The Ratio of the Two Distinct-Totient Counts

../

---

This lead's notes that cite no source are author-recorded research, not claims.

Conditional phase reduction

The precise remaining question is whether

$$s\longmapsto\frac{A(1/r;s)}{A(1;s)}\qquad(0\le s<1) \tag{1}$$

is constant. Here $r$ is the weight's argument, the least-preimage ratio $\ell(d)/d$ of a tail totient, and $A$ is the coefficient of the unrefereed totients manuscript. L65, an author-recorded conditional deduction at tier 0, proves uniform existence of $A(1/r;s)$, its uniform positive lower bound, and

$$V'(x)=\frac{xG_m}{\log x}\bigl(A(1/r;\theta(x))+o(1)\bigr). \tag{2}$$

Its proof extends the manuscript's count of the largest prime, using Proposition 5.3's identity for the least preimage and the comparison lemma 6.6. Lemma 7.3 supplies the uniform tail bound for $V-V'=\sum_{k\ge1}N_k$; in particular, $A(1/r;s)=A(1;s)-\sum_{k\ge1}A(f_k;s)$, uniformly in $s$. Each phase occurs at arbitrarily large real arguments, so (1) is constant if and only if $V/V'$ has a limit. If it is nonconstant, Proposition 417.2 gives a nondegenerate interval of limit points. L65 assumes no continuity of the coefficient.

L43, a further tier-0 conditional deduction resting on L65, proves that continuity from the same premises. Both $A(1;s)$ and $A(1/r;s)$ are continuous on $0\le s<1$, and each tends to $\lambda/\gamma$ times its value at $s=0$ as $s\to1$, where $\lambda/\gamma>1$ is the factor by which $G_m$ jumps at the end of a period. Hence $A(1;s)$ is not constant, and (1) is a continuous function on the circle $\mathbb R/\mathbb Z$. The limit points of $V/V'$ are exactly the values of the reciprocal of (1), and (1) is constant if and only if all its nonzero Fourier coefficients vanish. A limit for $V/V'$ therefore needs an exact cancellation between two nonconstant coefficients.

L44, also tier 0 and conditional on the manuscript, constructs the phase profiles of fixed finite-prime divisibility conditions on the global least preimage. It approximates (1) from above by one finite-prime weighted sum with error at most $1/(Y-1)$, uniformly in phase, when primes at most $Y$ are retained. The construction and remaining cancellation are stated below.

L103, a tier-0 deduction from the same manuscript applied at two cuts, shows that outside vanishing mass every tail totient factors through one prime tuple and a coprime cofactor, with a multiplicative least preimage. The phase observable in (1) therefore becomes a function of the smooth cofactor, averaged over a phase-dependent law. A proposed renewal comparison through that law still needs a weighted fiber estimate. Even if established with an error tending to zero, its degenerating kernel would not force constancy. The section "Tail uniqueness and the cofactor fiber" records the proposal, its gaps, and one quantitative estimate that would suffice.

Everything in this reduction depends on the manuscript's unrefereed structural and weighted estimates. The accepted fixed-dilation result for Problem 416 does not certify those estimates. Theorem 2.2 of the manuscript also implies $\liminf V/V'>1$ from $N_1(x)\asymp V(x)$, but does not decide (1). Neither the manuscript nor these deductions claim a solution of Problem 417.

Target and exact question

Problem 417, status open. With $V(x)=\#\{\varphi(m)\le x:m\ge1\}$, the number of totient values up to $x$, and $V'(x)=\#\{\varphi(m):1\le m\le x\}$, the number of distinct totients of the integers up to $x$: does $\lim_{x\to\infty}V(x)/V'(x)$ exist, and is it greater than $1$? The problem page records the formulation and status evidence; no solution is claimed here. Its sources on $V'$ are the order of magnitude $V\asymp V'$ that Pollack, Pomerance and Treviño attribute to Ford, and the closing remark of Maier and Pomerance that a lower bound for $V(x)-V'(x)$, the number of totients up to $x$ all of whose preimages exceed $x$, is a harder question than the order of $V$.

Route and established inputs at their actual scope

The route is the section "Problem 417" of the connections page of the Problem 416 folder, where every statement below is proved; write $R(x)=V(x)/V'(x)$ and $V_0(x)=V(x)-V'(x)$.

