bibkey: medina2015fibonacciregularity
authors: L. A. Medina; E. Rowland
year: 2015
title: p-regularity of the p-adic valuation of the Fibonacci sequence
doi: null
url: https://arxiv.org/abs/0910.2907
claim: Theorem 1.4 restates Lengyel's Fibonacci valuation formulas for every prime, retaining the initial rank valuation and the separate formulas for two and five.
strata_touched: []
license: citation-only
triage: anchor

Fibonacci valuations for every prime

Verified locator

L. A. Medina and E. Rowland, p-regularity of the p-adic valuation of the Fibonacci sequence, The Fibonacci Quarterly 53(3) (2015), 265–271. The arXiv record https://arxiv.org/abs/0910.2907 gives the journal reference; Theorem 1.4 (Lengyel) is on page 2 of arXiv:0910.2907v4 (26 January 2015), https://arxiv.org/pdf/0910.2907v4 . The proposed DOI 10.1080/00150517.2015.12427843 returns HTTP 404 from Crossref and doi.org (checked 2026-09-16); the attested locator is arXiv.

Claim and scope

Write alpha(p) for the least positive index with p dividing F_n. For n>=1, Theorem 1.4 states v_5(F_n)=v_5(n), and

• v_2(F_n)=v_2(n)+2 when n is zero modulo six;
• v_2(F_n)=1 when n is three modulo six;
• v_2(F_n)=0 in the other four residue classes modulo six.

For a prime p outside {2,5}, v_p(F_n)=v_p(n)+v_p(F_alpha(p)) when alpha(p) divides n, and v_p(F_n)=0 otherwise. The initial valuation v_p(F_alpha(p)) is not assumed to be one.

These formulas imply the square-transport equivalence in FPD.1 of Problems/wall-sun-sun-golden-unit-lift.md for positive multipliers; the zero multiplier is handled separately by F_0=0. The full valuation formula is attributed to Lengyel, rather than claimed as a new consequence of the dossier. The paper’s computational WSS bound is historical.

HD. Actual-rank harmonic cancellation and a false WSS construction

This is a written-theory continuation of the existing WSS/Wieferich family. It keeps the original Fibonacci sequence and fixed golden unit. No new Problems entry or formal declaration is introduced. In particular, the following results are not a proof of WSS existence or a new infinite non-WSS prime family.

HD.1 Fixed objects and the independent question

Let p>5 be prime, epsilon=(5/p), N=p-epsilon and rho the least positive index with p dividing F_rho. Put h=v_p(F_rho)=v_p(F_N)>=1 and q_p=F_N/p modulo p. The rank bound rho|N and the classical valuation formula justify the equality of depths without assuming h=1. Let L_n be the original Lucas sequence, L_0=2,L_1=1.

All congruences involving rational numbers below are in Z_(p), the rationals with denominator prime to p. For 1<=k<rho the integer F_k is a p-unit. Define the actual rational numbers

$$S_{\rho }=\sum _{k=1}^{\rho -1}\frac{L_k}{F_k},B_{\rho }=\sum _{k=1}^{\rho -1}\frac{(-1{)}^k}{F_k^2},A_{\rho }=\sum _{k=1}^{\rho -1}\frac{1}{F_k{F}_{\rho -k}}.$$

The attempted construction was to use unexpectedly strong divisibility of S_rho to force an initial WSS exception. HD.5 shows exactly why that single-test construction fails. HD.4 gives a necessary nonvanishing condition for genuine maximal-rank WSS primes.

HD.2 A complete torsion comparison at the actual initial depth

Lemma HD1. For every p>5 and its actual rho,h,

$$\boxed{B_{\rho }\equiv -\frac{5({\rho }^2-1)}{12}(\mathrm{mod}{p}^h).}\text{\hspace{1em}(HD1)}$$

Also, with

$$C_{\rho }=\sum _{k=1}^{\rho -1}\frac{(-1{)}^k{L}_k}{F_k^3},$$

one has C_rho=0 modulo p^h.

