighted laws. All operations act on one party's input only, with classical messages and retained local qubits: this is finite-round LOCC. Its two decision effects are fully separable.

4. The residual Chernoff optimum. Write r(t)=sum_i P_i^t Q_i^(1-t)=2^(t-1)(a^t+e^t). Its logarithm is strictly convex. At its stationary point the tilted probability of the small outcome equals u=log(2a)/d, because the tilted mean of log(P_i/Q_i) must vanish. Hence t*=log((1-u)/u)/d and C=-log r(t*)=log 2-H(u).
Here 0<t*<1/2. Indeed u<1/2 is equivalent to 4ae<1. Also log a>=-1/65535 and d<16 log 2 give u>1/32. The small-outcome tilted probability at t=1/2 is sqrt(e)/(sqrt(a)+sqrt(e))<1/128. This probability decreases strictly with t, and at t=0 equals 1/2. These inequalities establish existence and the asserted interval for t*. The expression for C follows by inserting (1-u)log a+u log e=-log 2 into the entropy formula for the tilted distribution.

5. Upper bound on policy error. In the majority branch use min(v,w)<=v^t* w^(1-t*). After summing final outcomes, each flag type contributes at most binom(n,k)a_k^t* b_k^(1-t*) r(t*)^n. For x=k/n>=1/2, its flag factor is bounded by exp(n f(x)), where
f(x)=H(x)+t*[x log p+(1-x)log(1-p)]+(1-t*)[x log(1-p)+(1-x)log p].
Its derivative is log((1-x)/x)+(2t*-1)log 4<=0. Thus f(x)<=f(1/2)=log(4/5). In the minority branch use the reversed powers; its type function is f(1-x), with the same bound. There are n+1 types, including the boundary convention exactly once. The equal-prior factor gives p_e<= (n+1)exp(-nE)/2, where E=log(5/4)+C.

6. Matching lower bound, including odd n. Retain only the flag type k=floor(n/2). If n is even, a_k=b_k=(2/5)^n. If n is odd, a_k=(2/5)^n/2 and b_k=2(2/5)^n. In either parity the weighted overlap is at least (2/5)^n/2 times the unweighted overlap S_n=sum_z min(P^n(z),Q^n(z)). This remains true in the minority branch where the two laws are reversed. The central binomial coefficient is at least 2^n/(n+1), so p_e>= (4/5)^n S_n/(4(n+1)).
Let R=(1-u,u), and choose a mode l=floor((n+1)u) of its binomial count. It satisfies |l-nu|<=1 and its type probability is at least 1/(n+1). On this type L_z=log(P^n(z)/Q^n(z))=d(nu-l), so |L_z|<=d. The exact tilted identities are
P^n(z)=r(t*)^n R^n(z) exp((1-t*)L_z),
Q^n(z)=r(t*)^n R^n(z) exp(-t*L_z).
Their minimum is at least exp(-d)r(t*)^n R^n(z). Summing this one type gives S_n>=exp(-d)r(t*)^n/(n+1). This proves the claimed lower bound for every n and both parities. Taking logarithms of the two-sided bounds proves convergence to E.

7. Strict separation and scope. Since C>=-log r(1/4), and r(1/4)=2^-3/4(a^1/4+1/16),
C>(3/4)log 2-log(17/16)>(3/4)(6931/10000)-1/16>457/1000.
Together with Section 2 this proves the gap exceeding 3/1000. Optimal LOCC and SEP errors are no larger than the policy error, so their lower exponents are at least E. The states have trace one and least eigenvalue e/5>0. Since [A,B] has nonzero off-diagonal entry (a-e)^2/2, [rho,sigma] is nonzero.
No source theorem about destructive single-visit adaptation is used. The policy communicates the first-pass information while retaining every local qubit, then revisits the parties. Accordingly the supplied distinction between Hayashi's Section V one-way scope and Section IX two-way/SEP question is respected; no independent historical novelty or primary-source verification is claimed here. No converse for optimal LOCC or SEP is proved.

