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{"id": "B1", "name": "equal_variables_extremum", "type": "solver_lemma", "domains": ["algebra"], "level": "standard", "track": "olympiad_algebra", "description": "MinOverSet/MaxOverSet of symmetric expression with symmetric constraint.", "dataset_count": 3480, "dataset_fraction": 0.0585, "dataset_as_root": 2115, "description_latex": "\\min/\\max f(x,y) \\text{ s.t. symmetric constraint}", "added_at": "2026-01-18"}
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{"id": "QF_PSD_ORBIT", "name": "qf_psd_orbit", "type": "dataset_only", "description": "count of unordered solutions of a symmetric PSD form Q(x) = R under variable-permutation ordering.", "dataset_count": 363, "dataset_fraction": 0.0061, "dataset_as_root": 191, "description_latex": "|\\{x_1 \\le \\dots \\le x_n : x^T A x = R\\}|"}
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{"id": "PRODUCT_OF_SUMS", "name": "product_of_sums", "type": "dataset_only", "description": "product of two independent sums, each reducible via its own antilemma.", "dataset_count": 218, "dataset_fraction": 0.0037, "dataset_as_root": 77, "description_latex": "\\prod_i \\sum_j a_{ij} \\text{ expansion}", "added_at": "2026-02-07"}
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{"id": "ZERO_BINOM_N", "name": "binom_n", "type": "spawner", "value": 0, "domains": ["COMB"], "complexity": 2, "description": "C(n, n) - 1 = 0", "dataset_count": 208, "dataset_fraction": 0.0035, "dataset_as_root": 14, "description_latex": "\\binom{n}{n} - 1 = 0", "added_at": "2026-01-29"}
{"id": "SUM_PRIMES", "name": "sum_primes", "type": "solver_lemma", "domains": ["number_theory"], "level": "standard", "track": "olympiad_nt", "description": "SumOverSet(SolutionsSet(n, And(n >= lower, n <= upper, IsPrime(n)))).", "dataset_count": 193, "dataset_fraction": 0.0032, "dataset_as_root": 76, "description_latex": "\\sum_{\\substack{p \\le n \\\\ p \\text{ prime}}} p", "added_at": "2026-02-06"}
{"id": "LEMMA_MOBIUS_SQUAREFREE", "name": "mobius_squarefree", "type": "spawner", "value": 0, "domains": ["NT"], "complexity": 3, "description": "μ(n)² = 0 when n has a square factor", "dataset_count": 181, "dataset_fraction": 0.003, "dataset_as_root": 56, "description_latex": "\\mu(n)^2 = [n \\text{ is squarefree}]", "added_at": "2026-01-31"}
{"id": "ONE_FACTORIAL_0", "name": "factorial_0", "type": "spawner", "value": 1, "domains": ["COMB"], "complexity": 2, "description": "0! = 1", "dataset_count": 180, "dataset_fraction": 0.003, "dataset_as_root": 57, "description_latex": "0! = 1", "added_at": "2026-01-29"}
{"id": "ONE_BINOM_N", "name": "binom_n", "type": "spawner", "value": 1, "domains": ["COMB"], "complexity": 2, "description": "C(n, n) = 1", "dataset_count": 174, "dataset_fraction": 0.0029, "dataset_as_root": 32, "description_latex": "\\binom{n}{n} = 1", "added_at": "2026-01-29"}
{"id": "ONE_BINOM_0", "name": "binom_0", "type": "spawner", "value": 1, "domains": ["COMB"], "complexity": 2, "description": "C(n, 0) = 1", "dataset_count": 171, "dataset_fraction": 0.0029, "dataset_as_root": 29, "description_latex": "\\binom{n}{0} = 1", "added_at": "2026-01-29"}
{"id": "LEMMA_DIVISOR_PARITY", "name": "divisor_parity", "type": "spawner", "value": 0, "domains": ["NT"], "complexity": 2, "description": "τ(n) mod 2 = 0 when n is not a perfect square", "dataset_count": 146, "dataset_fraction": 0.0025, "dataset_as_root": 30, "description_latex": "\\tau(n) \\bmod 2 = [n \\text{ is a perfect square}]", "added_at": "2026-01-31"}
{"id": "EULER_TOTIENT_SUM", "name": "euler_totient_sum", "type": "dataset_only", "description": "totient summatory function: sum of phi(k) over k = 1..n.", "dataset_count": 133, "dataset_fraction": 0.0022, "dataset_as_root": 42, "description_latex": "\\sum_{k=1}^{n} \\varphi(k)", "added_at": "2026-02-10"}
{"id": "STARS_BARS", "name": "stars_bars", "type": "dataset_only", "description": "compositions count: number of positive integer tuples (x1,...,xk) with sum S.", "dataset_count": 117, "dataset_fraction": 0.002, "dataset_as_root": 89, "description_latex": "|\\{(x_1,\\dots,x_k) \\in \\mathbb{Z}_{\\ge 1}^k : x_1+\\dots+x_k = S\\}| = \\binom{S-1}{k-1}"}
