R98/M26 — Mixed-time extraction removes the Ramsey bottleneck
26 October2026. Prospective main theorem candidate; current-agent review only.

THEOREM. In the signed ordered Poisson q-spin ensemble of R94, for every
alpha>0 there are q0(alpha), d0(alpha), and, for each fixed q>=q0 and d>=d0,
a constant kappa(q,d,alpha)>0 such that uniformly over deterministic QAOA
angle lists and depths p<=kappa log n,
                  E H(output)/n <= alpha+n^(-1/8)
for all sufficiently large n. Both parities q are covered. Angles may depend
on n,p,q,d, but not on the instance. The expectation includes disorder and
Born outcomes. This is a sparse theorem, not a dense growing-depth transfer.
The original fixed-depth dense candidate of R94 is not counted solved again.

Inherited inputs. R94 specifies M~Pois(dn) ordered tuples in [n]^q, each with
an independent Rademacher sign, H=d^(-1/2)sum J_e chi_e. Its exact Born-marginal
sharing, tree gluing and marked-Poisson identities are retained. R95 proves,
for a fixed sufficiently large C=C(q,d)>=2, uniformly in p and angles,
 Var[H(output)/n]<=C^(4p+10)/n. The radius-r geometric event for Q deterministic
slices has failure at most K Q^3 C^(2r+8)/(C_r^2 n), with C_r=sum_(j=0)^r mu^j,
mu=dq^2, when n>=Q. K can be absorbed into a fixed enlargement of C.
These finite-n estimates, including unbounded Poisson degrees and repeated
occurrences, are dependencies, not new concentration results in this note.

1. Interpolate a constant number of branches, retaining the shared graph.
Choose T=m^2, where m is fixed below. Each occurrence has independent signs
J_e^0,J_e^1,...,J_e^T. Set A_l=[floor(ln/Q)], R_l=[n]\A_l. At node (l,t),
use J_e^t if E_e intersects A_l, and J_e^0 otherwise. At l=0 all nodes are
one root. All other nodes use the same deterministic angles. Endpoints have
independent signs conditional on the shared graph, not independent graphs.

2. Exact mixed-time landscape estimate.
Consider m specified nodes, at most one root and otherwise DISTINCT branch
labels. Their times l_s need not agree. For fixed strings x^1,...,x^m and a
random tuple E, put W_E=sum_s J_e^(l_s,t_s) chi_s(E). Two distinct branches
share a sign precisely when E avoids BOTH A_(l_s),A_(l_t). Since these sets
are nested, this is E subset R_max(l_s,l_t). A root/nonroot pair obeys the
same formula. Thus
 E W_E^2=m+sum_(s!=t) [n^(-1)sum_(i in R_max(l_s,l_t)) x_i^s x_i^t]^q. (1)
The covariance is not computed at min(l_s,l_t). For any subset R and pair
with normalized Hamming distance D in[a,b], b<1/2,
 -D <= n^(-1)sum_(i in R) x_i^s x_i^t <=1-D,
so its absolute value is at most1-a. Hence V2<=m[1+(m-1)(1-a)^q]. At l_s=Q
for every selected endpoint, V2=m without any lower-distance condition.

Conditional on E, W_E is symmetric: reversing all independent signs reverses
W_E. Also |W_E|<=m. The exact Poisson exponential formula therefore gives
 log E exp(lambda sum_s H_(l_s,t_s)(x^s))
 <=n lambda^2 V2/2+n lambda^4 m^4 cosh(lambda m/sqrt d)/(24d). (2)
No Gaussian replacement or conditional graph concentration is used.
Choose
 m>=max(2,32 log2/alpha^2),
 0<b<1/4 with h(b)<=alpha^2/64,
 a=b/(4m), eta=a/2,
 (m-1)(1-a)^q<=1/4,
 d>=max(1,alpha^2 m^3), lambda=alpha/2.
R94's algebra in (2) yields probability at most exp[-65nm alpha^2/192]
that all m strings have energy >=alpha n. There are at most
 exp[n(log2+(m-1)h(b))] <=exp[9nm alpha^2/192]
ordered m-tuples of strings whose pair distances are at most b n. Therefore
with c=7m alpha^2/24, each specified mixed-time tuple has probability <=e^-cn
of admitting high-energy strings with all pair distances in[a,b]; the same
bound applies to each endpoint tuple with pair distances only <=b.
There are at most (1+TQ)^m mixed-time node tuples, plus T^m endpoint tuples.
Consequently the probability of ANY such forbidden tuple is at most
                   [(1+TQ)^m+T^m] e^-cn.                  (3)
The union is taken before adaptive selection of a tuple. There is no claim
that the selected nodes have unconditional laws after this selection.

3. A deterministic metric lemma with only m^2 branches.
Let T=m^2 paths x_(l,t), l=0,...,Q, in any metric space have a common root,
step length <=eta, and no m endpoints with pair distances <=b. Suppose
2a<=b and (m-1)(a+eta)<=b. Then one can select the root and m-1 nodes on
DISTINCT branches, possibly at different times, with all pair distances
in(a,b].