REMAINING GAP
The optimal LOCC and SEP exponents, their convergence and their possible equality remain undetermined, even for this pair. Historical novelty and independent primary-source comparison are not established. The supplied certificate is independently executable but was not executed in this final turn.

CHECKER
from fractions import Fraction as F
from math import comb

e=F(1,65536); a=1-e; p=F(4,5)
A=[[a,0],[0,e]]; B=[[F(1,2),(a-e)/2],[(a-e)/2,F(1,2)]]
def block(X,Y,c,d):
 return [[c*X[i][j] if i<2 and j<2 else d*Y[i-2][j-2] if i>=2 and j>=2 else F(0) for j in range(4)] for i in range(4)]
def mul(X,Y):
 return [[sum(X[i][k]*Y[k][j] for k in range(len(Y))) for j in range(len(Y[0]))] for i in range(len(X))]
def tr(X): return sum(X[i][i] for i in range(len(X)))
rho=block(A,B,p,1-p); sig=block(B,A,1-p,p)
assert tr(rho)==tr(sig)==1 and e/5==F(1,327680)
rs=mul(rho,sig); sr=mul(sig,rho)
assert rs[0][1]-sr[0][1]==F(2,25)*(a-e)**2>0
ZA=[[1,0],[0,0]]; ZB=[[F(1,2)]*2 for _ in range(2)]
assert tr(mul(A,ZA))==tr(mul(B,ZB))==a
assert tr(mul(B,ZA))==tr(mul(A,ZB))==F(1,2)
assert sum(rho[i][i] for i in (0,1))==p
assert sum(sig[i][i] for i in (0,1))==1-p
print('states: trace=1, minimum eigenvalue=1/327680, nonzero commutator')
print('flag laws=(4/5,1/5),(1/5,4/5); final laws=P,Q or Q,P')
# Certified rational fourth-root intervals, never floating point.
def root4(x):
 lo=F(0); hi=F(2)
 for _ in range(100):
  m=(lo+hi)/2
  if m**4<=x: lo=m
  else: hi=m
 return lo,hi
al,ah=root4(a); tl,th=root4(F(2))
ql=tl*(al+F(1,16))/2
qh=th*(ah+F(1,16))/2
assert F(63,100)<ql<=qh<F(64,100)
print('residual coefficient r(1/4): 63/100 < q < 64/100')
for n in (1,2,3,4,8,16):
 err=F(0)
 for k in range(n+1):
  w=p**k*(1-p)**(n-k); v=(1-p)**k*p**(n-k)
  for l in range(n+1):
   P=a**(n-l)*e**l; Q=F(1,2)**n
   x,y=(P,Q) if 2*k>=n else (Q,P)
   err+=F(comb(n,k)*comb(n,l),2)*min(w*x,v*y)
 bound_lower=F(n+1,2)*(F(4,5)*ql)**n
 assert 0<err<bound_lower/2
 if n==1: assert err==F(8193,81920)
 print('n=%d: exact Bayes sum / certified lower endpoint of bound < 1/2; relative margin > 1/2'%n)
print('n=1 exact Bayes error=8193/81920')

EXPECTED CHECKER OUTPUT
states: trace=1, minimum eigenvalue=1/327680, nonzero commutator
flag laws=(4/5,1/5),(1/5,4/5); final laws=P,Q or Q,P
residual coefficient r(1/4): 63/100 < q < 64/100
n=1: exact Bayes sum / certified lower endpoint of bound < 1/2; relative margin > 1/2
n=2: exact Bayes sum / certified lower endpoint of bound < 1/2; relative margin > 1/2
n=3: exact Bayes sum / certified lower endpoint of bound < 1/2; relative margin > 1/2
n=4: exact Bayes sum / certified lower endpoint of bound < 1/2; relative margin > 1/2
n=8: exact Bayes sum / certified lower endpoint of bound < 1/2; relative margin > 1/2
n=16: exact Bayes sum / certified lower endpoint of bound < 1/2; relative margin > 1/2
n=1 exact Bayes error=8193/81920


Coordinator assessment: reports/authorization-history/20261016T084317Z-through161-chernoff-support/preflight161/round161-support-counterexample-closure-frozen-outcome.json
