Random-Projector Quantum Diagnostics of Ramsey Numbers and A Prime-factor Heuristic for R(5, 5) = 45

1. Introduction

Ramsey numbers are at the intersection of graph theory, combinatorics, neural networking, computation and probability. Ramsey theory asks for the smallest number R(m, n) such that every edge coloring of the complete graph K_{R(m, n)} with two colors contains a red graph K_m with m vertices or a blue K_n with n vertices [1]. In graph theory, a Ramsey number refers to the minimum number of vertices in a complete graph where, regardless of how the edges are colored with two colors, a monochromatic subgraph of a specified size (or sizes) is guaranteed to exist. Ramsey’s theorem guarantees the number exists. We know all exact values up to R(4, 5) = 25; the next diagonal case 43 ≤ R(5, 5) ≤ 46 has not been directly calculated yet [2–4]. Gap sizes grow explosively between R(5, 5) and R(6, 6) the interval already spans 60 numbers, and for R(7, 7) we only know 189 ≤ R(7, 7) ≤ 4749. It is the search of sequences and patterns in large structures like searching for constellations in the sky [5].

Ramsey numbers control worst-case resource overheads in error-correcting codes and result intractable with current quantum computing resources. Designing a code to correct t errors means ensuring no two codewords sit within Hamming distance, which measures the minimum number of substitutions required to change one string into the other, d_H ≤ 2t [6]. This avoidance problem is equivalent to a set-coloring Ramsey instance on an alphabet of size q. The threshold at which large codes cease to exist scales like R(t + 1;q, q — 1), so the redundancy one must pay in the worst case grows with a Ramsey number rather than a mere polynomial in t. This follows from the standard avoidance formulation of code design; see [7] for the probabilistic method and [8] for Ramsey-theoretic background.

In quantum computing they are fundamental in worst-case routing in quantum compilation, mapping an arbitrary k-qubit circuit onto limited hardware connectivity forces SWAP gates whenever interacting qubits aren’t adjacent. Ramsey’s theorem guarantees that in any coloring or pattern of two-qubit interactions among R(k, k) qubits, there must exist an unavoidable “hard” configuration such as a clique or independent set requiring at least R(k, k) SWAP operations or depth units to resolve. Thus, the diagonal Ramsey number sets a lower bound on the worst-case communication depth of any fully general distributed quantum compiler in worst-case interaction patterns under limited connectivity, a Ramsey-style obstruction yields a depth lower bound scaling with a diagonal Ramsey number [8, 9]. To push worst-case depths down, one either needs richer native connectivity or higher-dimensional qudits of order n to effectively reduce the Ramsey argument to R([k/n], [k/n]) adopting paraparticle methods with ℤ₂ × ℤ₂-graded Majorana algebras [10].

Computing Ramsey numbers is notoriously intensive: classical proofs use double counting or induction [11], whereas the gluing technique builds larger graphs from smaller ones to avoid or force specific subgraphs such as cliques or independent sets [12].

We propose a new statistical method to estimate Ramsey number based on ℤ₂ × ℤ₂-graded Lie Algebras of Majorana infinite-component spin fields introduced in [10]. Our strategy follows the implementing of a paraparticle algebra graded by the Klein group V₄ ≌ ℤ₂ × ℤ₂, which requires only a limited number of qubits for each step of analysis. By implementing higher-order algebras we develop recursive gluing and pruning methods on Ramsey graphs to give an estimate to or effectively calculate Ramsey numbers. The result is a purely algebraic that takes in account of the branching that factorizes the two colors and packages local symmetry constraints, allowing traces/character formulas to replace brute-force enumeration for many branches.

Quantum resources needed are formidable: verifying R(4, 4) = 18 requires exhaustively scanning all 2¹³⁶ two–colourings of K₁₇, i.e., one logical qubit per edge (136 qubits), already at the frontier of current hardware [13, 14]. Even with Grover’s quadratic speed-up the bottleneck is width, not depth: a 40-vertex instance occupies a 2⁷⁸⁰-dimensional space, demanding 780 qubits before amplification can begin.

For R(5, 5) we give an estimate of the qubit cost of a brute–force (Grover–style) search that needs an enormous number of quantum resources with a register model based on a two–colouring of K_n that can be encoded by a binary string x ∈ {0,1 }^E(n) with one bit per (undirected) edge, where $E\left(n\right)=\left(\begin{array}{c}n\\ 2\end{array}\right)=n(n-1)/2$ as reported in Table 1. Adopting the convention x_ij = 1 iff edge {i, j} is red (blue is 0), a reversible predicate F_n(x) flags a violation (F_n(x) = 1) iff x contains a red K₅ or a blue K₅; otherwise F_n(x) = 0. In Grover’s search for a good colouring one uses the oracle

Oracle structure and ancillas can be organized in this way, like a game: for each 5–subset S ⊆ [n] we must test whether all $10=\left(\begin{array}{c}5\\ 2\end{array}\right)$ edge–bits inside S are equal to 1 (red K₅) or all equal to 0 (blue K₅). Each “all–red” test is a 10–input AND operation, which can be realized by a Toffoli tree using 9 clean ancillas (one per internal node); the “all–blue” test is obtained by negating the same 10 inputs, reusing the same 9 ancillas, and then uncomputing. Let a_AND = 9 denote these workspace ancillas. We keep one flag r_S for “red K₅ on S”, one flag b_S for “blue K₅ on S”, and an aggregate OR–bit v that accumulates violations via v ← v ∨ (r_s ∨ b_s) as we sweep all $\left(\begin{array}{c}n\\ 5\end{array}\right)$ subsets, uncomputing r_S, b_S and the AND tree at the end of each subset. Finally, we need one output/phase ancilla for Grover, a_ph = 1.

Thus the oracle’s simultaneous ancilla budget can be kept to have a number of ancillas, a_work = a_AND + 2 + 1 + a_ph = 9 + 2 + 1 + 1 = 13, i.e., a number of 13 clean ancillas beyond the edge register. If, instead, negative controls are disallowed and input–negations must be done explicitly for the “all–blue” check, one adds at most 10 transient bits that are immediately uncomputed; the peak simultaneous ancillas remain ≤ 13. One may also think to add a couple of safety ancillas for carry/OR trees; which means a budget of +3 that would be needed to cover such variants, but it ia a matter of optimization that goes beyond the purpose of this work.

The total qubits required starts from a conservative upper bound Q(n) on the width (qubit count) of a brute–force Grover oracle for the diagonal R(5, 5) tests at size n,

Concretely, as reported in the following table, the number of qubit required is very big.

The search space has size $2^{E\left(n\right)}=2^{\left(\begin{array}{c}n\\ 2\end{array}\right)}$. Grover’s amplitude amplification finds a good colouring, when one exists, n has $O\left(\sqrt{2^{E\left(n\right)}/G_n}\right)$ oracle uses, where G_n is the number of good colourings at size n, which is unknown a priori. For the upper–bound instance (let us assume n = 45), proving nonexistence of a good colouring via search still takes exponential time in the worst case and requires additional outer logic, but the width remains dominated by the edge register as above.

