R95/M26 — quantitative sparse concentration and tree costs
26 October2026. Auxiliary extension of the frozen R94 candidate.
No fixed-depth original solution is credited again; no dense growing-depth claim.

1. Sparse model and a uniform local-size bound

Use exactly R94's Poisson(dn) ordered q-tuples, independent signs, normalization
H(x)=d^(-1/2)sum_e J_e chi_e(x), fixed q,d, and p QAOA layers. Angles can be
arbitrary deterministic functions of n,p, but not functions of the instance.
Repeated tuple coordinates and occurrences are retained. They only reduce
actual supports, so counting them as full hyperedges safely overestimates
light cones. All cost gates within one layer commute.

Exploration of a new vertex reveals a Poisson number of as-yet-unrevealed
incident occurrences with conditional mean at most dq. Each such occurrence
introduces at most q new vertices. Thus a breadth-first exploration is
stochastically bounded by a branching forest with offspring
Y=q Pois(dq), mean mu=dq^2. Conditioning on previously revealed absence only
reduces the unexposed Poisson intensity. A Palm root occurrence contributes
at most q initial vertices; the remaining graph is again the same Poisson
process. Collisions and repeated coordinates can be ignored for an upper bound.

For a depth-r single-root branching tree T_r, m_r=E T_r and s_r=E T_r^2
satisfy m_r<= (1+mu)^r and
 s_r=1+2mu m_(r-1)+mu s_(r-1)+E[Y(Y-1)]m_(r-1)^2
     <= C0 s_(r-1),
 C0=1+3mu+mu^2+dq^3.
Hence s_r<=C0^r. For k deterministic roots, Cauchy gives second moment at
most k^2 C0^r. The number of discovered edge occurrences is bounded by
one more exploration generation, up to a fixed q,d constant. In what
follows C>=2 is a fixed sufficiently large function of q,d; every occurrence
neighborhood of radius r has first/second moments <=C^(2r+4).
No maximum-degree assumption and no constant depending on the gate angles
is introduced.

2. Finite-n energy concentration with explicit depth dependence

Let X=H(output)/n, with randomness over both the instance and Born output.
There is a fixed C=C(q,d) such that, uniformly in every deterministic angle
list, every n and p,
                         Var(X)<=C^(4p+10)/n.                  (1)
A bound greater than the trivial scale is allowed when p is large; the
useful regime below is p=O(log log n).

Quantum-output part. Conditional on an instance, two diagonal term
observables have zero connected covariance whenever their backward light
cones are disjoint. Conjugation through each commuting cost layer grows
support by at most one hypergraph step, and the initial state is a product.
The absolute covariance is at most2 otherwise. Therefore conditional
Var(H) is at most (2/d) times the number of ordered occurrence pairs whose
p-layer cones meet. Their graph distance is at most2p plus fixed endpoint
slack. Campbell's identity and the Palm neighborhood estimate of section1
bound the expected pair count by dn C^(4p+8). Thus
 E Var(X|instance)<=C^(4p+9)/n after enlarging C.

Disorder part. Put F(instance)=E[H(output)|instance]. Adding one marked
occurrence changes the new term's expected contribution by at most1/sqrt(d).
An old term's expectation can change only if its backward p-layer cone
meets the added occurrence; its change is at most2/sqrt(d). If K is the
number of such old occurrences,
 |D_e F|<=(1+2K)/sqrt(d).
The same neighborhood second-moment bound controls E(1+2K)^2 by
C^(2p+8). The Poisson Poincare inequality gives
 Var F<= integral E[(D_e F)^2] intensity(de)
       <=dn * C^(2p+8)/d = n C^(2p+8).
Applying total variance proves(1), increasing the fixed constant once.
This is an ANNEALED inequality for the actual quantum output energy. It
requires neither an efficient classical output sampler nor overlap
concentration. The Poisson Poincare inequality is the standard add-one
variance inequality, not a new concentration principle.

