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The complete minimum-gap-two construction

Abstract

One physical order and one literal ear for minimum gap two

Theorem 1.1 (All three orientations, with unbounded remaining gaps).

Lean statement: D5/S3/Combinatorics/Circle/ThreeNegativeMinimumTwoJointLift.actualSource_three_negative_minimum_two_joint_lift

Proof. Machine-checked in Lean as D5/S3/Combinatorics/Circle/ThreeNegativeMinimumTwoJointLift.actualSource_three_negative_minimum_two_joint_lift (✓ std3). ∎

Source. Repository-derived.

Commentary.

For every natural a,b,c with 2<=a, 2<=b, 2<=c and min(a,min(b,c))=2, put k=a+b+c and u=1+2^a+2^(a+b). There exist one list sigma of Fin(k) labels and a natural j<=k such that OriginalOrder(k,u,sigma) and OriginalOrder(k+1,2u,earWord(k,sigma,j)). Each OriginalOrder requires permutation of every physical label, complete integer total zero, and every natural take-prefix in the closed interval from zero to half the corresponding modulus. The second word inserts the fresh physical label k at literal position j, preserving the first word’s entire list and the identity embedding of all its old labels.

Write r=2^a, s=2^b, t=2^c, N=rst-1, C=(N-1)/2, B=1+t+tr and L=1+s+st. The positive physical chains U_i=i, V_i=c+i and W_i=c+b+i have lengths c-1,b-1,a-1. Their old and new value pairs are (u2^i,2u2^i), (B*2^i,(2B-1)2^i), and (L2^i,(2L-s-1)*2^i). The negative physical labels alpha=c-1,beta=c+b-1,gamma=k-1 have old magnitudes H_alpha=(N-B)/2,H_beta=(N-L)/2,H_gamma=(N-u)/2 and new magnitudes K_alpha=2H_alpha+1,K_beta=2H_beta+s/2+1,K_gamma=2H_gamma+(u+1)/2. These are actual centered-source identities, obtained from the original chains at (a,b,c), the chains at (a,b,c+1), and the physical index shift by one. The full-period identity gives the actual fresh value u.

Each row specifies two triples of binary masks P,T, a negative order pi1,pi2,pi3, and an ear block. Each mask is nonnegative and below its own chain capacity, and the P and T masks have disjoint bit supports in that chain. The physical block Q is the complement of those supports and the three negative labels. Thus sigma=[P,pi1,Q,pi2,T,pi3] contains every old label exactly once. No condition distributes one member of each common depth to each block. The ear positions are zero, |P|+1, and |P|+|Q|+2 for blocks P,Q,T respectively; the last also equals k-1-|T|.

The constructor has thirty branches: seven two-small-coordinate axis rows, five rows with a=2 and b,c>=3, five with b=2 and a,c>=3, and thirteen with c=2 and a,b>=3. It retains the finite exceptional rows together with the unbounded families. The first a=2 family uses P=((t/s)(s/2-1),0,0), T=(0,floor((st-3s-4)/(5t+1)),1) when s<=t. The other a=2 branches distinguish s=2t, s=4t and s>=8t. The b=2 branches distinguish t>=2r, t=r, r=2t and r>=4t, with the exceptional pair (r,t)=(16,8). The c=2 branches retain the small r or s families and distinguish r=s,2r=s,r=2s,r>=4s and s>=4r in the remaining range. Distinct powers of two make this case division exhaustive.

For each row put x=o(P), z=o(T), xn=n(P)+u if the ear is P and n(P) otherwise, and zn=n(T)+u if the ear is T and n(T) otherwise. The old seam obligations are H_pi1<=x<=C and z<=H_pi3 with H_pi2+H_pi3-z<=C. The new seam obligations are K_pi1<=xn<=N and zn<=K_pi3 with K_pi2+K_pi3-zn<=N. The intermediate positive prefixes stay between their endpoint heights. Full integer totals vanish by the actual source-zero theorem and complete physical inventory, separately for k and k+1. These facts determine the remaining middle seams and prove both entire paths.

The exact floor and ceiling masks are part of the construction. In the final c=2 regimes the W ceiling is ceil((2r+2-9s/4)/(9s+1)), bounded strictly below r/64, and the V floor is floor((rs-r-2)/(4r+5)), bounded strictly below s/4. The T mask 25r/64 is checked against their actual bit support. In the first four a=2 rows, the ear is Q when K_gamma-n(T)<=st+s/2, and T otherwise. The estimate st+s/2>=u justifies the second branch. The allocation and arithmetic obligations concern unbounded exponents; finite centered-residue checks do not discharge them.

This ordinary constructor is derived within the repository research on Conjecture 3.1 of Damasdi, Frankl, Pach and Palvolgyi, Monochromatic configurations on a circle, arXiv:2504.10687v2. It is not a claim of literature novelty. It addresses exactly the minimum-gap-two base; full C3.1 and the remaining general three-gap frontier stay open. The separately unfinished source-bridge information audit is not certified by this construction.

References