Die Gesetze
27 laws settled by calculation — each kernel-checked, proof open to inspection.
For non-negative integers, the product of max and min recovers a·b, and the difference of their squares factors as (max − min)(a + b)
Refuted by: Exhibiting non-negative integers a, b where (max(a,b) - min(a,b)) * (a + b) differs from max(a,b)^2 - min(a,b)^2, or where max(a,b)*min(a,b) != a*b.
Read the proof →The sum of two squares is never congruent to 3 modulo 4
Refuted by: Some non-negative integers a, b with (a^2 + b^2) % 4 == 3, i.e. a sum of two squares congruent to 3 mod 4.
Read the proof →The 20 nonzero fourth powers of GF(81) form a maximal SET cap in AG(4,3) — no three distinct points sum to 0 (mod 3) — and the 9 nonzero seventh powers of GF(64) form a maximal EvenQuads cap in AG(6,2) — no four distinct points sum to 0 (mod 2) (Kable–Mills–Wright: subgroups of finite fields as cap sets)
Refuted by: three distinct indices i<j<k with set81[i]+set81[j]+set81[k] ≡ 0 (mod 3), or four distinct indices with the corresponding eq64 sum ≡ 0 (mod 2)
Read the proof →Minimal double blocking sets of size 3q−1 exist in PG(2,13) and PG(2,19): each printed set meets every line at least twice and every point of it lies on a 2-secant — the constructive half of Csajbók–Héger, refuting Hill's 1984 conjecture; for prime q > 13 the first double blocking sets of size < 3q
Refuted by: a line of PG(2,13) meeting B13 (resp. PG(2,19) meeting B19) in fewer than 2 points, or a point of either set lying on no 2-secant
Read the proof →The Alexeev–Mixon set {1, 2, 4, 8, 13} is a Sidon set extending to no perfect difference set of order 5 or 6 — the kernel-attested finite core of the disproof of Erdős Problem 707 (the $1000 Sidon-extension conjecture)
Refuted by: a repeated pairwise difference in {1,2,4,8,13} (not Sidon), or the set itself a perfect difference set mod 21, or an x0 < 31 making {1,2,4,8,13,x0} a perfect difference set mod 31
Read the proof →The Guo–Krattenthaler divisibilities (6n−1) ∣ C(12n,3n) and (6n−1) ∣ C(12n,4n) hold for every n = 1..8, and (66n−1) ∣ C(330n,88n) at n = 1 — kernel-attested instances of the all-n theorems of Guo & Krattenthaler (2014)
Refuted by: an n between 1 and 8 with (6n−1) ∤ C(12n,3n) or (6n−1) ∤ C(12n,4n), or 65 ∤ C(330,88)
Read the proof →The integral closure of the monomial ideal M_{3,2} = (y²z², x²z², x²y²) — the Veronese cap-sum ideal of Mafi–Naderi Thm 1.6 — strictly contains M_{3,2} and gains the embedded associated prime (x,y,z) at the witness xyz, while the unmixed M_{3,2} admits no such witness (Cor 1.7; kernel-attested finite core, t = 2)
Refuted by: a monomial x^(a,b,c), a,b,c < 5, in M_{3,2} but outside the cap-sum ideal; or failure of the (x,y,z) colon-witness at x^(1,1,1) for the closure; or an (a,b,c) < 5 exhibiting the same embedded-prime behaviour for M_{3,2} itself
Read the proof →At the Markoff special point (1,1,1) mod p, for the certified primes p ≡ ±2 (mod 5) — 7, 13, 17, 23, 43, 47 and the Mersenne primes 127, 524287, 2³¹−1 — the rotation's companion matrix satisfies A^{p+1} = I and A^{(p+1)/2} ≠ I, so 2^{ν₂(p+1)} divides the rotation order (for Mersenne p, ord = p+1 exactly); and ord(A) = π(p)/2 for p ∈ {7, 127} (Bellah–Dunn–Naidu–Wells, Thm 2.10 / Prop 3.3)
Refuted by: a certified prime p with A^{p+1} ≠ I or A^{(p+1)/2} = I over F_p, or pisano(p) ≠ 2·ord(A) at p ∈ {7, 127}
Read the proof →Five natural sequences — n³, n⁴, n!, Fibonacci, and the primes — are each NOT self-ordered: for each, a kernel-decided witness (m,n) has Dₙ ∤ P(m,n) (census refutations for Problem 16 of Cahen–Fontana–Frisch–Glaz)
Refuted by: one of the five certified witnesses failing — i.e. Dₙ dividing P(m,n) at the stated (m,n) for that sequence's value prefix
Read the proof →Eleven corner triples 1 ≤ a ≤ b ≤ c ≤ 9 — including (2,3,7) and (3,4,5), both smaller than the textbook (4,5,7) — define non-normal monomial ideals I = closure(x^a, y^b, z^c): each carries a kernel-decided witness x^u ∈ closure(I²) ∖ I² (normality census for Problem 41 of Cahen–Fontana–Frisch–Glaz)
Refuted by: one of the 11 certified witnesses failing — wt(u) below the 2L threshold, or the witness monomial landing in I² after all