Proof. Put phi=(1+sqrt(5))/2, psi=1-phi, delta=sqrt(5) and nu=phi/psi=-phi^2 in the unramified quadratic p-adic algebra O tensor Z_p. In the split case this algebra is a product of two p-adic fields; it is not treated as one field. Binet’s identity gives

$${\nu }^k-1=\delta F_k/{\psi }^k.$$

Thus nu modulo p has exact order rho, p does not divide rho, and v_p(nu^rho-1)=h. The valuation on the quadratic algebra denotes coefficientwise p-divisibility; the displayed identity gives the same depth in both split factors. Simple-root Hensel lifting for X^rho-1 produces a unique rho-torsion point zeta congruent to nu modulo p. Factoring nu^rho-zeta^rho shows v_p(nu-zeta)=h, since the remaining geometric sum reduces to rho*zeta^(rho-1), a unit. Every nu^k-1 and zeta^k-1 for 1<=k<rho is a unit.

In characteristic zero, the polynomial P(X)=(X^rho-1)/(X-1) gives

$$\sum _{k=1}^{\rho -1}\frac{1}{1-{\zeta }^k}=\frac{\rho -1}{2},\sum _{k=1}^{\rho -1}\frac{1}{(1-{\zeta }^k{)}^2}=\frac{(\rho -1)(5-\rho )}{12}.$$

Indeed P(1)=rho, P’(1)=rho(rho-1)/2 and P’‘(1)=rho(rho-1)(rho-2)/3. Apply P’/P and its derivative at one. Subtracting the first sum from the second yields

$$\sum _{k=1}^{\rho -1}\frac{{\zeta }^k}{({\zeta }^k-1{)}^2}=-\frac{{\rho }^2-1}{12}.$$

The exact identities

$$\frac{(-1{)}^k}{F_k^2}=5\frac{{\nu }^k}{({\nu }^k-1{)}^2},\frac{(-1{)}^k{L}_k}{F_k^3}={\delta }^3\frac{{\nu }^k({\nu }^k+1)}{({\nu }^k-1{)}^3}$$

now prove the claims. Rational expressions with unit denominators preserve congruence modulo p^h. For the second expression its sum at zeta is exactly zero: replacing k by rho-k negates the term, and a possible middle term has zeta^k=-1 and is zero. The resulting rational sums descend to Z_(p). This proof uses no sampled value of h.

HD.3 Exact factorization and cubic-depth remainder

Lemma HD2. The following identity and congruence hold for every p>5:

$$\boxed{S_{\rho }=F_{\rho }{A}_{\rho },S_{\rho }\equiv -\frac{2F_{\rho }}{L_{\rho }}{B}_{\rho }(\mathrm{mod}{p}^{3h}).}\text{\hspace{1em}(HD2)}$$

Moreover,

$$\boxed{A_{\rho }\equiv \frac{5({\rho }^2-1)}{6L_{\rho }}(\mathrm{mod}{p}^h).}\text{\hspace{1em}(HD3)}$$

Proof. The addition identity L_k F_(rho-k)+L_(rho-k) F_k=2F_rho, summed after division by F_k F_(rho-k), gives the exact identity. Also

$$2(-1{)}^k{F}_{\rho -k}=F_{\rho }{L}_k-L_{\rho }{F}_k.$$

The norm identity L_rho^2-5F_rho^2=4(-1)^rho makes L_rho a p-unit. Put z=F_rho/L_rho, R_k=L_k/F_k and b_k=(-1)^k/F_k^2. Then

$$A_{\rho }=-\frac{2}{L_{\rho }}\sum _k\frac{b_k}{1-z{R}_k}.$$

Every denominator here is a p-unit. Use the exact expansion

$$\frac{1}{1-zR}=1+zR+\frac{z^2{R}^2}{1-zR}.$$

The linear correction sums to z*C_rho, which has valuation at least 2h by HD1. The remaining correction also has valuation at least 2h. Thus A_rho+2B_rho/L_rho belongs to p^(2h)Z_(p). Multiplication by F_rho proves HD2. Substitution of HD1 gives HD3.

