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Exact growth from finite rank-one fibers

Abstract

Finite rank-one relation fibers determine Lee-ball images in arbitrary additive targets and every in-range ball of an actual composite maximum pair.

Theorem 1.1 (A nonzero displacement controls arbitrarily long finite fibers).

Lean statement: D5/S3/FiniteGroups/BajnokBeattyExtremalGrowth.lee_ball_image_card_of_displacement

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/BajnokBeattyExtremalGrowth.lee_ball_image_card_of_displacement (✓ 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.

Write Dᵣ={(x,y)∈Z²: |x|+|y|≤r}. Let f:Z²→G be additive for an arbitrary additive commutative group G, and let h=(h₁,h₂)≠0 satisfy f(h)=0. Assume every kernel vector of Lee norm at most 2r is an integer multiple of h. Set A=max(0,2r+1−|h₁+h₂|) and B=max(0,2r+1−|h₁−h₂|). Then |f(Dᵣ)|+⌊(AB+1)/2⌋=2r²+2r+1. The integer positive parts in A and B are taken before converting to natural numbers. Either width may vanish. There is no finiteness or surjectivity condition on G or f, no primitivity or sign condition on h, and no additional radius restriction.

All points in a fiber inside Dᵣ differ by integer multiples of h, because the difference of two diamond points has Lee norm at most 2r. Integer norm interpolation shows that the admissible multiples form a finite interval: if y−x=k h with k>0, then k(y−h)=(k−1)y+x, so the triangle inequality and homogeneity force y−h back into Dᵣ. Remove Cᵣ=Dᵣ∩(Dᵣ+h), retaining Rᵣ=Dᵣ∖Cᵣ. A retained point is the bottom of its fiber in the h direction. If two retained points share an image, a positive multiple forces the later one into Cᵣ; a negative multiple swaps their roles. Thus f is injective on Rᵣ even when a fiber has many points.

For surjectivity choose, in each nonempty finite fiber, a point w minimizing w·h. A point of Cᵣ has w−h in the same fiber, with dot product lower by h₁²+h₂²>0. A minimizer therefore survives the trimming. This finite minimum replaces a one-step deletion argument and requires neither a primitive displacement nor a bound on fiber length. Consequently Rᵣ maps bijectively onto f(Dᵣ), and |f(Dᵣ)|+|Cᵣ|=|Dᵣ|.

Under S(x,y)=(x+y,x−y), Cᵣ is the equal-parity part of the product of two integer intervals. The interval for t=h₁+h₂ or h₁−h₂ is [max(−r,t−r),min(r,t+r)], with width max(0,2r+1−|t|). Equal-parity coordinates have the integral inverse ((u+v)/2,(u−v)/2). In each interval the parity of r occurs ⌈width/2⌉ times and the other parity ⌊width/2⌋ times; for odd positive widths the initial endpoint has the parity of r, for either sign of t. The existing residue-count and filtered-product cardinality suppliers give |Cᵣ|=⌈A/2⌉⌈B/2⌉+⌊A/2⌋⌊B/2⌋=⌊(AB+1)/2⌋, including empty intervals. The inherited two-parity LeeCode gives |Dᵣ|=(r+1)²+r². These counts combined with the fiber bijection prove the balance.

Theorem 1.2 (A finite transversal in an arbitrary additive target).

Lean statement: D5/S3/FiniteGroups/BajnokBeattyExtremalGrowth.lee_ball_image_card_of_short_kernel

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/BajnokBeattyExtremalGrowth.lee_ball_image_card_of_short_kernel (✓ 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.

Write Dᵣ={(x,y)∈Z²: |x|+|y|≤r}. Let f:Z²→G be additive, where G is any additive commutative group, let B>0 and r<2B be natural numbers, and set h=(0,2B). Assume f(h)=0 and every relation z of Lee norm at most 2r is 0, h or −h. Then the finite image f(Dᵣ) has cardinality 2r²+2r+1 for r<B and 2B(2r−B+1) for B≤r. The target need not be finite and f need not be surjective.