{"id": "WILSON", "name": "wilson", "type": "dataset_only", "description": "Wilson's theorem: (p-1)! is congruent to -1 modulo prime p.", "dataset_count": 115, "dataset_fraction": 0.0019, "dataset_as_root": 2, "description_latex": "(p-1)! \\equiv -1 \\pmod{p}", "added_at": "2026-02-10"}
{"id": "LTE_SUM", "name": "lte_sum", "type": "solver_lemma", "domains": ["number_theory"], "level": "olympiad", "track": "olympiad_nt", "description": "MaxKDivides(Sum(Pow(a, n), Pow(b, n)), p) where n is odd.", "dataset_count": 114, "dataset_fraction": 0.0019, "dataset_as_root": 86, "description_latex": "v_p(a^n + b^n) \\text{ for odd } n, \\; p \\mid a+b", "added_at": "2026-01-20"}
{"id": "LIOUVILLE_MINUS_ONE", "name": "liouville_minus_one", "type": "dataset_only", "description": "Liouville lambda evaluates to -1 when Omega(n) is odd.", "dataset_count": 114, "dataset_fraction": 0.0019, "dataset_as_root": 43, "description_latex": "\\lambda(n) = (-1)^{\\Omega(n)} = -1 \\text{ when } \\Omega(n) \\text{ odd}", "added_at": "2026-02-10"}
{"id": "IDENTITY_DIV_SELF", "name": "div_self", "type": "spawner", "value": 1, "domains": ["ARITH"], "complexity": 1, "description": "a / a = 1", "dataset_count": 110, "dataset_fraction": 0.0018, "dataset_as_root": 73, "description_latex": "a / a = 1", "added_at": "2026-02-07"}
{"id": "LIOUVILLE_ONE", "name": "liouville_one", "type": "dataset_only", "description": "Liouville lambda evaluates to 1 when Omega(n) is even.", "dataset_count": 107, "dataset_fraction": 0.0018, "dataset_as_root": 15, "description_latex": "\\lambda(n) = (-1)^{\\Omega(n)} = 1 \\text{ when } \\Omega(n) \\text{ even}", "added_at": "2026-02-10"}
{"id": "POLY4_MIN", "name": "poly4_min", "type": "dataset_only", "description": "minimum of a quartic form F(x) (sum of 4th powers of linear forms) over x in [1,M]^n.", "dataset_count": 100, "dataset_fraction": 0.0017, "dataset_as_root": 49, "description_latex": "\\min_{x \\in [1,M]^n} F(x),\\; \\deg F = 4"}
{"id": "IDENTITY_POW_ZERO", "name": "pow_zero", "type": "spawner", "value": 1, "domains": ["ARITH"], "complexity": 1, "description": "a^0 = 1", "dataset_count": 93, "dataset_fraction": 0.0016, "dataset_as_root": 57, "description_latex": "a^0 = 1", "added_at": "2026-02-07"}
{"id": "HALFPLANE_COUNT", "name": "halfplane_count", "type": "dataset_only", "description": "count of lattice points (x,y) in [1,Nx] x [1,Ny] under a half-plane constraint ax+by <= C.", "dataset_count": 93, "dataset_fraction": 0.0016, "dataset_as_root": 64, "description_latex": "|\\{(x,y) \\in [1,N_x]\\times[1,N_y] : ax + by \\le C\\}|"}
{"id": "BIG_OMEGA_ZERO", "name": "big_omega_zero", "type": "dataset_only", "description": "Big Omega vanishes at n = 1 (no prime factors with multiplicity).", "dataset_count": 89, "dataset_fraction": 0.0015, "dataset_as_root": 30, "description_latex": "\\Omega(1) = 0", "added_at": "2026-02-10"}
{"id": "BIG_OMEGA_ONE", "name": "big_omega_one", "type": "dataset_only", "description": "Big Omega equals 1 at any prime p.", "dataset_count": 81, "dataset_fraction": 0.0014, "dataset_as_root": 11, "description_latex": "\\Omega(p) = 1", "added_at": "2026-02-10"}
{"id": "SUM_SQUARES_IDENTITY", "name": "sum_of_squares_equals_sum_of_products", "type": "solver_lemma", "domains": ["algebra"], "level": "olympiad", "track": "olympiad_algebra", "description": "a²+b²+c² = ab+bc+ca forces a=b=c; combined with linear constraint determines unique value.", "description_latex": "a^2+b^2+c^2 = ab+bc+ca \\iff (a-b)^2+(b-c)^2+(c-a)^2=0 \\implies a=b=c", "dataset_count": 81, "dataset_fraction": 0.0014, "dataset_as_root": 44, "added_at": "2026-02-24"}
{"id": "POLY_ORBIT_LEGENDRE_COUNT", "name": "poly_orbit_legendre_count", "type": "dataset_only", "description": "count of starting values a with exact minimal period r under iterated P mod p and Legendre-sum filter.", "dataset_count": 80, "dataset_fraction": 0.0013, "dataset_as_root": 80, "description_latex": "|\\{a \\in [0,N] : \\text{period}_P(a) = r,\\; \\textstyle\\sum_{i=0}^{r-1}\\bigl(\\frac{P^i(a)}{p}\\bigr) \\equiv 0 \\pmod m\\}|"}
