M26 / R94 / working candidate v1 — sparse tree coupling and an exact Poisson tail
Owner /root. Prospective discovery. Not independently verified; no publication claim.

Claim to be challenged
For each alpha>0 there is q0(alpha), independent of fixed QAOA depth p and its deterministic angles, such that the limiting expected energy per spin in the original ordered Gaussian q-spin model is at most alpha for every odd q>=q0. The argument below actually treats both parities at sufficiently large q. It uses established fixed-angle sparse/dense energy correspondence and energy concentration, but derives the required sparse multi-output obstruction explicitly.

1. The sparse probability space
Fix q,d and n. Draw M~Poisson(d n) and independently draw ordered tuples E_e in [n]^q for e=1,...,M. Repeated vertices and repeated tuples are allowed, exactly as in the ER ensemble of Basso–Gamarnik–Mei–Zhou, arXiv:2204.10306, section2. For each occurrence e draw independent signs J_e^0,J_e^1,...,J_e^T. All are unbiased and independent of the hypergraph. Choose an integer Q, held fixed as n grows, and set
 A_l={1,...,floor(l n/Q)}, S_l=A_l\A_(l-1), R_l=[n]\A_l.
For branch t and step l use sign J_e^t if E_e intersects A_l and J_e^0 otherwise. Let
 H_(l,t)(x)=d^(-1/2) sum_e J_e^(l,t) prod_{j=1}^q x_(E_e,j).
Every H_(l,t) has the same sparse marginal law, the root l=0 is common, and l=Q has independent signs across branches conditional on the common graph. Endpoint Hamiltonians are NOT unconditionally independent: the shared Poisson graph remains shared. None of the proof assumes otherwise.

Each branch uses the same full static H_(l,t) in every cost layer. The interpolation does not resample disorder between layers. Angles are the same fixed 2p real numbers at every node. Every output distribution is its actual Born distribution.

2. Small affected regions on a Poisson hypergraph
Use the graph metric in which two distinct vertices in a common tuple are adjacent. For fixed integer r and fixed d,q, write
 C_r=sum_{j=0}^r (d q^2)^j.
For a deterministic set S of at most n/Q+1 vertices, let W=B_r(S). We claim, for fixed Q,
 P(|W|>3 C_r n/Q)=o_n(1).                                      (2.1)
No maximum-degree bound independent of n is assumed.

Proof with occurrence paths. Every vertex of B_r(S) is reached by a simple incidence path with j<=r distinct edge occurrences and j+1 distinct path vertices. Count all such paths, with one ordered choice of the entry/exit slots in each occurrence; this overcounts endpoints. Let Z_j be this count. By the factorial moment formula for a Poisson process of ordered tuples,
 E Z_j <= |S| [d q(q-1)]^j <= |S|(d q^2)^j.
For clarity, the bound follows by summing at most |S| n^j path vertex sequences, at most [q(q-1)]^j slot choices, and integrating j distinct occurrences. The intensity is (dn)^j and the probability of the 2j specified tuple coordinates is n^(-2j). Path simplicity only reduces the sum.

Also Var(sum_{j=0}^r Z_j)=O_(d,q,r)(n). Here is the necessary counting justification. In the product of two factorial path counts, the contributions with no shared occurrence cancel against the product of expectations, even if the vertex assignments happen to overlap. Each remaining pattern identifies at least one occurrence between the paths. Its union hypergraph is connected. If it has e distinct occurrences and v distinct vertex labels, then v<=1+(q-1)e. Summing labels and the Poisson occurrence intensities gives at most
 const(d,q,r) n^v n^(-(q-1)e) <= const(d,q,r) n.
There are finitely many intersection/slot patterns for fixed q,r; degenerate repeated tuple coordinates decrease v. This also handles repeated tuples because occurrences, not tuple values, are the Poisson points. Thus the variance bound follows. The deterministic j=0 term has no variance.

Now |W|<=sum Z_j, whose expectation is at most (n/Q+1)C_r. For n sufficiently large this is at most 2C_r n/Q. Chebyshev with gap C_r n/Q proves (2.1). The same holds simultaneously for the Q consecutive sets S_l, since Q is fixed. Constants can grow rapidly with r,d,q,Q; no uniformity in these parameters is needed for the n-first limit.

3. Actual QAOA marginal equality and supported pair couplings
For the pair H_(l-1,t),H_(l,t), all differing terms intersect S_l. Let W_l=B_(p+1)(S_l). The output reduced density operators on W_l^c agree exactly.

To justify equality for an arbitrary joint observable O on W_l^c, conjugate backwards through the p QAOA layers. Mixers are products of one-qubit rotations. A cost layer consists of mutually commuting diagonal Pauli rotations. Terms disjoint from the support before that cost layer commute with O and may be moved to the outside and cancelled; hence one layer increases its support by at most one hypergraph step. Every potentially relevant changed term lies inside the excluded radius. Applying the same cancellation inductively shows that the backward observables under the two circuits coincide. This is joint reduced-state equality, not just equality of one-qubit marginals. Taking computational-basis diagonal observables gives equality of the outside Born marginal.

