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Fibonacci Full-Prefix Transport

Abstract

Actual integer carry histories separate Fibonacci composition labels, and every moving time in a full prefix has an operational matrix characterization.

Theorem 1.1 (Integer evolution and full-prefix diagonal collapse).

Proof. Machine-checked in Lean as D5/S3/Quantum/Transport/FibonacciPrefix.fibonacci_prefix_transport (✓ std3). ∎

Source. Repository-derived.

Commentary.

Here A(d)=ZMod(d)×ZMod(d), H(e)=ZMod(e)×ZMod(e), and Mat(d) is the full complex matrix algebra on A(d). All d,e≥2 are allowed, including d=e=2. The fixed constant phi is Real.goldenRatio=(1+sqrt(5))/2. The integer golden generator phiZ in L’s evolution maps to this real constant under the existing embedding.

F(d,t) is lowTrajectory d t, the t-fold iteration of (a0,a1)↦(a1,a0+a1) modulo d. The actual carry k(d,a,j) is the sum of the canonical representatives of F(d,j)(a), divided by d with natural integer division. s(d,N,a)=carryPrefix d N a retains every position j<N. L(d,j,a)=integerTrajectory d j a is the golden integer whose coordinates are those two canonical representatives.

The high-fibre permutation q(d,e,a,0) is identity; its next step sends (h0,h1) to (h1,h0+h1+k(d,a,t)) modulo e. J(d,e,t)=jointTrajectory d e t sends (a,h) to (F(d,t)(a),q(d,e,a,t)(h)). The shared CarryTransport source defines fibonacci, carry, digit, digitJoin and transport; the theorem identifies F(d,1) with fibonacci d and J(d,e,1) with transport d e. Its actual matrices are lowUnitary d=permMatrixHom(fibonacci d) and jointUnitary d e=permMatrixHom(transport d e), with the displayed adjoint-left/right factors giving the column-ket convention. R(d,e,t,B)=movingPullback d e t B is defined by the reindex algebra homomorphisms for the moving low observation and the joint pullback.

S(d,e,N)=prefixAlgebra d e N is the intersection, over every positive time t≤N, of the comap of the lowTensor range under R(d,e,t). The shared lowTensor hom is B↦B tensor high identity, so membership has its literal operational meaning. Delta(d)=diagonalAlgebra d is independently the range of Matrix.diagonalAlgHom. The displayed tensor is the Kronecker matrix representation.

Equal N-prefixes give the integer difference evolution for every j≤N. A distinct collision at N≥1 obeys phi^N≤phi^3(d−1)^2. Every positive permutation period P with F(d,P)=identity already gives injectivity of s(d,P). The strict real logarithmic criterion also gives injectivity, and equality of the entire infinite carry history forces equality of the labels.

n(d)=prefixThreshold d is natural floor(3+2 log_phi(d−1))+1. The theorem proves that its integer cast equals the ordinary integer floor expression plus one. Every N≥n(d) gives injectivity, and n(2)=4 is an exact symbolic result. The nonnegative logarithm and the golden constants are proved inside the theorem.

For every natural t and every unrestricted complex low matrix B, R is exactly (U^t)†((V^t B (V^t)†) tensor I)U^t. For every N, membership in S is equivalent to vanishing entries between unequal actual prefixes. Every admitted pullback at each 1≤t≤N equals B tensor I. The sequence S is antitone, its intersection over positive N is Delta, every positive low permutation period gives S_P=Delta, and S_N=Delta for every N≥n(d).

The estimate is independent of e≥2, sufficient rather than optimal, and asserts no coherence at every shorter prefix. It applies to the moving full-prefix algebra, not a single-endpoint algebra. The iterated low and joint permutations have the specified source carry action. Under the canonical digit split x=a+d h, the shared source theorem supplies transport d e (a,h)=(fibonacci d a,(h1,h0+h1+carry d a)); the proved identifications above therefore connect the operational E trajectory to the actual digit-split source for every d,e≥2, including the non-coprime case.

References