M24 candidate: Gaussian optimality for one versus finitely many modes
Root derivation, 22 October 2026. Accepted consolidation v1 after two
independent internal proof reviews. This version incorporates the measure
clarification below without changing the frozen algebraic argument.

Claim. For every finite-energy Gaussian state with Alice holding one mode
and Bob finitely many modes, E_F=E_F^G, including arbitrary real q/p cross
covariances and singular physical boundaries. This covers the complete M24 original under the named source dependencies. The proof imports the affine TWO-MODE extremality
theorem in Adesso arXiv:2608.01909v3, Theorem1/Eqs8–9, as a source theorem.
It does not claim that the full mixed state factors into a two-mode core.

Conventions. Order real quadratures by modes; J=direct_sum [[0,1],[-1,0]].
Use doubled covariances, vacuum I, so physical V satisfies V+iJ>=0 and pure
Gamma satisfies Gamma J Gamma=J. The source uses vacuum I/2; its covariance
must be half of ours. First moments can be removed by local displacements.
All entropies use natural logarithms.

1. An attained Gaussian optimum and its pure Schmidt form.
The inherited Gaussian-roof characterization (Wolf et al.,
quant-ph/0306177v2 Proposition1/Eq9) gives
 E_F^G(V)=min_{pure Gamma<=V} g((sqrt(det Gamma_A)-1)/2),
where g(x)=(x+1)log(x+1)-x log x. For one Alice mode it suffices to minimize
 a(Gamma)=sqrt(det Gamma_A).
The feasible set is nonempty: Williamson decomposition yields a pure
covariance below V. It is compact. Indeed Gamma<=V bounds its largest
eigenvalue; purity gives Gamma^{-1}=J^T Gamma J, so its smallest eigenvalue
is at least1/||V||. Closure then preserves positivity and purity. Thus a
minimizer Gamma_* exists. If its a=1, it is a product pure seed across
A:B; its displacement ensemble is separable and both roofs vanish.

Assume a>1 and, first, V+iJ>0. Apply local symplectic changes of coordinates
putting Gamma_* into its pure Gaussian Schmidt form: one two-mode squeezed
pair A,D and independent Bob vacuum spectators. This standard pure-state
normal form is an inherited tool. Write q=tanh r,0<q<1,
 c=cosh r, s=sinh r, Z=diag(1,-1),
 S_pair=[[cI,sZ],[sZ,cI]],    Gamma_*=S S^T,
where S acts as S_pair on A,D and identity on spectators. Thus
 a=c^2+s^2=(1+q^2)/(1-q^2).
The whitening S^{-1} is only an algebraic coordinate change in the proof;
it is not an allowed local operation on the physical input.

Set V'=S^{-1} V S^{-T}=I+D with D>=0. It remains strictly physical because
S is symplectic. Let K=ker D and P_K its real orthogonal projector.

2. Strict physicality supplies the missing KKT qualification.
We first prove K intersect J K={0}. If v and Jv belong to K and v!=0, the
nonzero complex vector z=v-iJv satisfies (I+iJ)z=0 and Dz=0. This contradicts
V'+iJ>0. (Changing the sign convention for J changes only the choice of
the complex sign.)

At the whitened pure covariance I, the tangent space of real symmetric pure
covariances is
 T={X=X^T: XJ+JX=0}.
Every such tangent is realized by Gamma(t)=exp(tX), since exp(tX/2) is
symplectic. Its Frobenius orthogonal complement in symmetric matrices is
the space commuting with J, equivalently fixed by X->JXJ^T.

The tangent X0=J P_K J^T-P_K lies in T and is strictly negative on K:
 for0!=v inK, v^T X0 v=||P_K J^T v||^2-||v||^2<0.
Equality would imply J^T v inK, excluded above. Thus the active semidefinite
constraint has a strict inward tangent direction. This argument is needed:
the nonconvex pure-covariance problem cannot simply inherit SDP Slater.

For completeness, finite-dimensional multiplier separation now gives a
real W>=0 supported in K such that
                         G+W belongs to T^perp,          (1)
where G is the derivative of a at I in whitened coordinates. Here is the
precise alternative used. No tangent X can simultaneously have
 Tr(GX)<0 and P_K X P_K<0 onK. Such an X would, for small positive t, make
V'-exp(tX)>0: the active kernel block gains a positive order-t term and
the off-diagonal Schur correction is order-t^2, while the nonkernel block
starts strictly positive. It would also lower a, contradicting optimality.
Separate the linear image {(Tr(GX),P_K X P_K):X inT} from the open product
of negative scalar and negative-definite matrix cones. The separator has
coefficients alpha>=0 and W_K>=0, not both zero, annihilates that linear
image, and hence alpha G+P_K W_K P_K is perpendicular to T. If alpha=0,
its pairing with X0 is strictly negative for W_K!=0, a contradiction.
Divide by alpha to obtain (1). This also covers K={0}: separation would
force G perpendicular to T, impossible for the entangled optimum below.
Consequently WD=0 and Tr(WD)=0. The sign is PLUS G in (1).

