A computer-assisted 23/33 + ε bound for the exceptional set in the binary Goldbach problem.
Abstract
Let \(E(X)\) denote the number of even integers not exceeding \(X\) which are not a sum of two primes. Building on the zero-packet framework of Zhao and the exceptional-set reduction of Pintz, we prove \[E(X)\ll_\varepsilon X^{23/33+\varepsilon},\] and hence \(E(X)\ll X^{69697/100000}\). The new ingredient is not a stronger zero-density theorem. It is an exact use of information already present in the fixed-class and unrestricted zero-density inequalities. First, an exhaustive early-or-late split retains the positive density charge supplied by each character class’s own first zero. Second, character classes are kept in their common first-zero order while the \(R\)- and \(T\)-parts of Zhao’s packet are recombined. A cap-and-mass majorization lemma then gives the exact quadratic maximum over the resulting enlarged polytope.
At the exponential coefficient \(A=33/10\), exact rational interval arithmetic proves, in every discretized non-near-Siegel branch, \[\mathcal Q_A<\frac{198479}{200000}=0.992395;\] the directed limiting value is at most \(0.992394547694\), with slack at least \(0.007605452306\). The near-Siegel branch has a separate positive gap depending on the fixed lower zero defect. A coefficient-stable form of the Pintz–Zhao bridge gives \[E(X)\ll_\varepsilon X^{23/33+\varepsilon}.\] The exact inequality \[\frac{69697}{100000}-\frac{23}{33} =\frac1{3300000}>0\] then gives the stated exponent. All transcendental comparisons in the finite certificate use outward rational enclosures; floating point is used only to select trial parameters which are subsequently verified exactly. Source identifiers and reproduction commands are included.
1. Introduction
Write \[E(X) := \#\{\,n\leq X:n\equiv0\pmod 2,\ n\neq p+p'\text{ for all primes }p,p'\,\}.\] Pintz proved \(E(X)<X^{0.72}\) for sufficiently large \(X\) (Pintz 2018). Zhao subsequently obtained \[E(X)=O(X^{7/10})\] (1.1) and, by the same zero-density architecture, the Linnik-type bound \(P(q)=O(q^5)\) (Zhao 2026). The implicit constants in these exceptional-set results are ineffective.
The purpose of this paper is to refine Zhao’s Goldbach packet while leaving its analytic zero-density inputs unchanged. The refinement has three parts.
Inside a fixed relative-conductor class, the first zero of that class contributes a positive \(D\)-term to Zhao’s fixed-class inequality. An exhaustive partition of its possible location yields a universal class cap without assigning an invalid upper position from a cumulative global count.
The unrestricted inequality orders the possible character classes by their first zeros. This gives rankwise \(R\)-caps, while the unrestricted estimates give total \(R\)- and \(T\)-mass budgets.
The two cap sequences refer to the same ordered classes. They must therefore remain coupled. An exact aligned majorization lemma maximizes \(\sum_i(R_i+T_i)^2\) and prevents the loss created by separately maximizing the square and cross terms.
The coefficient in Zhao’s packet is \(10/3\). We work at \[A=\frac{33}{10}.\] (1.2) This is a strengthening: lowering \(A\) increases every exponential zero weight.
Theorem 1 (Main theorem). Let \(E(X)\) be the binary Goldbach exceptional set defined above. For every \(\varepsilon>0\), \[\boxed{E(X)\ll_\varepsilon X^{23/33+\varepsilon}.}\] (1.3) The implicit constant is ineffective.
Corollary 2. One has \[E(X)\ll X^{69697/100000}=X^{0.69697}.\] (1.4)
The exact improvement in the base exponent over (1.1) is \[\frac7{10}-\frac{23}{33}=\frac1{330}.\] (1.5)
The proof is computer-assisted only in a finite collection of explicit inequalities. The analytic inputs are quoted from Zhao (Zhao 2026, Lemma 3.1, Lemma 3.3, (3.14)–(3.43)) and Pintz (Pintz 2018, sec. 2, Theorems I–K). The elementary majorization argument is proved in full in Section 5; the exact analytic-to-certificate correspondence is stated in Section 7.
Remark 3 (What the theorem does not prove). Theorem 1 still permits a power-sized exceptional set. It does not prove the binary Goldbach conjecture and it gives no effective threshold beyond which every even integer is Goldbach.
