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Actual Cloitre Deficit-Four Selector

Abstract

A qualified actual deficit-four root has a unique canonical periodic selector with a complete quadratic enclosure.

F is the Fibonacci sequence with F(0)=0 and F(1)=1. Put phi=(1+sqrt(5))/2 and G(n)=floor((n+1)/phi). C is the actual positive-index Cloitre sequence with C(1)=C(2)=1. Its legal domain is D(N)=[1,N-1], its inner map is T(N,x)=N-C(x), and its orbit X(N,i) starts at N-1. The prescribed depth is d(N)=C(N-1), the selected point is g(N)=X(N,d(N)), and C(N)=C(g(N))+C(N-g(N)) for N>=3. Put Q(m,t)=F(m-1)-C(F(m)-t) on the full natural closed block 0<=t<=F(m-2). The upper cap makes Q the exact nonnegative integer difference. It is distinct from the golden excess C(n)-G(n).

Theorem 1.1 (Actual selection and first deficit-four hit).

Lean statement: D5/S1/Recurrence/Invariants/CloitreActualDeficitFourSelector.full30_3

Proof. Machine-checked in Lean as D5/S1/Recurrence/Invariants/CloitreActualDeficitFourSelector.full30_3 (✓ std3). ∎

Source. Repository-derived.

Commentary.

Hyp24_1(U) includes the complete inherited Hyp21_1: the finite ratio condition 22877C(n)<=15225n for 16384<=n<=131071; the full golden base and equality classification for 1<=n<=65535; and prescribed periodic entry for 3<=N<=52. The golden base states G(n)<=C(n), with equality implying n=F(j) or F(j)+1 for some j>=2, n+1=F(j) for an odd j>=3, or n in {11,24,25,59}. For every positive n, the global bounds are 1<=C(n) and G(n)<=C(n)<=U(n)<=n. The upper function satisfies U(1)=1, the piecewise formula U(n)=min(n-F(j-2),F(j)) on F(j)<=n<F(j+1) for j>=3, and monotonicity and increments U(n)<=U(n+1)<=U(n)+1 on positive indices. For j>=2, U(F(j))=C(F(j))=G(F(j))=F(j-1); for j>=3, C(F(j)+1)=G(F(j)+1)=F(j-1)+1. For every q>=6 and t>=0, the right collar [F(q-1),F(q-1)+t] is legal and invariant under T(F(q)+t), captures every legal orbit, and contains every legal periodic point. For each N>=3, the earliest periodic entry of X(N,i) precedes or equals d(N). Hyp24_1 also includes C(F(j)-1)=F(j-1) for j>=5, and for j>=6 and 0<=b<=F(j-1), with N=F(j+1)-b, the collar [F(j)-b,F(j)] intersected with D(N) is invariant under T(N) and captures every legal orbit. Every legal periodic point x of T(N) satisfies max(F(j-1),F(j)-b)<=x and x<=min(F(j),F(j)+F(j-3)-b).

Two additional finite full-block conditions are required. For every v<=F(18), Q(20,v)<=2 exactly when v<=35; when v>35, 3<=Q(20,v)<=max(3,v-36). For every v<=F(19), Q(21,v)<=3 exactly when v<=45; when v>45, 4<=Q(21,v)<=max(4,v-46). These conditions and the inherited foundations are premises; no instance of them is asserted.

For every natural m>=22 and b<=F(m-2) with exact parent qualification Q(m,b)=4, put N=F(m)-b, z=F(m-1)-g(N), w=F(m-2)-(N-g(N)), P=floor((m-2)^2/3)+30-3m, Omega(r)=b-Q(m-1,r), and r0=b-4. The actual ordered routes are g(N)=F(m-1)-z and N-g(N)=F(m-2)-w, with z+w=b, z<=F(m-3) and w<=F(m-4). Their exact child deficits are Q(m-1,z)=4 and Q(m-2,w)=0, and w<=platformWidth(m-2), where platformWidth(k)=floor((2k-9)/3). The signed integer jump g(N)-T(N,g(N)) is w-4. Each term is cast to the integers before subtraction.

The complete quantitative enclosure is b<=4P<=F(m-4). For every r<=b, both lower natural-block domains hold: r<=F(m-3) and r<=F(m-4). The corresponding physical point satisfies 1<=F(m-1)-r<=N-1. The full first lower profile satisfies Q(m-1,r)<=floor(2r/3)<=b. Thus Omega(r)<=b and T(N,F(m-1)-r)=F(m-1)-Omega(r) on the entire interval.

There exists a natural tau<=b with Q(m-1,Omega^tau(r0))=4 and Q(m-1,Omega^i(r0))!=4 for every i<tau. Its actual selected gap is z=Omega^tau(r0), its physical endpoint is g(N)=F(m-1)-Omega^tau(r0), and the minimal period of g(N) under this same actual T(N) is tau+1. For every r<=b that is periodic under Omega and satisfies Q(m-1,r)=4, r=Omega^tau(r0). This uniqueness ranges over every periodic gap in the interval, including distinct candidate cycles. The case tau=0 is included.

The cap-three propagation starts from the two complete finite seeds. With Z(k)=3*k+floor((k-1)/3)-24 and W=Z(m-1), a qualified root lies either in b>=W+5 or at b=W+4 with m mod 3!=1. In the tail, periodic predecessors and the outside shelf force every first-child periodic deficit to be at least four; actual deficit addition then forces the selected pair (4,0). At the critical boundary, the prescribed absolute orbit alternates sides from its actual origin. Its depth F(m-1)-4 is odd, so the selected point is F(m-1)-W-1, giving z=W+1, w=3 and signed jump -1.

The positive-cap budget uses P(9)=13 and P(k)=floor((k-2)^2/3)+30-3*k for k>=10. The complete gap lengths at orders nine and ten are thirteen and twenty-one, so positivity supplies the bases. Actual ordered child deficits add, zero child deficits use the existing platform zero set, and the budget recurrence yields v<=Q(k,v)P(k) when Q(k,v)>0. Fibonacci domination gives 4P(m)<=F(m-4) for all m>=22.

Bounded coordinate conjugacy identifies the gap and physical period predicates in both directions. The actual selected point is periodic and Omega(z)=r0, so r0 lies on that actual cycle. Injectivity on all periodic points makes z the unique periodic deficit-four gap. If its minimal period is p, the first hit from r0 is exactly p-1; the finite interval has b+1 points, so tau<=b. Coordinate injectivity is used only within the bounded legal interval.

The exact parent qualification and the true full lower profile remain inputs. This finite walk does not acquire that qualification, identify an absolute clock-zero point of the orbit from N-1, or recover a transient entrance certificate. It needs no separate predecessor deficit, entrance clock, entrance point or phase label on this qualified domain. The profile-table size and walk length may grow with b; Fibonacci arithmetic and acquisition costs remain separate.

References