EXACT CLAIM

Fix 0<p<1, q=1-p, x,y in C, a=|x|²>0, and b>|y|². Write Y=|y|², B=b-Y>0, R=Re(conj(x)y), and L=ln(1/t). On the vacuum/one-photon subspace prepare independent one-mode inputs
rho_A(t)=[[1-at, x sqrt(t) sqrt(1-at)], [conj(x) sqrt(t) sqrt(1-at), at]],
rho_B(t)=[[1-bt, y sqrt(t)], [conj(y) sqrt(t), bt]],
with zero entries outside that subspace. For 0<t<min{1/a,B/b²}, these are physical finite-energy states; rho_A is exactly pure and rho_B is genuinely mixed. Retain c=sqrt(p)a_op+sqrt(q)b_op after beam-splitter mixing, obtaining rho_C(t). Define
H=pq[2BY+(a+Y)²-4R²].
Then, for every fixed allowed parameter choice, as t decreases to zero,
g^{-1}(S(rho_C(t)))-p g^{-1}(S(rho_A(t)))-q g^{-1}(S(rho_B(t)))
= Ht²+B²t²[q/(L-ln B)-q²/(L-ln(qB))]+O(t³).
In particular H>0, and the gap is also Ht²+pqB²t²/L+O(t²/L²). Thus EPnI is strict for all sufficiently small positive t in this family, with arbitrary complex relative phase. All remainder constants and the eventual strictness threshold may depend on the fixed parameters.

FULL FROZEN PROOF

Put P=sqrt(p), Q=sqrt(q), and u=x sqrt(t) sqrt(1-at), v=y sqrt(t). The first input has trace one and determinant zero, with positive diagonal entries on the stated interval, and is therefore pure. The second has determinant Bt-b²t²>0 and positive diagonal entries: indeed B/b²<=1/b. Its mean photon number is bt, and the first input's is at. Both are physical finite-energy states.

1. Exact retained output. Choose the complementary output phase so that the beam splitter sends
|10> -> P|10>-Q|01>,
|01> -> Q|10>+P|01>,
|11> -> sqrt(2)PQ|20>+(p-q)|11>-sqrt(2)PQ|02>.
Here the first slot is retained. Since the inputs have at most two photons together, the output is supported on |0>,|1>,|2>. Direct partial trace gives the following upper-triangular entries M_ij of rho_C:
M_22=2pqab t²,
M_11=(pa+qb)t+2PQ R t sqrt(1-at)-4pqab t²,
M_00=1-M_11-M_22,
M_01=P u(1-2qb t)+Q v(1-2pa t),
M_02=sqrt(2)PQ u v,
M_12=sqrt(2)PQ(Pat v+Qbt u).
The lower entries are their conjugates. These formulas retain every complex phase.

For compactness define
r=pa+qb+2PQ R,
k=-4pqab-PQ aR,
m=2pqab,
alpha=Px+Qy,
beta=-Px(a/2+2qb)-2pQay,
gamma=sqrt(2)PQxy,
delta=sqrt(2)PQ(Pay+Qbx).
Expansion of the exact entries yields
M_11=rt+kt²+O(t³), M_22=mt²,
M_00=1-rt-(k+m)t²+O(t³),
M_01=alpha sqrt(t)+beta t^(3/2)+O(t^(5/2)),
M_02=gamma t+O(t²), M_12=delta t^(3/2)+O(t^(5/2)).
The correction -Pax/2 in beta comes from the physical pure-state factor sqrt(1-at).

2. Characteristic coefficients and spectral control. Let sigma_2 be the sum of the three principal two-by-two minors. The displayed entries give
sigma_2=dt+wt²+O(t³),
d=r-|alpha|²=qB>0,
w=k+m-r²-2Re(conj(alpha)beta)-|gamma|².
Set e=w+d². Explicit simplification gives
e=pq a²+2pq aY-4pq R²+q²(Y²-2bY)
 =H+q(B²-b²).
For example, the cross term used in this simplification is
-2Re(conj(alpha)beta)=pa²+4pqab+4pq aY+PQ R(a+4qb+4pa).
Together with |gamma|²=2pq aY and r=pa+qb+2PQ R, this proves the identity directly.

The coefficient of t³ in det M is
rm+2Re(alpha delta conj(gamma))-|delta|²-r|gamma|²-m|alpha|²
=d(m-|gamma|²)-|delta-conj(alpha)gamma|².
But
m-|gamma|²=2pq aB,
delta-conj(alpha)gamma=sqrt(2)Pq xB.
Consequently this coefficient is 2pq²aB²-2pq²aB²=0. The determinant has no lower-order terms, so det M=O(t⁴). These remainder statements follow directly from convergent expansions of sqrt(1-at): every product entering the characteristic coefficients has an integer power of t. No eigenvalue analyticity assumption is required.