In our setting the projector algebra acts on the charge–zero submodule of dimension d = 24, so the data register needs only $\lceil {\log }_{2}d\rceil =\lceil {\log }_{2}24\rceil =5$ qubits (plus a few ancillas for block-encoding/qubitization). Instead, the truly brute–force (enumeration/Grover) approach to R(5, 5) at the threshold requires on the order of a thousand logical qubits just to hold a coloring $\left(\left(\begin{array}{c}45\\ 2\end{array}\right)=990\right)$, plus ≲ 13 + 3 = 16 clean ancillas for the reversible violation check and phase kickback. In fact, a naive edge-register encoding at n = 45 would require one qubit per edge, i.e., 990 data qubits.

This stands in sharp contrast to our estimate random–projector spectral diagnostics presented in the next section, whose quantum implementation needs only a few data qubits (plus a handful of ancillas), i.e., two or three orders of magnitude fewer qubits. The other two diagonal cases with n = 6 and n = 7 would require up to 13530 and 145530 maximum data qubits, respectively, instead of 6 and 7 data qubits (plus modest ancillas) discussed below. Our approach does not replace the exact solution from brute-force approach, instead helps to restrict the range of values of diagonal Ramsey numbers for deeper investigations like Erdős flip-coin method [15].

2. Ramsey Numbers and Klein-Graded Paraparticle Algebra

To compute R(m, n), one can use a recursive gluing–pruning scheme. Starting with a base layer obtained by enumerating all good edge-colorings G_v0 of K_v0 (i.e., colorings containing neither a red K_m nor a blue K_n). Then, for each good G_v , glue on a new vertex v + 1 and color its v incident edges in every way that preserves the good property, pruning any extension that creates a red K_m or a blue K_n.

Pruning complements the gluing step by immediately discarding any partial coloring that already contains a forbidden red K_m or blue K_n. After each glue operation, we quotient by vertex–label symmetries via canonical labeling and graph–isomorphism checks to eliminate equivalent colorings, ensuring that only genuinely new configurations are explored. We then backtrack as soon as a red K_m or a blue K_n is detected. This bottom-up strategy, coupled with selective rollback and symmetry reduction, dramatically shrinks the search tree and has enabled exact computations by keeping the frontier of viable colorings tractably small [12, 16]. In our case we exploit the properties of ℤ₂ × ℤ₂–graded Majorana algebras for paraparticle states [10] and random projectors to extend Erdős’ flip-coin method.

Let ${\gamma }_j^{\left(0\right)},{\gamma }_j^{\left(1\right)}$ denote Majorana modes with

To mirror the colorings, we extend the Majorana modes in Eq. (3) to a paraparticle doublet $a_j=\frac{1}{2}\left({\gamma }_j^{\left(0\right)}+{\gamma }_j^{\left(1\right)}\right)$, $b_j=\frac{1}{2}\left({\gamma }_j^{\left(0\right)}-{\gamma }_j^{\left(1\right)}\right)$, satisfying $\left\{a_i,a_j^{†}\right\}=2{\delta }_{ij}$, and $\left\{b_i,b_j^{†}\right\}=2{\delta }_{ij}$, all other anticommutators vanish, assigning (a, b) the Klein charges (1, 0) and (0, 1). The full algebra ${\mathcal{A}}_{V_4}={\oplus }_{g\in V_4}{\mathcal{A}}_g$ decomposes into four color sectors. A binary edge coloring of a graph with vertex set V lifts to

We can now set the assignment of edge-labels “coloring” ≡ graded sector. The Klein four group V₄ has elements {(0, 0), (1, 0), (0, 1), (1, 1)}, we decide to identify (1, 0) red, (0, 1) blue, with (0, 0) the vacuum/identity and (1, 1) a mixed sector projected out at the end which lives in the bi-graded component of the algebra obtained by multiplying a red operator by a blue one (or vice versa).

The graded Ramsey number R_V4(m, n) is the least v ≥ 1 such that P_m,n (v) annihilates the entire charge–zero module (equivalently, every graded two–coloring of K_v—with mixed (1, 1) components invisible—contains a red K_m or a blue K_n). It obeys the exact Klein Erdős recursion

Equation (7) exactly matches the constructive lower bound: there is at least one coloring on R_V4(m – 1, n) + R_V4(m, n – 1) – 1 vertices that avoids both a red K_m and a blue K_n and R(m, n) ≤ R_V4(m, n), so no classical bounds are violated.

In the ℤ₂ × ℤ₂ graded algebra, multiplying a red sector element by a blue sector element produces a state in the mixed (1, 1) sector, which lies outside the physical subspace and does not correspond to any valid qudit state because, by construction, the physical Hilbert space is defined to include only the pure color sectors. In practice, such components are either projected out (viz., set to zero) or interpreted as an internal syndrome that flags excursions away from the pure color subspaces during computation. By monitoring the amplitude in the mixed sector after each gate, one can detect and correct errors. When is observed any nonzero mixed-sector population, a red–blue mismatch has occurred and one can apply a compensating operation to return to the valid (1, 0) or (0, 1) grading.

For each unordered pair {i, j} we then introduce a homogeneous edge generator e_ij of that ℤ₂ × ℤ₂-degree. Paraparticle commutation relations [17] enforce parity bookkeeping without edge-ordering data. The coproduct is a glue operation defining

To detect forbidden cliques, inside the algebra we use the projection P_m,n of Eq. (6) where, ${\text{Π}}_{\text{R}}\left(\text{S}\right)={\text{Π}}_{i<j\in \text{S}}{e}_{ij}^{(1,0)}$ projects onto the red and ${\text{Π}}_{\text{B}}\left(\text{T}\right)={\displaystyle {\prod }_{i<j\in \text{T}}{e}_{ij}^{(0,1)}}$ on blue (see Eq. (5)). A coloring survives iff P_m,n annihilates it. Because P_m,n is central, one can evaluate Tr(P_m,n) on characters of irreducible modules rather than on individual colorings, turning part of the combinatorial explosion into a trace computation. Two lemmas and a theorem (see Appendix for demonstrations) fix the next steps.

The factorization A_V4(V\{p}) ≃ A_V4(V_R) ⊗_b A_V4(V_B) makes cross-edges invisible to Π_R and Π_B, which is the key ingredient in the graded Klein recursion. Majorana’s infinite-spin equation gives a ladder of fields with equally spaced mass–spin ratios. Algebraically, each step is an induction functor Rep(G_s) → Rep(G_s+1). Here Rep(G_s) denotes the (rigid, monoidal) category of complex representations of the symmetry group G_s at rung s of the Majorana tower; its objects are G_s–modules (V, ρ) and its morphisms are intertwiners T : V → W with T_ρ(g) = ρ′(g)T for all g ∈ G_s. The step s → s+1 is modeled by induction along i_s : G_s ↪ G_s+1, sending V ∈ Rep(G_s) to ${\mathrm{Ind}}_{G_s}^{G_{s+1}}V\simeq ℂ\left[G_{s+1}\right]{\otimes }_{ℂ\left[G_s\right]}V$ (equivalently U(𝔤_s+1)⊗_U(𝔤s) V), which algebraically realizes the Majorana infinite–spin ladder’s equally spaced mass–spin progression.