{"id": "OMEGA_ZERO", "name": "omega_zero", "type": "dataset_only", "description": "small omega vanishes at n = 1 (no distinct prime factors).", "dataset_count": 76, "dataset_fraction": 0.0013, "dataset_as_root": 10, "description_latex": "\\omega(1) = 0", "added_at": "2026-02-10"}
{"id": "OMEGA_ONE", "name": "omega_one", "type": "dataset_only", "description": "small omega equals 1 at any prime power p^k (single distinct prime).", "dataset_count": 65, "dataset_fraction": 0.0011, "dataset_as_root": 5, "description_latex": "\\omega(p) = 1", "added_at": "2026-02-10"}
{"id": "LTE_DIFF_P2", "name": "lte_difference_p2", "type": "solver_lemma", "domains": ["number_theory"], "level": "olympiad", "track": "olympiad_nt", "description": "MaxKDivides(Sub(Pow(a, n), Pow(b, n)), 2)", "dataset_count": 58, "dataset_fraction": 0.001, "dataset_as_root": 39, "description_latex": "v_2(a^n - b^n) = v_2(a-b) + v_2(a+b) + v_2(n) - 1", "added_at": "2026-01-20"}
{"id": "ABS_INEQ", "name": "abs_ineq", "type": "dataset_only", "description": "count of integers x in [1,N] satisfying an absolute-value inequality |ax - b| <= c.", "dataset_count": 56, "dataset_fraction": 0.0009, "dataset_as_root": 35, "description_latex": "|\\{x \\in [1,N] : |ax - b| \\le c\\}|"}
{"id": "SUM_AP", "name": "sum_ap", "type": "dataset_only", "description": "closed form for the general arithmetic-progression sum Sigma_{k=start}^{n} (a*k + b).", "dataset_count": 40, "dataset_fraction": 0.0007, "dataset_as_root": 32, "description_latex": "\\sum_{k=s}^{n} (a k + b) = a\\cdot\\tfrac{n(n+1) - s(s-1)}{2} + b(n - s + 1)"}
{"id": "QUADRATIC_INEQ", "name": "quadratic_ineq", "type": "dataset_only", "description": "count of integers x in [1,N] satisfying a quadratic inequality a x^2 + b x + c <= 0 with a > 0.", "dataset_count": 35, "dataset_fraction": 0.0006, "dataset_as_root": 25, "description_latex": "|\\{x \\in [1,N] : a x^2 + b x + c \\le 0\\}|"}
{"id": "IDENTITY_MUL_ZERO", "name": "mul_zero", "type": "spawner", "value": 0, "domains": ["ARITH"], "complexity": 1, "description": "a * 0 = 0", "dataset_count": 26, "dataset_fraction": 0.0004, "dataset_as_root": 8, "description_latex": "a \\cdot 0 = 0", "added_at": "2026-02-07"}
{"id": "IDENTITY_SUB_SELF", "name": "sub_self", "type": "spawner", "value": 0, "domains": ["ARITH"], "complexity": 1, "description": "a - a = 0", "dataset_count": 23, "dataset_fraction": 0.0004, "dataset_as_root": 8, "description_latex": "a - a = 0", "added_at": "2026-02-07"}
{"id": "ZERO_PHI_PRIME", "name": "phi_prime", "type": "spawner", "value": 0, "domains": ["NT"], "complexity": 2, "description": "phi(p) - (p-1) = 0 for prime p", "dataset_count": 18, "dataset_fraction": 0.0003, "dataset_as_root": 7, "description_latex": "\\varphi(p) = p - 1", "added_at": "2026-01-29"}
{"id": "IDENTITY_MOD_SELF", "name": "mod_self", "type": "spawner", "value": 0, "domains": ["ARITH"], "complexity": 1, "description": "a % a = 0", "dataset_count": 16, "dataset_fraction": 0.0003, "dataset_as_root": 2, "description_latex": "a \\bmod a = 0", "added_at": "2026-02-07"}
{"id": "TELESCOPE", "name": "telescope", "type": "dataset_only", "description": "telescoping sum Sigma_{k=start}^{n} (f(k+1) - f(k)) collapses to f(n+1) - f(start).", "dataset_count": 4, "dataset_fraction": 0.0001, "dataset_as_root": 2, "description_latex": "\\sum_{k=s}^{n} \\bigl(f(k+1) - f(k)\\bigr) = f(n+1) - f(s)"}
{"id": "POLY_PREPERIOD_COUNT", "name": "poly_preperiod_count", "type": "dataset_only", "description": "count of starting values with preperiod t and exact cycle length s under iterated polynomial map mod M.", "dataset_count": 2, "dataset_fraction": 0.0, "dataset_as_root": 2, "description_latex": "|\\{a \\in [0,N] : P^{t+s}(a) \\equiv P^t(a) \\pmod{M},\\; P^{t+k}(a) \\not\\equiv P^t(a) \\text{ for } 1\\le k<s\\}|"}