• Proposition 417.2 (author-recorded): the set of subsequential limits of $R$ is the closed interval $[\liminf R,\limsup R]$, so either the limit exists or $R$ has continuum many limit points. Inputs: $V'$ has unit jumps and $V'(x)\ge\pi(x)$, so the adjacent variation of $R$ tends to $0$, and Step 7 of the preprint's reconstruction runs verbatim; the finiteness of the upper endpoint is the cited order of magnitude $V\asymp V'$ (page 7 of the 2013 manuscript on its card, quoting Ford's §§4–5, not reread), or alternatively Lemma 4.1(iii) of that manuscript as reconstructed on the Problem 49 folder's page with its two imported counts.
• Theorem 417.3 (author-recorded): $V_0(x)\ge\eta V(x)$ for arbitrarily large $x$, for some $\eta>0$; hence $\limsup R>1$, and if the limit exists it exceeds $1$. Inputs: Ford's Theorem 8 (card, statement level) with the paper's witness $d=54$, whose preimages $81$ and $162$ are verified on the connections page, so that a positive proportion of totients have every preimage divisible by $3$ and hence least preimage at least $\tfrac32$ times the value; Ford's Theorem 1 with the derivative bound of Step 1 of the preprint's reconstruction, which sums $V$ along a geometric sequence of scales; and Chebyshev's bound. The refereed inputs suffice; the accepted doubling law of the Theorem 1.1 reconstruction is not needed and would only sharpen a constant.
• The reformulation $V_0(x)=\#\{v\le x\text{ totient}:f_{\min}(v)>x\}$ with $f_{\min}$ the least preimage, and the frame under which the limit exists and equals the reciprocal of $\int_0^1G(1/u)\,du$, where $G$ would be a limiting distribution function of $f_{\min}(v)/v$ over totients and the general-scale law $V(ux)\sim uV(x)$ is assumed.

Finite-range numbers (exploratory, from a sieve of Euler's function to $6.2\cdot10^7$, which covers every preimage of every totient up to $10^7$): $R(x)$ is $1.0991$, $1.1000$, $1.0996$ and $1.1000$ at $x=10^4,10^5,10^6,10^7$, every totient $v\le10^7$ has $f_{\min}(v)\le2.0414\,v$, and among the totients in $(5\cdot10^6,10^7]$ the proportions with least-preimage ratio at least $1.1$, $1.25$ and $1.5$ are $0.319$, $0.266$ and $0.194$. These are observations, not evidence for the limit.

The same three unconditional statements and the conditional one are derived independently on the Problem 49 connections page (Propositions 2, 3 and 4) from Lemma 4.1 as stated, with the seed totient $1$ for boundedness and the reversed-pair seed $2^{18}\cdot257$, all of whose preimages are even, for the limit superior; its premise (W) is (P1) in convenient-integer form.

Provisional premises

• (P1) The short-interval form of Ford's Theorem 8: totients in $(2x/3,x]$ all of whose preimages are divisible by $3$ number $\gg V(x)$ for all large $x$. Ford's Theorem 4 localizes the paper's multiplicity counts to intervals $(x,x+y]$ with $y\ge x^\theta$ by a modification of the proofs of Theorems 1 and 2, and Theorem 8 is a corollary of the proof of Theorem 2, but the held paper does not state the localized form. Under (P1), Proposition 417.4 of the connections page gives $\liminf R>1$.
• (P2) A limiting distribution of the least-preimage ratio over totients, together with the general-scale law of Problem 416. Under (P2) the limit exists with the value given by the frame; without it the frame is only a reformulation. The general-scale law is an accepted input recorded on the problem page; convergence of the distribution is not established here.

(P1) remains a possible route from the refereed literature to a uniform lower bound above one. It is not needed for the conditional phase test, whose manuscript already gives that conclusion. (P2) is stronger than what that test needs: constancy of the single weighted average (1) suffices, whether or not a whole limiting distribution is available.

Consequential gap

Whether (1) is constant. For the finite coefficients, let $p_{H,s}(d)$ be the probability of the union of the witness events for $d$ in the manuscript's Equation (2.7). Cancellation of the common prefactor gives

$$\frac{A_H(1/r;s)}{A_H(1;s)} =\frac{\sum_d p_{H,s}(d)/\ell(d)}{\sum_d p_{H,s}(d)/d}. \tag{3}$$

L65 proves uniform convergence of this quotient. Both the finite witness set and its union probabilities vary with $s$. The sum is over distinct totients $d$, and $\ell(d)$ is minimized over all preimages, not over a chosen witness. A proposed separation of small-prime arithmetic from the phase must respect both operations. The manuscript's common-prefix proposition does not itself make the tail law independent of phase. Applied also at the shorter cut $P$, it removes the union over tail prime tuples outside vanishing mass (L103). Deduplication and the global minimum then act only inside the distinct totients of the smooth cofactor, which is where two phases must be compared.

The earlier near-hit mechanism for Problem 416 also does not decide this: $R$ compares two functions at one scale, so consecutive ratios do not telescope. The accepted pair construction bounds least preimages above without supplying the phase cancellation needed in (3).

The finite-coefficient test

Write $q_H(s)$ for (3), $g(d)=d/\ell(d)$, $\beta=\gamma/\alpha_s$, and $\mu_{H,s}$ for the probability measure on tail totients proportional to $p_{H,s}(d)/d$, so that $q_H(s)$ is the $\mu_{H,s}$-expectation of $g$. The following observations are author-recorded and unreviewed; they locate where the phase enters (3) without deciding (1).