Read the proof →Steiner systems S(2,8,225) and S(2,9,289) exist: the printed difference families in ℤ₃×ℤ₃×ℤ₅×ℤ₅ and ℤ₁₇×ℤ₁₇ each hit every nonzero group element exactly once among their within-block differences, so each develops to its Steiner system — resolving two of the 129 undecided Handbook cases (Hetman 2026)
Refuted by: a repeated, zero, or missing within-block difference — i.e. the 224 (resp. 288) differences of blocks8 (resp. blocks9) failing to be exactly the nonzero elements of the group, once each
Read the proof →The 33-vector, 14-basis Cabello set is a Kochen–Specker set (admits no {0,1} coloring) — refuting the ≥16-basis conjecture (Cabello, PRL 135, 190203, 2025)
Refuted by: a {0,1}-assignment f of the 33 rays with f(u)+f(v) ≤ 1 for orthogonal u,v and exactly one 1 per orthonormal basis (a valid KS coloring)
Read the proof →A complex Hadamard matrix of order 94 exists (Szöllősi 2026) — kernel-attested structural core: the Example-1 circulant quadruple (A,B,C,D, order 47, {−1,1}) satisfies A A^T+B B^T+C C^T+D D^T = 188·I with A,B symmetric (Theorem 4)
Refuted by: a nonzero shift s∈[1,47) at which the summed periodic autocorrelations of A,B,C,D are nonzero (or ≠188 at s=0), or an asymmetry in A or B — breaking the Theorem-4 construction hypothesis
Read the proof →For every non-negative integer n, the expression (2n)^4 + 2*(2n)^3 + 4*(2n)^2 + 8*(2n) is divisible by 16.
Refuted by: Some non-negative integer n makes ((2n)^4 + 2*(2n)^3 + 4*(2n)^2 + 8*(2n)) % 16 nonzero.
Read the proof →For every non-negative integer n, the expression n^3 + 5*n is divisible by 3.
Refuted by: Some non-negative integer n makes n^3 + 5*n not divisible by 3.
Read the proof →For every non-negative integer n, the fourth power n^4 leaves remainder 0 or 1 when divided by 5 (never 2, 3, or 4).
Refuted by: Some non-negative integer n with n^4 % 5 equal to 2, 3, or 4.
Read the proof →For every non-negative integer n, the fourth power n^4 leaves remainder 0 or 1 when divided by 8 (never 2 through 7).
Refuted by: Some non-negative integer n with (n^4) % 8 equal to a value other than 0 or 1.
Read the proof →For every non-negative integer n, the expression n^4 + n^2 is divisible by 2.
Refuted by: Some non-negative integer n makes n^4 + n^2 leave a nonzero remainder modulo 2.
Read the proof →For every non-negative integer n, the expression n^4 + 4*n^3 + 5*n^2 + 2*n + 8 leaves remainder 8 when divided by 12; equivalently it is divisible by 12 only after subtracting 8, since n^4+4n^3+5n^2+2n = n(n+1)^2(n+2) is divisible by 12.
Refuted by: Some n >= 0 makes (n^4 + 4*n^3 + 5*n^2 + 2*n) % 12 nonzero, i.e. n*(n+1)^2*(n+2) not divisible by 12.
Read the proof →For every non-negative integer n, the expression n^4 + n^3 + n^2 + n + 1 is never divisible by 4; its residue modulo 4 always lies in {1, 2, 3}.
Refuted by: Some non-negative integer n makes n^4 + n^3 + n^2 + n + 1 divisible by 4, i.e. (n^4+n^3+n^2+n+1) % 4 == 0.
Read the proof →For every non-negative integer n, the product (2n)*(2n+1)*(2n+2) of three consecutive integers starting at an even number is divisible by 4.
Refuted by: Some non-negative integer n makes (2n)*(2n+1)*(2n+2) leave a nonzero remainder when divided by 4.
Read the proof →For every non-negative integer n, the residue of n^2 + 3*n modulo 6 is always one of 0, 4, or 2 (it is never 1, 3, or 5).
Refuted by: Some non-negative integer n makes (n^2 + 3*n) % 6 equal to 1, 3, or 5.
Read the proof →For every non-negative integer n, the expression n^2 + n + 1 is never divisible by 5; that is, n^2 + n + 1 modulo 5 always lies in {1, 2, 3} and is never 0 or 4.
Refuted by: Any non-negative integer n for which (n^2 + n + 1) % 5 equals 0 or 4 would refute the claim.
Read the proof →For every non-negative integer n, the residue of n^2 + n modulo 6 is always one of 0, 2, or 3 (it is never 1, 4, or 5).
Refuted by: Some non-negative integer n makes (n^2 + n) % 6 equal to 1, 4, or 5.
Read the proof →Addition of natural numbers is commutative
Refuted by: two naturals a, b with a + b ≠ b + a
Read the proof →Every successor of a natural number is positive
Refuted by: a natural n with n + 1 ≤ 0
Read the proof →Every power of two is positive
Refuted by: a natural n for which 2 ^ n = 0
Read the proof →