Since Gamma_{*,A}=aI, the derivative of sqrt(det Gamma_A) is I_A/2.
Therefore, in whitened coordinates,
 G=(1/2) S^T (I_A direct-sum0) S
  =(1/2)[[c^2 I,cs Z],[cs Z,s^2 I]] on the pair, zero elsewhere.
Equation(1) implies
             W-JWJ^T=-(G-JGJ^T).                        (2)

3. Rank and support: the witness lives on one pair.
The ranges of W and JWJ^T are contained in K and JK and are disjoint.
For two positive matrices with disjoint ranges, the rank of their difference
is the sum of their ranks. To see this, factor them as AA^T and BB^T with
full-column-rank factors; [A B] has independent columns and congruence of
diag(I,-I) preserves its rank and inertia. In particular the range of the
difference is exactly the sum of their ranges.

The right side of(2) has rank4 and support exactly on A,D, since cs>0.
Thus rank W=2, and both W and JWJ^T are supported on that same real
four-dimensional pair. There is no claim that V itself has this support
or no correlations with spectator modes.

Write the pair blocks of W as [[A,C],[C^T,B]]. Equation(2) requires
 A=alpha I, B=beta I,
 C=x I+y J_2-kappa Z,       kappa=cs/2>0.
This uses the elementary real2x2 basis I,J_2,Z,X: the I,J_2 components are
fixed under C->J_2 C J_2^T and the Z,X components change sign.
Positivity and nonzero kappa imply alpha>0,beta>0. Rank W=2 makes its Schur
complement zero, so C^T C=alpha beta I. Direct multiplication gives
 C^T C=(x^2+y^2+kappa^2)I-2kappa x Z-2kappa y X.
It follows that x=y=0 and alpha beta=kappa^2. Therefore
             W=[[u^2 I,-uv Z],[-uv Z,v^2 I]],
with u,v>0,uv=kappa. This is a positive EPR witness, not merely an arbitrary
stationary quadratic form. The rank argument eliminates the extra
number-conserving intermode terms which would obstruct that identification.

4. Transform back and identify the exact source affine witness.
In the local Schmidt frame before whitening, put W0=S^{-T} W S^{-1}.
It remains supported on A,D and Tr[W0(V-Gamma_*)]=0. Factoring W as the
Gram matrix of [uI,-vZ] gives
 W0=[[A0^2 I,-A0 B0 Z],[-A0 B0 Z,B0^2 I]],
 A0=uc+vs>0, B0=us+vc>0.
Because A0-qB0=u/c>0 and B0-qA0=v/c>0, the gain
             t=min(A0/B0,B0/A0)
satisfies q<t<=1. If necessary exchange the names of the two modes in
the two-mode inequality; its entanglement is invariant under this exchange.
With eta=max(A0^2,B0^2)>0,
 W0=eta K_t,
 K_t=[[t^2 I,-tZ],[-tZ,I]].

For any centered two-mode state with our doubled covariance V_R, the source
observable D_t=(t a-b^dagger)^dagger(t a-b^dagger) has mean
        <D_t>=(1/4)Tr(K_t V_R)+(1-t^2)/2.                (3)
For the pure Schmidt state this is
             d_t(q)=(1-tq)^2/(1-q^2).
The complementary-slackness trace above therefore gives EXACTLY
                    <D_t>_{rho_R}=d_t(q),               (4)
where rho_R is the actual retained pair after the LOCAL Schmidt-coordinate
changes and discarding the other Bob modes. The mixed input may retain
arbitrary cross-covariances with those discarded modes; (4) uses only the
support of W0 and its trace saturation.

Adesso2608.01909v3 Theorem1/Eqs8–9 asserts, for q<t<=1 and any finite-energy
two-mode state, the affine lower bound
 E_F(rho_R)>= E(q)-lambda_(q,t)(<D_t>_{rho_R}-d_t(q)),
 lambda_(q,t)=-2q log q/[(t-q)(1-tq)]>0,
 E(q)=g(q^2/(1-q^2)).
The source proof permits general pure-state probability measures (its
infinite-dimensional limit discussion); an integrated affine bound does
not presume existence of a finite optimal decomposition. Applying(4) gives
E_F(rho_R)>=E(q). Local-unitary invariance and monotonicity under discarding
Bob spectators then give
 E_F(V)>=E_F(rho_R)>=E(q)=E_F^G(V)>=E_F(V).
This proves equality under strict physicality. It also proves lossless
extraction for these entanglement values, without factorization of the state.
As a check on normalization, the KKT multiplier for objective a has
eta=q/[2(t-q)(1-tq)], and the source entropy multiplier is
lambda_(q,t)=-4 eta log q; the proof only needs the exact saturation (4).

