bibkey: li2026nonwieferich authors: Ruofan Li and Jiuzhou Zhao year: 2026 title: “Non-Wieferich property of prime ideals and a conjecture of Erdos” doi: 10.48550/arXiv.2601.12753 url: https://arxiv.org/html/2601.12753v1 claim: “Theorem 1.1 supplies eventual single-prime order growth. The displayed global product step (4.1), digit Theorem 1.2, and complexity formula in Theorem 1.3 require the corrections below.” strata_touched:

• D5/S0/Carrier/Ring
• D5/S3/Arith/GoldenPrimeSplitting license: citation-only triage: anchor

Joint orbit audit for a proposed Wieferich input

This is a source audit within the existing problem family Problems/wall-sun-sun-golden-unit-lift.md, not another open-problem entry. The primary version inspected is arXiv:2601.12753v1, submitted19January2026. The checked arXiv record lists only that version. The claims below concern its displayed hypotheses and equations. They do not attribute the correction to the authors, invalidate its local Theorem1.1, or resolve the Erdos or WSS existence questions.

1. The exact cyclic subgroup under CRT

Let K be a number field with integer ring O. Let alpha,beta belong to O, N(beta)>1, and suppose alpha is coprime to beta and not a root of unity. Factor the principal ideal as

$$(\beta )=\prod _{j=1}^h{\mathfrak{p}}_j^{g_j}.$$

Write p_j for the rational prime below p_j, e_j for its ramification index, f_j for its residue degree, and

$$o_j(m)={\mathrm{ord}}_{(O/{\mathfrak{p}}_j^{g_{j}m}{)}^{\times }}(\alpha ).$$

Let H_m be the subgroup generated by alpha in (O/beta^m O)^times. The Chinese remainder map is an isomorphism on the ambient rings and groups, but its restriction to this subgroup gives

$$H_m=\{({\alpha }^n,\dots ,{\alpha }^n):n\in \mathbb{Z}\}$$

inside the product of the local cyclic images. The exponent is shared. Consequently the exact order is

$$\boxed{\#H_m={\mathrm{lcm}}_{1\le j\le h}{o}_j(m).}\text{\hspace{1em}(JO1)}$$

Proof. A positive exponent returns the joint element to one if and only if it is divisible by every o_j(m). The least such exponent is their lcm. The subgroup equals the whole product of the local cyclic groups exactly when those orders are pairwise coprime. Indeed equality requires equality of the finite orders, so the lcm must equal their product, and conversely that equality forces surjectivity of the injective subgroup map.

Equation(4.1) of the inspected preprint uses the product without that coprimality condition. Distinct prime ideals may lie above the SAME rational prime. Their high-order principal-unit factors then share arbitrarily large powers of that rational prime, and the missing condition is substantive.

2. A counterexample in the original golden field

Take

$$K=\mathbb{Q}(\sqrt{5}),O=\mathbb{Z}[\varphi ],{\varphi }^2=\varphi +1,\alpha =2,\beta =11.$$

The factorization

$$(X-4)(X-8)\equiv X^2-X-1(\mathrm{mod}11)$$

has two distinct roots. Thus (11) is the product of two distinct unramified primes, each with residue field F11. All local hypotheses of the displayed Theorem1.2 hold. The element2 is coprime to11 and is not a root of unity. Choose the complete digit representatives

$$\mathcal{D}=\{a+b\varphi :0\le a,b\le 10\},\#\mathcal{D}=121.$$

This is a valid digit set in Definition1.2: (1,phi) is an integral basis and reduction modulo11 reduces its two integer coefficients separately. It includes zero.

Proposition JO2. In this digit system, for every n>=1 and every positive truncation length m, the frequency of the digit phi in the11-adic expansion of alpha^n is zero. Therefore its Cesaro mean over n and subsequent limit over m are also zero, contrary to the stated value1/121.

Proof. The ordinary integer base-eleven expansion of2^n is a valid 11-adic expansion in O, with all digits in {0,…,10} contained in D. Equivalently, repeated division of an integer by11 gives an integer quotient and a rational digit; the phi coefficient stays zero at every step. Uniqueness of the expansion in the complete representative set identifies these with the digits used in the source definition. The digit phi has coefficient one at phi and therefore never occurs. The conclusion holds before either limit is taken, so no interchange of limits is involved.

Proposition JO3. For every m>=1,

$$\#H_m=10\cdot 11^{m-1},\mathcal{C}(2)=\frac{1}{2},$$

where C is the block complexity defined in equation(1.8)’s preceding paragraph. The displayed formula of Theorem1.3 gives one for this example.

Proof. The embedding Z/(11^m) into O/(11^m O) is injective since O is free on (1,phi). Thus the order of2 in the larger ring is its ordinary integer order. It is10 modulo11, and

$$2^{10}-1=1023=11\cdot 93,11\nmid 93.$$

The binomial or geometric-sum lifting formula therefore gives ord_(11^m)(2)=10*11^(m-1). The number of length-m digit blocks arising from powers is exactly #H_m, because the representative expansion identifies blocks with residue classes. Since N(11)=121,

$$\underset{m\to \infty }{\lim }\frac{\log (10\cdot 11^{m-1})}{m\log 121}=\frac{1}{2}.$$

In the source formula both prime ideals have g_j=e_j=f_j=1 and p_j=11, so its numerator and denominator both equal2log11 and its result is one. Already at m=1, the true joint orbit under CRT is {(2^n,2^n):n in Z}, of order10. Each local image has order10, so their product has order100; for instance (1,2) is in that product but not in the joint image. This directly falsifies the product step.

The propositions concern the statements as written in v1. They make no claim that local Wieferich lifting is false or that digit equidistribution cannot hold under stronger independent hypotheses.

3. Corrected all-number-field block complexity

Let P be the finite set of DISTINCT rational primes among the p_j. Then under the assumptions in Section1, the correct formula is

$$\boxed{\mathcal{C}(\alpha )=\frac{\sum _{p\in P}\underset{j:p_j=p}{\max }\left(\frac{g_j}{e_j}\right)\log p}{\sum _{j=1}^h{g}_j{f}_j\log p_j}.}\text{\hspace{1em}(JO4)}$$

Proof. The local eventual-growth result gives

$$\log o_j(m)=\frac{g_j}{e_j}m\log p_j+O(1).$$

More precisely, the part of o_j(m) prime to p_j is bounded independently of m, and v_(p_j)(o_j(m))=g_jm/e_j+O(1). This also follows directly: raise alpha to its finite residue order and then to a fixed p_j-power so that its difference from one has valuation greater than e_j/(p_j-1). The binomial expansion makes each further p_j-power increase this valuation by exactly e_j. The least exponent needed to reach valuation g_jm then has the displayed slope, with a bounded rounding error. Such a finite starting point exists since alpha is not a root of unity; an algebraic identity alpha^r=1 in a completion would already hold in K.