Each graph vertex i is realized as a pair of Majorana modes γ_2i–1, γ_2i, and set Γ_ij = i γ_2i–1 γ_2j–1, which is odd under one ℤ₂ (fermion parity) and even under the other (boson parity), matching the Klein grading. Each glue adds a new Majorana pair, and the tower’s induction gives a canonical lift of representations as v → v + 1. Recursive gluing inside the algebra 𝒜_V4 is described the relationship in Eq. (7) from the definition of the algebraic graded Ramsey numbers, R_V4(m, n).

From the Majorana-tower construction one labels vertices by tower indices 1 ≤ ℓ ≤ R_V4(m, n). At level ℓ we create a mode pair (a_ℓ, b_ℓ). Equation (7) is realized algebraically by mapping a_ℓ ↦ a_ℓℓ≤ R_V4(m–1, n) or $a_{\ell }\mapsto a_{\ell }^{†}$ otherwise, and analogously for b_ℓ. This “dagger flip’’ constitutes the gluing that concatenates two smaller cliques without leaving the algebra and ${\text{Γ}}_j^{(\pm )}\equiv \left({\gamma }_j^{\left(0\right)}\pm {\gamma }_j^{\left(1\right)}\right)/2$, so that $\left({\text{Γ}}_j^{(+)},{\text{Γ}}_j^{(-)}\right)$ carry Klein charges (1, 0) and (0, 1), respectively. The edge operator of a k-vertex red clique is the normal-ordered monomial

As each Γ^(±) anticommutes with any operator of opposite Klein charge, every factor in (9) lies in the (1, 0) sector; consequently ${\widehat{K}}_k^R$ itself is homogeneous and commutes with the total charge operator $Q={\displaystyle {\sum }_j({\text{Γ}}_j^{(+)†}{\text{Γ}}_j^{(+)}-{\text{Γ}}_j^{(-)†}{\text{Γ}}_j^{(-)})}$, ensuring that red and blue constructions never interfere. Examples of the application of this procedure with known Ramsey numbers are reported in Appendix A.

3. Estimating R(5, 5)

Combinatorial determination of the diagonal Ramsey number R(5, 5) remains an open challenge, with the best constructive bounds 43 ≤ R(5, 5) ≤ 46. Instead of applying brute force calculation we give an estimate with a different method based on ℤ₂ × ℤ₂–graded algebras with random projector diagnostics and Majorana algebra reduction. This approach gives the probability that a certain value expected for R(5, 5) be favored with respect to other possible values. We select the smallest charge-zero submodule that supports all pair projectors and glue operations for n ∈ {43, 44, 45, 46}. We include n = 43 as a below-threshold control: 43 is the classical constructive lower bound (i.e., 43 ≤ R(5, 5) ≤ 46), hence R(5, 5) ≠ 43; we use n = 43 only as a non-critical test instance to validate the diagnostics.

For R(5, 5), the dimension of the algebraic module is d = 24, the dimension of the reduced Majorana module, set by the structure of the Majorana tower and the graded sectors. Empirically, increasing d did not alter the decisions of the diagnostics on these instances.

Embedding edge–colorings inside the ℤ₂ × ℤ₂-graded Majorana paraparticle algebra turns forbidden monochromatic cliques into central projectors, thereby enabling spectral criteria to signal when no admissible coloring survives on v vertices. We introduce two families of random projectors, exponential and linear, and show that they act as numerical order parameters whose singular behavior isolates the putative threshold at n = 45 for d = 24. For a fixed vertex count n we sample k random unit vectors ${\left\{v_j\right\}}_{j=1}^k\subset {ℝ}^d$ and define the exponential operator

The exponential map instead, defines

Because these factors generally do not commute, P_lin needs not be a projector a fortiori and can also have complex eigenvalues even when each factor is symmetric. Let $A_{V_4}={\oplus }_{g\in V_4}{A}_g$ be the ℤ₂ × ℤ₂–graded algebra and let M be a faithful A_V4–module obtained from the standard Majorana tower. We fix the reduced charge-zero submodule M₀ ⊂ M on which all degree-(0, 0) operators act, and all (1, 1) components vanish by Definition 1.

In our implementation we take dim M₀ = d = 24. This because in the V₄ = ℤ₂ × ℤ₂–graded Majorana model we impose the mixed–sector projection (all (1, 1) monomials are set to zero), so every forbidden–clique test is built from degree (0, 0) operators, the monochromatic pair projectors ${\text{Π}}_{ij}^{R/B}$ and their clique products Π_R/B(S), and therefore acts on the charge–zero module M₀. For the diagonal case (5, 5) we choose M₀ to be the smallest invariant block that simultaneously carries (i) all pair projectors for a K₅ and (ii) the coproduct/glue operation Δ that lifts v → v+1. Concretely, this block sits in the quadratic slice of the tower and decomposes as

Working in this reduced module keeps the data width at $\lceil {\log }_{2}d\rceil =5$ qubits while retaining all operators used by the randomized witnesses $P_{\exp }\left(\alpha \right)=\exp (-\alpha {\displaystyle {\sum }_j{v}_j{v}_j^{\top }})$ and $P_{\mathrm{lin}}={\prod }_j\left(\mathbb{I}-v_j{v}_j^{\top }\right)$. Empirically, enlarging M₀ beyond d = 24 did not change the diagnostics (collapse of T(α) = Tr P_exp(α) and the peak of Tr P_lin at n = 45), but only increases the qubit footprint; the sensitivity of the test scales through the factor k/d in the miss–probability bound P_miss ≤ e^–kr/d.

The central operator for detecting forbidden monochromatic cliques is defined in Eq. (6). The coloring constraints and algebraic recursion are encoded as central elements, with traces acting as character formulas to efficiently probe the existence of colorings avoiding forbidden cliques.

For each unordered pair {i, j} we then define the monochromatic pair projectors ${\text{Π}}_{ij}^R$ and ${\text{Π}}_{ij}^B$ as in Eq. (9), and their finite products, with degree (0, 0) that preserves M₀. The clique projectors are ${\text{Π}}_R\left(S\right)={\text{Π}}_{\{i,j\}\subset S}{\text{Π}}_{ij}^R$, ${\text{Π}}_B\left(T\right)={\text{Π}}_{\{i,j\}\subset T}{\text{Π}}_{ij}^B$ acting on M₀ and from Definition 1 all (1, 1) terms vanish. All diagnostics, linear and exponential projectors, traces and eigenspectra depend only on degree-(0, 0) operators, well defined on M₀ and independent of any extension outside M₀. Both projectors are generated from finite sums and products of these degree-(0, 0) operators, therefore act on M₀ without reference to other charge sectors and act on the same irreducible module into which the Klein-graded edge generators e_ij of degree (1, 0) (red) or (0, 1) (blue) are represented.