2. The zero packet and the exponent bridge
Packet notation
Let \(Q\) be a modulus and let \(\{\mathcal K_\nu\}_{\nu\in I}\) be any finite family of nonempty, pairwise-disjoint sets of Dirichlet characters modulo \(Q\). We call their union the relevant characters and call the \(\mathcal K_\nu\) packet classes. For fixed truncation height \(H\), let \(\mathcal Z_\nu\) be the multiset of zeros \(\rho=\beta+i\gamma\), counted with multiplicity, of the primitive \(L\)-functions inducing the characters in \(\mathcal K_\nu\) which satisfy \[1-\frac{H}{\log Q}\leq\beta\leq1, \qquad |\gamma|\leq H.\] Set \[\lambda_\rho=(1-\beta)\log Q, \qquad S_{\nu,A}=\sum_{\rho\in\mathcal Z_\nu}\mathrm e^{-A\lambda_\rho},\] and label defects within a nonempty class and the classes themselves by \[\lambda_{\nu,1}\leq\lambda_{\nu,2}\leq\cdots, \qquad \lambda_{1,1}\leq\lambda_{2,1}\leq\cdots.\] Thus \(\lambda_{1,1}\) is the distinguished smallest defect and \(\mu_\nu:=\lambda_{\nu,1}\) is the first defect of class \(\nu\). Define the quadratic packet \[\mathcal Q_A=\sum_\nu S_{\nu,A}^{\,2}.\] (2.1)
Definition 4 (Coefficient-\(A\) packet property). For fixed \(A>0\), write \(\mathsf Z(A)\) for the following assertion. For \(c>0\), put \[H_c:=\max\left\{5.68,\ 1.09\log\frac1c\right\}.\] For every \(c>0\) there exists \[\kappa=\kappa(A,c)>0\] such that, for every \(H\geq H_c\) and \(C\geq1\), there exists \[Q_0=Q_0(A,c,H,C)\] with the following property. For every modulus \(Q\geq Q_0\) and every finite family of nonempty, pairwise-disjoint character sets \(\{\mathcal K_\nu\}_{\nu\in I}\) modulo \(Q\), if the associated zero multisets satisfy \[\lambda_\rho\geq c \quad(\rho\in\mathcal Z_\nu), \qquad \max_{\chi,\chi'\in\mathcal K_\nu} \operatorname{cond}(\chi\overline{\chi'})\leq C,\] (2.2) for every \(\nu\in I\), then \[\mathcal Q_A\leq1-\kappa.\] (2.3)
Proposition 5 (Separation of defect and conductor parameters). Fix \(A>0\) and \(c>0\). Suppose that the limiting coefficient-\(A\) packet calculation obtained from Zhao’s finite argument closes, at zero source error, with a gap \(\kappa_0=\kappa_0(A,c)>0\), uniformly for every \(H\geq H_c\). Here uniformity means that all contributions above the finite detector levels are controlled by \(H\)-independent tail majorants and that the finite operations admit a common perturbation tolerance \(\delta=\delta(A,c)>0\). Then the analytic estimates underlying that calculation may be made uniform for every \(H\geq H_c\) and every fixed relative-conductor bound \(C\) in the precise order \[(A,c)\longmapsto(\kappa_0,\delta),\qquad \forall H\geq H_c:\ H\longmapsto\epsilon_*(A,c,H),\qquad \forall C\geq1:\ (H,C)\longmapsto Q_0.\] (2.4) In particular, one may take the final packet gap \(\kappa=\kappa_0/2\), independently of \(H\) and \(C\); those parameters affect only \(Q_0(A,c,H,C)\).
Proof. We distinguish throughout the target perturbation \(\delta>0\) tolerated by the finite packet calculation from the source conductor exponent \(\epsilon_*>0\) in Pintz’s density theorems. They are not the same parameter.
The relative-conductor hypothesis \(M\leq C\) is used to place one class in Zhao’s \(M=O(1)\) alternative. In that alternative, Lemma 3.1 uses \[k=2(\phi+3x+y+z),\qquad \phi=\frac13,\] (2.5) and the fixed-class form of Lemma 3.3 is \[(\Delta^2-\varepsilon_z)N+2\Delta D\leq1.\] (2.6) After this alternative has been selected, neither (2.5) and (2.6), nor Zhao’s displayed constant \(\mathcal C(x,y,z,\Lambda,\lambda_0)\) contains the numerical value of \(M\). The same is true of Zhao’s fixed-class integer consequences, Abel summation, the \(R\)-decomposition, and the two fixed-class \(D\)-alternatives. Thus \(C\) is absent from every limiting finite inequality.
By hypothesis, the exact certificate consists of finitely many rational inequalities, finitely many continuous operations whose denominators are strictly separated from zero, and tail inequalities which are uniform in \(H\). Its common target tolerance \(\delta=\delta(A,c)>0\) is such that perturbing all source inequalities by at most \(\delta\) changes the final packet ceiling by less than \(\kappa_0/2\). This choice precedes \(H\) and \(C\).
Now fix \(H\geq H_c\). At the source of Zhao’s fixed-class estimates, Pintz’s Theorems D and I are applied under \[\operatorname{cond}(\chi\overline{\chi'})\leq Q^{\epsilon_*}.\] (2.7) Pintz’s Theorem D, specifically his (4.3), and Theorem I, specifically his (4.34), enter Zhao’s Lemmas 3.1 and 3.3 with errors of the form \[O_H(\epsilon_*)+o_{H,\epsilon_*}(1) \qquad(Q\longrightarrow\infty);\] (2.8) in particular, the multiplier \(1+C_D(H)\epsilon_*\) in Theorem D is height-dependent. Choose \(\epsilon_*=\epsilon_*(A,c,H)>0\) sufficiently small that every \(O_H(\epsilon_*)\)-term, after its finitely many uses in the packet calculation, consumes less than \(\delta/2\).
Only now fix \(C\). If \[Q\geq C^{1/\epsilon_*},\] (2.9) then \(M\leq C\) implies (2.7). Enlarge \(Q_0\) until all the \(o_{H,\epsilon_*}(1)\)-terms in (2.8), together with the fixed-height zero-labeling and Deuring–Heilbronn errors, consume less than \(\delta/2\). The resulting finite packet ceiling is at most \(1-\kappa_0/2\).
