Supplement 1

Witness an integer layer

W = (1 1), b = 1, and s0 = 5 send (4, 7) and (5, 6) to the same y = 2 and ρ = 2. The witness distinguishes them. This page does not replace the 399 ledger.

A rank-losing integer layer can send two inputs to the same quotient and remainder. The retained witness separates them.

W, one row per linebias bdivisor s0input xsecond input

Collision before the witness, then the retained witness and the recovered input.
	First input	Second input
y	2	2
ρ	2	2
k	7	6
recovered x	4, 7	5, 6
serialized bytes	77	77

Rank 1. U = 1. D = 1 0. V = 1 -1 | 0 1. The identity U W V = D and det U, det V ∈ {±1} were checked for this factorization. For this chosen V, those witnesses are the second input coordinates. Omitting k leaves the shared quotient and remainder.

First witness record: {"schema":"INTEGER_LAYER_WITNESS/1","s0":"5","y":["2"],"rho":["2"],"k":["7"]}

Import witness record

No witness record has been imported.