# The supported coefficient: Dickman cancellation and the signed sieve budget **The full signed margin and twin-prime infinitude remain OPEN.** This completes the harmonic and elementary sieve-budget test proposed in [paired-factor-budget.md](paired-factor-budget.md) section 6. It does not estimate the exact complement favorably. The literature check below preceded the calculation; it supplies the simplification used here. The subsequent [joint-correction source audit](joint-correction-source-audit.md) checks the completion proposed in sections 5--6. It identifies the unrestricted logarithmic weight with an existing short Vaughan approximant, up to the paid exception. Reassembling the full joint correction returns the original open remainder; no signed bound is improved. The selected branch still cannot inherit the unrestricted weight's support. Read that audit before another completion attempt. ## 1. Novelty check and imported interfaces The owning subjects are harmonic Möbius sums, Dickman convolution, weighted sieves with switching, and prime-factor distributions of well-distributed sequences. Literature novelty is unestablished. | Primary source, read 2026-09-06 | Usable input and limit | |---|---| | [Soundararajan, arXiv:1005.3494v1](https://arxiv.org/pdf/1005.3494v1), p. 1 | The classical Dickman integral expansion and delay equation (4) below. We use these introductory identities, not the paper's asymptotic expansion theorem. | | [Drappeau--Mounier, arXiv:2606.30428v1](https://arxiv.org/pdf/2606.30428v1), Theorem 1.1, Lemma 2.2, Remark 2.4 | Rigorous integration of reciprocal monomials over fixed rational polytopes, allowing polynomial numerators. Precision complexity is polynomial for fixed dimension and polytope. The authors supply [Sage/LattE code](https://github.com/sarydrappeau/sieve_integral); its README was read, but code was not run or audited. | | [Matomäki--Zuniga Alterman, published version](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/986429394BA969687D48224E177E2F1C/S0305004125000258a.pdf/weighted_sieves_with_switching.pdf), section 4, (4.16)--(4.17) | Expanding their additive signed weight and using lower/upper sieves according to sign is established practice. Their weight differs from our subset coefficient. Only the previously matched linear-sieve input is used here, not their full weighted-sieve theorem. | | [Bharadwaj--Rodgers, published 2026](https://doi.org/10.4153/S0008439526101982), Proposition 2, Theorem 7, Lemma 8 | Shifted primes are 1/2 well-distributed; the full Poisson--Dirichlet law requires the stronger 1 well-distributed hypothesis. Lemma 8 gives unconditional factorial correlations only for test functions supported where the sum of factor exponents is strictly below 1/2. Section 6 checks the mismatch with our prime-partner term. | The C3 subset coefficient is polynomial on chambers cut out by rational linear inequalities in prime exponents. This matches the polynomial extension of the integration method. Its positive part can have a curved zero boundary inside a chamber: integrating the positive part requires additional enclosures or subdivision, not just calling a polytope routine. The signed total has the simpler analytic evaluation below, so this attempt does not build another general integration engine. **Source correction:** Drappeau--Mounier v1, p. 9, (21), prints the Dickman delay equation without the factor u on its left side. The correct equation is uD'(u)=-D(u-1), as in Soundararajan p. 1. The printed version would give D(2)=0; the correct value is 1-log 2>0. This is a scoped correction to that display. It does not establish a defect in their integration theorem, implementation or reported computations. ## 2. The complete supported harmonic coefficient Use T=log x and the C3 profile from [global-smooth-majorant.md](global-smooth-majorant.md). Put \[ a=11/50,\quad b=6/25,\quad U=1/2,\quad\delta=1/100,\qquad \phi(t)=\chi((t-a)/(b-a)). \tag{1} \] Here