For fixed \(A\), changing the exponential coefficient affects only fixed numerical weights and detector parameters. None of the limiting conductor reductions introduces the numerical value of \(C\). This proves the asserted order (2.4) and the quantifiers in Definition 4. ◻
Remark 6. Zhao states the lower-defect and relative-conductor restrictions using one parameter. Proposition 5 records the separation of their roles needed below. It is essential here that \(C\) is fixed as \(Q\to\infty\); the statement would be false with no further information if \(C=C(Q)\) were allowed to grow.
The Pintz–Zhao bridge
Lemma 7 (Conjugation and common-modulus domination). Let \(0<\theta<1/A\), \(Y=\lfloor X^\theta\rfloor\), \(q\leq Y\), and \(C_0\geq1\). Let \(\mathcal R\) be a submultiset of the natural ordered Cartesian product of two finite zero multisets: each ordered pair may occur only with the product of its zero multiplicities. Write its elements as \((\rho_i,\rho_j)\), with \(\rho_\ell=\beta_\ell+i\gamma_\ell\) belonging to a primitive character \(\chi_\ell^*\) of conductor \(r_\ell\mid q\), and suppose \[\operatorname{cond}\bigl((\chi_i^*\chi_j^*)^*\bigr)<C_0 \qquad ((\rho_i,\rho_j)\in\mathcal R).\] (2.10) Put \(Q=q\lfloor Y/q\rfloor\). After conjugating the second zero variable, let \(\mathscr X\) be the set of distinct primitive characters occurring in either coordinate, induce them to modulus \(Q\), and partition them into the connected components generated by \[\operatorname{cond}\bigl((\chi^*\overline{\chi'^*})^*\bigr)<C_0.\] (2.11) Fix any \(U\geq1\) for which the selected and conjugated zeros lie in the Zhao rectangle \[1-\frac{U}{\log Q}\leq\beta\leq1,\qquad |\gamma|\leq U.\] If \(\mathcal Z_\nu(U)\) denotes the full zero multiset of the characters in component \(\nu\) in that rectangle, and \(\lambda_\rho=(1-\beta)\log Q\), then \[\sum_{(\rho_i,\rho_j)\in\mathcal R} X^{-(1-\beta_i)-(1-\beta_j)} \leq \sum_\nu \left(\sum_{\rho\in\mathcal Z_\nu(U)} \mathrm e^{-A\lambda_\rho}\right)^2.\] (2.12) Every component has the fixed within-class bound \[\operatorname{cond}\bigl((\chi^*\overline{\chi'^*})^*\bigr) \leq C_{\mathrm{rel}}:=C_0^{K-1},\] (2.13) where \(K=\lvert\mathscr X\rvert\) (and the assertion is void when \(\mathscr X=\varnothing\)).
Proof. If \(L(\rho_j,\chi_j^*)=0\), then \(L(\overline{\rho_j},\overline{\chi_j^*})=0\). The involution \[(\rho_j,\chi_j^*)\longmapsto (\overline{\rho_j},\overline{\chi_j^*})\] (2.14) preserves \(1-\beta_j\), height, conductor, multiplicity, and the condition \(r_j\mid q\). It converts (2.10) bijectively into (2.11). This is the reindexing between Pintz’s (2.13) and (2.36)–(2.37).
Since \(q\leq Y\), one has \[q\mid Q,\qquad Y/2<Q\leq Y.\] (2.15) If \(\chi^*\) has conductor dividing \(Q\), its induction \(\chi_Q\) satisfies \[L(s,\chi_Q)=L(s,\chi^*) \prod_{\substack{p\mid Q\\p\nmid\operatorname{cond}\chi^*}} \left(1-\chi^*(p)p^{-s}\right).\] (2.16) The primitive \(L\)-function inducing \(\chi_Q\) is \(L(s,\chi^*)\), so the primitive packet zero multiset is unchanged by definition. The additional Euler-factor zeros of the imprimitive function lie on \(\Re s=0\) and do not enter that multiset. Moreover, \[\operatorname{cond}\bigl((\chi_Q\overline{\chi'_Q})^*\bigr) = \operatorname{cond}\bigl((\chi^*\overline{\chi'^*})^*\bigr).\] (2.17)
Conductor submultiplicativity along a simple path of at most \(K-1\) edges gives (2.13). For \(\delta_\ell=1-\beta_\ell\) and \(B=1/\theta>A\), (2.15) gives \[X^{-(\delta_i+\delta_j)} \leq Q^{-B(\delta_i+\delta_j)} =\mathrm e^{-B(\lambda_i+\lambda_j)} \leq\mathrm e^{-A(\lambda_i+\lambda_j)}.\] (2.18) Every retained ordered occurrence is therefore bounded by the corresponding ordered occurrence in its component square. The Cartesian-product multiplicity hypothesis ensures that no occurrence is used more often than it appears in that square. Enlarging the selected zeros to the full multisets \(\mathcal Z_\nu(U)\) only adds nonnegative terms, which proves (2.12). ◻
