One group and exact growth gaps for all odd square displacements
Abstract
All odd square displacements give one noncyclic group, equal diameter, and exact growth gaps.
Theorem 1.1 (All odd square displacements: one group, two markings, one exact peak).
Lean statement: D5/S3/FiniteGroups/BajnokBeattySquareDisplacementGrowth.same_group_square_displacement_growth
Proof. Machine-checked in Lean as D5/S3/FiniteGroups/BajnokBeattySquareDisplacementGrowth.same_group_square_displacement_growth (✓ std3). ∎
Source. Repository-derived.
Acknowledgement. B. Bajnok and W. K. Beatty (2024). On the Diameter of Undirected Cayley Graphs of Finite Abelian Groups. URL: https://arxiv.org/abs/2406.04045v1.
Commentary.
Let p,q,t range over all integers such that p >= 3, p is odd, q >= 3, q is odd, and q+1 <= 2t. Since q+1 is even, the last inequality is equivalent to (q+1)/2 <= t, with exact integer division. No primality is required of either p or q. Put m=2qt-1, D=(m^2-q^2)/2, H=Z/pZ x Z/(pD)Z, and s=((pm-1)/2).toNat. These hypotheses imply m is odd, q^2 < m, D > 0, and 2s+1=pm. In particular the absolute-value moduli in the formal product ZMod(p.natAbs) x ZMod((pD).natAbs) are precisely the positive integers p and pD.
For each of the two pairs (a,b)=(1,q^2) and (q,q), define v1=(p(m-a)/2,p(m+a)/2), v2=(p(m-b)/2,-p(m+b)/2), K=span_Z{v1,v2}, G=Z^2/K, and F:Z^2 -> G to be the actual quotient homomorphism. Write K1,G1,F1 for the first pair and Kq,Gq,Fq for the second. In the function-coordinate model Plane=Fin(2)->Z, let E:Plane -> Z x Z be the usual linear identification and let f1=F1 composed with E inverse and fq=Fq composed with E inverse. For every natural radius r, let L(r)={(u,v) in Z^2 : |u|+|v| <= r}, c1(r)=|f1(L(r))|, cq(r)=|fq(L(r))|, and gap(r)=(c1(r):Z)-(cq(r):Z). Thus these counts refer to the actual signed balls for the ordered coordinate generators, rather than to separately posited count functions.
There exist additive equivalences e1:G1 -> H and eq:Gq -> H with the following values for every z=(z0,z1) in Z^2. The first is e1(F1(z))=(-z0+z1 mod p, -qt*z0+(qt-1)*z1 mod pD). The second is eq(Fq(z))=((-2t-1)*z0+(2t-1)*z1 mod p, -(qt+(q-1)/2)*z0+(qt-(q+1)/2)z1 mod pD). The group H is noncyclic. Both actual quotients are finite of order p^2D, as supplied by the asymmetric quotient theorem; finiteness, cardinality and these equivalences are not assumptions.
The primitive mechanism explains why the underlying group is unchanged. For the asymmetric pair take x1=qt-1, y1=qt, u1=(m-q^2)/2 and v1=-(m+q^2)/2. The primitive first column has Bezout row (alpha,beta)=(-1,1), with alphax1+betay1=1 and x1v1-y1u1=-D. For the symmetric pair take xq=qt-(q+1)/2 and yq=qt+(q-1)/2, with second column (xq,-yq). The row (alpha,beta)=(-2t-1,2t-1) satisfies alphaxq+betayq=1, and xq*(-yq)-yqxq=-D. Indeed 2xq=m-q and 2yq=m+q. The row (t,1-t) instead gives txq+(1-t)*yq=(q-1)/2, so it is suitable only at q=3.
For either primitive column (x,y), the integer coordinate map has rows (alpha,beta) and (-y,x) and determinant alphax+betay=1. Its integral inverse sends (A,B) to (xA-betaB,yA+alphaB). Reducing the first coordinate modulo p and the second modulo pD kills both actual lattice columns. The inverse lift proves surjectivity of the descended map; the already established actual quotient order p^2*D proves bijectivity. The same coordinate-descent supplier handles both markings. The noncyclicity follows from the actual quotient noncyclicity and either equivalence.