chi is 1 below 0, 0 above 1, and 1-35t^4+84t^5-70t^6+20t^7 between them. In particular 0<=phi<=1, phi=1 on [0,a] and phi=0 on [b,infinity). The actual profile uses the logarithms of the floored cutoffs and tends uniformly to phi. With W_L=floor(x^b), define \[ \begin{split} \mathcal M_\delta(x)&=\{1x^\delta,\ P^+(m)\le W_L\},\\ c_x(m)&=1_{\mathcal M_\delta(x)}(m)\widehat F_L(m),\\ H_\delta^\pm(x)&=\sum_m\frac{(c_x(m))_\pm}{m}, \qquad H_\delta(x)=H_\delta^+(x)-H_\delta^-(x). \end{split} \tag{2} \] The signs mean positive and negative parts, both nonnegative. The term m=1 is excluded because n=mr would then be prime and G_L(n)=0. It must be restored temporarily, then subtracted in the calculation. There are at most 50 prime factors with multiplicity, and |c_x|<=2^50. The unrestricted harmonic mass of numbers supported on these primes is the Euler product product_(x^deltax^delta)p^(-2))=o(1), also after multiplication by 2^50. For the squarefree terms, prime Mertens and finite rectangle approximation give the limits H_delta and H_delta^plus/minus in (2). The limiting subset function is continuous; exponent cut boundaries have measure zero. These observations justify the same passage to the limit for the absolute value, not just for the signed coefficient. Define nu_(c,d)(dt)=1_(c=0)nu^(*k)/k!, whose k=0 term is delta_0. On [0,U] only finitely many terms occur. Write \[ \mathsf A_\delta=\exp_*(\nu_{\delta,b}),\quad C_\delta(v)=\mathsf A_\delta([0,v]),\quad \mathsf M_\delta=\exp_*(-\nu_{\delta,\infty}). \tag{3} \] Set C_delta(v)=0 for v<0. Its total mass is exp(log(b/delta))=24. The classical Dickman function D, extended as zero at negative arguments when needed, satisfies \[ \begin{split} &D(u)=1\quad(0\le u\le1),\qquad uD'(u)=-D(u-1)\quad(u>1),\\ &D(u)=\sum_{k\ge0}\frac{(-1)^k}{k!} \int_{\substack{t_i\ge1\\t_1+\cdots+t_k\le u}} \frac{dt_1\cdots dt_k}{t_1\cdots t_k}. \end{split} \tag{4} \] The k=0 integral is 1. Thus mathsf M_delta([0,v])=D(v/delta); it has an atom 1 at zero and density -D(v/delta-1)/v for v>delta, zero density between zero and delta. Expanding the **complete** subset coefficient before taking its sign gives \[ 1+H_\delta =[(\phi\,\mathsf M_\delta)*\mathsf A_\delta]([0,U]). \tag{5} \] For clarity, the k-prime integral on the left has integrand sum_(A subset {1,...,k})(-1)^|A| phi(sum_(i in A)t_i), and measure product dt_i/t_i divided by k!. Choosing j members of the subset and l=k-j outside it yields the factor 1/(j!l!) and exactly the convolution on the right. This use of a factorial is valid because the complete subset integrand is symmetric. Since phi vanishes for t>=b, replacing exp_*(-nu_(delta,b)) by mathsf M_delta inside phi changes nothing. This proves (5), including the restored empty configuration. ## 3. An explicit collapse with a small profile error Convolution exponentials cancel locally: \[ \mathsf M_\delta*\mathsf A_\delta =\exp_*(-\nu_{b,\infty}),\qquad [\mathsf M_\delta*\mathsf A_\delta]([0,U])=D(U/b). \tag{6} \] All changes of order are in finite sums of finite integrals on [0,U]. The signed measure mathsf M_delta has no positive mass away from zero, and phi(0)=1. Hence (phi-1)mathsf M_delta is nonnegative. Equations (5)--(6) give the exact formula \[ \boxed{H_\delta=D(25/12)-1+E_\delta},\qquad E_\delta=\int_a^U(1-\phi(t)) \frac{D(t/\delta-1)}{t}C_\delta(U-t)\,dt\ge0 . \tag{7} \] Here is an analytic bound on the entire error, without a dimension-50 integration. Equation (4) implies, for u>=1, \[ uD(u)=\int_{u-1}^u D(v)\,dv. \tag{8} \] Both sides have derivative D(u)-D(u-1) and agree at u=1. Positivity and monotonicity of D then give nD(n)<=D(n-1), so D(n)<=1/n! and D(u) tends to zero. Since C_delta<=24, integrate the negative derivative of D(t/delta) from a to infinity to obtain \[ 0\le E_\delta\le24D(22)\le\frac{24}{22!},\qquad -12 and D is decreasing, while D(2)=1-log 2. For the strict lower bound D(25/12)>0: it also follows directly from (4), since at u=25/12 only k=0,1,2 occur and 1-log(25/12)>0. The rational comparisons log 2>69/100 and 24/22!