Lemma 8 (Explicit-formula positivity ledger). Fix Pintz’s explicit-formula parameter \(\tau>0\), and let \(H_0,T_0\) be fixed truncation parameters. Let \(K\) be an upper bound for the number of selected zeros and let \(\eta>0\) be the small-singular-series cutoff in Pintz’s (2.12). Set \[\mathfrak s_0 :=2\prod_{p\geq3}\left(1-\frac1{(p-1)^2}\right)>0.\] (2.19) For an even \(m\in[X/2,X]\), let \(\mathcal C_X(m)\) be the retained zero–zero sum in Pintz’s (2.21), with its original \(X\)-weights and with the pole–pole pair removed. Assume that no retained pole–zero or zero–pole pair occurs. Then, for all sufficiently large \(X\), \[\begin{split} \frac{R_1(m)}{\mathfrak S(m)m} \geq{}& 1-(1+\omega_X)\mathcal C_X(m) -\frac{2\eta(K+1)^2}{\mathfrak s_0}\\ &-\frac{2C_{\mathrm{EF}}(\tau)}{\mathfrak s_0} \left( \mathrm e^{-c_{\mathrm{EF}}H_0} +\frac1{\sqrt{T_0}}+X^{-\tau} \right), \end{split}\] (2.20) where \(\omega_X=O_{H_0,T_0}(1/\log X)\). Consequently, if \(\mathcal C_X(m)\leq1-\kappa\) for a fixed \(0<\kappa\leq1/2\), then the parameters may be chosen in the order \[\kappa\longmapsto H_0,T_0 \longmapsto K \longmapsto\eta \longmapsto X_0\] (2.21) so that \[R_1(m)\geq\frac{\kappa}{2}\mathfrak S(m)m>0 \qquad(X\geq X_0).\] (2.22)
Proof. For even \(m\), the Euler product in Pintz’s (2.9) gives \(\mathfrak S(m)\geq\mathfrak s_0\). The pole–pole term in Pintz’s Theorem A is exactly \(\mathfrak S(m)m\). For every selected non-pole–pole singularity pair, Pintz’s beta-integral estimate (2.20) gives, uniformly over the fixed rectangle, \[\left| \frac{\Gamma(\rho_i)\Gamma(\rho_j)} {\Gamma(\rho_i+\rho_j)} \right| m^{-(\delta_i+\delta_j)} \leq (1+\omega_X)X^{-(\delta_i+\delta_j)}.\] (2.23) Indeed, if \(s=\delta_i+\delta_j\leq2H_0/\log X\), then \[m^{-s}\leq2^sX^{-s} =\left(1+O_{H_0}\!\left(\frac1{\log X}\right)\right)X^{-s},\] which is the precise use of \(m\geq X/2\).
By hypothesis the retained non-pole terms are exactly the zero–zero terms counted by \(\mathcal C_X(m)\). There are at most \((K+1)^2\) non-pole–pole pairs. For every pair failing one of Pintz’s three retention conditions, his (2.12) gives \(\lvert\mathfrak S(\chi_i,\chi_j,m)\rvert\leq\eta\); after increasing \(X_0\), the beta factor in (2.23) is at most two. Their normalized total is therefore at most the second error term in (2.20). Dividing the remainder in Pintz’s Theorem A by \(\mathfrak S(m)m\geq\mathfrak s_0X/2\) gives the final line of (2.20).
Choose \(H_0,T_0\) so that the first two explicit-formula terms contribute at most \(\kappa/16\) each. The log-free density bound then fixes \(K=O_\tau(\mathrm e^{2H_0})\). Choose \(\eta\) so that its term is at most \(\kappa/16\). Finally enlarge \(X_0\) until the \(X^{-\tau}\)-term is at most \(\kappa/16\) and \(\omega_X\leq\kappa/4\). Since \[(1+\kappa/4)(1-\kappa)\leq1-\frac{3\kappa}{4},\] (2.20) yields (2.22). ◻
Theorem 9 (Coefficient-stable bridge). Assume \(\mathsf Z(A)\). If \[0<\theta<\min\left\{\frac25,\frac1A\right\},\] (2.24) then \[E(X)\ll_{A,\theta}X^{1-\theta}.\] (2.25) Consequently, for every \(\varepsilon>0\), \[E(X)\ll_{\varepsilon,A} X^{1-\min\{2/5,\,1/A\}+\varepsilon}.\] (2.26) In particular, if \(A\geq5/2\), then \[E(X)\ll_{\varepsilon,A}X^{1-1/A+\varepsilon}.\]
Proof. It is enough to count exceptions in \(X/2<m\leq X\). Fix, independently of \(X\), \[0<\tau< \min\left\{\tau_0,\frac{2/5-\theta}{2},\frac{2}{45}\right\},\] (2.27) where \(\tau_0\) is the small absolute upper bound in Pintz’s Theorem A. Set Pintz’s preliminary short-interval parameter \(\varepsilon_0=2\tau\), and apply the explicit formula with \[\vartheta_{\mathrm P}=\theta+2\tau, \qquad \varepsilon_{\mathrm P}=\tau.\] (2.28) Its hypotheses hold because \[\tau<\vartheta_{\mathrm P}<\frac25<\frac49-\tau,\] and it supplies \[X^{\theta+\tau}\leq P\leq X^{\theta+2\tau}.\] (2.29) The fact that this one \(P\) works simultaneously for every \(m\leq X\), rather than being selected separately for each target, is the uniformity assertion in Pintz’s published Theorem 1 (Pintz 2023). Pintz’s minor-arc estimate (2.6) therefore leaves \[O(XP^{-1}\log^{10}X) =o(X^{1-\theta})\] (2.30) targets.