For every natural R, LeeCovers(F1,R) holds if and only if s <= R, and LeeCovers(Fq,R) holds if and only if s <= R. Here LeeCovers(F,R) means that for every g in the corresponding quotient there exists z in Z^2 with |z0|+|z1| <= R and F(z)=g. Thus both signed diameters are exactly s. For every natural r with r >= s, c1(r)=cq(r)=p^2*D. For every natural r, cq(r) <= c1(r), and cq(r) < c1(r) if and only if p(m-q)/2 <= (r:Z) and r < s.
For every natural r <= s put delta=pm-(2r+1)=2(s-r). Then delta is an even nonnegative integer. The complete gap regimes are: if delta >= qp then gap(r)=0; if p <= delta < qp then 2gap(r)=(qp-delta)^2+1; if delta < p then 2gap(r)=(q-1)^2pdelta. All divisions and equalities are discrete integer ones. Together with saturation beyond s these formulas describe every natural radius, including both transition boundaries and r=s.
To derive these regimes, write w_+=max(w,0) and T(r)=2r^2+2r+1. The imported actual counts give c1(r)+floor(((p-delta)+(q^2p-delta)++1)/2)=T(r) and cq(r)+floor(((qp-delta)_+^2+1)/2)=T(r). These are formulas for finite images of the two actual quotient maps. Subtraction cancels T(r). When delta >= qp both cap contributions vanish. In the middle regime only the symmetric cap remains. In the small regime both products are positive and the polynomial identity (qp-delta)^2-(p-delta)(q^2p-delta)=(q-1)^2p*delta gives the stated linear gap. Every positive width is odd, so product-plus-one is exactly divisible by two; a nonpositive width contributes zero.
Define M=(q-1)^2*p^2/2-(q-1)*p+1 and rstar=p(m-1)/2-1. The radius rstar is a nonnegative integer and rstar < s; its natural representative is rstar.toNat. For every natural r one has gap(r) <= M, and gap(r)=M if and only if (r:Z)=rstar. In particular there exists exactly one natural r with gap(r)=M. At q=3 this maximum is precisely 2p^2-2p+1.
The maximum is a discrete boundary comparison. Since p is odd and delta is even, the middle regime starts at delta=p+1. There, 2M-((qp-delta)^2+1)=(delta-p-1)((2q-1)p-delta-1). The first factor is nonnegative and the second is positive throughout that regime, so equality occurs exactly at delta=p+1. The largest small-regime gap is (q-1)^2p(p-1)/2, at delta=p-1. Its difference from M is (q-1)*(q-3)*p/2+1 > 0 for every allowed q, including q=3. The large regime and the saturated regime have zero gap. Finally delta=p+1 is equivalent to (r:Z)=p(m-1)/2-1, and the parameter lower bounds ensure that this integer is an available natural radius.
At the penultimate natural radius s-1, delta=2 and gap(s-1)=(q-1)^2*p > 0. Consequently there does not exist an additive equivalence e:G1 -> Gq carrying F1(1,0) to Fq(1,0) and F1(0,1) to Fq(0,1) simultaneously. Such an equivalence would identify every integral linear combination of the coordinate generators, hence every signed ball and its cardinality, contradicting the penultimate gap. The conclusion concerns a change of marking on the same noncyclic finite group.
This family supplies the common finite-group setting for the separate analytic profile with square parameter q^2: the pair (1,q^2) and the balanced pair (q,q) can now be compared without changing the underlying abstract group. The gap theorem asserts no q-parametric return-count formula, integer single-crossing threshold, continuous finite-count identity, or finite-count envelope for that analytic profile. Those connections require separate results. The comparison is a repository derivation from asymmetric_quotient_growth and the primitive-coordinate construction, with no claim of literature novelty.