Andoyo's equal Pisano sets question

Section 5, Question 5.3 asks:

Are there values of a₁, a₂, b₁, and b₂ for which χ(a₁,b₁)=χ(a₂,b₂)?

The preceding paragraph requires a₁,b₁,a₂,b₂∈ℤ, distinct ordered pairs, and neither pair equal to (0,0). Definitions 2.6 and 2.7 specify the least positive scalar period at every nonnegative index, the three index counts for Legendre values 1,−1,0, and their integer difference over all odd primes.

The frozen declaration D5/S3/Arith/AndoyoPisanoSetCollision.result answers the existential question affirmatively with the distinct nonzero integer seed pairs (0,1) and (1,1). Their complete Pisano sets agree over all odd primes, with the source's least positive scalar periods and all three Legendre index counts.

Preregistration: #22843. The bounded named source screen in the implementation assignment reports no exact settlement within its stated readings. The versioned preprint and journal text both explicitly pose Question 5.3. Broader web, citation, code and Reservoir coverage remains unmeasured; no global absence or publication priority claim follows.

Write fₙ=Fₙ(0,1) and gₙ=Fₙ(1,1). Their recurrences give gₙ=fₙ₊₁. Every positive scalar period of f is one of g. Conversely, a scalar period t of g gives fₜ₊₁≡f₁ and fₜ₊₂≡f₂; the recurrence gives fₜ≡f₀, including the missing zeroth index. The remaining indices follow by shifting. Thus the complete sets of positive scalar periods agree, and so do their least members.

Existence of a positive period follows from the invertible Fibonacci step on the finite pair space over ZMod p, using the original canonical PisanoMatrixBridge.period_iff_return. No common-period replacement is used.

Let T be the shared least period. For any of the three Legendre classes, write uₙ for its zero-or-one index indicator. Periodicity gives uT=u0, and the two decompositions of the sum over 0,…,T give

Cancellation proves equality of the two index counts. This includes the zero class and the wraparound endpoint and uses no T>1 assumption. The three equalities give equal K values for each same prime, hence equal sets over all odd primes.

The source correspondence would fail if the source excluded either witness, identified shifted seed pairs, or used a different period or counting convention. The proof mechanism would fail if reverse period transport omitted index zero, or if any Legendre-class count changed at the wraparound endpoint. The statement echo and the period and count equalities address these obligations for the literal integer-seed question; finite prime samples cannot establish it.

The source is Definitions 2.6 and 2.7 and Section 5, Question 5.3 in the versioned preprint and journal text, identified by andoyo2026pisanolegendrecollision and DOI 10.62072/acm.2026.09011. The clause correspondence is:

| Source clause | Lean counterpart | | --- | --- | | Integer initial values and recurrence from n=0 | F (a b : ℤ), with branches 0, 1, and n+2 | | Positive scalar period for every n≥0 modulo p | ScalarPeriod p a b t | | Least such positive period | pi; least_period proves positive membership and minimality for (0,1); pi_equal proves the same minimum for (1,1) using both directions of periods_equal | | All indices 0,…,π−1, with repeated values counted repeatedly | signCount is the cardinality of a filtered Finset.range of indices | | Legendre classes 1,−1,0 | The three arguments of signCount in K | | Integer subtraction | Each count is cast to ℤ separately in K | | All odd primes | chi quantifies over every natural prime p with p≠2 | | Distinct nonzero ordered seed pairs | The three inequalities in result, witnessed by (0,1) and (1,1) | | Equality of complete Pisano sets | The final equality in result |

Theorem: the established collision follows from cyclic phase invariance of index counts over the shared least period, including the zero class. It is an existence result for (0,1) and (1,1), not a classification of all collisions. The neighboring research candidate is the source's Question 5.2: for integers a,b with (a,b)≠(0,0), does every k∈χ(a,b) satisfy |ϑ_(a,b)(k)|=ℵ₀? It asks for infinitely many odd primes at each attained balance; the collision theorem supplies no such infinitude or distribution result. PR #22894 is published; merge and delivery remain pending.

The literature absence is bounded to the source screen described above. The information-escape audit remains unfinished as disclosed in #22843; the issue is not completed registration evidence. Classification of other seed-pair collisions and the neighboring Question 5.2 are outside the proved result. This dossier makes no world-priority claim.