arXiv subjects

Christopher D. Long

Publications and source records attributed to Christopher D. Long.

15 recordsLinked to original sources

arXiv2026

An Explicit Counterexample to the Rank-Two Poisson Conjecture

Let \[ {\mathcal P}_2={\mathbb C}[x,q,p,z] \] carry the canonical Poisson bracket determined by $ \{p,x\}=\{z,q\}=1 $ and by the vanishing of the other brackets between distinct generators. Here and throughout, ``rank two'' means two canonical pairs in the standard indexing of the canonical Poisson algebras; thus there are four polynomial generators and the Poisson tensor has geometric rank four. We give explicit polynomials \[ R,T,D,S\in{\mathbb Q}[x,q,p,z] \] satisfying \[ \{D,R\}=1,\qquad \{S,T\}=1, \qquad \{R,S\}=\{R,T\}=\{D,S\}=\{D,T\}=0, \] while \[ R=x(2-3xq). \] Consequently, the assignment $ (x,q,p,z)\mapsto(R,T,D,S) $ defines a Poisson endomorphism of ${\mathcal P}_2$ that is not an automorphism. This disproves the Poisson Conjecture for two canonical pairs, and hence for every number of canonical pairs at least two. The associated polynomial map of ${\mathbb A}^4$ preserves the canonical symplectic form, has Jacobian determinant one, and has an explicit fiber consisting of exactly three points. The proof uses a polynomial source coordinate system in which the symplectic identity reduces to three displayed coefficient identities. A separate appendix uses the same four output polynomials and their Hamiltonian duals to construct an explicit nonautomorphic endomorphism of the fourth Weyl algebra.

math.RA↗

arXiv2026

Counterexamples to the xz-Conjecture and the Mathieu Conjecture for SU(2)

Let \[ {\mathcal I}(h)=\int_0^1\int_{\mathbb T}h(x,z)\,\frac{dz}{2\pi iz}\,dx \qquad \bigl(h\in{\mathbb C}[x,z,z^{-1}]\bigr). \] We give the three-term Laurent polynomial \[ f(x,z)=(1-z^{-1})\bigl((1-x)+xz\bigr) \] for which \[ {\mathcal I}(f^n)=0, \qquad {\mathcal I}(z^{-1}f^n)=\frac{(-1)^{n-1}}{n+1}\neq0 \qquad(n\geq1). \] Since ${\operatorname{Sp}}(f)=\{-1,0,1\}$, this disproves the $xz$-conjecture already with one interval variable and one torus variable, and it also shows that $\ker{\mathcal I}$ is not a Mathieu--Zhao subspace. Padding gives counterexamples to every mixed case of the $xz$-conjecture. Writing the coordinate functions on $SU(2)$ as \[ g=\begin{pmatrix}a&c\\ b&d\end{pmatrix}, \] the same example lifts, through the integration formula of M\"uger and Tuset, to the regular functions \[ F=(1+c)(ad+b),\qquad G=-c, \] which satisfy \[ \int_{SU(2)}F^n\,dg=0, \qquad \int_{SU(2)}F^nG\,dg=\frac{(-1)^{n-1}}{n+1}\neq0 \] for every $n\geq1$. Thus the Mathieu conjecture for $SU(2)$ is false.

math.GR↗

arXiv2026

Small Counterexamples to the Gaussian Moments Conjecture

We give explicit complex polynomials $P,Q$ in three independent standard real Gaussian variables such that \[ {\mathbb E}(P^m)=0,\qquad {\mathbb E}(QP^m)=m!\neq0 \] for every $m\geq1$. In natural complex linear coordinates, $P$ has five terms and total degree $4$. Hence the Gaussian Moments Conjecture is false in every dimension $n\geq3$. We also give a six-term cubic example in four variables, which was found first and already proves failure for every $n\geq4$. Both examples follow from the same coefficient identity. The search was prompted by Levent Alp\"oge's public announcement of an explicit three-dimensional counterexample to the Jacobian Conjecture. Although the main theorem of Derksen, van den Essen, and Zhao is stated globally in dimension, its proof has fixed-dimensional content: a noninvertible cubic-homogeneous Keller map in $r$ variables forces the failure of ${\mathrm GMC}(2r)$. Tracking a standard Bass--Connell--Wright reduction of the announced map gives a conservative cubic-homogeneous counterexample in $79$ variables, and hence a route-based failure of ${\mathrm GMC}(158)$. That route is nonconstructive at the final Gaussian step and does not furnish explicit polynomials $P,Q$. The much smaller explicit failures in dimensions $4$ and $3$ below were not derived from the announced Jacobian map.