The pairing and quotient-sum expansion are classical. In the published version of Ballot’s paper listed in HD.7, Lemma 23 proves the maximal-rank specialization modulo p^4. HD2 exposes the p^(3h) control while retaining arbitrary actual depth and allowing nonmaximal ranks. It is presented as an ordinary deduction from the displayed identities, not as a claim that this proof mechanism is new.

HD.4 Exact depth separation and a genuine necessary WSS condition

Theorem HD3. If rho<N, then

$$\boxed{v_p(S_{\rho })=h.}\text{\hspace{1em}(HD4)}$$

If rho=N and h>=2, then

$$\boxed{v_p(B_{\rho })=1,v_p(A_{\rho })=1,v_p(S_{\rho })=h+1,}\text{\hspace{1em}(HD5)}$$

with the exact nonzero first digits

$$\boxed{\frac{B_{\rho }}{p}\equiv \frac{5ϵ}{6},\frac{S_{\rho }}{p{F}_{\rho }}\equiv -\frac{5}{6}(\mathrm{mod}p).}\text{\hspace{1em}(HD6)}$$

Proof. One has 2<rho<=p+1. If p divides rho^2-1 then rho is p-1 or p+1. Combining either possibility with rho|p-epsilon forces rho=N. Thus in the nonmaximal case HD3 is a unit, so A_rho is a unit and HD2 gives HD4.

In the maximal case,

$${\rho }^2-1=p(p-2ϵ),v_p({\rho }^2-1)=1.$$

Under h>=2, HD1 fixes B_rho modulo p^2 at a quantity of valuation one, and B_rho/p=5epsilon/6 modulo p. HD3 likewise makes A_rho have valuation one. Also L_N=2epsilon modulo p, since phi^N=epsilon modulo p and psi^N=epsilon modulo p. Dividing HD3 by p therefore gives A_rho/p=-5/6 modulo p. Multiplication by F_rho gives all remaining statements. Nonzero constants are legitimate because p>5.

In particular the following exclusion is unconditional:

$$\boxed{\rho =p-ϵ\text{and}{B}_{\rho }\in p^2{\mathbb{Z}}_{(p)}⟹h=1.}\text{\hspace{1em}(HD7)}$$

This specifies an arithmetic class disjoint from WSS. No infinite size, density, or newly unbounded prime family for that class has been proved.

HD.5 The false-positive branch and a corrected two-test criterion

Theorem HD4. For maximal rank rho=N, the following two alternatives are disjoint, and they are exactly the causes of S_rho in p^3 Z_(p):

$$\boxed{S_{\rho }\in p^3{\mathbb{Z}}_{(p)}⟺(h\ge 2)\text{or}(h=1\text{and}{B}_{\rho }\in p^2{\mathbb{Z}}_{(p)}).}\text{\hspace{1em}(HD8)}$$

Equivalently,

$$\boxed{p\text{ is WSS}⟺S_{\rho }\in p^3{\mathbb{Z}}_{(p)}\text{and}{B}_{\rho }\notin p^2{\mathbb{Z}}_{(p)},\rho =p-ϵ.}\text{\hspace{1em}(HD9)}$$

Proof. HD1 always makes B_rho divisible by p in this rank regime. If h>=2, HD5 proves the first alternative and excludes the second. If h=1, HD2 holds modulo p^3, and its multiplier -2F_rho/L_rho has valuation exactly one. Thus S_rho is zero modulo p^3 precisely when B_rho is zero modulo p^2. This proves both directions and disjointness. For nonmaximal rank the separate criterion is simply S_rho in p^2Z_(p), by HD4. None of these equivalences alone supplies an existence argument.

