bibkey: renault2013periodrankorder authors: Marc Renault year: 2013 title: The Period, Rank, and Order of the (a,b)-Fibonacci Sequence Mod m doi: 10.4169/math.mag.86.5.372 claim: Theorems 1-2 give least-common-multiple period and rank laws and prime-power lifting with the actual initial delay retained; they do not assume absence of WSS primes. strata_touched: [] license: citation-only triage: anchor

Golden prime clocks, original delays, and interlevel reciprocity

Primary source and exact role

Marc Renault, The Period, Rank, and Order of the (a,b)-Fibonacci Sequence Mod m, Mathematics Magazine 86 (2013), 372-380. https://doi.org/10.4169/math.mag.86.5.372 Author-hosted published text: https://webspace.ship.edu/msrenault/fibonacci/RenaultPeriodRankOrderMathMag.pdf

The Preliminaries on printed pages 373-374 identify matrix order with pair period and scalar return with the zero rank. Theorem 1 on page 374 proves the least-common-multiple laws. Theorem 2 on page 376 retains the initial prime-power plateau before subsequent lifts multiply the period by p. The GPC period lift also follows directly from a unit binomial expansion. These are the period-theoretic inputs; the block and reciprocity identities below are proved in the problem owner.

For the standard sequence, the two common Fibonacci matrices are conjugate by the swap of their coordinates. This preserves every residue order. The exact real eigenpairs recorded by the existing FibonacciEigen Scribe identify the same integer substitution, but do not decide its p-adic lift. The full golden-clock matrix at layer L is Q^(L+2); the scalar floor map and real fractional-part observation are kept distinct from the matrix modulo a prime square.

Mathematical consumer in the existing WSS owner

Problems/wall-sun-sun-golden-unit-lift.md, PCL.1-PCL.7, proves the following ordinary specializations for the original recurrence:

• With stride s and external-period valuation beta_p, the extra exponent is max(0,a_p-h_p-max(beta_p,v_p(s))). Coupling and stride delays combine by their maximum. The simultaneous square lift of a squarefree modulus sees exactly the non-WSS sinks in its increasing prime-period graph.
• The actual quotient F_(p^b)/F_(p^(b-1)) supplies larger primes of exact rank p^b and period 4p^b. This constructs arbitrarily deep auxiliary period masking, not new original WSS depths. The explicit cofactor F_(p^b) gives the same period without requiring its factorization.
• The two ternary families C_j=L_(3^j)^2+1 and B_j=L_(3^j)^2+3 have exact native block-power periods. Lemma PCL7A includes their rank and actual valuation proofs locally; no unmerged TBN or DCE appendix is required.
• Their base periods contain only two and three, absent from their prime supports. Native multiplicities nevertheless hide h_p. Replacing a block product by its radical makes the square-lift ratio exactly the product of simple factors. That radical still needs independent control.
• The block products have an exact K+1-step period-iteration trajectory to 24, whereas the transient exponent of the largest prime P is max(a-n*h_P,0). A common terminal fixed point does not decide WSS.

The B-block period formula also occurs in GPC. PCL supplies an independent specialization without relying on GPC’s reciprocity appendices. Independent priority of the block and masking specializations is not claimed.

Other classical sources retained from the prepared PCL package

B. Benfield and O. Lippard, Fixed Points of K-Fibonacci Sequences, arXiv:2404.08194v2, Theorems 2.1-2.2, 2.6 and 2.11: https://arxiv.org/html/2404.08194v2 The source attributes ordinary Fibonacci fixed-point classification and convergence to Fulton-Morris (1969). PCL proves the displayed special trajectories, not a new general fixed-point theorem. The separate conjecture about generalized (a,-1) sequences is not used.

T. Ross, Z. Shen and T. Cai, The p-adic Valuations of Mobius Duals of Lucas Sequences, arXiv:2512.03481v1, Proposition 3.2 and the classical attribution in Theorem 3.3: https://arxiv.org/html/2512.03481v1 The regular-sequence valuation input retains h_p and credits classical Lucas theory. It does not assert a simple primitive factor or a WSS example.