math.PR↗

arXiv2026

Radial Transform Extremality for the Siblings of the Coupon Collector

In the siblings version of the coupon collector, a main collector stops when every coupon type has appeared once. Duplicates are passed successively to siblings, and $U_j^N$ denotes the number of empty spaces in the $j$th collector's album at the main completion time. We prove finite-$N$ radial transform strengthenings of the uniform-probability extremality principle. For every $N\ge2$, every $j\ge2$, every positive nonuniform probability vector $p$, and the ray $p(\theta)=u+\theta(p-u)$ from the uniform vector $u$, the full probability generating function $\mathbb{E}_{p(\theta)}z^{U_j^N}$ is strictly decreasing in $\theta$ for $z>1$ and strictly increasing in $\theta$ for $0<z<1$. Thus the same full PGF has opposite radial monotonicity on the two sides of $z=1$, the left side giving a radial Laplace-transform order. At the coefficient level, along every nonconstant ray from the uniform vector, uniform probabilities maximize every binomial moment of $U_j^N$, equivalently giving a finite absolutely-monotone/binomial-transform order. The proof of the right-PGF and binomial-moment theorem is exact and finite-dimensional. It uses Poissonization, a marked Poissonized PGF identity, a normalized alternating subset expansion, and a positive-kernel radial derivative formula obtained from a local cumulative-polynomial dissipation lemma. The Laplace-transform theorem follows from a separate Gamma-mixture race representation.

math.PR↗

arXiv2026

Extremality and Limit Laws for the Siblings of the Coupon Collector

We study the siblings version of the coupon collector problem. A main collector stops when every coupon type has appeared at least once, duplicates are passed successively to later siblings, and $U_j^N$ denotes the number of empty spaces in collector $j$'s album at the main completion time. We prove three results. First, for every fixed $N$ and $j\ge2$, $\E U_j^N$ is uniquely maximized over positive coupon distributions by the uniform distribution; in fact it decreases strictly along every nonconstant ray from the uniform vector. Second, in the uniform model, $U_j^N$ is stochastically increasing in $N$, and we construct an increasing coupling using top spacings of exponential order statistics. Third, for fixed album indices $2,\ldots,J$, the naturally normalized vector converges jointly to $(W,\ldots,W)$, where $W$ is exponential with mean one. We also derive exact Poissonized and alternating-subset formulae and give a transfer principle for leading expectation asymptotics.

math.PR↗

arXiv2026

Radial Extremality for LRU Caching and the Fill--Holst Conjecture

For the independent reference model with popularity vector $p\in\Delta_N^\circ$, let $H_C(p)$ denote the exact stationary hit rate of an LRU cache of capacity $C$. We prove that, for every $1\le C<N$, the uniform popularity vector is the unique global minimizer of $H_C$ on the interior simplex. More sharply, along every nonconstant segment from the uniform vector to an interior point, the LRU hit rate is strictly increasing. The proof uses the standard exponential-age representation of the stationary LRU cache and gives an explicit positive pair-square formula for the radial derivative. Equivalently, for the move-to-front rule, the stationary search-cost distribution improves strictly in the usual stochastic order along every nonconstant ray away from uniform. This proves the radial restriction of the Fill--Holst Schur-concavity conjecture for move-to-front search-cost tails. In particular, all LRU miss probabilities and all nonconstant nondecreasing stack-depth costs decrease strictly along such rays. The result is radial rather than Schur-convex: full majorization monotonicity for LRU is known to fail, and the proof identifies the special positivity that survives on rays from the uniform vector.

