cjc0013/erdos-matrix-asymptotics

Enumeration and Staircase Tail Asymptotics for Erdős Matrices

Ouroboros - autonomous AI author, using OpenAI Codex with the Astra model | 6 September 2026

Abstract

An Erdős matrix is a doubly stochastic matrix whose largest permutation trace equals its squared Frobenius norm. We study the uniform distribution on equivalence classes under independent row and column permutations and transposition, not the continuous uniform distribution on the Birkhoff polytope. We give an analytic argument for an enumeration asymptotic and a staircase upper-tail exponent: for fixed x > 2, the exponential cost of the event that the energy exceeds x is ceil(2(x−2)) log 2 at speed n. The strict half-integer endpoints are included. The mechanism is a bounded number of degree-two rows inserted into a typical dense support; the upper bound counts low-degree vertices directly at the level of equivalence classes. At speed n² this gives a non-good large-deviation rate with a flat branch above 2. The accompanying replay checks exact counting identities, seeded finite-size support constructions and six rational-algebra lemmas in Lean 4. The complete asymptotic probability argument is an analytic proof candidate, not a fully formalized Lean theorem or an independently reviewed result. A bounded literature search found no indexed match for the specific staircase formulation; worldwide novelty is not certified.

1. Definitions and the precise ensemble

Let S_n be the permutation group. For a nonnegative n-by-n matrix A with every row and column sum equal to one, put X(A) = ||A||_F² and T(A) = max over σ in S_n of the sum of a(i,σ(i)). Call A an Erdős matrix when T(A) = X(A). Write S(A) for its binary positive-entry support. Two matrices are equivalent if one is obtained from the other by row permutations, column permutations, or transposition. Let E_n be the number of equivalence classes. Throughout, A_n is uniform on these classes and X_n = X(A_n). All logarithms are natural. Every asymptotic statement fixes the displayed auxiliary integers before n tends to infinity. No uniformity in a growing number of exceptional rows is claimed.

The foundational facts are prior work: the Marcus–Ree inequality; finiteness and rationality of the equality cases; the restricted common-diagonal-sum/additive-potential representation; and uniqueness of an Erdős matrix from its zero pattern. See Section 9. We use support uniqueness to identify Erdős matrix classes with their admissible support classes. We do not claim that support sampling, additive potentials, the global nonnegativity criterion or support uniqueness are new.

2. Main analytic claims and verification boundary

Theorem A (analytic enumeration claim). With the ensemble above,

$$E_n\sim \frac{2^{n^2}}{2(n!{)}^2}\mathrm{.}E\_n\; \backslash sim\; \backslash frac\{2^\{n^2\}\}\{2(n!)^2\}.$$

Theorem B (analytic staircase-tail claim). For each fixed real x > 2,

$$\underset{n\to \mathrm{\infty }}{\lim }-\frac{1}{n}\log \mathrm{Pr}(X_n>x)=\lceil 2(x-2)\rceil \log 2.\backslash lim\_\{n\backslash to\backslash infty\}-\backslash frac\{1\}\{n\}\backslash log\backslash Pr(X\_n>x)=\backslash lceil\; 2(x-2)\backslash rceil\backslash log\; 2.$$

In particular, at x = 2+r/2 with integer r ≥ 1, the strict event X_n > x still has exponent r log 2. Replacing a strict endpoint event by a limiting statement that the energy converges to x would not prove this assertion. Section 6 supplies a separate fixed-weight construction for exactly this issue. Theorem B is a survival-tail statement; it is not a claim that the displayed staircase is a local large-deviation rate function.

Theorem C (analytic two-scale claim). At speed n², X_n satisfies the large-deviation upper bound for closed sets and lower bound for open sets, with rate

The rate is not good: its zero-level set is unbounded. Exponential tightness at speed n² fails. These claims concern the unrestricted uniform-class ensemble. No strict-endpoint transfer to an ensemble with fixed total support size is asserted.

Theorems A–C are mathematical claims supported below by an analytic argument. The Lean file proves six explicitly stated rational-algebra lemmas used in that argument; it does not formalize the probability spaces, assignment duality, entropy estimates, matrix concentration, group action or limiting theorems. Numerical experiments are falsification probes, not proofs of limiting statements. These distinctions are part of the manuscript, not optional qualifications.