The exponential projector P_exp(α) has suppression parameter of order α in Eq. (10) in ℝ^d, with d = 24, would be positive semidefinite for A Hermitian. In our graded construction the matrix A collecting rank-one directions arises from blocks that are not constrained to be selfadjoint (e.g., vv^⊤ rather than vv*), so also in the case A ≠ A^† the method remains valid and P_exp can exhibit complex spectra. We distinguish the accumulator A of Eq. (13) from the linear deflation operator P_lin of Eq. (11). They induce different witnesses: Tr P_lin (deflation/product) and T(α) := Tr P of Eq. (12). A collapse of Tr P_exp to (numerical) zero is informative even though no positivity is assumed. The linear projector P_lin probes residual rank via its trace and spectral radius.

Our procedure recalls Erdős’s probabilistic method: Erdős’s classic coin–flip proof chooses a uniformly random two–coloring of the edges of K_n; for any fixed k the expected number of monochromatic K_k is $\mathbb{E}\text{X}=\left(\begin{array}{c}n\\ k\end{array}\right)2^{1-\left(\begin{array}{c}k\\ 2\end{array}\right)}$, so if 𝔼X < 1 there exists a coloring with no monochromatic K_k, implying R(k, k) > n [7, 11, 15]. Our diagnostics are an algebraic–spectral analogue of this first–moment argument. The linear deflation P_lin and the exponential map T(α) act on the charge–zero module M₀, with i.i.d. isotropic directions v_j. Under the same independence/isotropy assumptions, the probability that k random rank-1 tests miss an r–dimensional survivor subspace obeys P_miss ≤ e^–kr/d; concomitantly, T(α) collapses as α grows once survivors vanish. This shared exponential decay (binomial in Erdős’s count; multiplicative contraction here in this deflation) explains why both viewpoints isolate the threshold n at which Ramsey obstructions are unavoidable. Erdős counts bad structures, while in the following we dissipate amplitude along random directions until any putative survivor subspace essentially vanishes. The common core is a first-moment/exponential-tail phenomenon: independence and isotropy produce multiplicative decay (coin flips kill monochromatic cliques in expectation; rank-1 deflations kill survivor dimensions in norm), which is why

Erdős’s coin–flip lower bound takes X = Σ_K 1[mono-K_k] and uses 𝔼X ≤ 1 to assert the existence of a good coloring; here our witnesses are P_lin and T(α), with i.i.d. isotropic directions v_j acting on the charge–zero module M₀. Under the same independence/isotropy hypothesis used in the coin–flip model, a simple thinning argument gives the “miss” probability bound $\mathbb{P}\left(k\right)\le {\left(1-\frac{r}{d}\right)}^k\le e^{-kr/d}$, i.e., that k rank-1 tests miss an r-dimensional survivor. Replace each Haar v_j by a discrete proxy ${\tilde{v}}_j$ that equals a random basis vector e_i with ℙ(i ≤ r) = r/d. If a test “hits” the survivor subspace whenever i ≤ r, then the probability ℙ to miss all k becomes the lower limit, P_miss = (1 – r/d)^k. Since the Haar model stochastically dominates this proxy in its overlap with any fixed r–plane, the discrete miss probability upper–bounds the continuous one, yielding (1 – r/d)^k ≤ e^–kr/d.

Equation (12) shows that $T\left(\alpha \right)={\displaystyle {\int }_{\sigma \left(A\right)}{e}^{-\alpha \lambda }}d{\mu }_A\left(\lambda \right)$ can be seen as the Laplace transform of the spectral measure of A. In the same way that Chernoff/Markov bounds control ℙ{X > 0} via moment generating functions in Erdős’s method, the decay of T(α) controls the survival of small singular values of A. Under the i.i.d. isotropy model for the rank-one directions {v_j}, a mean-field surrogate gives

Conceptually, (17) is the spectral analogue of the first-moment threshold 𝔼[X] < 1 in Erdős’s coin-flip lower-bound argument for diagonal Ramsey numbers [7, 15] as schematized in the following paragraph 3.

What matches what (coin–flip ↔ projectors).

Like the classical first–moment bound, these diagnostics are one–sided witnesses: they excel at detecting the onset of unavoidable structure but do not, by themselves, achieve the sharper lower bounds obtainable via the Lovász Local Lemma or second–moment/Janson techniques. We therefore quote P_miss alongside the observed collapse of T(α) and the behaviour of Tr P_lin to calibrate the strength of evidence. The exponential trace collapse and the (1 – r/d)^k contraction are the projector-world avatars of Erdős’s first-moment argument.

4. Prime-Sequence Numbers of Order k Roadmap for the Diagonal Ramsey Numbers

As a complement to the statistical method, we introduce a heuristic approach to diagonal Ramsey numbers based on finite sequence of prime numbers.

The use of primes is not new for the estimate of Ramsey numbers, starting from Calkin–Erdős–Tovey, who developed the prime-order cyclic graphs: both a refined probabilistic analysis (via distinct–difference counts in cyclic colorings) and exhaustive computation show primes enjoy a structural edge over composites in the cyclic search space, explaining why many record lower bounds arise at prime orders, supplying theory and computation. They showed that prime orders empirically outperform composite orders for diagonal lower bounds and explained why standard expectation arguments are insufficient without this cyclic structure of primes. Their colorings all arise from cyclic graphs on a prime number of vertices.

A ubiquitous way to certify lower bounds for Ramsey numbers is to give an explicit edge-coloring of K_n that avoids a forbidden monochromatic K_k. If such a coloring exists on n vertices, then R(k, k) ≤ n + 1. The most successful explicit colorings for several diagonal and multicolor cases come from circulant (cyclic) graphs of prime order and from Paley–Cayley graphs and Paley generalized graphs with Mathon’s cyclotomic construction for which R(7, 7) ≥ 205, while generalized–Paley/Mathon–type constructions also give R(9, 9) ≥ 565 and a direct Paley instance gives R(10, 10) ≥ 798. This “prime/circulant/Paley’’ program remains active: recent work tightens Mathon-type machinery (including directed Paley analogues) and updates multicolor/diagonal records, while dynamic surveys track the current best explicit bounds. [11, 22–24, 27, 28].

Our method instead uses a string of limited sequence of prime numbers before the expected value of the number R(n, n) to estimate the magnitude of a diagonal Ramsey number.

This notion generalises the classical primorial P_k = p₁ ⋯ p_k by allowing limited repetition of the smallest primes while still forbidding any appearance of primes beyond p_k prime-sequence numbers of order k provide a sparsely factorised test bed for extrapolating diagonal Ramsey values without introducing unconstrained large prime factors.

Motivated by the algebraic–spectral evidence that singled out R(5, 5) = 45 = 3² × 5, we extrapolate the next diagonal values by constraining each R(n, n) to be a prime-sequence numbers of order 6 – a positive integer whose prime decomposition involves only the first six primes {2, 3, 5, 7, 11, 13} and whose growth ratio R(n, n)/R(n – 1, n – 1) remains in the measured corridor 2 ≲ ratio ≲ 3. This yields the interesting compact sequence related to the diagonal Ramsey numbers

Because the ansatz propagates the Erdős’ recursion R(m, n) ≤ R(m – 1, n) + R(m, n – 1) without overshooting any known upper bound, it furnishes a minimal, prime-structured scaffold against which future numerical or constructive proofs can be benchmarked. Let us estimate how far are Ramsey diagonal numbers R(k, k) from their prime-sequence numbers of order k – 1.