By JO1, the exponent of each rational prime p in #H_m is the maximum of its exponents in the o_j(m). Contributions from local groups above a different rational prime are bounded. Thus

$$\log \#H_m=m\sum _{p\in P}\underset{j:p_j=p}{\max }(g_j/e_j)\log p+O(1).$$

The number of blocks equals #H_m, and log N(beta)=sum_j g_j f_j log p_j. Divide by m log N(beta) and let m tend to infinity. This proves existence of the limit and JO4. The finite constants and local Wieferich thresholds may depend on alpha and beta; no uniform estimate over varying primes is asserted.

Corollary JO5. If beta is a rational integer b>=2 and [K:Q]=d, then

$$\boxed{\mathcal{C}(\alpha )=1/d}$$

for every eligible alpha. In particular, the ORIGINAL golden unit phi has block complexity1/2 in every rational integer base b>=2.

Proof. At a prime ideal above p|b, its exponent in (b) is g_j=e_j*v_p(b), so the maximum g_j/e_j is v_p(b). The numerator of JO4 is log b; its denominator is log|N_(K/Q)(b)|=d log b. Golden phi is a unit and not a root of unity, so it meets the assumptions for every b.

This supplies no new WSS zero-set restriction. In the golden case the initial depths affect bounded offsets in the order-growth formulas, while the normalized complexity1/2 is unchanged. A growth dimension cannot distinguish whether the original first exceptional lift occurs.

4. A sufficient repair of the digit-equidistribution hypothesis

Assume the displayed Theorem1.2 local hypotheses e_j=f_j=1, and ADD that the rational primes p_j are pairwise distinct. Then its digit conclusion holds for every complete representative digit set containing zero.

Proof. For sufficiently large m, the local cyclic orders have form c_jp_j^(g_jm+t_j), where c_j is independent of m and prime to p_j. In their lcm, any p_j-factor from another constant c_i is eventually dominated by the growing p_j-part of its own group. All other factors remain constant. JO1 therefore gives, for every sufficiently large m,

$$\#H_{m+1}/\#H_m=\prod _j{p}_j^{g_j}=N(\beta ).$$

The reduction H_(m+1)->H_m is surjective. Its kernel has size N(beta), which equals the entire ambient reduction kernel, so every retained length-m block has all N(beta) possible next digits. If A_m(c) counts the total occurrences of a fixed digit c among all length-m blocks in H_m, then

$$A_{m+1}(c)=N(\beta )A_m(c)+\#H_m.$$

Starting at a sufficiently large M and dividing by m#H_m yields the limit1/N(beta). This supplies a sufficient added hypothesis, not a classification of every possible digit system for repeated rational primes. Even under this repair, finite initial Wieferich depths are not determined by the limiting digit distribution.

Prior-art and delivery boundary

The local valuation growth, cyclic-order lcm law and digit counting are classical mechanisms. The source-specific counterexample and the corrected maximum-over-primes formula are the deductions documented here. Exact identifier/title searches with correction and erratum did not locate a published correction in the checked scope. This does not establish priority, and no external author response or journal acceptance is claimed.

No new Lean declaration, Scribe, CI file or automatic resolved-problem marker is attached. The existing WSS target remains open. JO2 and JO3 refute the displayed global statements of this preprint, not separately posed open conjectures; the two failures have the same missing joint-orbit constraint and are not counted as two solved open problems.

5. The original golden unit: curved joint orbit and a complete digit theorem

The counterexample with alpha=2 does not imply that the original golden unit has biased high digits. The following calculation keeps the common exponent and proves the canonical-digit statement for phi, including inert primes. It uses no assumption that the initial WSS depth is one.

Fix a prime p>5. Put O=Z[phi], phi^2=phi+1, and use precisely the digit set

$${\mathcal{D}}_p=\{a+b\varphi :0\le a,b<p\}.$$

Write eps=(5/p), N=p-eps, h=v_p(F_N)>=1, and let tau be the order of phi in (O/pO)^times. For m>=1 let H_m be the actual cyclic subgroup generated by phi modulo p^m. Norms in the integral basis are

$$Q(x,y)=x^2+xy-y^2,\nabla Q(x,y)=(2x+y,x-2y).$$

All appearances of h below refer to the initial depth of the ORIGINAL phi. A different digit representative set is not covered by this theorem.

5.1 Exact finite orbit, including arbitrary initial depth

Lemma GD1. For every m>=1,

$$\#H_m=\tau p^{\max (0,m-h)}.\text{\hspace{1em}(GD1)}$$

For m>=h it is a disjoint union of tau equal-sized norm slices. For 0<=i<tau, the i-th slice consists of all z=x+y*phi modulo p^m such that

$$z\equiv {\varphi }^i(\mathrm{mod}{p}^h),Q(x,y)\equiv (-1{)}^i(\mathrm{mod}{p}^m).$$

Each slice has exactly p^(m-h) points.

Proof. Golden Frobenius gives phi^N=eps modulo p, so tau divides 2N and p does not divide tau. Taking norms shows tau is even. The exact identity

$$(L_N-2\mathrm{eps})(L_N+2\mathrm{eps})=5F_N^2$$

and L_N=2eps modulo p show v_p(L_N-2eps)=2h. In the basis (1,sqrt(5)), which is an integral basis locally at p, the identity

$${\varphi }^N-\mathrm{eps}=(L_N-2\mathrm{eps}+\sqrt{5}{F}_N)/2$$

therefore has coefficientwise p-valuation h. This argument works also in the split quadratic algebra: valuation here means divisibility by the rational ideal p^e O, or the minimum of the two local valuations. The factor phi^N+eps is a unit, so phi^(2N)-1 also has depth h. Since 2N/tau is prime to p, the geometric sum identifies v_p(phi^tau-1)=h.

For any p-integral z with v_p(z)>=1, binomial expansion gives v_p((1+z)^p-1)=v_p(z)+1. An exponent prime to p preserves that valuation. It follows that phi^tau has exact order p^max(0,m-h) modulo p^m, proving GD1. Its norm is one. The norm-one elements congruent to one modulo p^h number p^(m-h): at each precision, the nonzero norm gradient imposes one nonzero linear equation on two new base-p digits, giving exactly p lifts. The cyclic subgroup is contained in this kernel and has the same size, so the two sets coincide. Multiplication by phi^i proves the slice statement. The tau residues are already distinct modulo p. No WSS nonexistence assumption is used in the order or cardinality argument.

