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Exact Clifford Leaf Observation Fibers

Abstract

The Clifford leaf product has exactly six distinct canonical phases.

Let Q(a,b)=aa+ab-bb on the real coordinate plane, and let C be its Clifford algebra with vv=Q(v)1. Write A and B for the canonical images of (1,0) and (0,1). Sources are the existing ordered Boolean leaf trees; alpha is true and beta is false. The existing substitution sends alpha to beta and beta to (beta,alpha).

The quadratic form Q and ordered leaf product E are those of FixedHistoryComposition. The free-magma homomorphism E sends alpha to A and beta to B, and multiplies the leaves in source order.

Definition 1.1 (Canonical observation).

Lean statement: D5/S3/Arith/FibonacciAtomic/CliffordLeafOrbit.X

Formalization. D5/S3/Arith/FibonacciAtomic/CliffordLeafOrbit.X (✓ std3).

Source. Repository-derived.

Commentary.

X(j)=E(rho^j(alpha)) for every natural index j.

Definition 1.2 (Six chronological phases).

Lean statement: D5/S3/Arith/FibonacciAtomic/CliffordLeafOrbit.phases

Formalization. D5/S3/Arith/FibonacciAtomic/CliffordLeafOrbit.phases (✓ std3).

Source. Repository-derived.

Commentary.

The six values are A, B, BA, A+B, -B, AB, in that order.

Definition 1.3 (Reader on the canonical image).

Lean statement: D5/S3/Arith/FibonacciAtomic/CliffordLeafOrbit.Factors

Formalization. D5/S3/Arith/FibonacciAtomic/CliffordLeafOrbit.Factors (✓ std3).

Source. Repository-derived.

Commentary.

A target g factors when a function on the range of X takes X(j) to g(j) for every natural j. The reader receives only the algebra element.

Theorem 1.4 (Exact fibers and canonical successor).

Proof. Machine-checked in Lean as D5/S3/Arith/FibonacciAtomic/CliffordLeafOrbit.result (✓ std3). ∎

Source. Repository-derived.

Commentary.

Here c is the existing composition, M(a,b)=(b,a+b), T(j)=rho^j(alpha), and P denotes the displayed six-element phase list. For every type Y and every g from the natural numbers to Y, a reader exists exactly when g is six-periodic. This includes the successor target g(j)=X(j+1).

The Clifford square and polar relations give AA=1, BB=-1 and AB+BA=1. The ordered source recursion gives X(j+2)=X(j+1)X(j). These relations produce the six phases. A two-by-two real matrix representation separates all six, so equality of observations is precisely equality of indices modulo six.

Let t2=(alpha,alpha) and t4=(t2,t2). Both leaf products are 1, while their substituted products are -1 and 1. No reader on the full source image can therefore perform substitution. The unequal trees p=((alpha,alpha),alpha) and q=(alpha,(alpha,alpha)) have equal composition (3,0), equal ordered leaf labels, and equal Clifford observations. Here leafLabels(t) is the list of the indexedEquiv leaf-position function. For each Y and target g, a reader can fit g on all indices j<6, because these observations are distinct.

The Clifford construction and its universal property are standard; see Lundholm and Svensson, Clifford algebra, geometric algebra, and applications, arXiv:0907.5356v1, sections 2.1-2.3.

References