<1/1000 suffice for the last strict inequality. The harmonic-limit derivation also works for any continuous fixed limiting profile between 0 and 1 with the same two plateaus, provided the actual profiles tend to it uniformly. It concerns a harmonic coefficient, not the sign of the actual shifted correlation. No Poisson--Dirichlet law for shifted primes is used to obtain it. ## 4. What the existing nonnegative sieves actually deliver For m in (2), take r prime with x/(2m)W_L and is the unique prime factor of n exceeding W_L, so each n has one representation. Let R_delta be the actual contribution to the full residual from these n with P^-(h)>W_R=floor(x^(1/20)). Define B_c^r and B_c^p as the sums of c_x(m)log r over these pairs, restricted respectively to rough h and prime h. Define B_plus/minus by replacing c_x with its nonnegative parts. The proof of [paired-factor-budget.md](paired-factor-budget.md) section 3 applies to each coefficient divided by 2^50. In particular, Pan--Ding through Wu's Lemma 2.3 allows m<=x^(1/2) and D=x^(9/20). Its maximum over y pays partial summation of log r=log(mr)-log m, using also the bounded coefficient c_x(m)log m/T. The main term is x/(2m varphi(d)). Removing (m,d)=1 now costs O_delta(x^(1-delta)T): sum_(p|m)1/(p-1)=O_delta(x^(-delta)) and sum_m |c_x(m)|/m=O_delta(1). Even moduli have zero actual and model mass. Thus the same density h(d)=1/varphi(d) for odd squarefree d, zero for even d, and arbitrary fixed logarithmic precision hold. For the previously specified rough lower, rough upper and prime upper sieves, put \[ \ell=10\log3,\quad u=40/3,\quad v=40/9. \qquad \begin{cases} TB_\pm^r\ge\ell C_2xH_\delta^\pm+o_\delta(x),\\ TB_\pm^r\le u C_2xH_\delta^\pm+o_\delta(x),\\ 0\le TB_\pm^p\le v C_2xH_\delta^\pm+o_\delta(x). \end{cases} \tag{10} \] These are nonnegative sieve applications. The error absorbs the harmonic limits in (2); it has not been quantified for a shrinking margin. The global exceptional-prime-power bound O_epsilon(x^(39/40+epsilon)), with epsilon<1/40, pays the exceptions to G_L(mr)=-F_L(m)log r. On rough h the exact right coefficient is G_R(h)=Lambda(h)-log h. Proper-power partners and replacing log h by T cost o_delta(x), using uniqueness and the absolute rough upper bound in (10). Consequently \[ R_\delta=T(B_c^r-B_c^p)+o_\delta(x). \tag{11} \] Apply (10) term by term to (11), retaining the direction of prime subtraction: \[ R_\delta\ge C_2xK_\delta+o_\delta(x),\quad K_\delta=(\ell-v)H_\delta^+-uH_\delta^- =uH_\delta-(u-\ell+v)H_\delta^+<-\frac{80}{9}. \tag{12} \] The last inequality uses (9), H_delta^plus>=0, and u-ell+v>0. Thus **this lower bound plus the main term cannot establish positivity with the complement omitted**. Computing the positive and negative harmonic integrals more accurately cannot repair that particular estimate. This is an INSUFFICIENT BOUND, not a proof that R_delta is negative or that signed aggregation is impossible. In contrast, the earlier two-family note supplied an actual negative upper bound on its smaller regional residual. Do not interchange these conclusions. ## 5. Exact remaining consumer Write R_rest,delta=Rhat-R_delta, by exact subtraction, and group the signed prime-partner correction with that complement: \[ \mathcal C_\delta(x)=R_{\mathrm{rest},\delta}(x)-TB_c^p(x). \quad S(x)=C_2x+TB_c^r(x)+\mathcal C_\delta(x)+o_\delta(x). \tag{13} \] The complement includes multiple large left factors, m outside (2), nonrough right partners and the small-prime tail. The exceptions have only been replaced in (11) with a paid error; the definition of the complement uses the actual residual. It is neither discarded nor assumed nonnegative. For example, (10) makes the following a sufficient **OPEN** fixed-margin input, on an unbounded common sequence of dyadic scales, for some eta>0: \[ \frac{\mathcal