5.2 Square-root cancellation on a single actual norm slice

For any h>=1, m>=2h, a p-adic unit c, and z0 modulo p^h with Q(z0)=c modulo p^h, set

$${\mathcal{S}}_m(c,z_0)=\{z\in (\mathbb{Z}/p^m{)}^2:z=z_0(\mathrm{mod}{p}^h),Q(z)=c(\mathrm{mod}{p}^m)\}.$$

The value of c modulo p^m suffices. Let e_M(t)=exp(2pii*t/M).

Theorem GD2. If (u,v) is a primitive frequency, meaning that at least one of u,v is not divisible by p, then

$$\boxed{\left|\frac{1}{\#{\mathcal{S}}_m}\sum _{(x,y)\in {\mathcal{S}}_m}{e}_{p^m}(ux+vy)\right|\le p^{h-\lfloor m/2\rfloor }.}\text{\hspace{1em}(GD2)}$$

When m=2r, the unnormalized sum is either zero or has magnitude exactly p^r. The nonzero alternative occurs precisely when the stationary normal condition in the proof has a solution in the prescribed low residue ball.

Proof. Put r=floor(m/2) and s=ceil(m/2). Partition the slice by its residues z modulo p^s. Choose one norm-c lift z* of each such residue to precision p^m. All other lifts are z*+p^s w, with w modulo p^r satisfying

$$\nabla Q(z^{\ast })\cdot w=0(\mathrm{mod}{p}^r).$$

The quadratic correction vanishes because 2s>=m. This kernel is a free rank-one module with p^r elements. Character orthogonality makes its contribution zero unless

$$(u,v)=t\nabla Q(z^{\ast })(\mathrm{mod}{p}^r)$$

for a scalar t. Primitivity forces t to be a unit. The matrix of the gradient is J=[[2,1],[1,-2]], with J^2=5I and Q(Jw)=5Q(w). Consequently every surviving residue satisfies

$$5c{t}^2=Q(u,v)(\mathrm{mod}{p}^r),z^{\ast }=J(u,v)/(5t)(\mathrm{mod}{p}^r).$$

If Q(u,v) is divisible by p there is no such unit t. Otherwise the square root equation has at most two roots modulo p^r. Its unit roots lift uniquely from modulo p because its derivative 10ct is a unit; any two are negatives. They produce opposite points z and -z, which cannot both lie in the prescribed class modulo p^h since c is a unit and p is odd. There is thus at most ONE surviving norm point modulo p^r. It has p^(s-r) norm-c lifts modulo p^s. Each surviving inner sum has magnitude at most p^r. The whole unnormalized sum is bounded by p^s. Divide by #S_m=p^(m-h) to get GD2. When s=r, at most one inner sum survives; when it does survive, its character is constant on that affine lift fibre and its magnitude is p^r. This proves the sharp alternative at even precision.

Applying GD2 to the equal-sized slices of GD1 yields exactly the same bound for the normalized character sum over H_m. For a frequency divisible by p^v in both coordinates, first reduce to H_(m-v). Reduction is surjective with equal fibres, so a nonzero frequency of conductor p^ell has bound p^(h-floor(ell/2)) whenever ell>=2h. No local direct-product independence is used anywhere in this argument.

5.3 All fixed-length high digit words are uniform on average

Fix r>=1 and m>=r. For z=x+y*phi modulo p^m, choose its two coefficient representatives in [0,p^m). Its high r-digit word is the pair

$$\left(\lfloor x/p^{m-r}\rfloor ,\lfloor y/p^{m-r}\rfloor \right)\in \{0,\dots ,p^r-1{\}}^2.$$

This is exactly a word of length r in the alphabet D_p, at digit positions m-r through m-1, counted from the least significant digit. It is not a statement about the leading digits of the positive integers F_n. Let P_(p,m,r)(a,b) be the proportion of H_m with the displayed word (a,b).

Theorem GD3. For m>=2h+r-1 and 0<=a,b<p^r,

$$\boxed{\left|P_{p,m,r}(a,b)-p^{-2r}\right|\le p^{h-\lfloor (m-r+1)/2\rfloor }(2+m\log p{)}^2.}\text{\hspace{1em}(GD3)}$$

The bound may be replaced by its minimum with one. In particular, for every fixed p and r, every high r-digit word has limiting frequency p^(-2r). This holds whether h=1 or h>=2.

Proof. Put M=p^m and L=p^(m-r). The condition on either coefficient is membership in an interval I_a=[aL,(a+1)L) in Z/M. With the normalized finite Fourier transform, the zero coefficient of its indicator is p^(-r). For k nonzero modulo M its Fourier coefficient vanishes if p^r divides k, by summing a complete geometric progression. Otherwise

$$|\widehat{1_{I_a}}(k)|\le \min \{p^{-r},1/(2\min (k,M-k))\}.$$

Indeed the numerator of the geometric sum has magnitude at most two, and sin(pik/M)>=2min(k,M-k)/M. Summing the harmonic bound proves

$$\sum _{k\mathrm{mod}M}|\widehat{1_{I_a}}(k)|\le 2+\log M.$$

Expand the two interval indicators by Fourier inversion. The zero pair contributes p^(-2r). Every other frequency pair with a nonzero coefficient has common p-valuation v<=r-1: at least one of its nonzero coordinates is not divisible by p^r. Its conductor exponent ell=m-v is therefore at least m-r+1>=2h. The character bound following GD2 is at most p^(h-floor((m-r+1)/2)). Multiply by the product of the two Fourier l1 bounds to obtain GD3. The exponential decay dominates the squared logarithm for fixed p,r,h, proving the limit.

For every fixed m the distribution on H_m is exactly the Cesaro limit of the word along the powers phi^n, since one full order enumerates H_m. Thus the order of limits is explicitly

$$\underset{m\to \infty }{\lim }\underset{X\to \infty }{\lim }\frac{1}{X}\#\{1\le n\le X:\text{the word at positions }m-r,\dots ,m-1\text{ of }{\varphi }^n\text{ equals }(a,b)\}=p^{-2r}.\text{\hspace{1em}(GD4)}$$

The limits are not interchanged. There is no conclusion about individual large exponents n, about most of their digits up to their real size, or about normality of a concatenation of Fibonacci numbers.