Invoke Pintz’s Theorem B with \(\varepsilon'=\tau\). If its exceptional real zero exists, then \[E(X)\ll X^{3/5+\tau}=O(X^{1-\theta}),\] because \(\theta+2\tau<2/5\). We may henceforth work in the complementary alternative. The classical primitive zero-free region (Davenport 2000, Ch. 14), together with the absence of that single possible exceptional real zero, supplies a fixed \(h=h(\tau)>0\) such that every relevant bounded-height zero satisfies \[1-\beta\geq\frac h{\log X}.\] (2.31) The constant \(h\) is chosen from the absolute zero-free-region constant and Pintz’s Theorem B before any truncation height below is chosen. For every later fixed height \(U\), \[\log(r(U+2))\leq(2/5)\log X+O_U(1).\] Thus taking \(U=H_Z\) below affects only \(X_0\), not \(h\); there is no circular dependence.
Set \(c=h\theta/2\), and let \(\kappa=\kappa(A,c)>0\) be furnished by \(\mathsf Z(A)\), decreased if necessary so that \(\kappa\leq1/2\). Choose \(H_0,T_0,K,\eta\) in the order (2.21), decreasing \(\eta\) further so that \(\eta\leq1\). Write \[C_0=\eta^{-3}, \qquad C_{\mathrm{rel}}=C_0^{K-1}.\] (2.32) Here Pintz’s selected labelled-zero multiset is conjugation-stable, and the number of distinct primitive characters is no larger than the number of labelled zeros. Thus the log-free density estimate supplies one \(K\geq1\) which bounds both quantities (even if the selected set is empty). These are fixed numbers. If one writes \(\eta=\varepsilon_{\mathrm{cut}}/K^2\), then \(\eta^{-3}=K^6/\varepsilon_{\mathrm{cut}}^3\); this is the reciprocal forced by Pintz’s (2.13) and (2.24), rather than the opposite parenthetical quantity printed in his (2.34).
Pintz partitions the targets into at most \(2^K\) subsets according to which selected primitive conductors divide \(C_1(\eta)m\); see his (2.26). For one subset let \(q\) be their least common multiple, with the convention \(\operatorname{lcm}(\varnothing)=1\), and put \(Y=\lfloor X^\theta\rfloor\). For the empty subset the retained zero sum is empty. If \(q>Y\), Pintz’s divisibility count (2.29) gives \(O_{\eta,K}(X^{1-\theta})\) targets in that subset. We may therefore suppose \(q\leq Y\) and form the modulus \(Q\) in Lemma 7.
The retained critical sum contains no pole–zero pair once \(X\) is large. Indeed, such a pair would force the zero character to have primitive conductor \(<C_0\). There are only finitely many such primitive \(L\)-functions, and none vanishes at \(1\); their bounded-height zeros eventually lie outside \(\beta\geq1-H_0/\log X\). We may therefore apply Lemma 7 to the retained zero–zero pairs. These pairs form a submultiset of the natural ordered zero product, with the multiplicities required there.
Pintz’s horizontal truncation and (2.15) give \[(1-\beta)\log Q\leq\theta H_0, \qquad |\gamma|\leq T_0.\] Thus the selected zeros lie in a fixed Zhao rectangle with \[H_Z:=\max\left\{ \theta H_0,\ T_0,\ 5.68,\ 1.09\log\frac1c \right\}.\] (2.33) Moreover, \(\log Q\geq(\theta/2)\log X\) for all sufficiently large \(X\), and (2.31) gives \[\lambda_\rho=(1-\beta)\log Q\geq c.\] (2.34) The same bound holds for every additional zero of the selected primitive characters in this full Zhao rectangle. In particular, \(H_Z\geq H_c\), so the moving cutoff required in the near-Siegel branch is contained in the rectangle before \(\mathsf Z(A)\) is invoked.
All parameters in \[Q_0(A,c,H_Z,C_{\mathrm{rel}})\] are now fixed, and \(Q>Y/2\to\infty\). Hence, for sufficiently large \(X\), \(\mathsf Z(A)\) and (2.12) give \[\mathcal C_X(m) \leq \sum_\nu \left(\sum_{\rho\in\mathcal Z_\nu(H_Z)} \mathrm e^{-A\lambda_\rho}\right)^2 \leq1-\kappa.\] (2.35) Lemma 8 now yields \(R_1(m)>0\) outside the minor-arc exceptions and the large-\(q\) subsets. Summing their bounds over the at most \(2^K\) subsets proves the desired count: indeed, on every remaining target, \[R_1(m)\geq\frac{\kappa}{2}\mathfrak S(m)m \geq\frac{\kappa\mathfrak s_0}{4}X,\] which dominates Pintz’s minor-arc bound \(\lvert R_2(m)\rvert\leq X/\sqrt{\log X}\). Hence \(R(m)=R_1(m)+R_2(m)>0\) for sufficiently large \(X\). This proves (2.25) on \([X/2,X]\), and dyadic summation proves it up to \(X\).
Finally put \(\alpha=\min\{2/5,1/A\}\). Given \(\varepsilon>0\), choose once and for all a fixed \(\theta<\alpha\) with \(\alpha-\theta<\varepsilon\). Then \[X^{1-\theta}\leq X^{1-\alpha+\varepsilon},\] which proves (2.26). No parameter is allowed to vary with \(X\) in this endpoint passage. ◻
Remark 10 (Logical scope of the bridge). Theorem 9 is conditional only on the packet property \(\mathsf Z(A)\). It does not itself prove the conductor-uniform packet statement. The quantifier order in Definition 4 is used precisely after \(H_Z\) and \(C_{\mathrm{rel}}\) have been fixed and before the final modulus threshold is imposed.