Exact counterexample to the one-test construction. At p=11, epsilon=1, rho=10, F_10=55 and q_11=5. Direct rational summation gives

$$S_{10}=\frac{602943}{30940}=\frac{3\cdot 11^3\cdot 151}{30940},$$

$$B_{10}=-\frac{5782018759}{34462209600}=-\frac{11^2\cdot 2273\cdot 21023}{34462209600},A_{10}=\frac{54813}{154700}.$$

Both displayed denominators of S and B are units at eleven. Therefore v_11(S_10)=3 and v_11(B_10)=2, although h_11=1. This is an actual Fibonacci example, not a changed generator or artificial residue vector. The additional divisibility of S comes from cancellation in A. At the second diagnostic example p=1559, rho=1558, exact computations modulo p^5 similarly give q_p=36, v_p(S_rho)=3 and v_p(B_rho)=2.

HD.6 Connection to the latest torsion-state draft and remaining target

The actual SJC.2-SJC.5 text in #8343 was read at c5e9479e82bfc38925453bdf95309ebf3c166c43. It writes multiplication by v=phi^2 as a torsion phase and the original drift -q_p, and proves that a rephasing cannot remove its accumulated drift. The torsion comparison in HD1 uses nu=phi/psi=-phi^2 instead. Its order rho is the Fibonacci zero rank; it must not be confused with the order of v in SJC. Both constructions keep the actual mixed-characteristic algebra and use multiplicative torsion sections, not a ring section modulo p^2.

This continuation evaluates arithmetic functions on the complete rho orbit to test whether a high-order cancellation independently selects a WSS prime. HD8 proves that it does not: a second non-WSS cancellation branch exists. Requiring B_rho to vanish even more strongly actually excludes the maximal-rank WSS branch, by HD7.

The remaining sufficient target in this formulation is a prime of maximal rank with S_rho zero modulo p^3 and B_rho nonzero modulo p^2, or a nonmaximal-rank prime with S_rho zero modulo p^2. The formulas here do not force either event, prove their infinitude, or make them cheaper to find than the original Fibonacci quotient. The primewise joint arithmetic of the two sums still needs an independent estimate or an explicit witness. WSS existence is not declared completed.

HD.7 Sources, finite checks and scope

Christian Ballot, The Congruence of Wolstenholme for Generalized Binomial Coefficients Related to Lucas Sequences, Journal of Integer Sequences 18 (2015), Article 15.5.4, published 19 May 2015: https://cs.uwaterloo.ca/journals/JIS/VOL18/Ballot/ballot14.html . The official 22-page PDF, printed page 16, Lemma 23, supplies the related maximal-rank quotient-sum congruence modulo p^4; its numbering differs from the earlier arXiv version 1409.8629. Lemma 22 and Theorem 24 record related higher congruences and Lucasnomial consequences. The publication text and displayed formulas supply the cited comparison; no claim here depends on visual rendering.

The valuation input remains Lengyel’s theorem recorded in the original Medina-Rowland note above. Root-of-unity power sums, Hensel lifting and Lucas addition identities are classical. Repository keyword searches for Wolstenholme/Ballot found no matching indexed entry, but this is not an exhaustive statement about every unmerged branch. Bounded external searches did not establish global priority for the all-depth partition HD8. No first-ever arithmetic theorem, external conjecture resolution, or new unbounded WSS exclusion family is claimed.

The final exact checker uses the original recurrence modulo p^5, batch inversion, independent Fibonacci doubling and integer matrix powers. For the 427 primes from 7 through 3000, it checks HD1-HD4, the cubic-depth identity, and HD8-HD9 where the rank is maximal (163 primes). All actual initial depths are one. It independently constructs the rational p=11 counterexample. The two harmonic false positives in this range are 11 and 1559. Earlier exploratory checks of B_rho modulo p^2 covered 854 maximal-rank primes at most 20000 and gave the same two examples; this is not a new WSS search bound or a proof of their absence beyond that range. Finite tests do not certify the hypothetical h>=2 cases, whose proofs retain h symbolically. These written claims are not Lean/Scribe artifacts and are not kernel-certified.

DF. A phase-preserving two-scale filter with no harmonic false-positive branch

This is a written-theory continuation of HD in the SAME WSS/Wieferich problem family. It retains the original Fibonacci/Lucas sequences. The new observation combines two samplings of that sequence; it does not replace the golden unit by a freely chosen Hensel lift. The result removes the cancellation that produced the actual p=11 and p=1559 false positives. It is not a proof that an original WSS prime exists.

