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Metadata 
slug: wall-sun-sun-golden-unit-lift
bibkey: shi2026second
doi: 10.48550/arXiv.2603.25343
triage: wall
motivation_gids:
  - D5/S0/Carrier/Ring
  - D5/S0/Carrier/Conj
  - D5/S0/Carrier/Norm
  - D5/S0/Carrier/Units
  - D5/S1/Scale/Units
  - D5/S1/Scale/UnitGroup
  - D5/S1/Scale/Fibonacci
  - D5/S3/Arith/FibonacciRank
  - D5/S3/Arith/GoldenApparition
  - D5/S3/Arith/GoldenPrimeSplitting
  - D5/S3/Arith/GoldenPell

Wall-Sun-Sun primes as a golden-unit lift problem

Problem

Let pi(m) be the Pisano period, the least positive period of the Fibonacci recurrence modulo m. The primary problem is whether there exists a prime p with pi(p) = pi(p^2). The paper also records the stronger conjecture that infinitely many such primes exist.

Quoted from the introduction of arXiv:2603.25343v1:

“A natural question was asked by Wall in his paper: Can there be a prime (p) such that (\pi(p)=\pi(p^2))?”

“It is known that up to (10^{14}), there are no such primes (cf. [16]). Still, using heuristics and probabilistic arguments, some authors conjecture the existence of infinitely many primes (p) satisfying (\pi(p)=\pi(p^2)) [7, 11].”

The same paper identifies the classical case with d = 5 and says there are no known WSS(5) primes.

Candidate formal statement, after defining pisanoPeriod:

Existence:  ∃ p : Nat, Nat.Prime p ∧ pisanoPeriod p = pisanoPeriod (p^2)
Stronger:   Set.Infinite {p | Nat.Prime p ∧ pisanoPeriod p = pisanoPeriod (p^2)}

The paper states the difficulty:

“The question of Wall for these sequences is related to certain deep arithmetic properties of real quadratic fields.”

It makes this precise for its generalized recurrence: equality of the periods modulo p and p^2 corresponds, subject to stated hypotheses, to failure of p-rationality of the associated real quadratic field. This is why the global existence question is not a routine finite-period exercise.

Motivation

  • Multiplication by phi on the basis (1, phi) is the Fibonacci matrix. Frozen Scale/Fibonacci already expresses powers of phi in Fibonacci coordinates.
  • GoldenApparition and FibonacciRank control the first Fibonacci zero modulo a prime and the p ± 1 Frobenius index; GoldenPrimeSplitting supplies the split/inert division according to 5 modulo p.
  • The period can therefore plausibly be re-expressed as an order of the reduced golden unit or Fibonacci matrix. Equality at p and p^2 is then an exceptional failure of the usual order multiplication by p under lifting.
  • The first reachable theorem is not existence. It is an exact bridge among the pair recurrence period, the order of the Fibonacci matrix, and the order of phi in an appropriate golden algebra modulo p^e for e = 1, 2.

Gap

  • GoelPisanoRatioRefutation already defines the frozen pisanoPeriod and proves the internal pair-period helpers. The new PisanoMatrixBridge theorem exposes the missing public bridge to the Fibonacci matrix order for every modulus n > 1.
  • GoldenApparition works modulo a prime; there is no golden algebra modulo p^2 and no Hensel or p-adic order-lift theorem.
  • There is no p-rational field or p-adic logarithm machinery.
  • PID/UFD facts and the global unit classification alone do not decide the exceptional local lift.

Route

  1. Use the public bridge PisanoMatrixBridge.pisanoPeriod_eq_matrix_order to identify pi(m) with the order of [[1,1],[1,0]] over ZMod m; its proof tracks (F_n, F_{n+1}) through the reduced golden unit.
  2. For r = pi(p), write the first lift as A^r = I + pB (mod p^2). Prove pi(p^2) = pi(p) if and only if B = 0 (mod p), and that otherwise the period acquires the expected factor p.
  3. Use the frozen split/inert and apparition results to reduce the required congruence to a Fibonacci/Lucas quotient modulo p, separately in the two Legendre-symbol cases.
  4. Only after those bridge theorems exist should a theorist choose between a conditional nonexistence theorem for a prime class, a density heuristic, or the global Wall question. Do not jump from a finite scan to existence.