Corollary GD4. For the canonical digit set, the source’s averaged digit conclusion is true for alpha=phi, beta=p, for EVERY prime p>5. Moreover the average frequency in the first m positions is 1/p^2+O_(p,h)(1/m).

Proof. Its Cesaro average over the exponent is the average of the m individual digit distributions. The finitely many levels below 2h contribute O_(p,h)(1/m). The errors at subsequent levels have a finite sum by GD3 with r=1. This proves the stated rate. In particular the split and inert cases are both covered, without replacing the true orbit by a product. The analogous statement for each fixed word length follows in the same way.

5.4 The remaining joint constraint and the WSS information boundary

For every m>=h and every retained point (x,y) modulo p^m, there are exactly p possible next digit pairs (a,b), out of the p^2 alphabet symbols. They form the affine line

$$(2x+y)a+(x-2y)b\equiv \frac{c-Q(x,y)}{p^m}(\mathrm{mod}p),\text{\hspace{1em}(GD5)}$$

where c=1 or -1 is determined by the point’s norm modulo p. This follows by expanding Q(x+p^m a,y+p^m b) to the next precision. The actual orbit contains every one of these lifts by GD1, and a uniform orbit point gives uniform mass to each lift. Thus exact joint constraints persist at EVERY level, despite the marginal uniformity in GD3. In entropy language, the next digit has conditional entropy log p given all lower digits, whereas its marginal entropy tends to 2log p. Their mutual information tends to log p. This uses ordinary finite Shannon entropy, not a new assumption.

Consequently the block complexity is still 1/2, as in JO5. At base eleven, the original phi has asymptotically uniform high digits, whereas alpha=2 from Section2 never has a nonzero phi-coordinate digit. Both have block complexity 1/2. Complexity alone therefore determines neither digit marginal. The norm-conic argument supplies the additional cancellation for phi.

To make the role of initial depth explicit, fix an integer a>=0 and replace the generator by gamma=phi^(p^a). Its residue order is still tau and its principal depth is h+a by the same binomial proof as GD1. It has norm -1. The proofs GD1-GD4 apply with that depth and the corresponding residue balls. Hence arbitrarily large initial lifting delays coexist with the same limiting frequency for every fixed word, within this actual family of golden units. These altered generators are NOT original WSS samples. They demonstrate the failure of an inference from such digit uniformity alone to depth one.

For the ORIGINAL phi, GD3 requires m>=2h+r-1 and its error constant depends on h. It cannot be specialized at a fixed small precision while silently assuming that condition. It supplies no uniform bound on h as p varies, and no new constraint forcing or excluding q_p=0 in an unbounded prime family. For example, if h>=2, H_2 has only tau<=2(p+1) points. Its one-digit marginal at precision two consequently has total variation distance from uniform at least 1-2(p+1)/p^2. This finite-level sparsity is fully compatible with GD3: the latter concerns sufficiently high precision for that fixed h.

5.5 Established tools, target boundary and source roles

The Fibonacci-coordinate and norm identities are those of the existing D5/S1/Scale/Fibonacci.lean; the finite lift and norm-one descriptions agree with CG.1-CG.3 in the existing WSS dossier. The present argument adds an explicit conic character estimate and its canonical high-word consequence. It does not change the prior counterexample, restore the false general CRT product step, or assert the digit theorem for arbitrary representative sets.

Finite-character orthogonality, Hensel lifting of simple roots, and local stationary phase are classical. For related p-adic oscillatory estimates, see K. M. Rogers, A van der Corput lemma for the p-adic numbers, Proceedings of the American Mathematical Society 133 (2005),3525-3534, https://arxiv.org/abs/math/0311014 . GD2 is proved here directly by exact finite fibres and a quadratic congruence, without invoking an unverified uniform stationary-phase theorem. For the antecedent averaged-digit problem, see Taylor Dupuy and David E. Weirich, Journal of Number Theory158(2016),268-280, DOI10.1016/j.jnt.2015.05.022, and the stated source v1. Existing Fibonacci p-adic distribution theory also includes Bragman-Rowland, https://arxiv.org/abs/2202.00704 ; its whole-residue density is different from the high-word frequency GD4.

These are ordinary mathematical proofs of a positive original-unit subcase and a limit on a proposed WSS inference. No new externally posed open problem, WSS existence result, unbounded WSS exclusion family, or kernel-certified Lean declaration is claimed. Global priority for the combined formulation is unconfirmed; its standard analytic mechanisms are explicitly credited.

6. Exact golden conic transforms at every sufficiently high precision

Keep the original golden norm Q(x,y)=x^2+xy-y^2. The following refines GD2 at odd precision, supplies the exact phase rather than only an upper bound, and counts the active primitive frequencies. It remains in the same Wieferich problem family and the same actual coordinate carrier.

6.1 An exact chart, with its inverse and its quadratic normal phase

Put

$$J=\left(\begin{array}{cc}2& 1\\ 1& -2\end{array}\right),V(x,y)=(x-2y,-2x-y).$$

Work first in any commutative residue ring R=Z/M. Assume that two and c=Q(z) are units. For a parameter t with 1-5t^2 a unit, define

$${\Gamma }_z(t)=\frac{(1+5t^2)z+2tV(z)}{1-5t^2}.\text{\hspace{1em}(ES1)}$$

All inverses here are of explicitly specified units. No inverse of the modulus or of a nonunit is used.

Theorem ES1. The point Gamma_z(t) has norm c. If Q(w)=c, put

$$A=\frac{Jz\cdot w}{2c},B=\frac{z_2{w}_1-z_1{w}_2}{2c}.$$

Then w=Az+BV(z) and A^2-5B^2=1. Whenever 1+A is a unit, the unique chart parameter of w is t=B/(1+A); its denominator inverse is (1+A)/2. For every scalar lambda,

$$\lambda Jz\cdot {\Gamma }_z(t)=2c\lambda +\frac{20c\lambda t^2}{1-5t^2}.\text{\hspace{1em}(ES2)}$$

If t^3=0 in R, these exact identities simplify to

$${\Gamma }_z(t)=(1+10t^2)z+2tV(z),\lambda Jz\cdot {\Gamma }_z(t)=2c\lambda +20c\lambda t^2.\text{\hspace{1em}(ES3)}$$

Proof. Direct polynomial identities give Q(V(z))=-5c, Jz dot V(z)=0, and det(z,V(z))=-2c. The latter is a unit, so the displayed inverse coordinates give w=Az+BV(z), and the norm is c(A^2-5B^2). For a chart point those coordinates are A=(1+5t^2)/(1-5t^2) and B=2t/(1-5t^2). Conversely, substituting A^2-5B^2=1 and e(1+A)=1 gives 1-5(Be)^2=2e, proving the asserted inverse denominator and reconstruction. Every chart representation satisfies (1+A)t=B, which proves uniqueness. Pairing the chart with lambda Jz gives ES2. If t^3=0, then t^4=0 and (1-5t^2)^(-1)=1+5t^2; expansion yields ES3. These are identities in rings with zero divisors as well as in fields.

