One physical permutation at literal cut zero
Abstract
The native ceiling yields both complete physical source orders
For all natural a,b,c,d with a>=2, d>=a+1, c>=a+d+1 and b>=c+1, put k=a+b+c+d and A=1+2^a+2^(a+b)+2^(a+b+c). The private joint construction produces one list sigma of Fin k labels satisfying OriginalOrder for the old source and for the enlarged source on earWord k sigma 0. The public Endpoint exposes this same four-data, four-proof-binder family with a cut witness equal to zero.
The power identities x=2^a, t=2^d=2xR, z=2^c=2xtZ and y=2^b=2zlambda connect the four independent gap parameters to Selection. Its bounded signed H is converted to a natural number only after nonnegativity and the upper bound are known. NativeMasks then partitions the actual W-positive labels using h=lambdaH, m=(z-1)lambda and low=lambda-1. Source supplies the full permutation and both rows of exact bank masses.
For the old row, terminal identities and SeamBounds put the entire interleaved word in the strip [0,N/2], where N=xyzt-1. For the enlarged row, the fresh value is A and the old-label tail lies in [0,N] starting from A. The fresh gate supplies 0<=A<=N. The four-strip argument handles every intermediate prefix, and the source zero totals close the last bank. The physical lift then retains all natural take-prefix bounds, including empty, full and overlong takes, and exact zero totals.
The enlarged list inserts the one fresh Fin.last k label at literal cut zero and retains all old labels. The construction uses actual centered source values and does not require distinct numerical values or orbit primitivity. It proves the separated gap cone only; the unrestricted b-maximum joint lift and full Circle J/C3.1 remain open.