DF.1 The explicit observable and its exact depth

Fix a prime p>5, epsilon=(5/p), N=p-epsilon, rho=r(p), and the actual h=v_p(F_rho)=v_p(F_N)>=1. For a positive integer a with a=1 modulo rho, put

$${\mathcal{H}}_{\rho }(a)=\sum _{k=1}^{\rho -1}\frac{L_{ak}}{F_{ak}},{\mathcal{D}}_{\rho ,a}=a^2{\mathcal{H}}_{\rho }(1)-{\mathcal{H}}_{\rho }(a).$$

Every denominator is a p-unit: rho dividing ak would imply rho dividing k because gcd(a,rho)=1. These are ordinary rational numbers in Z_(p). Assume additionally p does not divide a(a-1). Such a scale can always be chosen without knowing h or q_p:

$$a=\{\begin{array}{ll}2\rho +1,& \rho =p-1,\\ \rho +1,& \rho \ne p-1.\end{array}\text{\hspace{1em}(DF1)}$$

Theorem DF1 (cubic-depth congruence and exact valuation).

$$\boxed{{\mathcal{D}}_{\rho ,a}\equiv \frac{5a(a-1)({\rho }^2-1)}{6L_{\rho }}{F}_{\rho }(\mathrm{mod}{p}^{3h}).}\text{\hspace{1em}(DF2)}$$

Consequently, with e=1 if rho=N and e=0 otherwise,

$$\boxed{v_p({\mathcal{D}}_{\rho ,a})=h+e.}\text{\hspace{1em}(DF3)}$$

In particular, the exceptional cancellation in HD.5 has been removed for EVERY prime in the theorem, rather than only excluded by a second condition on B_rho. The observable is nonzero as a rational number.

Proof. The rank bound gives 2<rho<=p+1 and p does not divide rho. It follows that v_p(rho^2-1) is one precisely at rho=N and is zero otherwise, as proved in HD.4. The proposed a has a=1 modulo rho; a-1 is either rho or 2rho and is a p-unit. The only possible nonunit rho+1 is p, at rho=p-1, and the alternative 2rho+1=2p-1 is a unit. Thus DF1 supplies the stated scale in every case.

The proof of DF2 is given in DF.2-DF.3 below using an integral Taylor expansion and a torsion point only as a proof coordinate. Its displayed leading term has valuation h+e, because L_rho,6,5,a,a-1 are p-units. For h>=1 and e<=1, h+e<3h. Hence the remainder cannot change that valuation, proving DF3. No actual depth is replaced by one.

DF.2 Explicit elimination of the unknown quadratic coefficient

Work in O_p=Z[phi] tensor Z_p, and set delta=sqrt(5) and nu=phi/psi. For split p this algebra is the product of two unramified local rings; coefficientwise p-divisibility is used throughout. Let zeta be the unique rho-torsion lift of nu modulo p. HD.2 proves that

$$x=\nu /\zeta -1\in p^h{O}_p,v_p(x)=h.$$

Define the actual rational function

$$G_{\rho }(X)=\sum _{k=1}^{\rho -1}\frac{X^k+1}{X^k-1}.$$

All denominators at zeta are units. Therefore G_rho(zeta(1+X)) has an integral power series on X in pO_p. Pairing k and rho-k gives its constant coefficient zero. Write its first terms as

$$G_{\rho }(\zeta (1+X))=AX+B{X}^2+X^{3}E(X),E(X)\in O_p[[X]].$$

Here the same letter B does not denote the rational sum B_rho of HD. It is an unknown Taylor coefficient, and no nonvanishing of it is assumed. The linear coefficient is known exactly:

$$\boxed{A=\frac{\rho ({\rho }^2-1)}{12}.}\text{\hspace{1em}(DF4)}$$

Indeed, differentiation gives A=-2 sum_(k=1..rho-1) k*zeta^k/(zeta^k-1)^2. The unweighted term is unchanged by k -> rho-k, so its weighted sum is rho/2 times its unweighted sum. The exact root-of-unity identity in HD.2 now gives DF4. A possible middle term at even rho obeys the same identity.