At R=Z/p^m, p>5 and 1<=h<=m, fix a norm-c point z. ES1 gives a bijection

$$p^{h}R⟷\{w:Q(w)=c,w=z(\mathrm{mod}{p}^h)\},h\ge 1.\text{\hspace{1em}(ES4)}$$

Indeed, the forward map stays in that ball. Conversely A=1 and B=0 modulo p^h, so 1+A is a unit and the recovered parameter lies in p^hR. Thus the slice has exactly p^(m-h) points, and the chart covers ALL of them, not only a selected family of lifts.

6.2 Exact vanishing criterion and the unique normal point

Fix p>5, h>=1, m>=2h, c prime to p and z0 modulo p^h with Q(z0)=c. Let S_m=S_m(c,z0) be the same actual norm slice as in GD2. Let xi=(u,v) modulo p^m be primitive, meaning xi is nonzero modulo p, and write

$${\mathcal{F}}_m(\xi )=\sum _{z\in S_m}{e}_{p^m}(\xi \cdot z),e_N(a)=\exp (2\pi ia/N).$$

Theorem ES2. The sum is nonzero if and only if

$$\exists {\lambda }_0\in (\mathbb{Z}/p^h{)}^{\times }:\xi ={\lambda }_{0}J{z}_0(\mathrm{mod}{p}^h).\text{\hspace{1em}(ES5)}$$

When this holds, there is a unique pair (lambda,z*) modulo p^m such that

$$5c{\lambda }^2=Q(\xi ),z_{\ast }=(5\lambda {)}^{-1}J\xi ,z_{\ast }=z_0(\mathrm{mod}{p}^h).\text{\hspace{1em}(ES6)}$$

It satisfies Q(z*)=c and xi=lambda Jz*. If ES5 fails, the sum is zero. If it holds, put r=floor(m/2) and

$$G_p(a)=\sum _{s\mathrm{mod}p}{e}_p(a{s}^2).$$

The entire complex value is

$$\boxed{{\mathcal{F}}_m(\xi )=\{\begin{array}{ll}{p}^r{e}_{p^m}(2c\lambda ),& m=2r,\\ p^r{e}_{p^m}(2c\lambda )G_p(5c\lambda ),& m=2r+1.\end{array}}\text{\hspace{1em}(ES7)}$$

Proof of existence and uniqueness of the normal point. A unit norm makes one coordinate of Jz0 a unit, so ES5 determines lambda0 uniquely. Applying Q and using Q(Jw)=5Q(w) gives the square equation modulo p^h. Its derivative 10c lambda0 is a unit, hence each next base-p digit of lambda is uniquely determined up to precision p^m. Formula ES6 then provides z*, with the required norm and normal vector. If a normal point exists modulo p^r, reducing it gives ES5. Globally the square equation has at most two roots, with opposite signs and opposite points. Since h>=1 and c is a unit, at most one lies in the prescribed residue ball.

Proof of the sum formula. As in GD2, partition the slice by its residues modulo p^ceil(m/2). In the remaining affine norm-lift fibre, character orthogonality kills every term unless xi is proportional to the norm gradient modulo p^r. All surviving points are therefore congruent to z* modulo p^r. ES4 parametrizes their ENTIRE contribution by t in p^r R. No enumeration of all residues is used to justify this reduction.

If m=2r, t^2=0, so ES2 gives the constant phase 2c lambda. There are p^r parameters. If m=2r+1, then r>=1 and t^3=0, since 3r>=2r+1. Write t=p^r s, with s modulo p^(r+1). ES3 gives the phase 2c lambda+20c lambda p^(2r)s^2. Its second part depends only on s modulo p, and each such residue has p^r lifts. The sum is therefore p^r e_(p^m)(2c lambda)G_p(20c lambda). The substitution s -> 2s in the finite field makes G_p(20c lambda)=G_p(5c lambda), proving ES7.

Finally, for a nonzero modulo p, exact additive-character orthogonality and the invertible substitution (x,y)=(s-t,s+t) give

$$|G_p(a){|}^2=\sum _{x,y\mathrm{mod}p}{e}_p(axy)=p.$$

Thus ES7 is nonzero in every case satisfying ES5. This proves both directions of the claimed vanishing criterion, including odd precision.

6.3 Sharp magnitude and the complete primitive-frequency support

Corollary ES3. For every primitive xi and m>=2h,

$$\boxed{|{\mathcal{F}}_m(\xi ){|}^2\in \{0,p^m\}.}\text{\hspace{1em}(ES8)}$$

For the normalized transform on the slice the nonzero squared modulus is p^(2h-m). Exactly

$$\boxed{(p-1)p^{2m-h-1}}\text{\hspace{1em}(ES9)}$$

primitive frequencies have that nonzero value. Equivalently, the fraction of primitive frequencies that survive is p^(1-h)/(p+1). The total squared normalized transform over primitive frequencies is (p-1)p^(m+h-1).

Proof. ES8 follows from ES7 and the Gauss-sum energy identity. There are exactly (p-1)p^(h-1) unit normal vectors lambda0 Jz0 modulo p^h. Each has p^(2(m-h)) lifts as a frequency modulo p^m. ES5 shows that these and only these survive. Multiplication gives ES9. Divide by p^(2m)-p^(2m-2) for the proportion, and multiply by p^(2h-m) for the energy. These formulas apply to a SINGLE prescribed norm slice; phases from different slices may cancel.

In particular the original GD2 bound improves from p^(h-floor(m/2)) to the sharp value p^(h-m/2). At odd precision the previous bound lost a factor sqrt(p). The old bound is valid but need not be attained. For p=7,h=1,m=3,c=1,z0=(1,0), the normal frequency (2,1) has exact squared sum 343 on a slice of 49 points. Its magnitude is 7*sqrt(7), not the former upper bound 49.