• Scale covariance of a continuous relaxation. Multiply every tail coordinate $v_h=\log\log Q_h$, $P\le h<H$, by $c=\alpha_{s'}/\alpha_s$. The bands of (2.5) at phase $s$ become exactly those at phase $s'$, the simplex conditions of (2.5) are homogeneous, and every threshold $\beta\rho^hD(h,\eta)$ in (2.7) is unchanged, since $\beta$ becomes $\beta/c$ while $D$, linear in the $v_l$, becomes $cD$. Replacing unrestricted prime reciprocal sums by the Mertens measure $dv_h$ would leave a factor $c^{-P}$, from the prefactor $(\gamma/\alpha_s)^H$ and the Jacobian $c^{H-P}$. That common factor would cancel in a quotient. However, this map supplies no primes with the rescaled coordinates, no correspondence of distinct totient values, and no transport of the minimum over all preimages. The smooth cofactor and its bound also remain arithmetic. Mertens' theorem alone gives no error estimate for this weighted, deduplicated transport. The covariance is exact for the real inequalities and thresholds; extending it to (3) is the missing construction. After the factorization (6) below it must compare the restricted pair law and its cofactor fibers, whose smoothness bound $Q_P$ moves with the rescaled coordinates.
• Pieces and drift. For fixed $H$ the phase enters the witness conditions (2.5) only through the bands for $\log\log Q_h$ and the bound for $\log a$, so $[0,1)$ splits into finitely many intervals on which every witness set is constant (the first step of L43's proof). On such an interval only $\beta$ varies, with $d\beta/ds=-\lambda\beta$, and differentiating (3) gives $q_H'(s)=-\lambda\beta\,c_H(s)$, where $c_H(s)=\mathrm{Cov}_{\mu_{H,s}}(g,\partial_\beta\log p_{H,s})$. For a single witness $\eta$, $\partial_\beta\log p=-\sum_{h\ge H}\rho^hD(h,\eta)$, which is dominated by the largest tail primes, while $g$ is governed by the smallest primes of the least preimage. The rest of the variation of $q_H$ is a finite set of jumps where a witness enters or leaves a band or the cofactor bound. Constancy needs the integrated drift and the jumps between any two phases to cancel in the limit, including the jumps at the lowest band and at the cofactor bound, next to the smallest primes.
• Memoryless factorization. If all witnesses of $d$ share their primes $Q_h$ for $H'\le h<H$, then $D(h,\eta)=D'(h)+D''(h,\eta)$ with $D'$ the common contribution of those primes. Put $S'=\sum_{h\ge H}\rho^hD'(h)$. The memoryless property of the exponential variables gives exactly that $p_{H,s}(d)$ equals $e^{-\beta S'}$ times the probability of the union over $\eta$ of the events $E_h\ge\beta\rho^hD''(h,\eta)$ for all $h\ge H$. This factors a common part of the penalty. By itself it does not factor the sum over distinct $d$, the weight $1/d$, or $g(d)$, all of which retain the full arithmetic fiber. When all witnesses share the whole tuple ($H'=P$), $Q_P$ does not divide the cofactor, and the least preimage splits as in L103, all three factor through the cofactor totient; L103 shows that these conditions hold outside vanishing mass.
• Descent supplies no mixing law. At the nominal lower scale one has $\psi(\alpha_sH\rho^{-H})=H+s+o(1)$, so the central grid does not average over phases. The actual bands have width, and the cofactor is constrained by both size and smoothness. The proposed transition through the cofactor fiber, relation (9) below, would shift the phase by the relative position of $Q_P$ in its band. Under the coefficient-transfer premise stated below, that position concentrates as $P$ grows. Lemma 6.4 expressly bounds each phase without comparing witness sets at nearby phases, and Proposition 5.3 supplies no such comparison. A continuous simplex model omits deduplication and the global minimum; its fluctuations or Fourier coefficients therefore give no estimate for (1).
• Grouping the cofactors still requires a union over distinct values. Fix $H,s$ and a prime tuple $\mathbf Q$. Let $\mathcal A_{\mathbf Q,s}$ be its admissible cofactors in (2.5), and put $d_{\mathbf Q}(a)=\varphi(a\prod_hQ_h)$. The event in (2.7) depends only on $\mathbf Q$; denote it by $\mathcal E_{\mathbf Q}$ and put $\mathcal T_s(d)={\mathbf Q:d=d_{\mathbf Q}(a) \text{ for some }a\in\mathcal A_{\mathbf Q,s}}$. The exact weight is therefore

$$p_{H,s}(d)=\Pr\!\left(\bigcup_{\mathbf Q\in\mathcal T_s(d)} \mathcal E_{\mathbf Q}\right).$$

A fixed tuple does not make the cofactor-to-value map injective. For any admissible tuple of odd primes, $a=1$ and $a=2$ are admissible at sufficiently large $H$, and give the same totient and the same event, since $\varphi(2w)=\varphi(w)$ for odd $w$. A sum over cofactors with weights $1/\varphi(a)$ counts this contribution twice. More generally, the same duplication holds for every admissible odd $a$ for which $2a$ also meets the size bound. This exact duplication does not by itself give a lower bound for its share of the limiting mass; it does rule out identifying the cofactor sum with the coefficient without correction.