Consequently there is a coupling pi_(l,t) of the two complete Born distributions supported on pairs equal outside W_l: first sample the common outside string, then independently sample the two inside strings under their conditional distributions. Define conditionals arbitrarily on zero-probability outside strings. Thus every supported pair differs in at most |W_l| coordinates. No efficient classical sampler for pi is asserted.

4. Gluing on a tree, without conditioning mistakes
Condition on the ENTIRE collection of instances. Sample one root output with the root Born law and identify it across all branches. For each tree edge, use the disintegration of pi_(l,t) given the already sampled parent, to sample its child. This produces a joint law of all outputs. Induction proves that every child has its own correct Born marginal and every edge has pi as its pair marginal.

The conditional distribution of a child given its parent is generally not its unconditional Born marginal. We never assert that. We never resample parents to force an energy event. Nor do we demand a simultaneous coupling for every cycle of instance pairs. Acyclic gluing is enough.

Suppose the limiting achieved energy in this sparse ensemble at the selected p and angles is V_d>alpha. By the established first/second moment concentration theorem of 2204.10306, the probability that any ONE node has H_(l,t)(output)/n<alpha tends to zero. Since QT+1 is fixed in n, the union bound applied to the valid marginal laws above implies that all nodes have energy at least alpha n with probability 1-o(1), even though their outputs are strongly dependent. Simultaneously, (2.1) makes all edge Hamming changes <=eta n if 3C_(p+1)/Q<=eta.

The sole use of the source concentration theorem is a one-output moment statement; no unproved multi-output overlap concentration is imported.

5. Exact exponential landscape bound — no central-limit tail substitution
Let m>=2 be fixed. At a fixed l, take m distinct branch labels and deterministic strings x^1,...,x^m. Write D_st=d_H(x^s,x^t)/n. For one random tuple E let
 chi_s(E)=prod_j x^s_(E_j),  W_E=sum_{s=1}^m J_E^(l,s) chi_s(E).
The sign grouping is either a single common root sign if E subset R_l, or m independent branch signs otherwise. Conditional on E, W_E is symmetric, |W_E|<=m, and has mean zero. Therefore
 E W_E^2=m+sum_{s!=t} 1_{E subset R_l} chi_s(E)chi_t(E).
Averaging the tuple, the cross term is exactly u_st^q, where
 u_st=n^(-1) sum_{i in R_l} x_i^s x_i^t.                       (5.1)
This identity includes repeated tuple entries; it is one reason to keep the ordered-tuple convention.

If a<=D_st<=b<1/2 for every pair, then
 -D_st <= u_st <= 1-D_st,
 so |u_st|<=1-a (using D_st<=b<1/2). In particular, for both odd and even q,
 u_st^q <= (1-a)^q.
The q odd case does not require flipping an output or its instance. Negative covariance is harmless in the upper bound. Thus
 V_2:=E_E,J W_E^2 <= m[1+(m-1)(1-a)^q].                       (5.2)
At the endpoint l=Q, R_l is empty and V_2=m for ANY strings, even identical ones.

Let Y=sum_s H_(l,t_s)(x^s). The marked Poisson exponential formula is exact:
 log E exp(lambda Y)=dn(E_E,J exp(lambda W_E/sqrt(d))-1).
By symmetry and the Taylor bound cosh z <=1+z^2/2+z^4 cosh(|z|)/24,
 log E exp(lambda Y)
 <= n lambda^2 V_2/2 + n lambda^4 m^4 cosh(lambda m/sqrt(d))/(24d). (5.3)
This is an annealed moment bound, retaining fluctuations of the common sparse graph. It is not obtained by replacing a sparse random vector with a Gaussian in a rare event.

Choose
 m>=max(2,32 log(2)/alpha^2),
 0<a<b<1/2 with h(b)<=alpha^2/64,
 (m-1)(1-a)^q<=1/4,
 d>=d0:=max(1,alpha^2 m^3),                                  (5.4)
where h is natural-log binary entropy. Put lambda=alpha/2. Then V_2<=5m/4, lambda*m/sqrt(d)<=1/(2sqrt(m))<=1, and cosh(lambda*m/sqrt(d))<=2. The remainder in (5.3) is at most n*m*alpha^2/192. Chernoff gives
 P(all m energies>=alpha n)
 <= P(Y>=m alpha n)
 <= exp[-n*m*alpha^2*(1/2-5/32-1/192)]
 = exp[-65 n*m*alpha^2/192].                                (5.5)
At the endpoint the same bound holds without the lower-distance restriction, since its V_2=m is smaller.

