One order for both physical sources
Abstract
A literal first ear throughout the strict ascending d=2 chamber
Theorem 1.1 (All independent coordinates 2<=a<b<c).
Lean statement: D5/S3/Combinatorics/Circle/FourNegative/AscendingD2/Endpoint.actualSource_strict_ascending_d2_joint_lift
Proof. Machine-checked in Lean as D5/S3/Combinatorics/Circle/FourNegative/AscendingD2/Endpoint.actualSource_strict_ascending_d2_joint_lift (✓ std3). ∎
Source. Repository-derived.
Commentary.
For every three independent natural numbers a,b,c satisfying 2<=a<b<c, let k=a+b+c+2 and A=1+2^a+2^(a+b)+2^(a+b+c). There is one permutation sigma of the original Fin(k) physical labels such that OriginalOrder(k,A,sigma) and OriginalOrder(k+1,2A,earWord(k,sigma,0)). The fresh label k is inserted before every old label. Each OriginalOrder includes the exact complete integer total zero and the half-modulus strip for every natural take-prefix, including the empty prefix and every overlong take. No coprimality, primitive period, favorable phase, or externally supplied mask is assumed. Labels with equal numerical values remain distinct physical labels.
Put R=2^a,T=2^b,Z=2^c,N=4RTZ-1,M=2N+1 and A=RTZ+RT+R+1. The old and new values at each original label i are f_i=centered_N(A2^i) and g_i=centered_M(2A2^i). The fresh new value is A. The positive chains are U=[0], V=[2,…,c], W=[c+2,…,c+b], X=[c+b+2,…,c+b+a]. Their four negative successors are U-=1,V-=c+1,W-=c+b+1,X-=a+b+c+1. A chain coefficient multiplies 2^h at its h-th positive label.
The paired positive chain coefficients (old,new) are U=(A,2A), V=(4RT+4R+5,8RT+8R+9), W=((4R+5)Z+1,(8R+9)Z+1), X=(5TZ+T+1,9TZ+T+1). The doubled negative pairs are U-=(-4RTZ+4RT+4R+6,-8RTZ+8RT+8R+10), V-=(-4RTZ+(4R+5)Z+2,-8RTZ+(8R+9)Z+2), W-=(-4RTZ+5TZ+T+2,-8RTZ+9TZ+T+2), X-=(-3RTZ+R(T+1)+2,-7RTZ+R(T+1)+2). Four cyclic shifts give these exact centered representatives. The already established full-period source identity gives total f=0 and total g=-A; adding the fresh label gives total zero.
For a>=4 write r=2^(a-4),s=2^(b-a-1),t=2^(c-b-1), so R=16r,T=32rs,Z=64rst and r,s,t>=1. Select X mask mass 5r=2^(a-2)+2^(a-4) for block 1; block 2 takes the X complement of mass 3r-1 and W mask (8r+1)s=2^(b-2)+2^(b-a-1). Block 3 takes the W complement (8r-1)s-1 and V mask (16rs+1)t=2^(c-2)+2^(c-b-1). Block 4 takes the remaining V labels and U=0. Append the terminal negative labels X-,W-,V-,U- to these four positive blocks respectively. These masks and complements have disjoint binary supports in their actual physical chains.
For a=2 or a=3 and b>=4 put r=2^(a-2),u=2^(b-4),v=2^(c-b-1), so r is 1 or 2, T=16u,Z=32uv. Block 1 takes X mass r and W mass u. Block 2 takes X mass r-1 and W mass 4u. Block 3 takes W mass 3u-1 and V mass (8u+1)v. Block 4 takes the remaining positives, including U. Again the negative order is X-,W-,V-,U-. The two X masks are complementary. The W masks u and 3u-1 partition the low b-2 positions, and 4u is the separate highest bit. Thus these are literal disjoint masks for every b>=4, including r=1 where the second X mask is empty.
For a=2,b=3,c>=7 put z=2^(c-7), so Z=128z. The three selected (V,W,X) mask triples are (8z,0,1), (z,2,0), (32z,1,0). They give the literal blocks [c+5,c-2,X-], [c+3,c-5,W-], [c+2,c,V-], followed by the remaining V positives, U, and U-. The V selections are the distinct binary positions c-4,c-7,c-2; all are valid when c>=7.
The prefix argument uses the exact doubled negative values above. If P,Q,S are the old positive masses of the first three blocks, their successive doubled peaks are 2P,2P+2f_X-+2Q,2P+2f_X-+2Q+2f_W-+2S; adding the corresponding doubled negative gives each following doubled seam. In the new coordinate the same expressions begin with 2A and use the new masses and g. Every seam is nonnegative and every peak is at most twice its strip capacity: 2RTZ-1 for the old coordinate, 4RTZ-1 for the new. Substitution of each displayed mask reduces these exact inequalities to polynomials with nonnegative coefficients in r-1,s-1,t-1, or u-1,v-1, or z-1. The last peak is exactly -f_U- or -g_U- by the complete total identities, and the final seam is zero. Each block has nonnegative increments before its terminal negative, so its interior prefixes lie between its entering seam and its peak. This establishes every prefix in both coordinates, not just the four block endpoints; 0<=A<=N includes the initial ear.
The three remaining cases a=2,b=3,c=4,5,6 have explicit physical permutations. For c=4, sigma=[9,2,8,7,10,4,6,5,0,3,1], with old prefixes [0,649,798,99,773,24,620,957,102,651,949,0] and new prefixes [0,549,1710,2007,540,1854,81,1269,1926,207,1305,1899,0]. For c=5, sigma=[8,2,9,10,3,11,5,7,6,0,4,1], with old prefixes [0,1346,1495,92,1381,1679,162,1354,2027,316,1377,1973,0] and new prefixes [0,1061,3687,3984,1045,3358,3952,387,2763,4076,637,2759,3947,0]. For c=6, sigma=[9,3,10,11,4,12,6,8,7,0,5,2,1], with old prefixes [0,2690,2988,177,2746,3342,289,2673,4018,595,2680,3872,4021,0] and new prefixes [0,2085,7335,7929,2046,6663,7851,702,5454,8079,1200,5370,7746,8043,0]. Direct finite integer arithmetic establishes these three cases; the unbounded cases use the uniform masks and inequalities.
This result concerns the strict ascending d=2 three-coordinate chamber of the physical four-gap problem associated with Circle Conjecture 3.1 in Damasdi, Frankl, Pach and Palvolgyi, Monochromatic configurations on a circle, arXiv:2504.10687v2. The unrestricted four-gap J problem and Circle C3.1 remain open. No claim of external mathematical novelty is made.