One exact grouping orders the finite set of tuples and replaces $\mathcal E_{\mathbf Q}$, for each $d$, by $\mathcal E_{\mathbf Q}\setminus \bigcup_{\mathbf Q'\prec\mathbf Q,\ \mathbf Q'\in\mathcal T_s(d)} \mathcal E_{\mathbf Q'}$. These disjoint events sum to $p_{H,s}(d)$, but their probabilities now depend on the competing tuples for $d$. Inside each tuple the sum must still be over distinct values $d_{\mathbf Q}(a)$, with the weight $1/d$ and the global minimum $\ell(d)$. Even the product identity $d_{\mathbf Q}(a)=\varphi(a)\prod_h(Q_h-1)$ needs coprimality: (2.5) allows $Q_P\mid a$. Proposition 5.3 at the cut $H$ concerns the long prefix before this retained tail, as $x\to\infty$ with $H$ fixed; by itself it does not assert uniqueness of the tail tuple under $\mu_{H,s}$ as $H\to\infty$. A phase-independent cofactor law must consequently be proved for this deduplicated, weighted measure, or accompanied by a uniform error for replacing it. L103 supplies the uniqueness and coprimality outside vanishing mass, and turns the duplication of $a$ and $2a$ into a sum over distinct cofactor totients $e$; the phase-independent cofactor law remains unproved.

The finite-prime construction

The unrefereed, non-arXiv note An explicit bound for Erdős problem 417, posted on 10 October 2026 (PDF), claims $\limsup V/V'<2.195202$ from Theorem 2.1 and Proposition 5.3 of the unrefereed manuscript. It compares the mean of $v/\ell(v)$ with $V'/V$ and bounds prime divisibility of $\ell(v)$ by deleting a prime power. Both pages were read, including the proof and numerical procedure; the rational prime sum was not rerun, and no independent acceptance is recorded.

L44 carries the needed prime estimates into the coefficient itself. For fixed squarefree $k$, define

$$C_k(x)=\#\{v\le x:v\in\mathcal V,\ k\mid\ell(v)\},\qquad b_{H,k}(s)=\sum_d\mu_{H,s}(d)\mathbf 1_{k\mid\ell(d)}.$$

It proves $b_{H,k}\to b_k$ uniformly in $s$, where $b_k$ is continuous on the circle and $C_k(x)/V(x)=b_k(\theta(x))+o(1)$. The construction uses the common-prefix identity only after all its primes exceed those dividing $k$; it then applies the manuscript's bounded-weight approximation. It neither chooses a preferred witness nor replaces the global minimum.

For integers $Y\ge2$, put $P_Y=\prod_{p\le Y}p$ and

$$Q_Y(s)=\sum_{k\mid P_Y}\frac{\operatorname{mobius}(k)}k b_k(s), \qquad T_Y=\sum_{p>Y}\frac1{(p-1)^2}\le\frac1{Y-1}.$$

Then L44 gives the explicit phase-uniform estimate

$$0\le Q_Y(s)-q(s)\le T_Y,\qquad \limsup_{H\to\infty}\sup_s \mathbb E_{\mu_{H,s}}\bigl[w_Y(d)-g(d)\bigr]\le T_Y, \tag{4}$$

where $w_Y(d)=\prod_{p\le Y,\,p\mid\ell(d)}(1-1/p)$. The order of limits is fixed $Y$ first, then $H\to\infty$. Thus large primes in the weight can be discarded uniformly; the missing input is cancellation in the single finite-prime weighted sum $Q_Y$. Constancy of every $b_k$ would suffice but is stronger than needed, and no limiting distribution of the whole least-preimage ratio is required. The marginal bounds $b_2\le1/2$ and $b_p\le p/(p-1)^2$ impose no equality between phases.

Why prime deletion does not give a dilation identity

Even the prime $2$ exposes the missing arithmetic comparison. For a totient $u$, let $e(u)$ be its least even preimage, which always exists, and let $o(v)$ be the least odd preimage of $v$, with value $+\infty$ if there is none. An even global minimum for $2u$ must be divisible by $4$; halving it gives an even preimage of $u$. Conversely, doubling an even preimage of $u$ gives a preimage of $2u$. Hence, exactly,

$$\ell(2u)=\min\{o(2u),2e(u)\},\qquad C_2(x)=\sum_{\substack{u\le x/2\\u\in\mathcal V}} \mathbf 1_{\,2e(u)<o(2u)}. \tag{5}$$

The two finite candidates have opposite parity, so there is no tie. Formula (5) explains the bound $C_2(x)\le V(x/2)$ and its failure to be an identity: for $u=4$, $e(4)=8$ and $o(8)=15$, so the summand is zero. These minima follow by evaluating $\varphi(n)$ for $n\le15$; no claim about an absent preimage is needed. The reciprocal least-preimage ratio itself is not monotone under doubling: $g(2)=2/3<g(4)=4/5$ but $g(8)=8/15<g(4)$. Thus monotonicity of this elementary arithmetic operation supplies no phase coupling. Estimating the indicator in (5) with the required phase control remains open here.