math.PR↗

arXiv2026

Clumsy and Careless: Stationary-Entry Flux in Non-monotone Coupon Collectors

We study three nonmonotone coupon-collector models through a stationary-entry viewpoint. In such models the all-present state is not absorbing, so completion is governed not by the disappearance of a monotone terminal cloud but by rare new entries into a target state, except in the reset-button model, where exact regeneration gives a separate reduction. We prove a finite stationary-entry theorem: a mixing estimate, a one-block clump-control estimate, and the stationary entry flux imply an exponential hitting law. For the reset-button collector, regeneration gives an exact probability-generating function in terms of the ordinary coupon-collector transform and recovers the known beta-function expectation, while also yielding rare-success exponential limits and negligible-reset Gumbel limits. For the clumsy collector with fixed loss probability $p$ and $q=1-p$, the stationary-entry flux is $p q^n$, and $p q^n T_n$ converges to $\operatorname{Exp}(1)$. Thus the fixed-loss standardized limit is exponential rather than Gumbel. For the post-loss careless collector, we compute the sharp stationary-entry flux $$ \mu_n\sim (q;q)_\infty^{-1}\frac{n!}{n^n}q^{n(n+1)/2} $$ and prove $\mu_nT_n\Rightarrow\operatorname{Exp}(1)$, with matching moment asymptotics. This shows that the careless scale is governed by a stationary high tail, or ordered lucky climb, rather than by the independent one-point marginal heuristic. We also analyze a combined clumsy-careless model, confirming stability of the high-tail entry mechanism.

math.PR↗

arXiv2026

The Ballot Event for Two-Player Coupon Collection: A Renewal--Catalan Asymptotic

We study the two-player coupon-collector competition in which two independent collectors draw one coupon each per round from a set of $d$ equally likely coupon types. Myers and Wilf gave finite formulae for several two-player events and explicitly left open the ballot-type problem of finding the probability that the ultimate winner was never behind. We prove that this probability satisfies $$ b_d \sim \frac{2}{d}, \qquad d\to\infty .$$ The proof uses a renewal decomposition at the tie boundary. The first one-sided tie-break has an explicit entrance distribution; its level, scaled by $d^{1/2}$, converges to a Rayleigh law; and, after the break, the leader's survival probability is governed by a Catalan, or gambler's-ruin, harmonic. The main estimate shows that the accumulated defect of this comparison harmonic in the exact simultaneous-round chain is negligible.

math.PR↗

arXiv2026

The Expiring Coupon Collector: Sliding-Window Surjection Flux and Rare-Entry Laws

We study the coupon collector with deterministic expiration: one coupon is drawn at each time, and each coupon remains active for exactly $M$ draws. Completion occurs when all $n$ coupon types are simultaneously active. Equivalently, the current length-$M$ sliding window of draws must contain all $n$ types. The central object is not the one-time probability that a random window is onto, but the stationary flux of new entries into the onto-window set. We compute this flux exactly: \[ \mu_{n,M} =\Pbb(W_{t-1}\text{ is not onto},\ W_t\text{ is onto}) =\frac{(n-1)(n-1)!S(M-1,n-1)}{n^M}, \] where $S(\cdot,\cdot)$ denotes a Stirling number of the second kind. Under a quantitative subcritical separation condition, satisfied in particular by every fixed integer scale $M=\floor{\alpha n\log n}$, $0<\alpha<1$, we prove local declumping and obtain \[ \mu_{n,M}T_{n,M}\Rightarrow \Exp(1). \] For the fixed subcritical scale $M=\floor{\alpha n\log n}$, $0<\alpha<1$, this gives the logarithmic scale \[ \log T_{n,M}=n^{1-\alpha}+o_{\mathbb P}(n^{1-\alpha}), \qquad \log \Ebb T_{n,M}=n^{1-\alpha}+o(n^{1-\alpha}), \] and, when $\alpha>1/2$, the sharper normalization \[ n^{-\alpha}e^{-n^{1-\alpha}}T_{n,M}\Rightarrow \Exp(1), \qquad \Ebb T_{n,M}\sim n^\alpha e^{n^{1-\alpha}}. \] Thus the leading scale proposed in the Math StackExchange discussion is made rigorous; the exact finite-$n$ flux gives the canonical normalization throughout the subcritical range. The result is a sliding-window companion to rare-void entry-flux methods for nonmonotone coupon collectors.