Falsifier

The existential Wall question has no honest finite falsifier. A proof that no prime can satisfy the equality would refute it; a proof of finiteness would refute only the stronger infinitely-many conjecture.

The proposed bridge is finitely falsifiable: find a prime p for which the directly computed pair period disagrees with the order of the Fibonacci matrix/golden unit, or for which A^pi(p) = I (mod p^2) disagrees with pi(p^2) = pi(p).

Evidence

Implement three independent exact calculations for every prime p < 10^6, excluding and separately reporting ramified and small cases:

  1. direct pair-state Pisano periods modulo p and p^2;
  2. fast-doubling checks of F_r mod p^2 and F_{r+1}-1 mod p^2 at r = pi(p);
  3. matrix exponentiation of A^r mod p^2 and the first-lift matrix B mod p.

Receipt fields should include p, legendreSym 5 p, rank, pi_p, pi_p2, F_r mod p^2, F_(r+1)-1 mod p^2, and agreement of all three formulations. This is bridge validation, not evidence that the global existential is false.

Triage

wall. The repository is unusually close to the mod-p side, but the decisive p to p^2 lift is exactly the missing deep layer.

ASSUMED-UNVERIFIED

  • Whether the open problem was resolved after arXiv v1 is unverified; this records the paper’s statement, not the entire later literature.
  • The order of the reduced golden unit matches the chosen Pisano-period convention without a factor of 2 or a special case; that must be proved, not assumed.
  • A useful local quotient criterion can be stated entirely with the current GoldenInt coordinate model.
  • Any novelty of the proposed bridge lemmas is unassessed and belongs to the theorist’s search step.

PH. 固定黄金 Pell 曲线与初始 WSS 深度的高度预算

Full argument, assumptions and unresolved boundaries.

CG. 原黄金单位在连续、有限与整数几何中的同一缺陷

Full argument, assumptions and unresolved boundaries.

FPD. Recurrence-level square transport and maximal-prime descent

Full argument, assumptions and unresolved boundaries.

THS. Exact theta sieving and the principal-orbit coefficient

Full argument, assumptions and unresolved boundaries.

ROC. Fixed-golden orders, mixed-conductor kernels, and zero-parameter dihedral forms

Full argument, assumptions and unresolved boundaries.

SGN. Spectral Galois arithmetic, exact congruence ideals, and packet norms

Full argument, assumptions and unresolved boundaries.

STL. Saturated spectral trace lattices and exact loss under reduction

Full argument, assumptions and unresolved boundaries.

CTG. Complete-tower gluing and exact rational Eisenstein congruence depth

Full argument, assumptions and unresolved boundaries.

HCR. One-prime Hecke reconstruction and the exact observable subfield

Full argument, assumptions and unresolved boundaries.

SJC. Predictive completion of the original golden orbit at square precision

Full argument, assumptions and unresolved boundaries.

TBN. A fixed-golden split tower and simultaneous initial-depth balances

Full argument, assumptions and unresolved boundaries.

DCE. Primitive depth vectors from a solved exponential equation

Full argument, assumptions and unresolved boundaries.

GPC. Golden prime clocks, exact interlevel jets, and quadratic reciprocity

Full argument, assumptions and unresolved boundaries.

CF. Carlitz degree-five exactness within the same Wieferich family

Full argument, assumptions and unresolved boundaries.

CX. Global characteristic support and two additional Carlitz families

Full argument, assumptions and unresolved boundaries.

PCL. Golden-clock sampling, prime-period coupling, and native block periods

Full argument, assumptions and unresolved boundaries.

GCR. Cubic reciprocity on the original golden depth vectors

Full argument, assumptions and unresolved boundaries.