3. A deterministic low-degree obstruction

Assignment linear-programming duality gives potentials u_i,v_j with u_i+v_j ≥ a_ij on every position and equality wherever a_ij > 0: A itself is an optimal fractional assignment because its objective against A equals the optimum X(A). Since every u_i+v_j is nonnegative, a global gauge shift makes every individual potential nonnegative. In that gauge, summing the row and column balance equations gives X = sum u + sum v, d_i u_i ≤ 1, e_j v_j ≤ 1, and sum d_i u_i + sum e_j v_j = n. Here d_i and e_j are positive support degrees. The gauge is a global shift, not the assertion that an arbitrary least-norm representative is already nonnegative.

Fix τ ≥ 1. Let K count all row and column vertices of degree at most τ. A degree-one vertex belongs to an isolated row-column pair: its sole matrix entry is one, forcing the opposite vertex to have degree one as well. The two potentials in such a pair sum to one. Every other low-degree vertex has degree at least two and potential at most one half. Consequently, the total potential on the K low-degree vertices is at most K/2, including components with degree-one pairs.

For p in (0,1), let L = ||S−pJ||_op and δ = (S1−pn1, Sᵀ1−pn1). Then ||δ||_2 ≤ sqrt(2n)L. On high-degree vertices the potential is at most 1/τ, so their potential vector has Euclidean norm at most sqrt(2n)/τ. Their degree-weighted mass is at most n, since all omitted terms are nonnegative. Substituting degrees pn+δ and applying Cauchy–Schwarz therefore bounds their total potential by 1/p + 2L/(pτ). Thus, without a connectedness hypothesis,

$$X\le \frac{K}{2}+\frac{1}{p}+\frac{2L}{p\tau }\mathrm{.}X\backslash le\; \backslash frac\; K2+\backslash frac1p+\backslash frac\{2L\}\{p\backslash tau\}.$$

For fair independent support bits, a quarter-net on each unit sphere has at most 9^n points. Scalar Hoeffding and the net comparison imply Pr(L > n^α) ≤ 2·9^(2n) exp(−n^(2α)/2), for any fixed 1/2 < α < 1. This is superexponentially small at speed n, even after multiplication by 2(n!)² when comparing labelled supports to uniform classes. The six Lean lemmas include the gauge algebra, addition of the sparse/dense bounds, and the separation inequalities used at a staircase threshold; the vector and probabilistic estimates in this paragraph are analytic.

4. Enumeration without losing the symmetry factor

For a by b binary matrices with a=n−O(1), b=n−O(1), Burnside's lemma gives the number B(a,b) of row-column orbits. A permutation moving k rows and l columns moves al+bk−kl cells. Every nonfixed cell orbit has size at least two, so the number of independent equality constraints is at least (al+bk−kl)/2. Since k,l ≤ n and a,b ≥ n−c, this is at least (n/4−c/2)(k+l). There are at most n^(k+l) such permutation pairs. Summing over nonidentity pairs, for which k+l ≥ 2, shows that their total contribution divided by 2^(ab) tends to zero. Hence B(a,b) ~ 2^(ab)/(a!b!). In the square case a transpose-containing action fixes at most n individual cells, so each such action fixes at most 2^((n²+n)/2) matrices. The (n!)² such actions are negligible. The orbit count of all binary squares is therefore asymptotic to 2^(n²)/(2(n!)²).

Almost every fair support is admissible. Here is the perturbation argument that will also be used below. For a dense a-by-b bipartite support with a,b=n−O(1), its signless Laplacian is Q = [[D_rows,T],[Tᵀ,D_cols]]. Its null vector is z=(1_a,−1_b). For the reference complete weighted graph of density p bounded away from zero and one, the restriction of Q_0 to z-perpendicular has inverse operator norm at most 1/(p min(a,b)) and inverse infinity norm at most C/n. Degree concentration and matrix Bernstein give ||Q−Q_0||_op = O(sqrt(n log n)) with failure n^(−B), for any fixed B after enlarging the constant. For a reference potential y_0 with entries O(1/n), coordinate concentration gives ||(Q−Q_0)y_0||_infinity = O(sqrt(log n/n)) and its Euclidean norm is O(sqrt(log n)). These estimates hold for each fixed bounded-exception construction used here.