5. All physical boundaries via local additive noise.
For any physical finite V, set V_epsilon=V+epsilon I,epsilon>0. This is
strictly physical and is obtained by independent local Gaussian random
displacements, so E_F(V_epsilon)<=E_F(V). From the strict result,
 E_F^G(V_epsilon)=E_F(V_epsilon)<=E_F(V).
The Gaussian roof converges to E_F^G(V) as epsilon decreases to zero.
Here this follows directly from the finite covariance optimization, not
from an assumed infinite-dimensional continuity theorem for E_F. The
Gaussian minima increase monotonically and are bounded by the value at V.
For epsilon<=1 all optimal pure covariances lie in the compact positive
set bounded above by V+I, with the same reciprocal eigenvalue bound as in
Section1. Any convergent minimizing subsequence has pure limit Gamma<=V;
the continuous one-mode entropy at that limit forces the limiting minimum
to be at least E_F^G(V). Hence equality of the limit holds. Taking the
limit in the last displayed inequality and using E_F(V)<=E_F^G(V) proves
the claim on every boundary as well. Each finite-mode, finite V remains
finite-energy throughout. No uniform dimension limit is taken.

6. Novelty and review obligations.
The R48 q/p-block theorem supplies a special-case convex SDP route. The
present argument addresses arbitrary cross q/p covariance via the PURE
manifold tangent cone, strict-physicality qualification, rank-two witness,
and its exact EPR identification. It is not a continuation of the old
q/p reduction by asserting a nonexistent local q/p normal form. The R48
explicit no-local-qp example is included in the current claimed domain.
Pure Schmidt reduction, Gaussian-roof characterization, LOCC monotonicity,
and the entire two-mode affine extremality theorem are inherited tools.
The current source's two-mode and bisymmetric results do not themselves
state this general one-versus-many theorem. Literature novelty remains
unverified beyond bounded primary-source and internal-history comparison.

The two independent reviews specifically examined: (i) the tangent separation and
non-abnormal multiplier, (ii) the disjoint-range rank argument, (iii) the
EPR gain condition q<t including swapped mode names, (iv) the doubled
covariance factors in(3), and (v) the boundary limiting argument. Exact
algebraic controls are supplementary and will not substitute for this
proof. Both internal reviews support this as a complete-original candidate.
External review and worldwide novelty clearance remain outstanding.


APPENDIX: EXACT PROBABILITY-MEASURE INTERFACES

Clarification of inherited probability-measure and local-monotonicity interfaces
Applies to the general-one-versus-many v1 draft; no changed algebraic claim.

The M24 original uses arbitrary pure-state probability measures, not only
finite ensembles. The source affine inequality is valid pointwise on all
finite-energy pure two-mode states. For any ensemble with the fixed finite-
energy barycentre, the component energies are finite almost everywhere by
Tonelli, and their integrated energy is the barycentre energy. Their Alice
entropies are integrably bounded by the thermal maximal-entropy function
and Jensen's inequality. The quadratic EPR expectation is also integrable:
its positive operator is bounded by a fixed multiple of total number plus
a constant at fixed gain. Hence the source affine inequality can be
integrated over any such ensemble. Taking the infimum yields its mixed-
state lower bound for the exact measure convention of the original.

Local discarding cannot increase this E_F. Apply the fixed local Gaussian
coordinate change to each pure component, then measure the discarded Bob
modes in a joint number basis. The conditional retained states are pure;
their Alice reduced states average to that of the component. Entropy
concavity gives a mean retained entanglement no larger than the component
entanglement. Countable refinement followed by integration is legitimate
by nonnegativity and the energy bounds above; the resulting barycentre is
the retained two-mode state. Infimize over input measures. This supplies
the specific monotonicity used in Section4 without assuming a finite
optimal ensemble or entropy continuity for arbitrary infinite-dimensional
states.

The boundary noise in Section5 is itself a product of local random
displacements. A product unitary acts on each pure component with exactly
the same entanglement; refining the measure by the displacement distribution
produces the noisy barycentre and the same mean entanglement. Its covariance
is V+epsilon I. This directly proves E_F(V+epsilon I)<=E_F(V) for the
original measure convention, independent of an abstract LOCC theorem.

The inherited Gaussian-roof covariance characterization also holds for
arbitrary Gaussian pure-state measures. A short justification is useful
here. Measuring any fixed quadrature in a Gaussian mixture produces a
positive mixture of one-dimensional normal laws. If a set of positive
ensemble measure had component variance at least epsilon larger than the
target Gaussian variance, restrict further to a positive-measure subset
with means bounded by a finite M. Its moment-generating function has a
lower bound proportional to exp((target_variance+epsilon)t^2/2-M|t|),
contradicting the target Gaussian moment-generating function at large t.
Thus every fixed quadrature's component variance is at most the target's
almost everywhere. A countable dense set of rational directions, followed
by continuity of quadratic forms, gives Gamma_component<=V simultaneously
almost everywhere. Every component's entanglement is therefore at least
the minimum over pure Gamma<=V. Conversely a fixed feasible pure Gamma
with a Gaussian displacement measure of covariance (V-Gamma)/2 realizes
the exact target and that entanglement. This is the standard inherited
positive-Gaussian-mixture argument, not a new result or extra round credit.