6.4 At most four critical contributions on the original golden orbit

Return to the original phi, its residue order tau and its ACTUAL initial WSS depth h, as in GD1. Assume m>=2h. Its full orbit H_m is the union of the tau residue slices specified there, with norms either one or minus one.

Theorem ES4. For a primitive frequency xi, its unnormalized transform over H_m is the sum of the terms ES7 indexed by

$${\mathcal{K}}_m(\xi )=\{(c,\lambda ):c\in \{1,-1\},\lambda \text{ a unit},5c{\lambda }^2=Q(\xi )(\mathrm{mod}{p}^m),(5\lambda {)}^{-1}J\xi (\mathrm{mod}{p}^h)\in H_h\}.$$

This set has at most four elements. If p=3 modulo four it has at most two. If p divides Q(xi), it is empty. In particular, putting C_p=2 for p=3 modulo four and C_p=4 otherwise,

$$\boxed{\left|\frac{1}{\#H_m}\sum _{z\in H_m}{e}_{p^m}(\xi \cdot z)\right|\le \frac{C_p}{\tau }{p}^{h-m/2}.}\text{\hspace{1em}(ES10)}$$

The bound may be capped by one. No independence of the slices is asserted.

Proof. Apply ES2 to each actual slice. A unit square equation has at most two roots for each of the two possible norms. If -1 is a nonsquare modulo p, only one of Q(xi)/5 and -Q(xi)/5 can be a unit square. A nonunit Q(xi) allows neither norm. Thus at most C_p slices can contribute, even when tau is much larger. Each nonzero sum has magnitude p^(m/2), and #H_m=tau*p^(m-h). The triangle inequality gives ES10, with any inter-slice cancellations retained in the exact formula.

Using the same interval Fourier expansion as GD3 consequently gives for m-r+1>=2h

$$\boxed{|P_{p,m,r}(a,b)-p^{-2r}|\le \frac{C_p}{\tau }{p}^{h-(m-r+1)/2}(2+m\log p{)}^2.}\text{\hspace{1em}(ES11)}$$

Indeed, every nonzero frequency with a nonzero interval coefficient has conductor exponent at least m-r+1. Reduction of the actual orbit is surjective with equal fibres, so ES10 applies at that conductor. The two interval Fourier l1 norms are each at most 2+m log p, as proved in GD3. This proves ES11 without an exchange of limiting operations.

6.5 What this does and does not resolve

ES1 supplies the exact finite-ring parametrization used in the sum proof. ES7 determines phase and magnitude, not merely a decay estimate. ES9 shows that conditioning on another low digit changes the surviving normal-frequency support. ES10 uses the global quadratic geometry to replace a sum over tau possible slices by at most four actual critical contributions. These are refinements of the existing golden-unit digit subcase of the source problem, within the same proof family.

They do not produce a WSS prime. In particular ES10-ES11 still require m>=2h, or its stated conductor version. No estimate independent of the unknown h is thereby available at modulus p^2. The spectral support of a slice defined using p^h cannot independently prove what that h is. The exact phase formula keeps the initial choice of the original phi rather than replacing it by a freely adjusted generator.

The exact chart, inverse, and quadratic phase identities ES1 are retained here as ordinary mathematics. Their proofs are algebraic normalization from the displayed inverse certificates, so they are not delivered as a new Lean declaration. The Fourier-sum, support-count, and original-orbit statements ES2-ES4 likewise remain ordinary proofs rather than claims of Lean kernel certification.

Rational parametrization of a nonsingular conic, quadratic Gauss sums and p-adic stationary phase are classical mechanisms. Related primary literature includes Keith M. Rogers, A van der Corput lemma for the p-adic numbers, Proc. Amer. Math. Soc.133(2005),3525-3534, https://arxiv.org/abs/math/0311014 ; and Djordje Milicevic and Sichen Zhang, Distribution of Kloosterman paths to high prime power moduli, https://arxiv.org/abs/2005.08865 . The present formulas are a direct specialization and refinement for the actual constrained golden norm orbits. No first-ever claim for stationary phase, Gauss evaluation, or conic parametrization is made, and no separately posed external open problem is counted as newly solved.

7. No cancellation between the actual slices: rank, all precisions and onset

This section resolves the remaining possible complete cancellation in ES4. It also treats the earlier precisions m<2h, to which ES4 did not apply. Keep p>5 prime, the ORIGINAL golden unit phi, its order tau modulo p, and its exact initial depth h=v_p(F_(p-(5/p))). Put rho=r(p), the least positive Fibonacci zero index. Write M=p^m and e_M(t)=exp(2pii*t/M). All frequencies in this section are for the actual two-coordinate orbit H_m, with its one shared exponent, rather than a product of local images. Let

$$T_m(\xi )=\sum _{z\in H_m}{e}_{p^m}(\xi \cdot z),{\hat{\nu }}_m(\xi )=T_m(\xi )/\#H_m.$$

7.1 The scalar stabilizer has exactly tau/rho elements

Lemma RS1. For every m>=1, the scalar subgroup

$$U_m=H_m\cap (\mathbb{Z}/p^m\mathbb{Z}{)}^{\times }$$

has order kappa=tau/rho in {1,2,4}. Here scalars are embedded as (a,0) in the actual golden algebra. If kappa=2, U_m={1,-1}; if kappa=4, it is {1,-1,i,-i} for a root i^2=-1 modulo p^m. In the latter case p=1 modulo4. For every 1<=d<=h, the tau points of H_d give exactly rho distinct projective directions, and each direction contains kappa of those points. The same statement holds after applying the invertible matrix J.

Proof. The scalar condition on phi^n is precisely F_n=0 modulo p^m, by its original Fibonacci coordinates. The rank and valuation formulas give its least positive index as rhop^max(0,m-h). The order from GD1 is taup^max(0,m-h). The kernel of the map from this cyclic group to its projective orbit therefore has size tau/rho. A scalar in H_m has norm 1 or -1, so its fourth power is one. Over Z/p^m, four is a unit and the fourth roots of unity lift uniquely from the field. Their group is cyclic of order dividing four. This proves the possibilities and the descriptions. Reduction from U_m to U_d is bijective: it is injective on these prime-to-p torsion elements and both groups have the displayed cardinality. Two unit points of H_d lie on the same scalar ray precisely when their quotient is in U_d. Hence there are tau/kappa=rho rays, each of size kappa.

This keeps the residue order tau and the Fibonacci rank rho separate. Examples are (p,rho,tau,kappa)=(7,8,16,2),(11,10,10,1),(13,7,28,4). The claim that -1 always belongs to H_m would be false in the second example.