This axial API is a specialization of the preceding balance, with h=(0,2B). Its three allowed short relations 0,h,−h are mapped to the multiples 0•h,1•h,(−1)•h. Both rectangle widths are max(0,2r+1−2B). For r<B the cap is empty. Otherwise its cardinality is 2(r−B)²+2(r−B)+1, and the balance gives 2B(2r−B+1). Geometrically max(|y|,|y−2B|)=B+|y−B| also identifies the overlap with (0,B)+Dᵣ₋ᴮ. The shared finite-fiber theorem supplies the actual proof; no separate axial trimming argument is needed. The original radius premise is retained in this public API and supplied by the actual maximum-pair consumer below.

Theorem 1.3 (Every actual composite maximum pair has the same ball growth).

Lean statement: D5/S3/FiniteGroups/BajnokBeattyExtremalGrowth.sharp_pair_growth

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/BajnokBeattyExtremalGrowth.sharp_pair_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 s≥3 be natural, n=2s+1 composite, p its least prime factor, and B=(n−p)/2. The inherited parameter theorem supplies n=p(2t+1) with t≥1, so B=pt and A=p(t+1). Let G be a finite noncyclic additive commutative group in any universe, with distinct a,b, signed radius-s cover by a,b, and Nat.card G=MaximumOrder(s). For every natural r≤s, the signed Cayley ball {x•a+y•b: |x|+|y|≤r} has cardinality 2r²+2r+1 for r<B and 2B(2r−B+1) for B≤r. These hypotheses concern the actual pair; they do not select a particular model group or marking.

For the actual coefficient map, Smith normal form supplies one common divisor c≥2 together with containment of its relation kernel K in cZ². Its actual index is Nat.card G. Since p²≤n, the maximum order is greater than 2s². The inherited lattice upper bound therefore forces odd c dividing n and c²≤p². The least-factor inequality gives p≤c, so c=p. Keeping this same containment is essential for the compression that follows.

Apply S(x,y)=(x+y,x−y), put M=S(K), and let L={z:pz∈M}. The inherited square lift, residue rectangle and scaled-index laws show that L covers the square of side m=n/p and has actual index m²−1. NearSquareLatticeRigidity.near_square_rigidity at deficit one forces its two displacement parameters to be one. Its basis is (m,−ε),(−1,εm), with ε=±1. Taking integral preimages in the original S-image gives the two diagonal relations and, for every kernel vector, coordinates (A(u−v),B(u+v)), or the coordinate swap. No marked-group isomorphism is needed.

Here A+B=2s+1, A−B=p≥3, and s<2B. A nonzero first coordinate in the displayed diagonal form costs at least A+B or 2A, both greater than 2s. With first coordinate zero, the second coordinate is a multiple of 2B; every multiple beyond ±2B costs at least 4B>2s. Consequently the only possible short nonzero relations are (0,±2B), or their swaps, and the corresponding relation is actually present. Coordinate swap preserves Dᵣ. Applying the preceding general counting theorem therefore proves the ball formula for every r≤s.

Consecutive nested balls yield sphere sizes without an additional assertion: sphere zero has size one; for 1≤r≤s the size is 4r when r<B, 4B−1 when r=B, and 4B when r>B. At r=B exactly one pair is identified. At r=s the second branch is 2AB=Nat.card G, as required by the cover. The radius restriction is essential: the known s=4 product ZMod 3×ZMod 12 is already exhausted at radius four, so radius five has 36 points whereas the extended expression would give 48.

Bajnok–Beatty arXiv:2406.04045v1 supplies the Cayley-diameter problem and the known lower families, not an attribution of the cap-transversal proof to that source. BajnokBeattyFiniteUpper, LeeSquareCompression, NearSquareLatticeRigidity and BajnokBeattySharpMaximum supply the kernel containment, compression, rigidity, maximum and nontruncating arithmetic used in this derivation. This is a formal consequence of those inputs with the finite-fiber counting mechanism above; no current-literature priority or novelty is asserted.

References