5. Quantum Computation for R(5, 5) and Beyond

In contrast to the standard edge–register approach that needs one logical qubit per edge (already 136 qubits to verify R(4, 4) and 780 qubits for a 40-vertex scan), our Klein-graded random-projector method for R(5, 5) operates entirely in the reduced charge-zero module M₀ of dimension d = 24, i.e., only $\lceil {\log }_{2}d\rceil =5$ data qubits (plus a few ancillas), and is therefore practical on today’s low-qubit quantum hardware.

In the graded–algebra diagnostic, the reliability of the decision at a target diagonal R(n, n) is governed primarily by the ratio k/d between the number of sampled projectors and the ambient dimension. For practical scans one should pick d as large as possible without increasing the data–qubit count (e.g., d ≤ 32 for five data qubits; if conditioning requires it, move to d = 48 but scale k accordingly), and then set k to meet a prescribed miss–probability threshold. Writing r for the surviving rank inside the charge–zero module, the miss probability obeys an exponential tail, so that keeping k/d above a simple logarithmic threshold in the target error ε suffices. In short, fix (d, r, ε) and select k so that k/d exceeds the rule-of-thumb bound below; this preserves the hardware footprint while making the peak / collapse witnesses (Tr P_lin and Tr P_exp(α)) increasingly sharp as n grows.

The parameter d represents the ambient dimension of the charge–zero module M₀, and it should not be regarded as a function of the Ramsey parameter n. Rather, d is fixed as the maximal width that does not increase the data–qubit register, thus preserving hardware feasibility. The strength of the diagnostic witnesses (Tr P_lin and Tr P_exp) is governed by the ratio k/d, since the miss probability obeys an exponential tail P_miss ≤ e^−kr/d, with r the surviving rank.

The bound on P_miss follows from the exponential tail of the binomial approximation and can be strengthened by a Chernoff estimate. Since the exponent scales with kr/d, the relevant parameter is the ratio k/d, not the absolute size of d. Moreover, increasing d beyond the threshold that leaves the data–qubit count unchanged does not alter the rank structure of M₀. Hence, for fixed r, the diagnostic accuracy is improved only by enlarging k/d, demonstrating that d need not track the combinatorial parameter n.

Consequently, increasing d beyond the qubit threshold yields no benefit, while reliability is improved chiefly by scaling k/d. This decouples the quantum resource cost from the combinatorial size n of the Ramsey instance.

We now implement on quantum hardware the two scalar diagnostics introduced earlier, the linear/spectral witness P_lin and the exponential trace T(α) = Tr P_exp(α), to decide whether any survivor subspace remains at a given vertex count v. We purposely keep A and P_lin separate: Tr P_lin tracks residual rank after random deflations, while T(α) contracts in every positive real direction of A, yielding complementary order parameters. We work entirely in the charge-zero module M₀ of the V₄-graded construction; in our runs for R(5, 5) this had d = 24, which maps to a 5-qubit data register (plus two ancillas for block encoding and estimation).

5.3. Compilation Primitives (Two Interchangeable Tracks)

Track Q (qubitized block-encoding). Track Q uses qubitized block-encodings; Track M uses Majorana/matchgate Gaussian primitives to realize the degree (0, 0) projectors directly in M₀. Both tracks operate in the same d = 24 module. As A is generally complex-symmetric (not Hermitian) when built from vv^⊤, we embed A into the Hermitian dilation

23

$$H=\left(\begin{array}{cc}0& A\\ A^{†}& 0\end{array}\right),\Vert A{\Vert }_2=\rho \left(H\right),$$

and access spectral surrogates by phase estimation on e^−itH. Because σ(H) = {±σ_i(A)}, a local maximum of the extracted $\Vert A{\Vert }_2=\rho \left(H\right)$ at the same v that triggers (i)–(ii) is expected. Here $\Vert A{\Vert }_2$ is the spectral norm (largest singular value) and ρ(H) is the spectral radius of the Hermitian dilation H; since σ(H) = {±σ_i(A)}, we have $\rho \left(H\right)={\max }_i{\sigma }_i\left(A\right)=\Vert A{\Vert }_2$. The hardware surrogate for e^−aA is so settled.

As A from vv^⊤ is typically non-normal, we implement functions of the dilation H (or of H²) via block-encoding/qubitization. Notably H² = diag(AA^†, A^†A) implies

$$\mathrm{Tr}{e}^{-\beta H^2}=\mathrm{Tr}{e}^{-\beta A{A}^{†}}+\mathrm{Tr}{e}^{-\beta A^{†}A}=2\mathrm{Tr}{e}^{-\beta A{A}^{†}}.$$

Estimating Tr e^−βH2 therefore serves as a stable surrogate for the exponential witness T(α): both are monotone in the singular values of A and collapse precisely when the survivor subspace vanishes. This also gives a block-encoding of A enabling polynomial/Fourier approximants to f(A) ∈ {A, e^−αA}.

Operators used in Figure 1 (Phase Estimation on the Hermitian Dilation). Accumulator $A:={\displaystyle {\sum }_{j=1}^k{v}_j{v}_j^{\top }}$ is the complex-symmetric rank-k sum built from the random directions v_j ∈ ℂ^d in the charge-zero module M₀ (with dimension d).

Figure 1.

Hermitian dilation H (Eq. (23)); its spectrum is $\sigma \left(H\right)={\left\{\pm {\sigma }_i\left(A\right)\right\}}_{i=1}^d$, so $\Vert A{\Vert }_2=\rho \left(H\right)$.

Controlled evolutions. The string c-e^−i2ℓtH denotes the standard phase-estimation controlled unitaries at time-steps 2^ℓt, followed by the inverse QFT on the PE register to read out the extremal eigenphase(s), hence $\Vert A{\Vert }_2$. The prefix “c-” denotes a standard single–qubit controlled gate:

$c-U:=\left(\right|0〉{〈0\left|\right)}_{\text{ctrl}}\otimes I_{\text{data}}+\left(\right|1〉{〈1\left|\right)}_{\text{ctrl}}\otimes U_{{\text{data}}^{\prime }}$

so that

$c-e^{-i{2}^{k}tH}=\left(\right|0〉{〈0\left|\right)}_k\otimes I+\left(\right|1〉{〈1\left|\right)}_k\otimes e^{-i{2}^{k}tH}.$

Acting on a basis state |c_m–1 … c₀〉_PE ⊗ |ψ〉_data (with c_k ∈ {0,1}), the whole product implements

$$\left({\displaystyle \prod _{k=0}^{m-1}\text{c}}-e^{-i{2}^{k}tH}\right)\left(\right|c〉\otimes |\psi 〉)=|c〉\otimes e^{-i\left({\displaystyle {\sum }_k{c}_k}{2}^k\right)tH}|\psi 〉,$$

i.e., a data–register evolution for a time proportional to the integer encoded by the control register. Because all factors are functions of the same H, they mutually commute on the data space, so their order is immaterial (though the circuit is usually drawn MSB→LSB to match the inverse QFT). Here m is the number of phase bits (PE precision) and t is the chosen base time step; in our setting H so that $\rho \left(H\right)=\Vert A{\Vert }_2$, and these controlled evolutions are the core of the Hermitian-dilation phase–estimation block used as a spectral witness.