7.2 The two and four scalar phases cannot sum to zero

For A a unit modulo an odd integer M>=3 and sigma in {1,-1}, set

$$D_{\sigma }(A)=e_M(A)+\sigma e_M(-A).$$

If i^2=-1 modulo M, also set

$$C_{\sigma }(A,i)=e_M(A)+e_M(-A)+\sigma (e_M(iA)+e_M(-iA)).$$

Theorem RS2. Every displayed D_sigma and C_sigma is nonzero.

Proof. Put x=e_M(A) and y=e_M(iA). A character of an odd cyclic additive group never takes the value -1: its M-th power is one, whereas (-1)^M=-1. Thus D_+(A)=0 would give x^2=-1, impossible. The equation D_-(A)=0 would give e_M(2A)=1, hence 2A=0 modulo M, impossible for a unit A. For the four-term sums, the exact factorizations are

$$xy{C}_{+}=(x+y)(xy+1),xy{C}_{-}=(x-y)(xy-1).\text{\hspace{1em}(RS1)}$$

In the plus case, x=-y would give e_M((1-i)A)=-1, and xy=-1 would give e_M((1+i)A)=-1; both are impossible. In the minus case, x=y or xy=1 would imply i=1 or i=-1, respectively, by faithfulness and the unit A. Either contradicts i^2=-1 in an odd ring of size at least three. This proves the two signs without a numerical lower-bound approximation.

The companion Lean source proves this statement for the actual canonical ZMod.stdAddChar, every odd modulus M>=3, and both signs. The proof derives all exclusions from the modulus, its faithful character and the displayed inverse of A. It does not assume the desired noncancellation statement.

7.3 The complete primitive-frequency support, at every precision

Theorem RS3. If m<=h, then T_m(xi) is nonzero for EVERY frequency, including nonprimitive frequencies. If m>h, put

$$d_m=\min (h,m-h).$$

For a primitive frequency xi, the following are equivalent:

$$\boxed{T_m(\xi )\ne 0⟺\xi \mathrm{mod}{p}^{d_m}\in \{\lambda Jz:\lambda \in (\mathbb{Z}/p^{d_m}{)}^{\times },z\in H_{d_m}\}.}\text{\hspace{1em}(RS2)}$$

Every surviving frequency has exactly kappa contributing slices. Therefore

$$\boxed{\#\{\xi \text{ primitive}:T_m(\xi )\ne 0\}=\rho (p-1)p^{2m-d_m-1}(m>h).}\text{\hspace{1em}(RS3)}$$

Proof before activation. If m<=h, GD1 gives #H_m=tau, prime to p. If a sum of tau p^m-th roots of unity vanished, its integer coefficient polynomial would be divisible by Phi_(p^m). Evaluating at one would give p|tau, since Phi_(p^m)(1)=p. This is impossible. Division by the monic cyclotomic polynomial is in Z[X], so the weight divisibility follows without assuming an independence model for the summands.

Proof in the linear regime h<m<=2h. Choose an actual center z in any of the residue slices of H_m. In the chart ES1, t is divisible by p^h, so t^2=0 modulo p^m. The ENTIRE slice is therefore z+2tV(z). Additive character orthogonality makes its sum zero unless xi dot V(z)=0 modulo p^(m-h). Since (z,V(z)) is a basis with unit determinant -2Q(z), that condition is exactly xi=lambda Jz modulo p^(m-h), with lambda a unit. If it holds, the slice sum is p^(m-h)e_M(xi dot z).

By RS1, each active ray contains exactly kappa slice centers modulo p^h: reduction to p^(m-h) has the same scalar kernel. Choose the other actual centers to be uz, u in U_m. All their phases are uA, where A=xi dot z is a unit because A=2lambda Q(z) modulo p. Their total is

$$T_m(\xi )=p^{m-h}\sum _{u\in U_m}{e}_M(uA).\text{\hspace{1em}(RS4)}$$

For kappa=1 it is a single nonzero term; for kappa=2 or4 it is one of the plus-sign sums in RS2. Thus it cannot vanish.

Proof in the curved regime m>=2h. ES2 selects the unique normal point in each active slice. If (c,lambda,z*) is one such triple, all other selected points are precisely uz*, with u in U_m. Indeed equal frequency normals imply the two points are scalar multiples, and both points lie in the actual group H_m. The other triples have norm u^2c and normal multiplier lambda/u, so their phases are uA, A=2c lambda. At even precision the full sum is

$$T_m(\xi )=p^{m/2}\sum _{u\in U_m}{e}_M(uA).\text{\hspace{1em}(RS5)}$$

At odd precision it is

$$T_m(\xi )=p^{(m-1)/2}{G}_p(5c\lambda )\sum _{u\in U_m}\left(\frac{u}{p}\right)e_M(uA).\text{\hspace{1em}(RS6)}$$

The elementary identity G_p(au)=(u/p)G_p(a) follows by counting square roots in F_p, and |G_p(a)|^2=p as already proved in ES2. For kappa=2, the last sum is D_sigma with sigma=(-1/p). For kappa=4, (-1/p)=1, and it is C_sigma with sigma=(i/p). RS2 again excludes complete cancellation. The formulas agree with the linear regime at m=2h.

Finally H_(d_m) has rho distinct normal rays. Each ray contains (p-1)p^(d_m-1) primitive vectors modulo p^(d_m), and each such vector has p^(2(m-d_m)) frequency lifts modulo p^m. These rays are disjoint, proving RS3. A nonprimitive frequency reduces, with equal group fibres, to the same formula at its actual conductor. Thus all frequencies at all precisions are determined, not only those of conductor p^m.

Corollary. At every m>=2h the improved whole-orbit bound is

$$|{\hat{\nu }}_m(\xi )|\le \frac{1}{\rho }{p}^{h-m/2}.$$

There are exactly kappa terms of magnitude p^(m/2) in the unnormalized sum, and kappa/tau=1/rho. In the linear range h<m<=2h the analogous bound is 1/rho. Applying the interval Fourier argument of ES11 therefore replaces its prefactor C_p/tau by 1/rho, with the same conductor condition. No lower bound for the magnitude is inferred merely from nonvanishing.

7.4 The first spectral zero and the original modulus p squared

Let j_0 be the least precision at which SOME Fourier coefficient of H_m is zero, and put j_0=infinity if no such precision exists.