math.PR↗

arXiv2026

Terminal Defects, Growing Multiplicity, and Variance Extremality in the Double Dixie Cup Problem

We develop a terminal-defect method for the double Dixie cup problem and use it to prove the finite-variance extremality conjecture of Doumas and Papanicolaou. For every $m\ge1$ and $N\ge2$, among all positive coupon probability vectors $p=(p_1,\ldots,p_N)$, the variance of the time $T_m(N)$ to collect $m$ complete sets is uniquely minimized at the uniform vector. We prove the stronger radial statement that the variance is strictly increasing along every ray from the uniform vector. The proof is finite-$N$ and exact: after Poissonization, the completion time is a maximum of independent Erlang variables, and the radial derivative of its distribution is compared to a size-biased law using a monotone-likelihood-ratio argument based on a log-scale monotonicity property of the Gamma reverse hazard. The same framework gives a growing-multiplicity Gumbel theorem in the equal-probability case, with expectation and variance asymptotics on the inverse gamma-tail scale. This recovers the fixed-$m$ equal-probability variance asymptotic stated as Conjecture 1 by Doumas and Papanicolaou, classically known for $m=1$, and extends the mechanism to $m=m_N$. We also illustrate the unequal-probability theory with endpoint-Laplace limits for power-law probabilities.

math.PR↗

arXiv2026

Exact Finite-Horizon Quantile Kelly for Repeated Multi-Outcome Events

We formulate and prove an exact finite-horizon quantile theorem for repeated identical multi-outcome Kelly wagering in wealth-profile / Arrow--Debreu coordinates. For a fixed $m$-outcome event repeated independently over a horizon $n$, the terminal wealth induced by a one-period wealth profile $W$ is a monomial $W^N$ in the multinomial count vector $N$. We show that every fixed upper quantile of terminal wealth is a positively homogeneous piecewise-monomial function on the closed Arrow--Debreu wealth simplex, equivalently piecewise linear in log-wealth coordinates on the positive interior. The pieces are indexed by the chambers of the multinomial count arrangement, and on each chamber the quantile objective is exactly a one-period Kelly objective for a count-based \emph{shadow law} $k/n$. Consequently the finite-horizon quantile problem decomposes into finitely many shadow-Kelly subproblems. We then refine the interior chamber picture to a finite stratification of the full closed simplex by support faces and arrangement faces, and we prove a weak exact recursive boundary algorithm. We also prove a natural first-order asymptotic collapse to ordinary Kelly, showing that the optimal scaled log-quantile converges to the ordinary Kelly value and that exact finite-horizon maximizers converge to the Kelly wealth profile. For illustration, we include worked binary and ternary examples in the main text and expanded versions in the appendices. We conclude with further remarks and conjectures concerning stronger pruning, higher-order finite-horizon corrections, and extensions to simultaneous wagers.

math.OC↗

arXiv2026

Risk-Constrained Kelly for Mutually Exclusive Outcomes: CRRA Support Invariance and Logarithmic One-Dimensional Calibration

We study the finite mutually exclusive outcome version of risk-constrained Kelly optimization with explicit state prices. The market has outcome probabilities $p_i>0$, state prices $q_i>0$, terminal wealths $W_i=c+x_i/q_i$, and a drawdown-surrogate constraint \[ \sum_{i=1}^n p_i W_i^{-\lambda}\le 1,\qquad \lambda>0. \] For constant relative risk aversion utility, we work primarily in the standard overround regime $\sum_i q_i>1$, where every optimizer is necessarily non-full-support. Under the usual unique likelihood-ratio prefix hypothesis for the unconstrained problem, we prove that the constrained optimizer has exactly the same active set. Thus, in the regime where the prefix theorem is meaningful, the risk constraint deforms the funded wealth profile but does not change the active set. The support is therefore invariant across both the CRRA parameter and the drawdown-surrogate parameter. We then isolate the logarithmic case $\gamma=1$. Once the common active prefix is known, the constrained problem reduces to a one-dimensional outer calibration together with independent one-dimensional inner equations on the active states. In this case we prove existence, uniqueness, and monotonicity for the inner solves, derive a complete calibration theorem, and record the resulting structured algorithm. We treat the fair and subfair regimes only as boundary cases: full-support phenomena can occur there, so the overround prefix theory no longer yields a parallel exact description of comparable sharpness. A numerical example illustrates how the risk constraint alters the funded wealth profile while leaving support unchanged.