3. A uniform geometric event for growing p

R94's paths change disorder on deterministic consecutive sets S_l of
size at most n/Q+1. Let W_l=B_r(S_l), r=p+1, and
C_r=sum_(j=0)^r mu^j. Occurrence-path counting gives E|W_l|<=|S_l|C_r.
Adding an occurrence can change |B_r(S_l)| only inside an r+1 neighborhood
of that occurrence, regardless of whether it connects to the old ball.
The Poisson Poincare inequality and section1 therefore imply
 Var|W_l|<= n C^(2r+8).
When n>=Q, Chebyshev and a union over the Q slices show
 P(exists l: |W_l|>3C_r n/Q)
 <= Q^3 C^(2r+8)/(C_r^2 n),                               (2)
up to a fixed factor absorbed in C. This gives a concrete substitute for
R94's unspecified fixed-radius O(n) variance constant. It does not assert
that the neighborhoods of different slices are independent.

4. Pay for every tree node and every forbidden tuple

Fix alpha,m,a,b,q,d as in R94's EXACT marked-Poisson landscape estimate.
In particular q>=q0(alpha), d>=d0(alpha), and m,a,b do not depend on p.
Choose eta=(b-a)/4 and
 Q=ceil(16 C_(p+1)/(b-a))+1,
 T=(2(Q+1))^((Q+1)m+2).
These deterministic choices satisfy both the Hamming-step and Ramsey
requirements. The number of tree nodes is 1+QT, not merely Q.
The union of forbidden landscape events has probability at most
 (Q+1) binom(T,m) exp[-c_alpha n],                          (3)
with c_alpha=7m alpha^2/24 from R94. Conditional tree gluing still uses
unconditional Born marginals at every node; p may vary with n without
changing this finite-instance identity.

Let C1=2+mu, and choose a fixed
 kappa<1/[4 log(C1)].
For p<=kappa log log n, C_(p+1)<=C1^(p+2), so Q is at most a fixed
multiple of (log n)^(1/4), and
 log T=O(Q log Q)=o(log n).
Consequently QT=n^o(1), C^(4p+10)=n^o(1), the bound(2) is n^(-1+o(1)),
and (3) tends to zero exponentially. All constants may depend on fixed
q,d,alpha, but NOT on angles or n.

If the common mean energy per spin of the actual node marginals were at
least alpha+epsilon_n, (1), Chebyshev and the union over ALL nodes would give
 P(any node has energy below alpha n)
 <=(1+QT) C^(4p+10)/(n epsilon_n^2).                       (4)
Taking epsilon_n=n^(-1/8) makes (4) tend to zero. Combining(2)-(4) would
produce the forbidden paths and Ramsey clique with probability tending
to one, contradicting R94's deterministic extraction. Hence, for all
sufficiently large n, uniformly over deterministic angles and
p<=kappa log log n,
                  E[H(output)/n] < alpha+n^(-1/8).         (5)
This is a sparse-model quantitative extension conditional on the reviewed
R94 landscape/extraction lemmas, whose statements are retained unchanged.
The branching and variance argument supplies the previously missing
finite-n concentration input. No Gaussian CLT is used for an exponential
rare-event bound; that remains the exact Poisson calculation in R94.

5. Quantifiers, sources and limits

The finite-n comparison in(5) is stronger than the fixed-p sparse statement,
but it does NOT prove an analogous growing-depth Gaussian mean-field result.
The established sparse/dense correspondence is a fixed-angle, fixed-depth
limit theorem. Uniformity when p=p(n), and the order of d,n limits, require
additional analysis. R94's fixed-depth original candidate remains the same
candidate; its originality and independent validity are not upgraded here.

The local marginal principle and tree/Ramsey strategy are attributed to
Chou--Love--Sandhu--Shi arXiv:2108.06049v3 and the source tree methodology
arXiv:2609.10838v1, as in R94. Basso--Gamarnik--Mei--Zhou arXiv:2204.10306
still supplies only the fixed-depth sparse/dense correspondence used there.
Poisson Poincare/Campbell/Palm identities are standard probability tools;
their use here is the explicit QAOA locality composition. No claim that
(5) is absent from all existing sparse local-algorithm literature is made.

Original M26 is not counted solved again. Retain(1)-(5) as a bounded
auxiliary extension, with the dense growing-depth and priority gaps stated.