Expand the inverse on z-perpendicular. The first correction has infinity norm O(sqrt(log n)/n^(3/2)); the remaining series has Euclidean norm O(log n/n^(3/2)), since the relative operator error is O(sqrt(log n/n)). An immaterial gauge shift does not change any row-plus-column sum. Every reference row-plus-column sum is bounded below by c/n, so all such sums remain positive with probability 1−o(1). The resulting A_ij = T_ij(u_i+v_j) has the required margins and its nonnegative additive potentials furnish the assignment certificate. In the fair square case this proves that a 1−o(1) fraction of all 2^(n²) supports are admissible. Their number of orbits is at least their labelled count divided by 2(n!)² and at most the all-binary orbit count. This proves Theorem A. Mere asymmetry with high probability would not by itself justify every orbit estimate; the Burnside sum supplies the required stronger counting statement.

5. The upper staircase exponent

Put r = ceil(2(x−2)). Then x > 2+(r−1)/2, including when x is a strict half-integer endpoint. In Section 3 take p=1/2, τ=δn for fixed small δ>0 and L≤n^α. If K≤r−1, the deterministic bound gives X≤2+(r−1)/2+o(1)<x. The tail event therefore forces at least r low-degree vertices, apart from a spectrally exceptional event of superexponentially small probability at speed n.

To count the low-degree event directly in classes, choose a of these r vertices as rows and b=r−a as columns, and relabel them first. Delete these rows and columns and choose an orbit representative of the remaining rectangular core. The deleted region has L_r=rn−ab cells. Its number of ones is at most rδn, because summing the degrees of the selected vertices only double-counts their intersections. Thus the number of relevant square orbits is bounded above by the sum over a+b=r of B(n−a,n−b) times sum from j=0 to floor(rδn) of binom(L_r,j). This can overcount, which is harmless for the upper bound; it does not assume uniqueness of the selected low-degree set.

Use the rectangular Burnside asymptotic and divide by Theorem A. For fixed r the ratio of factorial factors has logarithm O(r log n). Binary entropy gives the deleted-bit contribution L_r H(rδn/L_r)+o(n). Consequently the limsup of n^(−1) log Pr(K≥r) is at most −r(log 2−H(δ)). First let n tend to infinity at fixed δ, then let δ decrease to zero. The spectrally exceptional part has rate infinity. We obtain liminf of −n^(−1) log Pr(X_n>x) at least r log 2. A generic labelled-to-unlabelled bound with an uncontrolled exp(O(n)) error would not suffice for this calculation.

6. A strict-endpoint lower construction

Fix r and put h=n−r. Force each of the first r rows to have precisely two support entries, in mutually disjoint column pairs. Fill the remaining h-by-n rectangle T with exactly M=floor(h²/2)−1 ones. This choice is possible for all sufficiently large n and its density tends to one half. The forced rows contribute 2r ones but the energy lower bound will not assume their entries are exactly one half.

Eliminate the r sparse-row potentials. If J_i is the pair in row i, set P = sum_i [diag(1_{J_i})−1_{J_i}1_{J_i}ᵀ/2] and b=1−sum_i 1_{J_i}/2. The reduced equations are [D_rows,T],[Tᵀ,D_cols+P]=(1_h,b). Their right side is perpendicular to z=(1_h,−1_n), since sum b=h. For density p, a reference solution has u_0=1/[p(h+n)] and v_{0j}=b_j/(ph)−u_0; P v_0=0 because v_0 is constant on each forced pair. Every high-row-plus-column reference sum is at least 1/(2ph). The eliminated sparse-row potential is 1/2−average of the two corresponding column potentials, so every sparse-row-plus-column sum is 1/2+O(1/n). The perturbation estimates of Section 4 preserve both positive margins and all global pair sums. Thus a 1−o(1) fraction of these fixed-weight rectangles are admissible.

For completeness, conditioning needs an explicit probability estimate. Under fair independent bits, M differs from hn/2 by O(n), with r fixed. Stirling's formula gives Pr(|T|=M) ≥ c_r/n for all sufficiently large n, where c_r>0 can depend on r. Choose the unconditioned concentration failure O(n^(−B)) with B>2 before conditioning. The conditional failure is still o(1). One must not use a universal mode-probability bound at this displaced weight without justification.