Tail uniqueness and the cofactor fiber

L103, a tier-0 conditional deduction, applies Proposition 5.3 at the cut $H$ and again at the cut $P=\lfloor\log\log H\rfloor$, whose common prefix ends at $Q_P$ and so contains every retained tail prime. Outside a set of data whose normalized mass is at most $\epsilon_H+\epsilon_P$ as $x\to\infty$, where $\epsilon_K$ is the rate of Proposition 5.3 at the cut $K$, every witness of the tail totient $d$ has the same prime tuple $\mathbf Q$, the prime $Q_P$ does not divide the cofactor, and

$$d=e\prod_{h=P}^{H-1}(Q_h-1),\qquad \ell(d)=\ell(e)\prod_{h=P}^{H-1}Q_h,\qquad e=\varphi(a)$$

for every witness $(\mathbf Q,a)$. Such $d$ are called factorized. L103 also gives $q(\theta(x))$, up to an error tending to zero with $H$, as the mean of $g(e)$ over the factorized data, with the manuscript's weights $1/(d\prod_{i\le R}(p_i-1))$. Distinct values and the global minimum now enter only through the cofactor totient $e$.

On a factorized $d$ the union in (2.7) has one event, which depends only on $\mathbf Q$, so exactly

$$\frac{p_{H,s}(d)}d=\frac{W_s(\mathbf Q)}e,\qquad W_s(\mathbf Q)=\frac{\exp(-\gamma S_{\mathbf Q}/\alpha_s)} {\prod_{h=P}^{H-1}(Q_h-1)},\qquad S_{\mathbf Q}=\sum_{h\ge H}\rho^hD(h,\mathbf Q).$$

Passing L103 from the manuscript's data to the coefficient's measure $\mu_{H,s}$ needs Lemma 6.4 and Proposition 6.5 for the indicator of the factorized set, which is not a function of $\ell(d)/d$. Their proofs bound the absolute difference between prime sums and volumes for each tail totient separately and sum these bounds over all tail totients and witnesses, so they appear to apply verbatim to every weight in $[0,1]$ that depends only on $d$ and the phase. With that reading of the two proofs as an additional premise, the $\mu_{H,s}$-mass of the nonfactorized tail totients tends to zero uniformly in $s$, and

$$q_H(s)=\frac{\sum_{(\mathbf Q,e)}W_s(\mathbf Q)\,g(e)/e} {\sum_{(\mathbf Q,e)}W_s(\mathbf Q)/e}+o_H(1) =\sum_{\mathbf Q}\pi_{H,s}(\mathbf Q)\,m_{H,s}(\mathbf Q)+o_H(1), \tag{6}$$

uniformly in $s$, where the sums run over the pairs given by factorized tail totients. Here $m_{H,s}(\mathbf Q)$ is the mean of $g(e)$ with weights $1/e$ over the cofactor totients $e$ paired with $\mathbf Q$, and $\pi_{H,s}(\mathbf Q)$ is proportional to $W_s(\mathbf Q)$ times their total weight. The pairs do not form a product set, since whether $e\prod_h(Q_h-1)$ is factorized depends on all its witnesses.

In (6) the phase enters in three places: the bands and simplex conditions of (2.5) on $\mathbf Q$; the threshold factor $\gamma/\alpha_s$ in $W_s(\mathbf Q)$; and the cofactor conditions $P^+(a)\le Q_P$ and $\log a\le\exp(2\alpha_sP\rho^{-P})$, which define the fiber. The scale covariance above matches the continuous regions by rescaling the coordinates $v_h$. It does not yet match the discrete factorized pairs: the fiber restriction also affects their total weights. Both the laws $\pi_{H,s}$ and the fiber means $m_{H,s}$ require comparison, including a uniform estimate for the prime-sum error with this restriction.

The fiber at its own scale. Put $Y=Q_P$. One possible route would replace the restricted cofactor fiber by all totient values below an appropriate power of $Y$, with harmonic weight $1/e$. In that unrestricted sum, partial summation would involve the counting coefficients $G_{m(t)}A(1;\theta(t))$ in the variable $\log\log t$. If their mass concentrated within $O(1/P)$ of the upper double-logarithmic scale, the continuity of $q$ (L43) would suggest the following fiber identification:

$$m_{H,s}(\mathbf Q)=q\bigl(\psi(\log\log Q_P)\bigr)+o(1) \qquad(P\to\infty). \tag{7}$$

Equation (7) is a proposed estimate, not an established harmonic inner limit. It requires control of nonfactorized pairs inside the fibers, counted with their multiplicity, which the distinct-value mass bound of L103 does not give; the cofactor smoothness and size restrictions near $Y$; and the weighted concentration just described. An error tending to zero after averaging with $\pi_{H,s}$, uniformly in $s$, would suffice in place of a pointwise estimate for every fiber. None is proved here. With $\log\log Q_P=(1+\epsilon_P)\alpha_sP\rho^{-P}$ and $|\epsilon_P|\le0.1$ by the band, a direct calculation from $\psi(B)=(\log B-\log\log B)/\lambda$ gives

$$\psi(\log\log Q_P)=P+s+\frac{\log(1+\epsilon_P)}\lambda +O\Bigl(\frac{\log P}P\Bigr). \tag{8}$$