C_\delta(x)}{C_2x} \ge -1-\ell H_\delta^++uH_\delta^-+\eta. \tag{14} \] It would give S(x)>=eta*C2*x/2 eventually on that sequence, after the o(x) errors. This is a sufficient condition, not a necessity theorem or a new arithmetic bound. The broader logarithmically shrinking consumer remains allowed, but cannot be inferred from unspecified o(x) convergence here. Fixed delta's paid tail is likewise only a fixed multiple of x. ## 6. Why the 2026 prime-factor theorem does not fill this input On a prime partner h=p, the left input is n=p+2. This is the shifted prime sequence in Bharadwaj--Rodgers Proposition 2, with shift a=-2. Its level 1/2 factorial-correlation result, Lemma 8, evaluates tests whose selected factor product is below x^(1/2-epsilon) for some fixed epsilon>0. The proof uses that exact restriction on p. 11 to replace divisor counts by their means in (16). Symmetrizing the test function first justifies the ordered-factor rewrite there. Our marked prime cofactor has r>x/(2m)>=sqrt(x)/2, and the support contains regions where r>x^(1/2+epsilon). A direct test marking r does not satisfy Lemma 8. Merely selecting m's small factors does not assert that the remaining cofactor is prime or rule out additional factors. The full law in Theorem 7 would require the unproved stronger distribution hypothesis for shifted primes (their Conjecture 5). Theorem 11's endpoint upper estimate is also not a signed evaluation of (13). None of these statements directly supplies (14). This is a checked interface mismatch, not an impossibility claim for a special weighted cancellation. The algebraic completion of (13) has now been checked in the [joint-correction audit](joint-correction-source-audit.md), using the relevant prior art first. It returns the existing open Vaughan kernel. An additional arithmetic estimate would be needed for a new signed bound. A formal rewriting of (13), a one-input Dickman calculation, or further cell volumes alone does not supply that estimate. ## 7. Checks, falsifiers and source custody [supported-coefficient-dickman-validation.js](supported-coefficient-dickman-validation.js) uses exact rational arithmetic to check finite Möbius convolution including repeated primes and the empty term, convolution-exponential cancellation, logarithm/factorial comparisons and the sign-budget algebra. The embedded output is a VERIFIED finite certificate, not verification of an asymptotic theorem. The derivations above justify the limits, the Dickman density sign and the source transfer. Decisive falsifiers are a lost m=1 term, unjustified factorial symmetrization, a reversed signed-measure density or prime subtraction, an unbounded cofactor coefficient, or use of Lemma 8 outside its support. These checks were performed. No effective onset, full shifted main term or favorable complementary estimate was computed. PDF versions and SHA-256 hashes used: - Soundararajan v1, 2010-05-19: p. 1 statement inspected as image and text; 1e7237739cf2fe29ef1c5442ab43d229472323cbd2a83fa3cf8c8c77a18047b3. - Drappeau--Mounier v1, 2026-06-29: sections 1, 2.2, 2.4--2.6 and Theorem 3.1 statement read; pp. 2, 5, 9, 12 inspected as images; 63df14b8f638beee6613cb0f9a020ea28cb06e4a2752fbbd6fcb19737ceb4e9a. - Matomäki--Zuniga Alterman published: section 4, printed pp. 360--361 (PDF pp. 10--11), inspected as text and images; prior Lemma 2.5 source match retained; 9cd4cf5d5b51bb903aa613dd8d33c07358bfc7f0bb5bdab9e2f56c6dd60c8d0b. - Bharadwaj--Rodgers published online 2026-04-17: definitions, Proposition 2, Conjecture 5, Theorem 7, Lemma 8, Theorem 11 and Lemma 8's proof read; pp. 3, 6, 7, 8, 11 inspected as images; 2fbd36e2454a67315c88dfed0a9f69854eb6026779f8c49503372d8c1e98ca64. Wu/Pan--Ding custody and its exact import are in the paired note. The cited 2026 computational method and classical identities are positive prior-art matches; this review is not a literature-exhaustion claim or a claim of a new general theorem.