Primitive signed lattice levels on the whole prefix class
Abstract
The whole oriented legal-prefix kernel class has primitive signed integer coordinates, with a cumulative converse.
A triple w:Fin 3 -> List Bool consists of three arbitrary finite chronological native multipliers, including empty words. False denotes M=[[0,1],[1,1]] and true denotes J=[[0,1],[1,0]]. The endpoint A_i=prefixMatrix(w,i) is the product of the first i words, with original reading indices i=0,1,2,3. Thus A_0=I and A_(i+1)=futureProduct(w_i) A_i. The legal monoid theorem supplies nonnegative integral entries and determinant plus or minus one for every actual endpoint. Individual endpoint legality is never used to infer that a transition is executable.
On all real 2 by 2 matrices let B_w(X)i=trace(A(i+1)X)-trace(X), for i in Fin 3. G_+(w,N) means the unchanged OrientedKernelGenerator predicate: N is nonzero of actual rank one, the entire kernel of B_w consists exactly of real multiples of N, and the determinant polar with N is strictly positive on every entrywise positive rank-one source. N is an arbitrary real positive orientation, with no rationality assumption.
The coordinate domain P consists of quadruples p=(a,b,c,d) of nonnegative integers with gcd(a,b)=gcd(c,d)=1 and t=ac+bd>0. Write f(p)=(a,-b), row(p)=(c,-d), r(p)=(d,c), and F(p)=f(p) row(p)=[[ac,-ad],[-bc,bd]]. Vectors f and r are columns. These signed objects are auxiliary coordinates, and are not acquired positive sources. Zero coordinates and zero matrix entries are included.
Theorem 1.1 (Whole-class primitive representation and cumulative converse).
Proof. Machine-checked in Lean as D5/S3/Observer/CubicGain/PrimitiveKernelLattice.whole_class_primitive_lattice (✓ std3). ∎
Source. Repository-derived.
Commentary.
The two universal clauses are independent. For every actual word triple and every oriented real generator, there are p in P and a positive real scale such that N=scale F(p), the real kernel line is generated by F(p), and integer parameters m_0,m_1,m_2,m_3 satisfy m_0=0 and A_i f(p)=f(p)+m_i r(p) at every original index. Conversely, for every actual word triple, every p in P and every such integer parameter function, these endpoint equations and rank(B_w)=3 imply G_+(w,F(p)). Rank here is the real dimension of the linear map range. The converse retains actual cumulative words; neither lattice equations nor separate endpoint legality replace them.
The real generator is only a positive real multiple of the primitive integer outer product. For example, irrational rescalings remain in the same oriented line. No equality between an arbitrary real generator and an integer matrix is asserted. There are no strict-positivity assumptions on a,b,c,d individually, no word-length bounds, and no assumptions already encoding a rational kernel, primitive factorization or integer endpoint parameter.
First form the rational prefix-difference map using integral representatives of the actual endpoint entries. It has a four-dimensional rational source and three-dimensional target. The finite-dimensional kernel theorem produces a nonzero rational matrix in its kernel. The finite common-denominator theorem gives a nonzero integer matrix, and the finite trace expansion transports its kernel equation to the real map. The given kernel-line condition makes this matrix a nonzero real multiple of N. Change its sign if necessary to obtain a positive multiple. Lemma 3.1 is used on this live construction to supply all four signs and strictly positive trace. Actual rank one supplies determinant zero.
Write the oriented integral matrix as [[P,-Q],[-R,S]] with nonnegative natural entries, determinant equation PS=QR and P+S>0. If P=0, either Q=0 or R=0. In the first branch choose column (0,-1) and divide the pair (R,S) by its positive gcd; in the second choose row (0,-1) and divide (Q,S) by its positive gcd. These branches include simultaneous permitted zeros. If P>0, divide (P,R) by its positive gcd g, obtaining a coprime pair (a,b) with a>0. The determinant relation gives aS=Qb. Coprimality forces a to divide Q, so Q=aD and S=bD. Divide (g,D) by its positive gcd k, obtaining a coprime pair (c,d). The original integer matrix is k F(p), and its positive trace gives t>0. Comparing the two positive scales gives the required real scale for N.
For the endpoint lattice use integer Bezout coefficients alpha,beta with calpha+dbeta=1. If cX-dY=ac+bd, the integer m=beta*(X-a)+alpha*(Y+b) satisfies X=a+md and Y=-b+mc. Both identities follow by multiplying the Bezout identity and subtracting the level equation; the same functional proves uniqueness. This calculation divides by neither c nor d, so it includes c=0,d=1 and d=0,c=1. The trace equation for the primitive outer product is exactly this level equation on the integer column A_i f. The initial identity endpoint has parameter zero.
For the converse, multiply the recorded column equations by the signed row. Since row(p) r(p)=0, F(p) is in the prefix-difference kernel. Its positive trace makes it nonzero, and its outer-product form has actual matrix rank one. Rank-nullity and rank(B_w)=3 make that nonzero vector span the entire real kernel. On any entrywise positive full matrix S the polar is bdS00+bcS01+adS10+acS11. All coefficients are nonnegative, and ac+bd>0 makes at least one diagonal coefficient positive. Thus the polar is strictly positive, including on every actual positive rank-one source. The original supplied words already realize each chronological transition, with no inverse, reset, extra read or alternative history.
This representation concerns the whole fixed-sign rank-one prefix class. It supplies neither the later four-nonzero-entry normal forms nor a second positive source, an all-slope collision, global recovery, or a sharper read bound. Replacing supplied cumulative words by endpoint quotient inequalities requires the exact cumulative continuation criterion; the equations alone do not provide that replacement.