Common-field presentation cost of the same golden blocks

The GIR fields generated by cube roots of these actual B_j also support GTC, with full ordinary proofs and classical source roles in Library/Tower/first2017separablegenerators.md. It computes exact dyadic splitting, local generator counts and the sharp unavoidable two-part of all primitive-element indices in that same tower. Its explicit global witnesses attain the bounds. This is a tower-presentation invariant at two, not the block normalization index detecting original WSS factors. The existing PCL and GIR/GNT/GMI statements and their attribution remain unchanged; no WSS prime-family decision follows from this cross-reference.

Interlevel reciprocity in GPC.1-GPC.6

Problems/wall-sun-sun-golden-unit-lift.md keeps the actual blocks B_j=L_(3^j)^2+3 and r_j=3^(j+1), whose factor valuations are the original h_p by TBN. Its GPC section proves the following ordinary specializations:

1. Every nontrivial divisor of B_j has pair period and zero rank2r_j. For every a>=1, pi(B_j^a)=2r_j*B_j^(a-1), even when an individual factor has a longer initial p-power plateau. Thus normal composite clock growth cannot decide whether a factor is WSS.
2. For i<j, B_j=3-3^(2(j-i)-1)*B_i^2 modulo B_i^3. Hence at each actual p|B_i the later displacement B_j-3 has valuation exactly2h_p.
3. Quadratic reciprocity gives (B_j/B_i)=-1 and (B_i/B_j)=+1. The primewise refinement is product_(q|B_j)(p/q)^h_q=1 for every earlier p. It retains all earlier prime factors, including those of even depth.
4. If a later block has only one odd-depth prime Q, that Q must split completely in Q(sqrt(p):p divides an earlier block). In particular the remaining minimal all-WSS pattern P^2 Q^3 must satisfy all these simultaneous conditions, in addition to DCE’s Q=19 modulo40.
5. For fixed j, the comparison primes with those quadratic tests and Q=1 modulo r_j have positive Dirichlet density1/(2^t*phi(40r_j)), where t is the number of distinct earlier primes. This comparison set omits the exact-clock condition. The primes of exact pair period2r_j are exactly the finite factor support of B_j, so the density is not a WSS or exact-period existence theorem.

The cross-block formulas are derived in the owner from the actual Lucas product, then classical reciprocity. They are not attributed to Renault; the paper supplies the period-theoretic input. Independent priority for these particular block specializations has not been established.

Other classical inputs, with exact boundaries

NIST Digital Library of Mathematical Functions, Section27.9, https://dlmf.nist.gov/27.9 , supplies the Jacobi prime-factor definition, the sign formula27.9.1 and quadratic reciprocity27.9.3, including its extension to coprime odd composite denominators. A positive Jacobi symbol at a composite modulus is not treated as a square-root certificate.

A. V. Sutherland, MIT18.785 Lecture28 (2021), Theorem28.9 on printed page6: https://math.mit.edu/classes/18.785/2021fa/LectureNotes28.pdf The theorem and its page image were read. GPC uses only the cyclotomic Dirichlet-progression case to count the explicitly independent congruence classes. The class-field family here exists without a WSS assumption, but the target primes must still divide the specified integer B_j. No Chebotarev application replaces this actual divisibility requirement.

Formal and arithmetic boundary

The period, block-product and character identities are stated with ordinary proofs in the problem owner. The FibonacciEigen Scribe links this note as context without adding these claims to its formal declarations. Quadratic characters see depth parity and allow both h=1 and odd h>=3. The remaining obligation is a restriction on original depths or simple-factor support; these identities do not construct a WSS prime.

Actual sparse child observations

The sparse consumer uses the locators for comparison. Its proof premises are the frozen Fibonacci and actual-source contracts, rather than a new primary-text assertion attributed to Renault.

D5/S3/Arith/FibonacciAtomic/GlobalGcdSampling.lean uses the actual signed Fibonacci recurrence, rather than a supplied projective classifier. Its sparse_gcd_sampling reuses positive prime-power zero ranks and applies the frozen fibonacci_entry_point directly. Within a parent rank $R=r_{e-1}$ it uses

$$M^R=cI+fM,c=F_{R-1},f=F_R,p^{e-1}\mid f,f^2=0(\mathrm{mod}{p}^e).$$

The square-zero estimate is $e\le 2(e-1)$ and therefore includes $p=2,e=2$. Consecutive Fibonacci coprimality makes $c$ a unit. The resulting unit-normalized observations along the same parent are affine in the child index. For a primitive parent hit, actual rank growth forces a nonzero slope modulo $p$. At least $p-1$ distinct queried children determine its unique root; two omitted children admit distinct fixed signed initial states, realized through inverse integer Fibonacci action, with separation beyond every cutoff. Actual stagnation gives scalar threshold persistence for all signed initial states, including zero.