If these comparisons were established uniformly in phase, (6)–(8) would express $q$ through itself:

$$q(s)=\lim_{H\to\infty}\sum_{\mathbf Q}\pi_{H,s}(\mathbf Q)\, q\Bigl(s+\frac{\log(1+\epsilon_P(\mathbf Q))}\lambda\Bigr), \tag{9}$$

a proposed renewal relation through the cofactor fiber whose kernel is supported in $[\log(0.9)/\lambda,\log(1.1)/\lambda]\subset[-0.18,0.16]$.

What a degenerating renewal kernel would give. Ford's deviation estimate behind the manuscript's (3.7), applied at the single level $P$, bounds the proportion of values for which some preimage has $|\epsilon_P|>\eta$ by $O_\eta(e^{-c\eta^2P})$. Under the same reading of Lemma 6.4 and Proposition 6.5, a kernel obtained from these weighted data would therefore tend to the point mass at zero. In the unrestricted continuous simplex model of Lemma 3.4 the coordinate at level $P$ is about $\alpha\rho^{-P}$ times a sum of $P+1$ exponential variables, so $\epsilon_P$ has spread of order $P^{-1/2}$ and the kernel of order $1/(\lambda\sqrt P)$. This variance description has not been transferred to the restricted fibers. An identity of the form (9) for any kernels concentrating at zero is satisfied by every continuous function on the circle and decides nothing by itself.

One sufficient quantitative route is the following. Suppose that for phase-free probability kernels $\kappa_P$ on $[-0.18,0.16]$ with $\int(1-\cos2\pi nt)\,d\kappa_P(t)\gg_n1/P$ for every $n\ne0$, as for a spread of order $P^{-1/2}$, one had

$$q(s)=\int q(s+t)\,d\kappa_P(t)+r_P(s),\qquad \sup_s|r_P(s)|=o(1/P).$$

Taking Fourier coefficients, $\hat q(n)(1-\hat\kappa_P(n))=\hat r_P(n)$ with $|1-\hat\kappa_P(n)|\gg_n1/P$ and $|\hat r_P(n)|=o(1/P)$, so every nonzero Fourier coefficient of $q$ vanishes and $q$ is constant. A kernel that depends on the phase would need a further mixing argument. The premises supply no such rate. Theorem 2.1 and L65 carry none. The manuscript's coverage bound at a cut $K$ is $e^{-cP_K}$ with $P_K=\lfloor\log\log K\rfloor$. In its existing estimates, the smooth cofactor below $P_K$ costs $K_K=\exp(CP_K\rho^{-P_K})$, which the $K^{-2}$ saving of Proposition 3.8 must absorb. Applying that coverage estimate at a cut of order $m(Q_P)\approx P$ gives an upper bound of order $(\log P)^{-c}$, which does not supply the $o(1/P)$ error needed by this Fourier argument. This is a limitation of the available bounds, not a lower bound on the actual error or an obstruction to every possible coupling.

What the rectangle estimate can compare

Theorem 2.1 of the totient-iteration paper compares divisibility of $p-1$, for $x<p\le2x$, by one to three distinct large primes with harmonic weights. Its coefficient is independent of $p$ and of the last prime. Section 3, Proposition 3.2, sums over an auxiliary prime to allow arbitrary bounded weights on one or two large factors in its truncated regions, with error $O((\log\log x)^{-7})$. Its hypotheses and the auxiliary-prime deduction were read; its underlying analytic proof has not been independently examined here. A collision between products of several shifted primes is not an event about divisibility of a single $p-1$ without another reduction.

The direct identification of the primes in (3) with those of the rectangle estimate fails uniformly in phase. With $T=\log\log x=B$, Theorem 2.1 admits only prime factors $q>x^{cT^{-20}}$, that is $\log\log q\ge B-20\log B+\log c$, and Proposition 3.2 only factors above $x^{T^{-6}}$. Every retained tail prime satisfies $\log\log Q_h\le1.1(\lambda/\rho)(H-1)\rho^{-(H-1)}$ and every cofactor $\log a\le\exp(2(\lambda/\rho)P\rho^{-P})$, bounds depending on $H$ alone, while the prefix primes after $p_0$ satisfy $\log\log p_i\le0.61B$. For all large $x$ the only prime of a basic preimage in the admissible range is therefore $p_0$. Applying the estimate with $p=p_0$ observes large factors of $p_0-1$, whereas the remaining statistic is a function of the tail and its cofactor. In particular fixed primes $q\le Y$ eventually lie below the allowed range. This direct use supplies neither constancy of the $b_k$ nor a certified error for the finite cut. An indirect encoding or application at another scale is not excluded, but would owe an exact allowed observable, a comparison with (6), and a phase-uniform omitted-mass bound. None has been constructed. Also, the coverage error relevant to the proposed renewal argument comes from Proposition 3.3 and Ford's band deviations at the lowest retained levels; the shifted-prime collisions of Proposition 5.1 already cost less than every power of $H$. Improving only that collision bound would leave the coverage estimate unchanged.