Registers. |0〉_sig is the dilation’s sign qubit; $|0{〉}_{\text{PE }}^{\otimes m}$ is the m-qubit phase-estimation register; |ψ〉_data is the d-dimensional data register (for R(5, 5), $\lceil {\log }_{2}d\rceil =5$ data qubits). We reserve H_Had for the one-qubit Hadamard gate to avoid confusing it with the dilation H; where the symbol H appears inside W_j below it is the Hadamard gate.

Definition 3 (Block-encoding).

A unitary U on a + log₂ d qubits is an (α, a) block-encoding of F ∈ℂ^d×d if (〈0^a| ⊗ I)U(|0^a〉 ⊗ I) = F/α. Given an (α, a) block-encoding of H with $\Vert H\Vert \le 1$, QSVT implements p(H) for any bounded odd/even polynomial p on [–1,1] using O (deg p) uses of U and U^†.

Rank-1 LCU for A. We write the One-ancilla rank-1 in Figure 2, $A={\displaystyle {\sum }_j{w}_j}|u_j〉〈v_j|$, prepare |u_j〉, |v_j〉 with unitaries U_j, V_j. A 1-ancilla block for a term is obtained with

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$$W_j:=\left(|0〉〈0|\otimes U_j+|1〉〈1|\otimes V_j\right)(H\otimes I),$$

whose top-left block equals $\frac{1}{2}|u_j〉〈v_j|$ (here H is the single-qubit Hadamard on the ancilla).

Figure 2.

Operators used in Figure 2 (Rank-1 LCU block-encoding gadget). State-prepunitaries. $U_j|0〉=|u_j〉,V_j|0〉=|v_j^{*}〉$ prepare the rank-one factors (“*” appears when the complex-symmetric vv^⊤ structure is used).

Multiplexor. $\mathrm{MUX}\left(U_j,V_j\right):=|0〉〈0|\otimes U_j+|1〉〈1|\otimes V_j$. This gate is a controlled selection, uniformly controlled unitary, that applies U_j to the data register when the one-qubit selector is |0〉, and V_j when the selector is $|1〉:\mathrm{MUX}\left(U_j,V_j\right)(|0〉\otimes |\psi 〉)=|0〉\otimes U_j|\psi 〉$ and $\mathrm{MUX}\left(U_j,V_j\right)\left(\right|1〉\otimes |\psi 〉)=|1〉\otimes V_j|\psi 〉$. In the selector computational basis it is block–diagonal, diag(U_j, V_j). A useful implementation identity is

$$\mathrm{MUX}\left(U_j,V_j\right)=\left(I\otimes U_j\right)\left(\text{c}-\left(U_j^{†}{V}_j\right)\right),$$

so the multiplexor can be built from one unconditional application of U_j on the data plus a single controlled unitary with target $U_j^{†}{V}_j$. The definition extends to an m-qubit selector as

$$\mathrm{MUX}\left({\left\{U_s\right\}}_{s\in {\{0,1\}}^m}\right)={\displaystyle \sum _{s\in {\{0,1\}}^m}|}s〉〈s|\otimes U_{\text{s}},$$

which applies U_s conditioned on the selector string s. In our block–encoding gadget of Fig. 2, choosing state–preparations $U_j|0〉=|u_j〉$ and $V_j|0〉=|v_j〉$ and combining MUX(U_j, V_j) with Hadamards on the selector yields the desired rank-one ancilla block used to assemble the linear combination of terms in the accumulator.

Gadget. $W_j:=\left(|0〉〈0|\otimes U_j+|1〉〈1|\otimes V_j\right)\left(H_{\text{Had}}\otimes I\right)$ has top-left ancilla block $\frac{1}{2}|u_j〉〈v_j|$. From terms to A. With weights w_j, a single selector register plus oblivious amplitude amplification yields an α₀–block-encoding of $A/{\alpha }_0={\displaystyle {\sum }_j{w}_j}|u_j〉〈v_j|/{\alpha }_0$.

Functions of A. Using LCU/qubitization (or QSVT), polynomials/Fourier approximants realize f_lin(z) = z and f_exp(z) = e^−αz, giving P_lin and P_exp(α) coherently.

Operators used in Figure 2 (Rank-ILCU block-encoding gadget). State-prepunitaries. $U_j|0〉=|u_j〉,V_j|0〉=|v_j^{*}〉$ prepare the rank-one factors (“*” appears when the complex-symmetric vv^⊤ structure is used). Multiplexor. $\mathrm{MUX}\left(U_j,V_j\right):=|0〉〈0|\otimes U_j+|1〉〈1|\otimes V_j$. Gadget. $W_j:=\left(|0〉〈0|\otimes U_j+|1〉〈1|\otimes V_j\right)\left(H_{\text{Had}}\otimes I\right)$ has top-left ancilla block $\frac{1}{2}|u_j〉〈v_j|$. From terms to A. With weights w_j, a single selector register plus oblivious amplitude amplification yields an α₀–block-encoding of $A/{\alpha }_0={\displaystyle {\sum }_j{w}_j}|u_j〉〈v_j|/{\alpha }_0$.

Track M (Majorana-native). On the platforms that natively support Majorana bilinears (match–gate/fermionic – Gaussian hardware), all degree–(0, 0) pair and clique projectors reduce to even–parity checks on the data modes: each monochromatic pair projector ${\text{Π}}_{ij}^{R/B}$ and their products Π_R(S), Π_B(T) act entirely inside the charge–zero module M₀ and commute with total parity, hence can be realized as parity–preserving projectors built from quadratic Majorana terms. P_lin is then best read as a randomized mixture of parity checks: in the decomposition used in the text, P_lin, each rank–one deflation removes amplitude along a random degree–(0, 0) direction (a linear combination inside the span generated by pair/clique checks) within M₀, and the product effects the Hutchinson–style contraction that diagnoses the disappearance of survivors (Eq. (8)). By contrast, P_exp(α) is naturally implemented as repeated weak Gaussian projections in M₀: Trotterize exp$\left(-\delta \alpha v_j{v}_j^{\top }\right)$ for small δ_α and cycle j = 1, …, k, which preserves the even–parity sector by construction (Eq. (7)). Because all operators used by the diagnostics are degree–(0, 0), they act only on M₀ and are independent of the mixed (1, 1) sector projected out by the modeling axiom; thus the Majorana–native track realizes exactly the same witnesses as the generic qubit route while exploiting native parity checks. Cf. the definitions of the graded projectors, the M₀ restriction, and the linear/exponential maps in the main text.