The actual terminal criterion is defined by distinct phase images, with the mandatory/optional first-layer distinction, full coverage at stagnant layers, and at least $p-1$ children in every growth parent. Its failure on an arbitrary finite positive table yields the corresponding omitted phase or two omitted children; the hypothesis is not a small bound on $|S|$. Primitive hits have a single exact phase at every precision, and all signed threshold observations transport between congruent positive times. The root realization exposes its adjacent observation equal to one, its support at every lower precision, and its whole-time top support. Consequently the two-omitted-child growth construction gives complete gcd equality on every positive table avoiding those omitted phases, including off-parent readings, with exact late answers $p^e,p^{e-1}$.

The frozen GraftAffineClosure.result supplies the actual observation inverse and bounded residue lifts. The same earning proof uses them to realize each signed pair by a fixed natural source below $Q{p}^e$, simultaneously at every positive time, with complete gcd answer $Q\gcd (|y_k|,p^e)$. This holds for every $Q>0$, including $p\mid Q$; source primitiveness is asserted only when $p\nmid Q$. The lifted growth collisions choose sources before every cutoff.

The frozen PrimePhaseGcdSampling.prime_phase_gcd_sampling supplies the first-layer identification and collision clauses at its actual contract, with the mandatory-hit and optional-hit distinction and the ramified prime retained. For all signed pairs, including zero and saturated pairs, the minimum gcd on a positive table containing two different first-layer phases is the capped common content. This does not assume that divided signed pairs remain actual natural sources.

Two distinct omitted first-layer phases give fixed primitive signed collisions. In the optional-hit first layer, a single omitted phase collides with a primitive no-hit state. At a stagnant higher layer, inverse action realizes the same lower supports with a permanent top exit. All these constructions also yield fixed bounded natural sources for every $Q>0$, with exact late answers $Q{p}^e$ and $Q{p}^{e-1}$.

The public implication from failed $D$ to fixed whole-table collisions includes all four branches and the simultaneous bounded natural lifts. The same positive table identifies all actual natural futures at every $H>0$ exactly when $D$ holds at every complete prime-power factor of $H$. Each full gcd reading projects to its local reading. Recombination compares the factorization of the gcd readings, which are nonzero even when the raw observations are zero. For $H=1$ every reading is one, and an empty table suffices. A numerical minimum across different prime factors is not this recombination: at $H=6$, source $(5,1)$ and times $\{1,2,7\}$ have readings $2,3,2$ but capped source content one. The same shifted terminal condition is equivalent to its whole-tower condition and all four signed/natural identification domains. It supplies first-layer phase diversity and both signed and actual-natural local content minima. The native affine decoding equivalence is part of the same theorem. The shifted-to-unshifted quotient-child adapter, sharp horizons and query-cardinality consequences are separate statements. The sparse actual-state construction is not attributed to Renault’s paper. The citation-only license permits no republication of the article; this note contains a paraphrase and repository-specific mathematics, not a copy of its text.

Primary-text comparison uses Medina and Rowland, arXiv:0910.2907v4, Theorems 1.2 and 1.4 on printed pages 1–2, and Lengyel, The Order of the Fibonacci and Lucas Numbers, Fibonacci Quarterly 33(3), pages 234–239, especially the valuation theorem and proof on pages 236–237. Their baseline Fibonacci valuation statements are not arbitrary-seed sparse-identification theorems. Park, arXiv:1407.8086v1, Lemmas 3–4 and Theorem 19, treats a different generalized-recurrence setting, with explicit parameter restrictions; its prime-power counting Theorem 20 is not used: for the ordinary Fibonacci parameters $s=t=1$, the checked rank $r(27)=36$ exceeds $27+1$. Aka, arXiv:2508.08016v1, Theorem 1.1 and its proof on pages 1–3, concerns zero-free sequences with specified periods dividing $p-1$, excluding $p=2,5$, not this all-prime prime-power table criterion. None of these citations supplies a Lean proof or a global originality claim for the consumer.