Proof. Start with the root, selected set S of size k=1. Inductively maintain
pair distances >a and diameter <=(k-1)(a+eta). At most k-1 branch labels
have been used. There are T-(k-1) unused endpoints. If all belonged to the
union of the k closed radius-a balls about S, assign each to one containing
ball. Since T-(k-1)>k(m-1) for k<=m-1, some ball would contain m unused
endpoints. They would have mutual distance <=2a<=b, a contradiction.
Choose an unused branch whose endpoint is outside all these balls. Its root
is in the union. Let l be its FIRST exit. Then x_(l,t) has distance >a from
every point of S, and its predecessor is within a of some point z in S.
Its distance to z is at most a+eta, and to every other selected point is
at most (k-1)(a+eta)+(a+eta)=k(a+eta). Add it. Induct until k=m.
Every added branch is new. With our a,eta, the final diameter is at most
3(m-1)b/(8m)<b. Equality/ties at the ball boundary are handled by choosing
the first exit from CLOSED balls. The lemma is deterministic and requires
neither independent paths nor a common time for the selected nodes.

4. Realize paths with exact Born marginals.
Changing slice S_l=A_l\A_(l-1) only affects output qubits in
W_l=B_(p+1)(S_l). The joint outside Born marginals are identical for the
pair of static-Hamiltonian QAOA circuits: commuting cost gates grow the
backward support by at most one hypergraph step per layer. Couple each
edge by the common outside marginal, disintegrate given its parent, and
glue the star of paths as a tree, conditional on ALL instances. This gives
correct unconditional Born marginals for every node and differences confined
to W_l. No output is resampled conditional on being high energy.
Take Q=ceil(12 C_(p+1)/a)+1. On R95's geometric event,
 |W_l|/n<=3 C_(p+1)/Q<a/4<eta for every l.

5. Pay the logarithmic-depth costs explicitly.
Enlarge the fixed C to be at least2+mu. Constants K below depend on fixed
alpha,q,d only. Then Q<=K C^(p+2). Put
                    kappa=1/(40 log C).
For p<=kappa log n, C^p<=n^(1/40), Q<=K n^(1/40), so n>=Q eventually.
The geometric failure bound, dropping its denominator C_(p+1)^2>=1, is
 <=K Q^3 C^(2p+10)/n <=K' C^(5p+16)/n=O(n^(-7/8)).          (4)
For any fixed angle list, all node marginal mean energies are identical.
If that mean exceeded alpha+epsilon_n, epsilon_n=n^(-1/8), Chebyshev and
the ACTUAL node marginals give
 P(any node energy<alpha n)
 <=(1+TQ) C^(4p+10)/(n epsilon_n^2)
 <=K C^(5p+12) n^(-3/4)=O(n^(-5/8)).                      (5)
Finally (3) tends to zero exponentially because Q is polynomial in n and
m,T are fixed. With positive probability all paths would have high energies,
small steps, no close m endpoints and no forbidden mixed-time tuple.
The metric lemma constructs exactly such a forbidden tuple, contradiction.
All error bounds are uniform in the deterministic angles; no angle net or
simultaneous event for all angles is being asserted or needed.

6. Quantifiers and contribution boundary.
For each alpha choose m,b,a as above, then q0 from
q0>=log(4(m-1))/[-log(1-a)] and d0=max(1,alpha^2 m^3).
Neither depends on p. For fixed q>=q0,d>=d0 choose C and kappa. The eventual
n threshold is uniform over p<=kappa log n and all deterministic angles.
The only depth extension proved here is in the sparse signed model. The
fixed-depth sparse/dense transfer of Basso et al. is not silently made
uniform in p,d,n. The original fixed-depth dense claim follows already from
R94 and is not a second original solution. The extra lemma removes that
candidate's artificial equal-time Ramsey cost; it does not remove physical
light-cone growth, alter the graph distribution or allow learned angles.

Sources checked 26Oct2026:
- Chou, Love, Sandhu, Shi, arXiv:2108.06049v3: logarithmic-depth local quantum
  limitations and concentration are established; the displayed general
  MAX-k-XOR theorem explicitly uses even k. We claim no new generic locality
  or logarithmic-depth principle.
- Mints, Anschuetz, arXiv:2609.10838v1, AppendixB and its cited Lemma4.10:
  tree/Ramsey extraction is prior methodology for a quantum hypergraph model.
  This note substitutes a classical Hamming metric lemma where the exact
  mixed-time Poisson covariance is available; it is not a claim to improve
  their different quantum model without another hypothesis check.
- Mark Goh, arXiv:2404.06087 / DOI10.2478/qic-2025-0018: the accessible author
  repository paper states even-q logarithmic-depth obstruction and describes
  odd-q evidence as numerical. No blanket first logarithmic QAOA claim.
- Basso, Gamarnik, Mei, Zhou, arXiv:2204.10306: fixed-depth energy transfer.
The greedy packing idea is elementary metric reasoning; worldwide novelty
of this precise odd-large-q composition has NOT been established. This is
an evidence-linked theorem candidate pending an independent proof/source
review. It is a full prospective target outcome, not a finite-size test.