3. Coefficient stability of Zhao’s analytic inputs
\(\varepsilon_z>0\) denotes Zhao’s auxiliary analytic error. For every fixed positive choice of \(\varepsilon_z\), the source estimates below hold once the modulus exceeds a threshold depending on the fixed detector parameters; equivalently, the discarded source errors tend to zero as the modulus tends to infinity. The certificate first works at \(\varepsilon_z=0\) with strict rational margins and then chooses one common positive value in (7.22).
Fix a cutoff \(\Lambda\), and decompose one class as \[\begin{split} T_i&=\sum_{\rho\in\mathcal Z_i} \mathrm e^{-A\max(\lambda_\rho,\Lambda)},\\ R_i&=\sum_{\rho\in\mathcal Z_i} \left(\mathrm e^{-A\lambda_\rho} -\mathrm e^{-A\max(\lambda_\rho,\Lambda)}\right). \end{split}\] (3.1) Then \(S_{i,A}=R_i+T_i\) and \[\mathcal Q_A=\sum_i(R_i+T_i)^2.\] (3.2) When the cutoff is variable, we write explicitly \[T_i(L)=\sum_{\rho\in\mathcal Z_i} \mathrm e^{-A\max(\lambda_\rho,L)}.\]
Lemma 11 (\(T\)-stability). Suppose Zhao’s Lemma 3.1 gives \[\sum_j \mathrm e^{-k\max(\lambda_j,\Lambda)} \leq(1+\varepsilon_z)\mathcal C(x,y,z,\Lambda,\lambda_0)\] (3.3) with \(k\leq A\). Then \[T_A(\Lambda) \leq (1+\varepsilon_z)\mathrm e^{-(A-k)\Lambda} \mathcal C(x,y,z,\Lambda,\lambda_0).\] (3.4)
Proof. For every \(\lambda\), \[\mathrm e^{-A\max(\lambda,\Lambda)} = \mathrm e^{-(A-k)\max(\lambda,\Lambda)} \mathrm e^{-k\max(\lambda,\Lambda)} \leq \mathrm e^{-(A-k)\Lambda} \mathrm e^{-k\max(\lambda,\Lambda)}.\] Sum and apply (3.3). ◻
Zhao’s fixed-class tail gives \(100\mathrm e^{-2.22\Lambda}\) at \(A_0=10/3\) for \(\Lambda\geq5.2\). Reapplying Zhao’s Lemma 3.1 with the same detector parameters and using Lemma 11, at \(A\leq A_0\) we obtain the adjusted tail \[x=\frac{29}{1000},\quad y=\frac{83}{1000},\quad z=\frac{52}{1000},\quad k=2\left(\frac13+3x+y+z\right)=\frac{833}{750},\] for which \(A_0-k=1667/750>2.22\). Hence the adjusted tail is \[100\exp\left( -\left[2.22-\left(\frac{10}{3}-A\right)\right]\Lambda \right).\] (3.5)
Lemma 12 (\(R\)-stability). Zhao’s \(R\)-comparison remains valid at coefficient \(A\) provided the detector parameter satisfies \[x> \max\left\{\frac2A,\frac{4\lambda_0}{5}\right\}.\] (3.6)
Proof. The kernel support is \(0\leq u\leq2/x\). For \(d=\Lambda-\lambda_j\geq0\), Zhao compares \(\mathrm e^{ud}-1\) with \(\mathrm e^{Ad}-1\). For \(0\leq u\leq A\), \[d\longmapsto\frac{\mathrm e^{ud}-1}{\mathrm e^{Ad}-1}\] is nonincreasing. Thus \(2/x\leq A\) supplies the ratio monotonicity, while Zhao’s density inequality independently requires \(x\geq4\lambda_0/5\). Every certified detector is chosen strictly above both thresholds, which is also the strict support hypothesis in Pintz’s Theorem K. ◻
No interpolation in \(A\) is used. Every detector selected numerically is substituted back into (3.4) and (3.6), and Zhao’s \(R\)-formulas using outward rational arithmetic.
4. The positional fixed-class cap
Zhao uses the compactly supported kernel \[g(u)=\frac{(2-u)^3(4+6u+u^2)}{30} \quad(0\leq u\leq2),\] (4.1) and \(g(u)=0\) otherwise. Let \[G(s)=\int_0^2g(u)\mathrm e^{-su}\,du.\] For fixed \(x,\lambda_0\), define \[\psi(v)= \frac{G((v-\lambda_0)/x)}{G(-\lambda_0/x)}.\] (4.2) Put \(f_x(u)=xg(xu)\), \(F_x(s)=G(s/x)\), and \[\xi= \frac{\phi f_x(0)}{2F_x(-\lambda_0)} =\frac{8\phi x}{15G(-\lambda_0/x)}, \qquad \phi=\frac13.\] For a fixed character class \(i\) and detector level \(u\), write \[\begin{split} N_i(u)&=\#\{\,\rho\in\mathcal Z_i:\lambda_\rho\leq u\,\},\\ D_i(u)&=\sum_{\substack{\rho\in\mathcal Z_i\\\lambda_\rho\leq u}} \bigl(\psi(\lambda_\rho)-\psi(u)\bigr),\\ \Delta(u)&=\psi(u)-\xi. \end{split}\] (4.3) Since \(g\geq0\), \(G'(s)<0\), and hence \(\psi\) is decreasing.