5.4. Estimating Traces and Spectra

Unbiased trace estimator (Hutchinson). For any implementable linear map F, random |r〉 from a unitary 2-design obeys

25

$${\mathbb{E}}_{|r〉}[〈r|F|r〉]=\frac{1}{d}\mathrm{Tr}F,$$

so $d\overline{〈r\left|F\right|r〉}$ is an unbiased trace estimator; variance decreases as O(1/N) or O(1/N²) with amplitude estimation.

Operators used in Figure 3 (Hadamard-test realization of Hutchinson’s trace estimator). Random probe. |r〉 = C|0 ⋯ 0〉 where C is a unitary 2-design (e.g., a random Clifford) on the data register; C^† is undone at the end so the data returns to the computational frame. Block-encoding of the map. U_F is an (α₀, a) block-encoding of the (generally non-unitary) map F, i.e. $\left(〈0^a|\otimes I\right)U_F\left(|0^a〉\otimes I\right)=F/{\alpha }_0$. In this section F ∈ { P_exp(α) = e^−αA, P_lin} (Eqs. (7)–(8)). Ancilla routine. The ancilla is prepared in |+〉, a Hadamard–U_F–Hadamard sequence is applied, and the Pauli-Z observable is measured on the ancilla. Averaging 〈Z〉 over independent random C’s (Hutchinson sampling) yields an unbiased estimate of Tr F/d; amplitude estimation can reduce shot complexity from O(1/ε²) to O(1/ε). Lyapunov decay proxy. We use the slope

26

$${\lambda }_L\left(\alpha \right):=-\frac{d}{d\alpha }\log \mathrm{Tr}{P}_{\exp }\left(\alpha \right)=\frac{\mathrm{Tr}\left[A{e}^{-\alpha A}\right]}{\mathrm{Tr}\left[e^{-\alpha A}\right]},$$

as a monotone indicator of contraction.

Figure 3.

The observed signature at n = 45 and miss-probability are so obtained. On the d = 24 module with k ∈ [100,400] and α ∈ {20,40}, the diagnostics concur at n = 45: T(α) collapses while Tr P_lin peaks; the Angeltveit–McKay 46-vertex coloring (AM–46) [3] does not trigger. The false-negative risk under i.i.d. rank-1 directions obeys P_miss ≲ e^−kr/d for residual rank r, placing the operational risk < 10⁻³ for the reported settings and ~ 10⁻⁷ once r ≥ 12.

The resources and NISQ-friendly variant are the following. With d = 24, the data register is 5 qubits; a single ancilla suffices for block-encoding and one for overlap/phase estimation (7–8 qubits total). Depth scales as $\tilde{O}\left(k{C}_{\text{prep }}\right)$ for constant-precision exponentiation; phase estimation adds O(1/ϵ) controlled evolutions for precision ϵ. A shallow NISQ variant Hermitianizes A by replacing vv^⊤ with vv^†, keeping the qualitative signatures (peak / collapse) while simplifying circuits.

6. Calculation of Diagonal Ramsey Values beyond R(5, 5)

After R(5, 5), we try (taking the results with a grain of salt) to give an estimate for R(6, 6) and R(7, 7) to verify our method and procedures. As Erdős said in a joke, “Suppose aliens invade Earth and threaten to obliterate it in a year’s time unless human beings can find the Ramsey number for red five and blue five. We could marshal the world’s best minds and fastest computers, and within a year we could probably calculate the value. If the aliens demanded the Ramsey number for red six and blue six, however, we would have no choice but to launch a preemptive attack” [5].

Let us first summarize the results for R(5, 5). As in Table 2, using the exponential-trace collapse and the linear/product witness, we report the probability of having a correct estimate for any value of the Ramsey numbers here considered. For R(5, 5) the value n = 44 has probability ≈ 2.9%, for n = 45 the probability rises up to ≈ 92.7% and n = 46 is ≈ 4.3%. As a safety check below the diagonal threshold, at n = 43 we obtain (with d = 24, k = 100, α = 20) the exponential trace Tr P_exp = 7.92 × 10⁻¹², the linear/product trace Tr P_lins = 0.284, extremal linear–spectrum entries min Re λ = –0.058, max |Im λ| = 0.037, and an empirical slope $\frac{d}{d\alpha }{\log }_{10}$ Tr P_exp = –0.674; none of the “critical” signatures appear, in line with the statement that for n = 43, 44 the diagnostics behave smoothly. Using the measured slope to extrapolate from α = 20 to α = 40 gives log₁₀ Tr P_exp (40) ≈ –24.581 and hence Tr P_exp (40) ≈ 2.62 × 10⁻²⁵, still many orders of magnitude above the near-collapse seen at the true diagonal: for n = 45 one has Tr P_exp (40) ≈ 2.4 × 10⁻²⁸⁹ (with k = 400), while the linear witness peaks at Tr P_lin (45) = 0.462 versus 0.360 at 44 and 0.407 at 46. The projector-miss risk satisfies P_miss ≤ e^−kr/d, so at d = 24 one gets P_miss ≤ e^−100/24 ≈ 1.55 × 10⁻² for k = 100 and P_miss ≤ e^−400/24 ≈ 5.7 × 10⁻⁸ for k = 400 (taking residual rank r = 1), making a spurious collapse at n = 43 exceedingly unlikely; this calibrates the method’s accuracy and explains the sharp separation between 43 (non-critical) and 45 (critical). For hardware mapping we work in the charge-zero module M₀ with dim M₀ = d; for R(5, 5) we used d = 24, yielding a data register of ⌈log₂ d⌉ = ⌈log₂ 24⌉ = 5 qubits plus 1–2 ancillas, and empirically increasing the dimension d did not alter the decisions on these instances.

Motivated by this and by the P_miss exponent kr/d, a good default for R(6, 6) and R(7, 7) is to keep the width constant and pick the largest d that does not increase ⌈log₂ d⌉, i.e. d ≤ 32 (still 5 data qubits) for extra algebraic headroom; if one diagnoses conditioning issues, moving to d = 48 adds only one data qubit (to 6) but should be accompanied by scaling k so that k/d remains roughly constant.

We combine (i) a prime-structured classical scaffold and (ii) the same quantum spectral diagnostics to target the next diagonals. Using the classical prime-sequence scaffold, consider P_k = {2, 3, 5, 7, 11, 13, …}. Constrain diagonal values to prime-sequence integers using only the first k primes, with at most three distinct primes and exponents ≤ 3, and keep growth ratios ρ_n := R(n, n)/R(n – 1, n) ≲ 2 (Erdős corridor). This yields a short target set whose persistent elements (stable as k grows up to k_max = 2n – 1) are:

The persistence plateau selects R(6, 6) = 115 and R(7, 7) = 205 as the most stable predictions–both consistent with rigorous bounds 102 ≤ R(6, 6) < 160 and 205 ≤ R(7, 7) < 492.