Since a=1 modulo rho, nu^a=zeta(1+x)^a. Using the integral expansion (1+X)^a-1=aX+a(a-1)X^2/2+X^3J_a(X), one obtains

$$\boxed{a^2{G}_{\rho }(\zeta (1+X))-G_{\rho }(\zeta (1+X{)}^a)=a(a-1)A(X-X^2/2)+X^3{R}_a(X),R_a\in O_p[[X]].}\text{\hspace{1em}(DF5)}$$

This follows by substituting into A X+B X^2: the two B terms are +a^2 B X^2 and -a^2 B X^2 and cancel identically. The coefficient of X^2 that remains is -A*a(a-1)/2. Cubic and higher terms remain integral because every denominator has a unit constant term. In particular no division by a factorial divisible by p occurs.

By Binet, delta*G_rho(nu^a)=H_rho(a). Thus the left side of DF5 at X=x is D_(rho,a)/delta. The unwanted quadratic coefficient has been eliminated before any arithmetic value of q_p is selected.

DF.3 Return to the fixed original Fibonacci error

Put z=delta F_rho/L_rho. The norm identity makes L_rho a p-unit, so z belongs to p^h O_p. The exact identity is

$$(1+x{)}^{\rho }={\nu }^{\rho }=\frac{1+z}{1-z}.$$

To degree two, subtract one from each side and then subtract half its square. The left side becomes rho(X-X^2/2) modulo X^3; the right side becomes 2z modulo z^3. Consequently,

$$\rho (x-x^2/2)\equiv 2\delta F_{\rho }/L_{\rho }(\mathrm{mod}{p}^{3h}).\text{\hspace{1em}(DF6)}$$

This is a finite rational-function computation with unit denominators. An infinite logarithm or a characteristic-p substitute for O/p^2O is not required. Substitute DF4 and DF6 into delta times DF5. Since rho and delta are units and delta^2=5, the result is exactly DF2. The resulting congruence is between rational numbers; intersecting p^(3h)O_p with Q_p gives the stated Z_(p) congruence.

DF.4 A single corrected test at the original WSS threshold

Corollary DF2. For every permitted a,

$$p\text{ is WSS}⟺{\mathcal{D}}_{\rho ,a}\in p^{2+e}{\mathbb{Z}}_{(p)},e=1_{\rho =N}.\text{\hspace{1em}(DF7)}$$

At maximal rank and the deterministic choice DF1,

$$\boxed{\frac{{\mathcal{D}}_{N,a}}{p^2}\equiv -\frac{5}{3}{q}_p(\mathrm{mod}p),p\text{ is WSS}⟺{\mathcal{D}}_{N,a}\equiv 0(\mathrm{mod}{p}^3).}\text{\hspace{1em}(DF8)}$$

Proof. DF7 follows from DF3. At rho=N one has (rho^2-1)/p=p-2epsilon and L_rho=2epsilon modulo p. For epsilon=1, DF1 gives a=2p-1; for epsilon=-1 it gives a=p+2. In BOTH cases a(a-1)=2 modulo p. Divide DF2 by p^2. Its remainder has valuation at least3h-2>=1; F_N/p=q_p modulo p. The resulting coefficient is -5/3, a unit. This includes h>=2, when both residues vanish.

Actual false-positive repaired. At p=11, rho=10 and a=21,

$${\mathcal{H}}_{10}(1)=602943/30940\equiv 0(\mathrm{mod}11^3),$$

but exact rational summation gives

$$\boxed{441{\mathcal{H}}_{10}(1)-{\mathcal{H}}_{10}(21)\equiv 1210=10\cdot 11^2\not\equiv 0(\mathrm{mod}11^3).}\text{\hspace{1em}(DF9)}$$

Its valuation is exactly two, as required for the actual h_11=1. Likewise, at the prior false-positive prime 1559, rho=1558, a=3117, q_p=36 and D_(rho,a)/p^2=1499 modulo p, nonzero. Neither false positive survives the corrected observation.