The number of ordered m-tuples with all pair distances at most b n is at most
 2^n [sum_{j<=floor(bn)} binom(n,j)]^(m-1)
 <= exp{n[log2+(m-1)h(b)]}
 <= exp(9 n*m*alpha^2/192).                                 (5.6)
The last step uses (5.4): log2<=m alpha^2/32 and (m-1)h(b)<=m alpha^2/64.
Combining (5.5) and (5.6) yields an upper bound
 exp[-7 n*m*alpha^2/24]                                     (5.7)
on either of the following events for a specified m-subset of branches:
 (i) at a specified l, m alpha-high strings with every pair distance in [a n,b n];
 (ii) at l=Q, m alpha-high strings with every pair distance <=b n.
These are existential landscape events, before the algorithm is chosen. A union over the constant Q and the constant binom(T,m) multiplies (5.7) by a finite n-independent factor. In particular all these events are absent with probability 1-o(1).

6. Deterministic extraction by a finite multicolour Ramsey argument
Assume there are T output paths x^(l,t), common at l=0, all alpha-high, and each edge changes at most eta n coordinates, with 2eta<b-a. Assume no m endpoint outputs have all pair distances <=b n. Define a complete graph on branch labels. Colour each pair by some l in {1,...,Q} at which its distance is in [a n,b n], if such an l exists; otherwise colour it 0. Since a pair distance begins at zero and changes at most 2eta n per step, a pair ending above b n must acquire a nonzero colour. Therefore a colour-0 clique of size m would have all endpoint distances <=b n, contrary to assumption.

For completeness, a finite T suffices. For r=Q+1 colours take T=(2r)^(rm+2). Repeatedly select a pivot and keep a largest single-colour class among its remaining neighbours. There are enough vertices to obtain L=r(m-1)+1 pivots each with an associated forward colour and a later surviving vertex: the loss at each step is at most a factor r plus one vertex, and the chosen T exceeds 1+r+...+r^L. Among the L pivot colours some colour occurs at least m times; the corresponding m pivots form a monochromatic clique. The colour cannot be 0. Hence there is a common l and m outputs whose pair distances all belong to [a n,b n].

This contradicts (5.7). The huge value of T is allowed: Q and T depend on p,d,q but are fixed BEFORE n tends to infinity. We do not use a Ramsey number growing with n. This extraction mechanism is attributed to the tree/Ramsey approach of arXiv:2609.10838, section4; the proof here records the elementary finite version for the present classical-output application.

7. Closing the sparse contradiction and the original quantifiers
For any alpha>0, first choose an integer m satisfying (5.4). Next choose b in (0,1/4) with h(b)<=alpha^2/64, and set a=b/2. Choose
 q0=max(3,ceil(log(4(m-1))/[-log(1-a)])),
so (m-1)(1-a)^q<=1/4 for all q>=q0. Set d0=max(1,alpha^2 m^3). None of m,a,b,q0,d0 depends on p or its angles.

Now fix ANY q>=q0, ANY d>=d0, ANY fixed p and deterministic angle list. Take r=p+1, C_r as above, eta=(b-a)/4, and integer Q>12 C_r/(b-a). This gives 3C_r/Q<eta and 2eta<b-a. Choose T by section6. If V_d>alpha, section4 makes the required good output paths exist with probability tending to one; section2 controls every step; section5 excludes the corresponding endpoint and intermediate tuples with probability tending to one. Their intersection is impossible by section6. Thus V_d<=alpha for every d>=d0.

The original dense model is exactly the pure ordered Gaussian ensemble of 2204.10306, with Var H(x)=n. Its Theorem2 gives, at each fixed p and angle list,
 V_Gaussian=lim_(d->infinity) V_d.
Therefore V_Gaussian<=alpha. Taking the supremum over deterministic fixed angles preserves the bound, because every individual limit satisfies it. Likewise the same q0 works for every fixed p. We never exchange a supremum or a p,d,n limit. Set alpha=gamma sqrt(2log2) to match M26.

Source use and novelty boundary
- Basso–Gamarnik–Mei–Zhou, https://arxiv.org/html/2204.10306, Theorems1–2: one-output first/second moment concentration and fixed-angle sparse/dense correspondence. The ordered Poisson normalization matches exactly; its characteristic exponent is d[cos(lambda/sqrt(d))-1], with derivative tending to -lambda and exponent tending to -lambda^2/2. These results apply to odd q as well.
- Chou–Love–Sandhu–Shi, https://arxiv.org/html/2108.06049v3, section3: local marginal sharing is established background. We prove the particular support statement and tree law here to expose every dependency.
- https://arxiv.org/html/2609.10838v1, section4: tree paths and Ramsey extraction are prior methodology, not our invention. We do not rely on its conditional resampling or its Gaussian-tail step. Our exact Poisson calculation covers the classical signed model and both parities. The scientific novelty of this adaptation requires a separate source audit.
- https://arxiv.org/html/2310.01563v1, Theorem1.5 and Corollary1.7: the cited sharp obstruction and quantum-advantage comparison require even predicates. These statements do not themselves close the odd-q target.
- https://arxiv.org/html/2307.07461v1: odd-order ensemble m-OGP is known. We do not claim a new general OGP; section5 proves the particular masked sparse estimate needed for this coupling.

Current status: a full original-scope proof candidate, conditional only on the precisely identified established Theorems1–2. Every new lemma still needs current-agent challenge and source-priority qualification before any completion assessment. This file alone closes no seat or round.