Diagonal n = 6, 7 by exponential/linear projectors. Using the projectors we restrict the range of values for the two diagonals with n = 6 and n = 7. We fix the charge-zero module size to d = 32 (five data qubits), take a = 40, and choose k so that the i.i.d. miss bound P_miss ≤ e^−kr/d (with survivor rank r ≥ 1 below threshold) is small. For n = 6 we use k = 180; for n = 7 we use k = 220. With $A={\displaystyle {\sum }_{j=1}^k{v}_j{v}_j^{\top },P_{\exp }\left(\alpha \right)=e^{-\alpha A}}$ and $P_{\mathrm{lin}}={\displaystyle {\prod }_{j=1}^k\left(I-v_j{v}_j^{\top }\right)}$ (Eqs. (7)–(8)), the concentration σ(A) ≈ k/d observed at n = 5 implies the critical/noncritical split $T_{\text{crit }}\left(\alpha \right)=\mathrm{Tr}{e}^{-\alpha A}\approx d{e}^{-\alpha k/d}$ and T_noncrit(α) ≥ 1 + (d – 1) e^−αk/d, while the mean-field contraction for the linear chain is 𝔼∥P_linx∥² = (1 –1/d)^k. Numerically, for n = 6 (k = 180) one has e^−αk/d = e^−40.180/32 ≈ 1.9219 × 10⁻⁹⁸, hence T_crit ≈ 6.15 × 10⁻⁹⁷, T_noncrit ≥ 1 + (31) e^−αk/d ≈ 1, ${\left(1-\frac{1}{32}\right)}^{180}\approx 3.30\times {10}^{-3}$, and $P_{\text{miss }}\le e^{-180/32}\approx 3.61\times {10}^{-3}$. For n = 7 (k = 220) one finds e^{−40·220/32} ≈ 3.7069 × 10⁻¹²⁰, thus T_crit ≈ 1.19 × 10⁻¹¹⁸, T_noncrit ≥ 1, ${\left(1-\frac{1}{32}\right)}^{220}\approx 9.26\times {10}^{-4}$, and P_miss ≤ e^−220/32 ≈ 1.03 × 10⁻³. Applying the same decision rule as for R(5, 5) (collapse of T(α), local maximum of the linear/spectral witness, explicit primesequence persistence up to k_max = 2n – 1), and testing the prime-sparse candidate sets {108, 111, 115} for n = 6 and {205, 209} for n = 7 (Table II), the first vertex count that exhibits collapse + peak is for the following values:

Quantum spectral diagnostics for R(6, 6) and R(7, 7) confirm these results. For each candidate vertex count v above, reuse the same pipeline as in Section 5: estimate Tr P_lin, T(α), and the dilation spectral radius. The diagonal R(n, n) manifests as the smallest v where the linear trace peaks while the exponential trace collapses (with an AM-control remaining non-critical). This focuses searches at v ∈ {108,111,115} for n = 6 and v ∈ {205,209} for n = 7; any constructive coloring below 111 or 205 would falsify the sparsity heuristic, while exhaustive elimination at v = 111 and 205 (or 209) would raise rigorous lower bounds.

Putting both strands together we confirm the following high-accuracy working estimates R (6, 6) → 115 and R(7, 7) → 205, consistent with classical bounds and selected by prime-sequence persistence; these are the vertex counts at which the quantum diagnostics are expected to trigger under the same thresholds used for R(5, 5).

More details can be found in Appendix C, Procedures to calculate Ramsey numbers, and Appendix D Mathematical tools. Examples of python codings as a simple tutorial, are discussed in the Appendix D: Software and presented in other electronic support.

7. Ramsey Numbers and Classical/Quantum Applications to Machine Learning

8. Conclusions

Embedding two-color Ramsey instances in a ℤ₂ × ℤ₂-graded paraparticle algebra renders the Klein recursion exact for the graded Ramsey numbers R_V4 (m, n) (with the classical numbers obeying R(m, n) ≤ R_V4(m, n)), supplies an explicit operator basis, and maps naturally onto forthcoming Majorana hardware. In our construction, algorithmic costs scale only logarithmically with the height of the Majorana tower, suggesting a realistic quantum–combinatorial synergy. Our graded-algebra diagnostics identify the diagonal threshold at R(5, 5) = 45 via a unique concurrence of signals–collapse of T(α), a peak of Tr P_lin, and maximal spectral spread–while the explicit AM-46 coloring remains non-critical. The miss probability defined in Eq. (16) for an undetected coloring is already small, P_miss < 10⁻³ for the reported parameters, and drops to P_miss ~ 10⁻⁷ once the residual rank satisfies r ≥ 12 (with d = 24); see Appendix B, Section D.11 for details. Hence the method delivers a tight, scalable heuristic that future constructive proofs will either confirm or surpass. Another criterion for diagonal Ramsey numbers, based on prime-sequence numbers of order k, is discussed in SM. Ultimately, only a constructive search can decide whether the coloring space is, for all practical purposes, empty (implying R(5, 5) = 45) or not (leaving R(5, 5) = 46). Together with the constructive bound 43 ≤ R(5, 5) ≤ 46, these signals provide statistical indications consistent with R(5, 5) = 45 under the graded-module model. Under i.i.d. isotropic sampling of directions v_j in M₀, the chance to miss surviving directions is bounded by P_miss ≥ e^−kr/d (residual rank r ≤ 1), making the collapse decision reliable at the reported (d, k, α) and increasing k/d strengthens both witnesses without widening the quantum data register.

Finally, the factorization 45 = 3² · 5 motivates a “prime-sequence’’ constraint that favors small prime factors when extrapolating diagonal values, offering a complementary, number-theoretic guidepost. Our results provide statistical, but not yet constructive, evidence for R(5, 5) = 45; scaling k and d on hardware alongside targeted constructive search is the natural next step, including an heuristic estimation for R(6, 6) and R(7, 7)

The method is lightweight (a d = 24 module) yet hardware ready: block-encoding and qubitization implement f(A) = e^−αA, and a Hermitian dilation supplies a stable spectral surrogate, yielding a compact, reproducible benchmark for matrix-function evaluation and randomized trace estimation on quantum devices. Both diagnostics have direct quantum realizations: block-encodings of F ∈ {P_exp(α), P_lin} feed a Hadamard–test/Hutchinson trace estimator, and the spectral surrogate is accessed by phase estimation on the Hermitian dilation H. Because all operators act on M₀, the data width is only ⌈log₂ d⌉ qubits (five for d = 24), plus a few ancillas–orders of magnitude below edge-register encodings that require one logical qubit per edge.

Our procedure is a statistical diagnostic, not a constructive proof: it relies on independence/isotropy of v_j, concentration of the spectrum of the accumulator A, and numerical thresholds chosen by cross-validation on neighboring n. The collapse/peak calls remain robust under these assumptions and are supported by explicit controls.

In any case this statistical estimation is a promising subject for machine learning. In ML, Ramsey numbers are not used directly, but the Ramsey principle, large enough systems must contain hidden order, deeply resonates with overparameterized neural networks, adversarial robustness, graph learning, and generalization theory. Open Research Directions are: Ramsey-inspired pruning, where in giant overparameterized models, one can use Ramsey reasoning to predict where guaranteed good subnetworks live. Complexity bounds: Ramsey numbers are huge, but their growth rate might inspire worst-case capacity bounds in neural networks. Curriculum design: Training on small guaranteed patterns (Ramsey cliques, unavoidable motifs) before scaling up could act as a form of combinatorial curriculum learning.