The condition p not dividing a(a-1) is essential for the valuation claim. For example the tempting scale a=rho+1=11 at p=11 is excluded. It introduces an automatic index lift and does not have DF3’s depth. The valid scale 21 avoids this effect. Sampling the fixed sequence is not permission to identify a multiplier-produced depth with h_p.

DF.5 The repaired observable has a unique zero in the entire local ball

The construction has a stronger local normal-form property. For the fixed torsion point zeta define, for X in pO_p,

$$D_a(X)=\delta \{a^2{G}_{\rho }(\zeta (1+X))-G_{\rho }(\zeta (1+X{)}^a)\}.$$

Theorem DF3. For all X,Y in pO_p,

$$\boxed{v_p(D_a(X)-D_a(Y))=e+v_p(X-Y)(X\ne Y).}\text{\hspace{1em}(DF10)}$$

Moreover D_a maps pO_p bijectively onto p^(e+1)O_p. Its unique zero is X=0, and division by p^e makes it a distance-preserving coordinate. This statement is componentwise in the split algebra as well.

Proof. In DF5 put C=delta*a(a-1)*A, so v_p(C)=e and C/p^e is a unit in every factor. The divided difference of X-X^2/2 at X,Y is 1-(X+Y)/2, which is one modulo p. The divided difference of X^3 R_a(X) has coefficients with at least two factors from X,Y; it therefore belongs to p^2O_p on this ball. After division by p^e, where e<=1, that correction belongs to pO_p. The complete divided difference is a unit times p^e, proving DF10.

For surjectivity write X=pW. The function D_a(pW)/p^(e+1) has the form (C/p^e)W+pE_1(W), with integral coefficients converging on O_p. Its reduction is multiplication by a unit, and its derivative is the same unit modulo p. Every target has a unique first residue solution and a unique successive base-p digit solution. Completeness gives a unique preimage. The argument applies to each factor in the split case. Since D_a(0)=0, zero is the sole root. Thus the harmonic false-positive branch has been removed at all precisions locally, not merely from the two previously observed examples.

DF.6 Scope, prior art, and remaining arithmetic

The same construction admits controlled higher-depth diagnostics: replace nu by nu^c with c=1 modulo rho. Its depth becomes h_c=h+v_p(c), and replace H_rho(a) by sum L_(ack)/F_(ack) and F_rho/L_rho in DF2 by F_(c rho)/L_(c rho). The proof is unchanged. These are labelled sampled subsequences; h_c>=2 is NOT evidence of an original WSS prime. The theorem itself retains h symbolically.

Classical inputs are the Fibonacci rank/valuation theorem already cited in this note, root-of-unity logarithmic derivatives, and unit-denominator Taylor expansion. Ballot’s published 2015 paper, Lemma 23 and Theorem 24, supplies the nearby harmonic/Lucasnomial congruence context. The characteristic-zero cancellation of a quadratic error by two samplings is also a standard extrapolation mechanism. The present contribution is the explicit phase-preserving arithmetic construction, cubic-depth remainder, and exact all-depth exclusion of the prior harmonic false-positive branch. Bounded searches for Lucas harmonic sums with dilation, extrapolation, and the displayed scale combination did not establish its global priority; no first-ever or external-open-problem-resolution claim is made.

The construction is a correct observation of the original q_p, not an independent theorem forcing q_p to vanish. It supplies neither an original WSS example nor an unbounded prime-family exclusion. Direct Fibonacci doubling tests q_p using O(log p) modular operations; the harmonic filter uses O(rho) such operations after initialization, with batch inversion available. No faster WSS search is asserted. The remaining existence obligation is still a new primewise arithmetic constraint or a certified prime at which the original quotient vanishes. This written proof is not a Lean/Scribe artifact and carries no kernel-certified claim.