To extract eigenvalues rigorously, order them lambda_0>=lambda_1>=lambda_2>=0. Positivity follows from the physical construction. Since lambda_0>=M_00=1-O(t), the sum T=lambda_1+lambda_2 is O(t). The identity sigma_2=(1-T)T+lambda_1 lambda_2 first gives T=dt+O(t²). Thus lambda_1>=T/2>=dt/4 for sufficiently small t, while lambda_0>=1/2. Using det M=lambda_0 lambda_1 lambda_2=O(t⁴) gives lambda_2=O(t³). In particular lambda_1 lambda_2=O(t⁴). Returning to sigma_2 now gives
T=dt+(w+d²)t²+O(t³)=dt+et²+O(t³).
Hence the spectrum is
(1-dt-et²+O(t³), dt+et²+O(t³), O(t³)).
This argument also covers an identically zero third eigenvalue.

3. Entropy and its inverse, including remainders. For any nonnegative normalized three-point spectrum of the preceding form, with fixed d>0, Taylor expansion of the first two entropy terms gives
S=dt(L+1-ln d)+et²(L-ln d)-d²t²/2+O(t³L).
For clarity, the large-eigenvalue contribution is dt+(e-d²/2)t²+O(t³), and the second contribution is dt(L-ln d)+et²(L-ln d-1)+O(t³L). The remaining eigenvalue contributes O(t³L): the function -z ln z, continuously extended by zero, is increasing near zero and is O(t³L) uniformly for 0<=z<=Ct³. Thus a small third eigenvalue has not been discarded.

Let D=L-ln d and define z=dt+et²-d²t²/D. For sufficiently small t, D>0 and z>0. The expansion
g(z)=z(1-ln z)+z²/2+O(z³)
shows
g(z)=dt(D+1)+et²D-d²t²/2+O(t³L)=S+O(t³L).
Indeed z-dt=(e-d²/D)t² has bounded coefficient, so the quadratic Taylor remainder of z(1-ln z) about dt is O(t³). Both z and g^{-1}(S) lie in [dt/2,2dt] for small t: for the latter this follows by comparing S~dtL with g(dt/2)~dtL/2 and g(2dt)~2dtL. On this interval g'(s)=ln(1+1/s) is bounded below by a positive constant times L. The mean-value theorem therefore proves
g^{-1}(S)=dt+et²-d²t²/(L-ln d)+O(t³).
This establishes the claimed inversion remainder rather than only a formal logarithmic series.

For rho_B, its smaller eigenvalue epsilon satisfies epsilon(1-epsilon)=Bt-b²t², and consequently
epsilon=Bt+(B²-b²)t²+O(t³).
The same entropy-inversion calculation, now for a two-point spectrum, gives
g^{-1}(S(rho_B))=Bt+(B²-b²)t²-B²t²/(L-ln B)+O(t³).
The first input is pure, so g^{-1}(S(rho_A))=0. Apply the preceding output formula with d=qB and e=H+q(B²-b²), and subtract q times the mixed-input formula. The order-t terms cancel, and the result is exactly
Ht²+B²t²[q/(L-ln B)-q²/(L-ln(qB))]+O(t³).

4. Strictness and scope. Cauchy-Schwarz for the complex scalars gives R²<=aY. Therefore
H=pq[2BY+(a-Y)²+4(aY-R²)]>0:
if Y>0 the first summand is strictly positive, while if Y=0 the square equals a²>0. Furthermore, for D_B=L-ln B>0,
q/D_B-q²/(D_B-ln q)=q[pD_B-ln q]/[D_B(D_B-ln q)]>0.
Its expansion is pq/L+O(1/L²), proving the stated weaker asymptotic as well. Dividing the gap by t² gives a limit H>0, so the gap is strictly positive for all sufficiently small t. This is a local fixed-parameter result only, not a proof of unrestricted EPnI.

REMAINING GAP

The EPnI for arbitrary independent finite-energy bosonic inputs remains unresolved. This theorem concerns only one-mode vacuum/one-photon inputs in a fixed-parameter low-energy family with one exactly pure input and one genuinely mixed input. It supplies neither an arbitrary-input proof or counterexample, nor a uniform finite-brightness threshold, nor coverage of multimode correlations.