At a detector level \(u\), Zhao’s fixed-class inequality is \[(\Delta(u)^2-\varepsilon_z)N_i(u)+2\Delta(u)D_i(u)\leq1.\] (4.4)
Lemma 13 (Directed positional count). Take \(\lambda_0=a\) in (4.2) and (4.3). Assume every zero in one fixed class has defect at least \(a\), and that the class contains zeros at positions at most \(v_1,\ldots,v_m\leq u\). Then \[D_i(u)\geq \sum_{j=1}^m\bigl(\psi(v_j)-\psi(u)\bigr).\] (4.5) Let \(B\) be a nonnegative integer. If rational lower enclosures \(0<\Delta_-\leq\Delta(u)\) and \(0\leq D_-\leq D_i(u)\) satisfy \[(B+1)\Delta_-^2+2\Delta_-D_- -1>0,\] (4.6) then, after taking \(\varepsilon_z>0\) sufficiently small, \[N_i(u)\leq B.\] (4.7)
Proof. Every certified zero contributes at least \(\psi(v_j)-\psi(u)\) because \(\psi\) is decreasing. If \(N_i(u)\geq B+1\), the left side of (4.4) exceeds one by (4.6) for all sufficiently small \(\varepsilon_z>0\), a contradiction. ◻
Definition 14 (Certified Abel cap). Fix an interval \([a,b]\subseteq[0.92,\Lambda]\). Choose a finite rational mesh \[\Lambda=w_0<w_1<\cdots<w_L=5.2.\] At every \(w_\ell\), set the detector base parameter \(\lambda_0=a\), use the same-class first-zero position \(b\), and choose a rational detector \(x_\ell\) in Lemma 13 to certify \(N_i(w_\ell)\leq B_\ell\). Replace these raw bounds by the nondecreasing suffix-minimum envelope \[\widetilde B_\ell=\min_{j\geq\ell}B_j.\] This remains valid because \(N_i(w_\ell)\leq N_i(w_j)\leq B_j\) for \(j\geq\ell\). Relabel \(\widetilde B_\ell\) as \(B_\ell\). The resulting directed cap is \[\begin{split} c_A(\Lambda;a,b) :=\;&B_0\mathrm e^{-A\Lambda} +\sum_{\ell=1}^L(B_\ell-B_{\ell-1}) \mathrm e^{-Aw_{\ell-1}}\\ &+100\exp\left( -\left[2.22-\left(\frac{10}{3}-A\right)\right]5.2 \right). \end{split}\] (4.8) Every exponential in the certificate is replaced by an outward rational enclosure. The definition of \(c_A^\infty(\Lambda)\) is identical, except that the same-class first-zero charge is omitted and every zero has lower defect \(\Lambda\).
Proposition 15 (Universal early-or-late cap). Let \(\mu_i\) be the first zero in a high class, and suppose \(\mu_i\geq0.92\). Set \[\Lambda=\frac{733}{500}=1.466\] (4.9) and use the partition \[0.92,\ 1,\ 1.05,\ 1.10,\ 1.15,\ 1.20,\ 1.25,\ 1.30,\ 1.40,\ 1.466.\] (4.10) Let \(\mathcal P\) denote the nine consecutive closed intervals determined by this list. For the caps of Definition 14, every high class satisfies \[T_i\leq C_A(\Lambda) := \max\left\{ c_A^\infty(\Lambda), \max_{[a,b]\in\mathcal P} c_A(\Lambda;a,b) \right\}.\] (4.11)
Proof. If \(\mu_i\in[a,b]\), every zero in the class is at least \(a\), and the class itself supplies a zero no later than \(b\). Thus Lemma 13 applies with the positive charge \(\psi(b)-\psi(u)\). Abel summation of the resulting certified integer envelope gives \(c_A(\Lambda;a,b)\). If \(\mu_i>\Lambda\), the ordinary cap at lower defect \(\Lambda\) applies. The cases in (4.10), together with this late case, exhaust every possible \(\mu_i\). ◻
Remark 16 (Direction of information). The cumulative unrestricted zero count gives rankwise lower bounds for first zeros. It does not give an upper position for an individual class. In Proposition 15, the actual first zero selects its own interval; no upper position is inferred from a global count.
In (4.8), every unresolved jump is deliberately placed at the adverse left endpoint. The adaptive tolerance measures only the excess of this safe placement; it is not an unaccounted numerical error.
5. Aligned cap-and-mass majorization
The next elementary lemma is responsible for the decisive saving in the secondary branches.
Theorem 17 (Aligned cap-and-mass lemma). Let \[a_1\geq a_2\geq\cdots\geq0, \qquad b_1\geq b_2\geq\cdots\geq0,\] on a finite or countable index set. Suppose \[0\leq r_i\leq a_i,\quad \sum_i r_i\leq U, \qquad 0\leq t_i\leq b_i,\quad \sum_i t_i\leq V.\] (5.1) Define the left-greedy fills \[\begin{split} g_i^R&=\min\left\{a_i, \left(U-\sum_{j<i}a_j\right)_+\right\},\\ g_i^T&=\min\left\{b_i, \left(V-\sum_{j<i}b_j\right)_+\right\}. \end{split}\] (5.2) Then \[\sum_i(r_i+t_i)^2 \leq \sum_i(g_i^R+g_i^T)^2.\] (5.3) The right side is attained in the enlarged polytope (5.1).
