{
  "schema_version": 1,
  "problem_number": "PE-1012",
  "title": "Parity Rock–Paper–Scissors Games and the Alper Constant",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Two players choose numbers from {1, …, n}. Equal numbers draw, an odd difference is won by the smaller number, a nonzero even difference is won by the larger number, and a win with the number m pays 2m − h, where 0 ≤ h < 2 is fixed. For h = 1 this is the game of Problem 1012 of Project Euler; for h = 0 the payment is proportional to the winning number. For every h outside an explicit countable set E we prove that the game has a unique equilibrium, supported on 2k + 1 consecutive numbers ending at n or n − 1, and we solve it in closed form. When the support ends at n, the probability P(n) of the largest number is ((2k² + 1)R_k − 2(k² − 1))/(3(2k + 1)) for an explicit product R_k that depends only on the largest payment; otherwise P(n) = 0. The support widens at thresholds given by the zeros of these functions; for h ∈ {0, 1} the thresholds are cubic polynomials in k for each parity. For S_h(N) = Σ_{n=3}^{N} P(n) we prove S_h(N) = (3/2)^{4/3} N^{1/3} − (1/4) log N − A(h) − (3/2) τ(1 − τ) + O_h(N^{−1/3}), where τ ∈ [0, 1) is the relative position of N in its block, and we determine the next term. The constant A(h) plays the role that Euler's constant plays for the harmonic series. We call A = A(0) = 1.7771799879… the Alper constant, give a series and an integral representation, and certify 80 decimal places of A and of A(1) = 1.6464176534… with interval arithmetic. No expression of either constant in classical constants was found. The Alper function h ↦ A(h) is strictly increasing and convex between consecutive points of E and jumps down at each of them; it tends to A(1) as h → 1− and to A(0) as h → 2−, and A(h + 2) = A(h) for 0 ≤ h < 2, h ∉ E, when the games are extended to larger parameters.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.OC",
    "math.CA"
  ],
  "keywords": [
    "rock-paper-scissors",
    "zero-sum game",
    "Nash equilibrium",
    "skew-symmetric matrix",
    "beta-binomial distribution",
    "digamma function",
    "asymptotic expansion",
    "interval arithmetic",
    "Alper constant",
    "Alper function",
    "PE-1012"
  ],
  "manuscript_version_date": "2026-10-05",
  "publication_date": "2026-10-05",
  "publication_date_kind": "first public online release",
  "version": "1.1",
  "date_modified": "2026-10-05",
  "presentation_revision_only": false,
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/parity-rps-alper-constant/",
  "pdf_url": "https://eulersolve.org/papers/parity-rps-alper-constant/paper.pdf?v=043ab16757fe",
  "doi": null,
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "code_url": "https://github.com/AlperTheKing/alper-constant",
  "source_commit": "505bc9a8f270bd038eb6da0e7c1917883bb8e3d8",
  "related_problem": {
    "name": "Project Euler Problem 1012: Rock Paper Scissors",
    "url": "https://projecteuler.net/problem=1012",
    "eulersolve_url": "https://eulersolve.org/problem/1012/",
    "relation": "the case h = 1"
  },
  "scope_caveat": "Complete theorems for the family G_n(h): uniqueness, closed form and transition law of the equilibrium for every h outside a countable exceptional set, with explicit cubic thresholds for h = 0 and h = 1; rigorous asymptotics of S_h(N) with an explicit second-order term; the Alper constant A = A(0), defined as an Euler-type limit, with 80 certified decimals (a computer-assisted interval evaluation of proved error bounds); and the Alper function A(h), with its proved shape and certified values at steps of 0.1. Global periodicity is not claimed. The case h = 1 is Project Euler Problem 1012, which is cited; its solvers may have obtained parts of the h = 1 results, and no priority is claimed for that case. Not claimed: a closed form, irrationality or transcendence of the constants; the negative PSLQ searches do not exclude a closed form. Values of S_1(N) are deliberately not published. Unrefereed; no proof-assistant verification.",
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    "verification_report.md": {
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  "ai_use_disclosure": "AI assistance was used in the derivations, in writing and running the verification code, and in preparing the manuscript. The author remains responsible for all claims and the final text.",
  "review_disclosure": "Author reviewed (author-directed, AI-assisted technical review); unrefereed preprint, no independent human peer review or proof-assistant verification claimed.",
  "version_history": [
    {
      "version": "1.0",
      "commit": "487992b",
      "note": "first public version on GitHub, 5 October 2026"
    },
    {
      "version": "1.1",
      "commit": "a4b73a7",
      "note": "adds Section 10, the Alper function"
    }
  ]
}