Exact attained width for geometric circle sources
Abstract
Every positive geometric source has an explicit physical-label permutation whose exact attained width is at most the half-modulus capacity.
Theorem 1.1 (Exact width after relocating the low physical labels).
Proof. Machine-checked in Lean as D5/S3/Combinatorics/Circle/GeometricSourceRelocation.geometric_source_relocation (✓ std3). ∎
Source. Repository-derived.
Commentary.
Let r,m,d be arbitrary positive naturals, B=2^m, k=rm+d and t=(B^r-1)/(B-1), with the quotient taken in the naturals. Put a=d-1, A=2^a, C=2^(k-1)-1 and W=C-min(A-1,(t-1)/2), with integer subtraction and Euclidean division in the width. The literal word rotates the first k-1 entries of finRange(k) by a and appends its last entry. Thus its labels are a,…,k-2,0,…,a-1,k-1. It is a permutation of the original Fin(k) labels. Every natural-length prefix, including saturated takes beyond k, has integer sum in [0,W], the total is zero, W<=C, and some prefix of length at most k attains W. This includes r=1, m=1, d=1, their intersections and the empty low block.
Multiplication by two on residues modulo N=2^k-1 is a cyclic action because 2^k=1 modulo N. Each original label retains its position in that action even when values coincide. The unsigned representative U_i=s_i+N when the centered source s_i is negative, and U_i=s_i otherwise, is an integer potential: U_(i+1)-U_i=s_i, including the last-to-first edge. Contiguous physical sums therefore telescope in the integers; a modular relation alone would not control their heights.
The geometric quotient equals the r-term sum of powers of B. Write L_j=sum_(e<j) B^e and u_j=2At-(2A-1)L_j. The exact source identity 2^(jm+d)t=N L_j+u_j, together with representative range checks, gives the negative-label representatives at nu_j=a+jm for 1<=j<=r. The nonfinal returns satisfy At<=u_j<=2At, and all intervening powers of two remain in the same full physical envelope At<=U_i<=ABt for a<=i<=k-1. Its upper endpoint is attained at i=a+m; the last unsigned and centered representatives are (N+t)/2 and (t-N)/2. These bounds are derived from the source, with no annulus, unit, distinctness or true-period hypothesis.
The main block has heights U_(a+n)-At and attains M=ABt-At=C-(A-1) at n=m. Reinserting the low labels adds only nonnegative actual increments, ending at L=C-(t-1)/2 at n=k-1. The final physical label returns to zero. Both M and L are visited, so their maximum is exactly W. This is the attained width of the displayed word; no minimum over all permutations or global optimality is claimed.
The source reference is Damasdi–Frankl–Pach–Palvolgyi, Monochromatic configurations on a circle, arXiv:2504.10687v2, Conjecture 3.1, DOI 10.5070/C6.68671. This symbolic family construction is repository-derived and makes no literature-novelty claim. The earlier ThreeParentFamily sources form the sign-and-rotation subfamily r=3,m=2,d=R-2 for R>=4: modulo N, -2^(R-1)21 equals 112^(R-1)-1. Their sources overlap; their schedules and optimality statements are not identified here. The full arbitrary-source Conjecture 3.1 remains open. No unit-cycle, girth or full-demand conclusion outside this source family is asserted.