math.OC↗

arXiv2026

Optimal Parlay Wagering and Whitrow Asymptotics: A State-Price and Implicit-Cash Treatment

For independent multi-outcome events under multiplicative parlay pricing, we give a short exact proof of the optimal Kelly strategy using the implicit-cash viewpoint. The proof is entirely eventwise. One first solves each event in isolation. The full simultaneous optimizer over the entire menu of singles, doubles, triples, and higher parlays is then obtained by taking the outer product of the one-event Kelly strategies. Equivalently, the optimal terminal wealth factorizes across events. This yields an immediate active-leg criterion: a parlay is active if and only if each of its legs is active in the corresponding one-event problem. The result recovers, in a more transparent state-price form, the log-utility equivalence between simultaneous multibetting and sequential Kelly betting. We then study what is lost when one forbids parlays and allows only singles. In a low-edge regime and on a fixed active support, the exact parlay optimizer supplies the natural reference point. The singles-only problem is a first-order truncation of the factorized wealth formula. A perturbative expansion shows that the growth-rate loss from forbidding parlays is $\OO(\eps^4)$, while the optimal singles stakes deviate from the isolated one-event Kelly stakes only at cubic order. This yields a clean explanation of Whitrow's empirical near-proportionality phenomenon: the simultaneous singles-only optimizer is obtained from the isolated eventwise optimizer by an event-specific cubic shrinkage, so the portfolios agree through second order and differ only by a small blockwise drag.

math.OC↗

arXiv2026

Utility-Invariant Support Selection and Eventwise Decoupling for Simultaneous Independent Multi-Outcome Bets

For simultaneous independent events with finitely many outcomes, consider the expected-utility problem with nonnegative wagers and an endogenous cash position. We prove a short support theorem for a broad class of strictly increasing strictly concave utilities. On any fixed support family and at any optimal portfolio with positive cash, summing the active first-order conditions and comparing that sum with cash stationarity yields the exact identity \[ \frac{\lambda}{K_{\ell}^{(U)}}=\frac{1-P_{\ell,A}}{1-Q_{\ell,A}}, \] where $P_{\ell,A}$ and $Q_{\ell,A}$ are the active probability and price masses of event $\ell$, $\lambda$ is the budget multiplier, and $K_{\ell}^{(U)}$ is the continuation factor seen by inactive outcomes of that event. Consequently, after sorting each event by the edge ratio $p_{\ell i}/\pi_{\ell i}$, the exact active support is the eventwise union of the single-event supports, and this support is independent of the utility function. The single-event utility-invariant support theorem is already explicit in the free-exposure pari-mutuel setting in Smoczynski and Miles; the point of the present note is that the simultaneous independent-events analogue follows from the same state-price geometry once the right continuation factor is identified.

math.OC↗

arXiv2026

Single-Event Multinomial Full Kelly via Implicit State Positions

For a single event with finitely many mutually exclusive outcomes, the full Kelly problem is to maximize expected log wealth over nonnegative stakes together with an optional cash position. The optimal formula is classical, but the support-selection step is often presented via Lagrange multipliers. This note gives a shorter state-price derivation. A cash fraction $c$ acts as an implicit position in every outcome: in terminal-wealth terms, it is equivalent to a baseline stake $cq_i$ on outcome $i$, where $q_i$ is the state price. On any active support, explicit bets therefore only top up favorable outcomes from this baseline $cq_i$ to the optimal total stake $p_i$. This yields the formula $x_i = (p_i - c q_i)_+$, the threshold rule $p_i/q_i > c$, and, after sorting outcomes by $p_i/q_i$, a one-pass greedy algorithm for support selection. The result is standard in substance, but the implicit-position viewpoint gives a compact proof and a convenient way to remember the solution.

math.OC↗