An adaptive binary grain controls both the inventory and the new charge
Abstract
A physical joint order when the first two gaps are equal and the last is larger
Theorem 1.1 (The entire family 2<=a=b<c).
Lean statement: D5/S3/Combinatorics/Circle/ThreeNegativeEqualInnerLargeCJointLift.actualSource_three_negative_equal_inner_large_c_joint_lift
Proof. Machine-checked in Lean as D5/S3/Combinatorics/Circle/ThreeNegativeEqualInnerLargeCJointLift.actualSource_three_negative_equal_inner_large_c_joint_lift (✓ std3). ∎
Source. Repository-derived.
Commentary.
Let a,b,c be independent natural coordinates with 2<=a, a=b and b<c. Put k=a+b+c, u=1+2^a+2^(a+b), N=2^k-1 and C=(N-1)/2. There exist one permutation sigma of the physical indices 0,…,k-1 and one natural cut j<=k such that sigma and the word obtained by inserting the fresh index k at cut j satisfy the two prefix bounds simultaneously. The old values are f(i)=center_N(2^i*u); the new values are g(i)=center_(2N+1)(2^(i+1)*u), including g(k)=u. Every old prefix lies in [0,C], every new prefix lies in [0,N], and both complete integer totals are zero. Here center_m is the unique integer representative in [-(m-1)/2,(m-1)/2] for an odd modulus m. Every old index keeps its physical identity in the new word. This includes a=2 and arbitrarily large c-a, without a coprimality, numerical-period or distinct-value hypothesis.
Write r=2^a, q=2^(c-a), t=qr, N=r^2t-1, C=(N-1)/2, u=1+r+r^2, B=1+t+tr and L=1+r+rt. The actual physical positive 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 pairs are (u2^i,2u2^i), (B2^i,(2B-1)2^i) and (L2^i,(2L-r-1)*2^i). The negative labels alpha=c-1, beta=c+b-1 and gamma=k-1 have magnitudes H_alpha=(N-B)/2, H_beta=(N-L)/2, H_gamma=(N-u)/2 in the old source. In the new source these become K_alpha=2H_alpha+1, K_beta=2H_beta+r/2+1 and K_gamma=2H_gamma+(u+1)/2. The fresh value is u. The previously established paired-chain identities apply in this physical phase and give both exact total-zero identities.
Choose d=ceil(q/u), h=2^max(0,c-a-2a), z=2hfloor(r^2(q-1)/(2h*u)) and p=(r/2-1)*q+d. Thus h=1 for q<r^2 and h=q/r^2 for q>=r^2. The two natural mask triples are P=(p,0,0) and T=(z,0,r/2-1). They denote literal binary supports in the U,V,W chains; Q is the physical complement after removing the three negatives. Use sigma=[P,alpha,Q,beta,T,gamma], with each positive block in a fixed order. Insert the fresh label at the start of T if q=2 and at the start of Q otherwise. The respective literal cuts are j=k-1-popcount(z)-(a-1) and j=popcount(p)+1.
The grain must serve two purposes at once. The ceiling obeys 1<=d<=h<q, and 2h is a power of two dividing q. Below r^2 the ceiling is exactly one; above it, q=r^2*h and u>r^2 give d<=h. The floor gives 0<=z<q-1 and makes z a multiple of 2h. In p, the term (r/2-1)*q occupies bits at or above q, while d occupies bits strictly below 2h. Every bit of z lies between those bands, so the two U-supports are disjoint. Also p<t/2 and z<t/2; the other mask capacities hold, all W labels belong to T, and all V labels belong to Q. This is a disjoint exhaustive inventory of the original physical labels, rather than scalar mass feasibility alone.
The first old landing is epsilon=up-H_alpha=du+1-q, with 1<=epsilon<=u. Its new landing is A=2epsilon-1>=1. For q=2, d=1 and epsilon=u-1=(B-1)/2. For every larger dyadic q, q>=4 and epsilon<=u<=(B-1)/2. Hence the first old peak is at most C and the first new peak is at most N-1. The old last landing is delta=H_gamma-o(T)=r^2*(q-1)-uz, satisfying 0<=delta<2hu. The new last margin is D=K_gamma-n(T)=2delta+r^2. The r^2 term includes both the full W-chain charge and the negative gamma charge; omitting it would change the physical phase.
When q=2, h=d=1 and z=0. Consequently delta=r^2<=(L-1)/2 and D=3r^2>=u, so the fresh label belongs at the start of T. When q=4, 2u<=3r^2<4u forces z=2, giving D=3r^2-4r-4<qr^2+r/2. For 8<=q<r^2, h=1 gives D<5r^2+4r+4<8r^2+r/2<=qr^2+r/2. Finally, for q>=r^2, the inequalities 16u<=21r^2 and r^2<=q give D<4hu+r^2<=(25/4)q<qr^2<qr^2+r/2. These branches exhaust all powers q>=2. Thus every q>=4 permits the Q insertion, with D<=lambda_beta=qr^2+r/2. That same bound implies 2delta<=L-1, supplying the old tail bound as well.
The old seam heights are 0,o(P),epsilon,H_beta+delta,delta,H_gamma,0. For the Q insertion the new seams are 0,2o(P),A,A+u,K_beta+D,D,K_gamma,0. For the T insertion they are 0,2o(P),A,K_beta+D-u,D-u,D,K_gamma,0. In particular both sides of the inserted label appear. The selected inequalities put all old seams in [0,C] and all new seams in [0,N]. Every interior of a positive block is monotone between its seams. The disjoint exhaustive inventory supplies the permutation and the literal cut while keeping the same physical indices in both words. The exact total-zero identities also cover empty prefixes and every requested prefix length beyond the word.
The new inference is the unbounded adaptive grain selection that simultaneously separates the ceiling mask from the rounded tail mask and bounds the charged remainder. The argument reuses the paired chains, exhaustive physical inventory and monotonicity within positive blocks. This contributes to the investigation of Conjecture 3.1 of Damasdi, Frankl, Pach and Palvolgyi, Monochromatic configurations on a circle, arXiv:2504.10687v2; no literature novelty is asserted. The scope is exactly 2<=a=b<c. The full physical joint order J, the unequal-tail regimes and full Circle C3.1 remain open.