Prime Phase Gcd Sampling

Abstract

Sharp prime Fibonacci phase identification and simultaneous bounded natural counterexamples.

F(0)=0 and F(1)=1. All queried times are positive natural numbers. The signed reading takes the absolute value of the entire linear combination. The source reading uses actual nonnegative coordinates, not merely residue labels.

Definition 1.1 (Signed local reading).

$$\forall p\in \mathbb{N},\forall n,z\in \mathbb{Z},\forall k\in \mathbb{N},G\left(p,n,z,k\right)=\gcd \left(\mathrm{abs}\left(F\left(k-1\right)\cdot n+F\left(k\right)\cdot z\right),p\right)$$

Formalization. D5/S3/Arith/FibonacciAtomic/PrimePhaseGcdSampling.localGcd (✓ std3).

Source. Repository-derived.

Commentary.

The state coordinates n,z are integers and the modulus p is natural.

Definition 1.2 (Natural source reading).

$$\forall H,a,b,k\in \mathbb{N},g\left(H,a,b,k\right)=\gcd \left(F\left(k+3\right)\cdot a+F\left(k+4\right)\cdot b,H\right)$$

Formalization. D5/S3/Arith/FibonacciAtomic/PrimePhaseGcdSampling.sourceGcd (✓ std3).

Source. Repository-derived.

Commentary.

The source coordinates a,b and the modulus H are natural numbers.

Definition 1.3 (Prime-primitive signed state).

$$\forall p\in \mathbb{N},\forall n,z\in \mathbb{Z},\mathrm{Primitive}\left(p,n,z\right)\iff (\neg \mathrm{dvd}\left(p,\mathrm{abs}\left(n\right)\right)\vee \neg \mathrm{dvd}\left(p,\mathrm{abs}\left(z\right)\right))$$

Formalization. D5/S3/Arith/FibonacciAtomic/PrimePhaseGcdSampling.primitive (✓ std3).

Source. Repository-derived.

Commentary.

A state is p-primitive when at least one coordinate is not divisible by p.

For each prime p, write r=zeroRank(p), the least positive d with p dividing F(d). For a finite positive-time table S, let A be its phase image modulo r. Set T=r-1 when r=p+1 and T=r otherwise.

Theorem 1.4 (Sharp identification and fixed-source late separation).

$$\forall p\in \mathbb{N},(\mathrm{Prime}\left(p\right))⟹(\forall S\in \mathrm{Finset}\left(\mathbb{N}\right),(\forall k\in S,0<k)⟹(((\forall n,z,nPrime,zPrime\in \mathbb{Z},(\forall k\in S,G\left(p,n,z,k\right)=G\left(p,nPrime,zPrime,k\right))⟹(\forall k\in \mathbb{N},(0<k)⟹(G\left(p,n,z,k\right)=G\left(p,nPrime,zPrime,k\right))))\iff (\mathrm{if}\left(\mathrm{zeroRank}\left(p\right)=p+1,\mathrm{zeroRank}\left(p\right)-1,\mathrm{zeroRank}\left(p\right)\right)\le \mathrm{card}\left(\mathrm{image}\left(k\mapsto \mathrm{mod}\left(k,\mathrm{zeroRank}\left(p\right)\right),S\right)\right)))\wedge ((\forall a,b,aPrime,bPrime\in \mathbb{N},(\forall k\in S,g\left(p,a,b,k\right)=g\left(p,aPrime,bPrime,k\right))⟹(\forall k\in \mathbb{N},(0<k)⟹(g\left(p,a,b,k\right)=g\left(p,aPrime,bPrime,k\right))))\iff (\mathrm{if}\left(\mathrm{zeroRank}\left(p\right)=p+1,\mathrm{zeroRank}\left(p\right)-1,\mathrm{zeroRank}\left(p\right)\right)\le \mathrm{card}\left(\mathrm{image}\left(k\mapsto \mathrm{mod}\left(k,\mathrm{zeroRank}\left(p\right)\right),S\right)\right)))\wedge ((\mathrm{card}\left(\mathrm{image}\left(k\mapsto \mathrm{mod}\left(k,\mathrm{zeroRank}\left(p\right)\right),S\right)\right)<\mathrm{if}\left(\mathrm{zeroRank}\left(p\right)=p+1,\mathrm{zeroRank}\left(p\right)-1,\mathrm{zeroRank}\left(p\right)\right))⟹(\forall Q\in \mathbb{N},(0<Q)⟹(\exists n,z,nPrime,zPrime\in \mathbb{Z},\exists a,b,aPrime,bPrime\in \mathbb{N},(\mathrm{Primitive}\left(p,n,z\right))\wedge (\mathrm{Primitive}\left(p,nPrime,zPrime\right))\wedge (a<Q\cdot p)\wedge (b<Q\cdot p)\wedge (aPrime<Q\cdot p)\wedge (bPrime<Q\cdot p)\wedge (\forall k\in \mathbb{N},(0<k)⟹((g\left(Q\cdot p,a,b,k\right)=Q\cdot G\left(p,n,z,k\right))\wedge (g\left(Q\cdot p,aPrime,bPrime,k\right)=Q\cdot G\left(p,nPrime,zPrime,k\right))))\wedge (\forall k\in S,(G\left(p,n,z,k\right)=G\left(p,nPrime,zPrime,k\right))\wedge (g\left(Q\cdot p,a,b,k\right)=g\left(Q\cdot p,aPrime,bPrime,k\right)))\wedge (\forall B\in \mathbb{N},\exists t\in \mathbb{N},(B<t)\wedge (0<t)\wedge (\neg (t\in S))\wedge (G\left(p,n,z,t\right)=p)\wedge (G\left(p,nPrime,zPrime,t\right)=1)\wedge (g\left(Q\cdot p,a,b,t\right)=Q\cdot p)\wedge (g\left(Q\cdot p,aPrime,bPrime,t\right)=Q)\wedge (g\left(Q\cdot p,aPrime,bPrime,t\right)=\mathrm{div}\left(Q\cdot p,p\right))))))))$$

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

Source. Repository-derived.

Commentary.

The two identification statements include zero and nonprimitive states, hence constant saturated readings. Only the number of distinct queried phases matters; no answer at time zero or common content is supplied. The rank satisfies 3<=r<=p+1, and pairwise recovery gives distinct kernel directions. A unit scalar return preserves zero support, without requiring the Fibonacci step to return to the identity. Nonzero states have at most one zero phase. When r=p+1 every direction has one; otherwise an affine direction avoids all zero phases. Missing one phase in the proper case, or two in the full case, therefore gives indistinguishable primitive states. Their inverse-source residues (5n-3z,-3n+2z), scaled by Q, lie strictly below Qp and realize every positive-time reading simultaneously. Both states and both sources are fixed before the cutoff B. The later separating time alone varies. The local p,1 separation uses unscaled signed states even when p divides Q. Natural-source identification is modulo p, not a claimed general-modulus Qp identification criterion.

References

• Truth anchor: D5/S3/Arith/FibonacciAtomic/PrimePhaseGcdSampling.localGcd
• Truth anchor: D5/S3/Arith/FibonacciAtomic/PrimePhaseGcdSampling.prime_phase_gcd_sampling
• Truth anchor: D5/S3/Arith/FibonacciAtomic/PrimePhaseGcdSampling.primitive
• Truth anchor: D5/S3/Arith/FibonacciAtomic/PrimePhaseGcdSampling.sourceGcd
• Dependency: D5/S3/Arith/FibonacciAtomic/TimeSampling