Corollary RS4. The exact possibilities are

$$\boxed{j_0=\{\begin{array}{ll}h+1,& \rho <p+1,\\ \infty ,& \rho =p+1,h=1,\\ h+2,& \rho =p+1,h\ge 2.\end{array}}\text{\hspace{1em}(RS7)}$$

In particular the TOTAL number of zeros at modulus p^2 is

$$\boxed{Z_2=\{\begin{array}{ll}(p-1)p^2(p+1-\rho ),& h=1,\\ 0,& h\ge 2.\end{array}}\text{\hspace{1em}(RS8)}$$

Proof. For m<=h there are no zeros. At m=h+1, d_m=1, so the surviving fraction among primitive frequencies is rho/(p+1). It is below one exactly when rho<p+1. If rho=p+1 and h=1, d_m=1 at EVERY later level, so no primitive coefficient ever vanishes, nor does any lower-conductor coefficient. If rho=p+1 and h>=2, the next level m=h+2 has d_m=2, with surviving fraction 1/p, so zeros first appear there. At modulus p^2, imprimitive frequencies have conductor p or one and never vanish; subtract the RS3 support at h=1 from p^4-p^2 to obtain RS8.

For every split prime, rho<=p-1<p+1. Hence WSS is equivalent in that case to a zero-free full transform at modulus p^2. The same statement holds for inert primes of nonmaximal rank. It fails without the rank qualification: p=7 has rho=8=p+1 and h=1, yet its transforms are zero-free at EVERY precision. This is a genuine non-WSS example, not a hypothetical exception. For p=11, rho=10 and h=1, RS8 gives exactly2420 zeros at p^2.

These are exact spectral reformulations, not independent prime-family constraints. Evaluating the actual orbit may already require the same initial lifting information as the Fibonacci quotient. The theorem neither forces a spectral zero nor prohibits one independently of that arithmetic. It supplies no new integer WSS example or unbounded exclusion family.

7.5 Exact second and fourth moments above the curvature threshold

For kappa=1,2,4 put respectively

$$(E_{\kappa },C_{\kappa })=(1,1),(6,3),(36,9),C_{\kappa }=E_{\kappa }/\kappa .$$

Theorem RS5. For m>=2h, the primitive-frequency moments are

$$\boxed{\sum _{\xi \text{ primitive}}|{\hat{\nu }}_m(\xi ){|}^2=\frac{(p-1)p^{m+h-1}}{\tau },\sum _{\xi \text{ primitive}}|{\hat{\nu }}_m(\xi ){|}^4=\frac{C_{\kappa }(p-1)p^{3h-1}}{{\tau }^3}.}\text{\hspace{1em}(RS9)}$$

In particular the fourth moment is independent of m throughout this range. If E(H_m) counts ordered quadruples z1+z2=z3+z4 in the actual orbit, then

$$\boxed{E(H_m)-p^{2}E(H_{m-1})=C_{\kappa }(p-1)\tau p^{2m-h-1}(m\ge 2h).}\text{\hspace{1em}(RS10)}$$

Proof. The stationary parameterization (z*,lambda) with z* in H_m and lambda a unit counts each active frequency kappa times. Holding z* fixed, A=2Q(z*)lambda ranges over all units modulo M. Thus the moment problem reduces exactly to the unit average of the periods in RS5-RS6. For a difference v the unit-character sum is phi(M) if v=0 modulo M, -p^(m-1) if v is zero modulo p^(m-1) but not M, and zero otherwise. This follows by subtracting the sum over multiples of p from the full additive-character sum.

For U of size kappa, distinct u in U remain distinct modulo p. Hence in the second moment only u=v survives, giving kappaphi(M). In the fourth moment, equal pair sums give E_kappaphi(M). There are no hidden congruence collisions for p>5: for {1,-1,i,-i}, a nontrivial pair-sum difference is A+B i with integer |A|+|B|<=4 and A+B even. If it were zero modulo p, p would divide A^2+B^2, whose possible nonzero values in this parity/range are2,4,8,10,16. Only two and five can occur as prime divisors. Actual zero pair-sum relations are four zero-sum ordered pairs, four doubled points, and four mixed sums of multiplicity two, giving16+4+16=36. For {1,-1} the count is6, and for {1} it is1.

The quadratic-character signs in RS6 do not change these energy counts: unordered equal pairs contribute one, and all zero-sum pairs have the same sign product (-1/p). The common nonzero Gauss factor has magnitude sqrt(p). Multiplying by the count of stationary representations and the normalization |H_m|=tau*p^(m-h) gives RS9.

Finally character orthogonality gives sum_all |T_m|^4=M^2 E(H_m). Imprimitive frequencies reduce to level m-1; each sum has the extra fibre factor p, so their total fourth power is p^4*(M/p)^2 E(H_(m-1)). Subtract this from the full sum and substitute the second identity in RS9. This proves RS10. It concerns additive relations among the REAL orbit points, with no independent-coordinate replacement.

7.6 Formal scope and prior literature

GoldenConicFourierNoCancellation.lean and its authored Scribe supply the canonical-character noncancellation kernel RS2. The exact rank quotient, all-precision support, onset and moment identities RS1,RS3-RS5 are ordinary proofs in this section. No additional kernel acceptance is inferred from having a Lean representation of RS2. The earlier conic chart is unchanged.

The period/rank ratio in {1,2,4} is classical, including Vinson’s 1963 work and Ballot-Elia, Rank and period of primes in the Fibonacci sequence. A trichotomy, Fibonacci Quarterly45(2007),56-63, DOI10.1080/00150517.2007.12428243. It is not counted as a new arithmetic classification here. Vanishing sums of roots of unity and prime-power stationary phase are also classical subjects. For established general weight results see T. Y. Lam and K. H. Leung, On Vanishing Sums of Roots of Unity, Journal of Algebra 224(2000),91-109, DOI10.1006/jabr.1999.8089. For explicit prime-power exponential sums see S. J. Gurak, Kloosterman sums for prime powers in P-adic fields, JTNB21(2009),175-201, DOI10.5802/jtnb.665, and the Milicevic-Zhang source in Section6. The additive character in the formal source is Mathlib’s actual ZMod.stdAddChar; its injectivity is used, not supplied as an unproved numerical-observation assumption.

The new completion here is the exact scalar-stabilizer reduction of the original constrained golden orbit and the resulting disappearance of the previously unresolved cancellation possibility. No worldwide priority is asserted for this specialization, and no separately posed open problem is counted as newly solved. In particular the first-zero formulas distinguish the spectral effects of h but do not independently constrain its arithmetic value as p varies.