Finite ranges and finite cuts

The phase is $\theta(x)=\psi(\log\log x)-m$ with $\psi(B)=(\log B-\log\log B)/\lambda$, where $\lambda=\log(1/\rho)\approx 0.6114$ and $\rho\approx0.5426$ is the root of $\sum_ja_j\rho^j=1$. The function $\psi$ decreases for $B<e$ and increases for $B>e$, with minimum $1/\lambda\approx1.636$ at $B=e$, that is at $x=\exp(e^e)\approx 3.8\cdot10^6$. Hence $m(x)=1$ for $x$ between about $150$ and about $10^{213}$, and for $10^4\le x\le10^7$ the phase stays between $0.635$ and $0.675$. The finite-range values of $R(x)$ recorded above therefore sample one narrow window of the circle. They neither cover a whole period nor come with an asymptotic error small enough to certify phase dependence or its absence. The next period, $m=2$, runs from about $10^{213}$ to about $\exp(7.6\cdot10^7)$.

The available finite-cut estimates do not supply the certificate described in the stop verdict. The manuscript bounds its cut errors only by rates such as $e^{-cP}$, with $P=\lfloor\log\log H\rfloor$ and an unspecified constant $c$, for cuts that are sufficiently large in an unspecified sense. Already $P\ge2$ needs $H\ge e^{e^2}>1618$ retained levels, whose largest tail primes have double logarithm about $\alpha_s(H-1)\rho^{-(H-1)}$. No direct enumeration at such cuts is available here, and an evaluated difference would still have to exceed an unquantified cut error.

An incomplete continuous model

The continuous simplex does not include deduplication or global least preimages. One can illustrate possible smoothing by replacing a low coordinate by a sum $X_h$ of $h+1$ independent exponential variables of mean one. In this equal-weight model,

$$\mathbb E X_h^{it}=\frac{\Gamma(h+1+it)}{h!}.$$

Thus convolution in the logarithmic coordinate damps frequency $t$ by $|\Gamma(h+1+it)|/h!$. At a phase frequency the corresponding value is $t=2\pi n/\lambda$. This exact identity in the model supplies no estimate for (1): neither the observable $g$ nor the restricted law in (6) has been transferred to it. In particular the model gives no reason to assert that any Fourier coefficient of $q$ is nonzero, or to estimate its size.

Stop verdict and reopening conditions

The bounded structural test reaches neither conclusion, and this route is blocked. By L43 the exact missing identity is

$$\lim_{H\to\infty}\int_0^1 q_H(s)\,e^{-2\pi ins}\,ds=0 \qquad(n\ne0). \tag{10}$$

Equivalently, by the proved truncation (4), the exact remaining estimate is

$$\lim_{Y\to\infty}\left(\sup_s Q_Y(s)-\inf_s Q_Y(s)\right)=0. \tag{11}$$

This is a criterion, not an assumed decoupling theorem. A phase-independent limit for each fixed finite-prime pattern would be sufficient; cancellation of just the weighted sums in (11) would also suffice. Neither has been constructed. A disproof needs some $n\ne0$ for which the limit in (10) is nonzero. By uniform convergence it would follow from one cut $H$ whose integral in (10) exceeds $\sup_s|q_H(s)-q(s)|$ in modulus, or from two phases with $|q_H(s_1)-q_H(s_2)|>2\sup_s|q_H(s)-q(s)|$. The manuscript's cut errors tend to zero without certified numerical bounds, so the finite formula alone supplies no such inequality. At the finite-prime level, set $Q_{H,Y}(s)=\mathbb E_{\mu_{H,s}}w_Y$ and $\delta_{H,Y}=\sup_s|Q_{H,Y}(s)-Q_Y(s)|$. A concrete sufficient certificate is $Q_{H,Y}(s_1)-Q_{H,Y}(s_2)>T_Y+2\delta_{H,Y}$: the one-sided bound (4) then gives $q(s_1)>q(s_2)$. L44 proves $\delta_{H,Y}\to0$ for fixed $Y$, but gives no certified finite-cut value for it.

The two bounded tests of the cofactor interface end as follows. The direct rectangle application does not compare the phases, because its admissible primes miss every prime of a preimage below the largest. The coupling test succeeds only structurally: L103 gives tail uniqueness, coprimality and the multiplicative least preimage outside vanishing mass. Equation (6) also uses the explicitly stated extension of the prime-sum comparison; (7) and (9) remain proposals requiring additional fiber estimates. The exact blocked comparison is the difference of the fiber means of (6) at two phases,

$$\lim_{H\to\infty}\Bigl(\sum_{\mathbf Q}\pi_{H,s_1}(\mathbf Q)\, m_{H,s_1}(\mathbf Q)-\sum_{\mathbf Q}\pi_{H,s_2}(\mathbf Q)\, m_{H,s_2}(\mathbf Q)\Bigr)=0\qquad(0\le s_1,s_2<1), \tag{12}$$

which is equivalent to the constancy of $q$ under the stated premise for (6). The Fourier route through (9) would suffice with a phase-free kernel, the displayed lower bound on its Fourier damping, and an error $o(1/P)$ uniformly in phase. The available coverage bound of order $(\log P)^{-c}$ does not supply that error. No certified phase comparison or finite-cut separation has been obtained.