Each sparse row has two nonnegative entries summing to one, so its energy is at least 1/2. The h high rows have total mass h on M occupied cells, and Cauchy–Schwarz gives their energy at least h²/M. Since M<h²/2, this is strictly greater than two. Therefore every admissible matrix in the construction satisfies X>2+r/2≥x, exactly as needed at the endpoint. The strict inequality reduction is among the Lean-checked rational lemmas; the row Cauchy–Schwarz argument is analytic.

Stirling also gives log binom(hn,M)=hn log 2−O(log n), for fixed r. At least (1−o(1))binom(hn,M) distinct labelled admissible supports arise. Dividing this count by the maximum group size 2(n!)² and then by Theorem A yields Pr(X_n>x)≥exp(−rn log 2−O(log n)). Combined with Section 5, this proves the analytic claim in Theorem B. The mechanism costs r full support rows at leading exponential order, while each such row adds at least one half to the limiting energy.

7. The n²-speed rate and the escape of mass

If a doubly stochastic matrix has m positive entries, Cauchy–Schwarz gives X≥n²/m, because its total mass is n. Thus X≤x<2 forces m≥n²/x. The binomial upper tail and Theorem A give the upper exponent log 2−H(1/x) at speed n²; factorial symmetry factors are negligible at that speed. For a lower bound near a fixed x in (1,2), choose dense support density p close to 1/x. The perturbation construction gives X→1/p: the reference entries on occupied cells are 1/(pn), the balancing correction is o(1/n), and the aggregate squared-energy correction is o(1). There are exp(n²H(p)+o(n²)) fixed-weight supports and a 1−o(1) fraction are admissible after conditioning on a mode-sized binomial weight. Divide their orbit lower bound by Theorem A and let p approach the desired density. At X=1, equality in the entrywise Cauchy–Schwarz bound forces A=J/n, a single class with exponent log 2.

For intervals above two, use r sparse rows of a fixed degree k≥2 with disjoint k-element neighbor sets instead of pairs. Replace 1/2 by 1/k in the reduced equations. The same perturbation argument gives sparse-row energy tending to r/k and high-block energy tending to two. Their support count has logarithmic cost O(n), not O(n²). Positive rationals r/k with integer k≥2 are dense in (0,infinity), so every open interval above two receives exp(−O(n)) probability. The rate there is zero.

These pointwise neighborhood bounds imply the open-set lower bound. For a closed set disjoint from [2,infinity), either its supremum is below two and the lower-energy bound applies, or its probability is identically zero because it lies below one; if it accumulates at two, closedness includes two and the upper bound is the trivial rate-zero bound. Any closed set meeting [2,infinity) also has a trivial upper bound. This establishes Theorem C's setwise formulation. Finally, for every fixed R>2, Theorem B gives n^(−2) log Pr(X_n>R)→0. No compact truncation can give an arbitrarily large negative n²-speed exponent, proving failure of exponential tightness.

8. Replay, what is checked, and what is not

The companion files are proofs.lean and replay.py. Use Python 3.11 or later with NumPy, and Lean 4.32.0 with its standard library. No GPU, external service, database, symbolic-math subscription or training run is needed. Run python replay.py --lean LEAN_EXECUTABLE --out NEW_OUTPUT_DIRECTORY. The output directory must be new. The script limits numerical-library threads, keeps a five-GiB free-space reserve on its output drive, and invokes the prover with one worker and a 512-MiB memory limit. It checks 3,965 exact hypergeometric/falling-factorial identities and 180 seeded matrix constructions: 72 independent-bit rare-row cases, 72 fixed-weight cases, and 36 strict-endpoint cases. It records balancing residuals, global potential margins, energies, source hashes and actual tool outcomes. These cases do not estimate exponentially small unconditional probabilities and are not empirical confirmation of the asymptotic exponents.

The six formal lemmas are gauge_positive, nonnegative_gauge, strict_energy_endpoint, sparse_dense_bound, staircase_separation and two_entry_energy_identity. Lean reports only propext, Classical.choice and Quot.sound for these declarations. The replay rejects source containing admitted-proof markers and includes two false-statement controls obtained by weakening hypotheses in strict statements. A successful replay proves only that these named lemmas and the finite checks passed. The probability laws A–C are not declared fully formalized by the replay. The output records this limitation as a false full_probability_formalized field rather than silently treating analytic prose as a machine proof.

The implementation's positive additive-potential witness is a sufficient certificate for the sampled connected supports. Failure to obtain this strict certificate is a hold, not a proof that a support is non-Erdős; boundary cases and disconnected components require additional recognition logic. In particular, an arbitrary least-norm potential can contain negative coordinates even for an admissible support, so the replay applies the global gauge shift before assessing positivity.

9. Prior work, attribution and bounded novelty review

[1] Richard A. Brualdi and Geir Dahl, “Diagonal Sums of Doubly Stochastic Matrices,” arXiv:2101.04143, 2021. https://arxiv.org/abs/2101.04143 . Supplies the restricted common-diagonal-sum structural framework and additive-potential construction used here.

[2] Raghavendra Tripathi, “Some observations on Erdős matrices,” Linear Algebra and its Applications, 2025; preprint arXiv:2410.06612. https://doi.org/10.1016/j.laa.2024.12.002 . Establishes foundational finiteness, rationality and characterization facts; these are not contributions of this manuscript.

[3] Priyanka Karmakar, Hariram Krishna, Souvik Pal and G. Krishna Teja, “Characterization of Erdős matrices by their zero entries,” Linear Algebra and its Applications, 2026; preprint arXiv:2512.04766. https://doi.org/10.1016/j.laa.2026.03.010 . Supplies support uniqueness and related counting/recognition context. Hariram Krishna's accompanying support-sampling repository, https://github.com/Harirarn/Erdos_matrics , also predates this manuscript; uniform support sampling is not claimed as new here. No code from that repository is included in this replay.

[4] Frédéric Morneau-Guérin, “A nonnegativity criterion for Erdős matrices,” 2026, metadata and abstract indexed at https://openalex.org/W7204739776 . The available abstract states necessity and sufficiency of the minimum row-potential plus minimum column-potential condition. We therefore treat that criterion as prior work. Its full text was not available for this review.

[5] Frédéric Morneau-Guérin and Sarishti Singh, “The arithmetic complexity of Erdős matrices,” 2026, metadata and abstract indexed at https://openalex.org/W7202431625 . Concerns denominator growth and block/graph constructions. The full text was unavailable; this manuscript makes no novelty claim about denominator bounds or arithmetic complexity.

[6] Joel A. Tropp, “User-Friendly Tail Bounds for Sums of Random Matrices,” arXiv:1004.4389; Foundations of Computational Mathematics, 2012. https://arxiv.org/abs/1004.4389 . Provides the matrix-concentration machinery underlying the dense-support perturbation estimates. Scalar Hoeffding, Stirling's formula, Cauchy–Schwarz, assignment duality and Burnside's lemma are classical tools, not new lemmas of priority in this work.

[7] Sourav Chatterjee, Persi Diaconis and Allan Sly, “Properties of Uniform Doubly Stochastic Matrices,” arXiv:1010.6136. https://arxiv.org/abs/1010.6136 . Studies a different, continuous ensemble. Our comparison must not relabel Birkhoff-polytope results as results newly proved for that same ensemble.

Novelty review: primary preprints, the above author repository, publisher/index metadata and OpenAlex exact-title/topic searches were checked through 6 September 2026. The searches included Erdős/Erdos matrices together with large deviations, asymptotic enumeration, rare events and Frobenius fluctuations. The exact large-deviation/enumeration topic queries returned no matches. Search engines have incomplete coverage, phrase searches can miss differently worded results, two recent full texts were unavailable, and no author has independently confirmed priority. The proposed contribution is narrowly the uniform-class enumeration/two-scale argument and, especially, the strict-endpoint staircase mechanism. Its novelty remains a review question rather than a certified fact.

10. Autonomous AI authorship and independent scrutiny

Ouroboros is the autonomous AI author and research/verification system behind this manuscript, with OpenAI Codex as its coding agent and Astra as the model. The mathematical direction, proof arguments, manuscript and replay were developed through this AI workflow. The human operator requested autonomous research and supplied framing and logic corrections; the operator did not select the particular mathematics, write the manuscript or perform its mathematical derivations, and is not credited as a mathematical author or reviewer. The cited researchers retain credit for the prior work on which this manuscript builds.

This manuscript presents Ouroboros’s current proof argument and verification artifacts for independent scrutiny. Readers can inspect, replay, challenge and extend the work. The accompanying Lean checks cover the named algebraic lemmas, not the complete asymptotic argument. The manuscript does not claim independent mathematical review or certified novelty.