Arguments tried that do not close the gap:

• separate constancy of the two coefficients, which L43 excludes;
• fixed dilation and the pair constructions for Problem 416, which compare arguments at a single phase;
• the common-prefix proposition and Lemma 6.4, whose bounds hold at each phase;
• factorization over independent witnesses, which the union and the minimum over all preimages forbid;
• prime deletion, which gives upper bounds but leaves the global-minimum comparison in (5);
• tail uniqueness at two cuts (L103), which removes the union, the duplicated cofactors and the coprimality question but leaves the fiber;
• descent through the tail and the cofactor: the proposed renewal relation (9) still needs a weighted fiber comparison, and even a rateless version would be compatible with every continuous $q$ if its kernel tends to a point mass;
• the rectangle estimate of the totient-iteration paper, whose admissible primes miss the tail.

Reopen the constancy direction on a theorem that the distinct-totient structure at bounded levels is asymptotically invariant under rescaling the double-logarithmic grid, for example a renewal or mixing statement for the smallest prime factors of least preimages that is uniform in the phase, or on an exact identity expressing $\#\{v\le x:k\mid\ell(v)\}$ through values of $V$ at fixed dilates, or on a one-sided comparison of phases, such as a coupling of $\mu_{H,s}$ and $\mu_{H,s'}$ for $s\le s'$ under which $g$ does not decrease up to $o(1)$; by L43 any such monotonicity forces constancy, because $q$ returns to $q(0)$ at the end of the period. For the renewal route specifically, reopen on (9) with an error $o(1/P)$ uniformly in phase, for a phase-free kernel or with a mixing argument for a phase-dependent one. A sufficient implementation would control the coverage, fiber restriction and counting errors at a cut $K$ by $o(1/K)$; improving the shifted-product collision estimate alone does not do this. These rates describe one route, not a necessary condition for every possible proof. Reopen the nonconstancy direction on an evaluated phase gap exceeding the explicit prime truncation bound and a certified cut error, or on an exact asymptotic for a nonzero Fourier coefficient of $q_H$. The continuous model supplies no estimate of that coefficient. Finite-range stability of $R$ and finite-cut differences do not settle either direction. Either outcome remains conditional on the unrefereed manuscript, requires independent scrutiny of the whole catalog formulation before any solution claim, and leaves the status of Problem 416 unchanged.

Current review

Needs update. The coefficient reduction L65, the continuity result L43, the finite-prime reduction L44, the tail factorization L103, the finite-coefficient test with its grouped cofactor, cofactor fiber and rectangle-estimate interfaces, the sections on finite ranges and on the incomplete continuous model, and the stop verdict above are author-recorded and outside the scope of the existing review. The earlier route was independently reviewed on 2026-09-28 in the review record the Problem 417 review, whose subject is the Problem 417 section of the connections page together with this lead, and graded in the grade, which records this lead as reviewed. The graded verdicts of the statements the route rests on, in the review's numbering, are these. Lemma 417.1 (S1): refutation-failed, its three parts re-derived. Proposition 417.2 (S2): refutation-failed under (W), the order of magnitude $V\asymp V'$ that the finite upper endpoint needs, with the Problem 49 folder's Lemma 4.1 route as the alternative. Theorem 417.3 (S3): refutation-failed, the witness $54$ re-enumerated and the sum of $V$ along the scales re-derived under the derivative bound on its proved range. Proposition 417.4 (S4): refutation-failed as a conditional on (P1). The frame without a support bound (S6(a)), as this page states it: refutation-failed. The finite-range numbers (S7(a)): refutation-failed, every figure reproduced. This page's route, inputs, premises and gap (S8): refutation-failed, with its cross-reference to the Problem 49 connections page checked by the grader. Two sentences of the section that this page does not carry were refuted-as-stated: the support clause $G(t)=1$ for $t\ge2$ of the frame's hypothesis (S6(b); Ford's witness $2^{18}\cdot257$ at $k=4$ gives a positive proportion of totients with least-preimage ratio above $2$) and the sentence placing every element counted by $V_0(x)$ in $(x/2,x]$ (S7(b); witness $v=5888$ with least preimage $11985$ at $x=11900$). The grade's corrections C1, C2 and C3 were applied to the connections page (the range of the imported derivative bound, which no proof uses; the frame's hypothesis (H2) and its display, now without a support bound; and the confinement sentence), so its current text differs from the reviewed text at those places; no correction changes the reviewed assertions retained here. The premises (P1) and the distributional part of (P2) remain provisional. No tier is assigned by those reviews, and the problem's status is open.