Proof. We first prove the bilinear statement \[\sum_i r_it_i\leq\sum_i g_i^Rg_i^T.\] (5.4) Decreasing rearrangement preserves feasibility. Indeed, if the \(k\)-th largest coordinate of \(r\) exceeded \(a_k\), then at least \(k\) original coordinates would exceed \(a_k\), while only the first \(k-1\) cap positions can do so. The same holds for \(t\). The rearrangement inequality gives \[\sum_i r_it_i\leq\sum_i r_i^\downarrow t_i^\downarrow.\] For every \(k\), \[\sum_{i\leq k}r_i^\downarrow \leq\min\left\{U,\sum_{i\leq k}a_i\right\} =\sum_{i\leq k}g_i^R.\] (5.5) Abel summation against the decreasing nonnegative sequence \(t^\downarrow\) gives \[\sum_i r_i^\downarrow t_i^\downarrow \leq\sum_i g_i^Rt_i^\downarrow.\] Apply the corresponding \(b,V\) prefix majorization against the decreasing sequence \(g^R\) to obtain (5.4).
Apply (5.4) to two copies of \(r\), two copies of \(t\), and to \(r,t\). Adding the resulting bounds proves (5.3). The greedy pair itself is feasible, so the bound is attained. The countable case follows by truncation, using the finite mass budgets. ◻
If an explicit finite prefix is followed by a repeated tail cap, the tail is included before taking the right-to-left suffix maximum. This produces the least nonincreasing coordinatewise majorant and avoids lowering a raw tail cap. Each residual mass is then filled by complete tail caps and at most one remainder.
6. Ordering the character classes
For a chosen unrestricted detector, write \[N(u)=\#\{\,\rho:\lambda_\rho\leq u\,\}, \qquad D(u)=\sum_{\lambda_\rho\leq u} \bigl(\psi(\lambda_\rho)-\psi(u)\bigr),\] where the sums run over all packet classes. In the active branch, the at most two zeros certified at or below \(0.92\) are called the designated low zeros, and the classes containing them are the low classes. Every remaining nonempty class is a high class and has first defect at least \(0.92\). Order these high classes by their first defects.
In the active branch, let \[0.92=u_0<u_1<\cdots<u_J=1.24, \qquad u_j-u_{j-1}=\frac1{200}.\] (6.1) The unrestricted form of Zhao’s Lemma 3.3 is \[(\Delta^2-\xi-\varepsilon_z)N(u)+2\Delta D(u)\leq1-\xi.\] (6.2) For an active box \(a\leq\lambda_{1,1}\leq b\), use \(a\) for every adverse lower-defect or exponential estimate, and use \(b\) only for the positive positional term.
If (6.2) certifies \(N(u_j)\leq B_j\), subtract the explicitly known low zeros. The number of additional high classes beginning by \(u_j\) is then bounded by a nondecreasing integer envelope \(H_j\). Ranks \[H_{j-1}<i\leq H_j\] may be assigned first-zero lower endpoint \(u_{j-1}\). Zhao’s fixed-class \(R\)-formula supplies a raw cap at that endpoint. A right-to-left suffix maximum, formed together with the repeated \(u_J\)-tail, gives a decreasing coordinatewise majorant.
The unrestricted estimates give total budgets \[\sum_iT_i\leq V_T, \qquad \sum_iR_i\leq V_R.\] (6.3) Because Zhao introduces the unrestricted \(R\)-formula under \(N\geq3\), \(V_R\) is replaced by the maximum of that unrestricted bound and \[2\left(\mathrm e^{-0.92A}-\mathrm e^{-A\Lambda}\right)\] (6.4) to cover \(N\leq2\). The ordered caps and (6.3) are now exactly the hypotheses of Theorem 17.
Allocation-compatible low classes
In the active range, Zhao’s alternatives allow at most two low zeros. There are exactly three allocations: \[\begin{array}{c|c|c} \text{allocation}&\text{global known positions} &\text{same-class positions}\\ \hline \text{two classes, two zeros}&(b,0.92)&(b)\ \text{and}\ (0.92)\\ \text{one class, one zero}&(b)&(b)\\ \text{one class, two zeros}&(b,0.92)&(b,0.92). \end{array}\] (6.5) The two-class case never assigns \((b,0.92)\) to a single class. Applying Lemma 13 separately to each identified low class gives its \(T\)-cap; the compatible fixed \(R\)-cap then gives the exact low-class square. The high-class square is bounded by Theorem 17.
Proposition 18 (Exhaustion of Zhao’s alternatives). Let \(H\geq H_c\). The following rows cover every \(c\leq\lambda_{1,1}\leq H\). The middle column is the applicable consequence of Zhao’s Lemma 2.4; the last column records where the corresponding packet is certified.
| range of \(\lambda_{1,1}\) | zero alternative | certificate branch |
|---|---|---|
| \([c,0.01]\) | \(N(\max\{5.68,1.09\log(1/\lambda_{1,1})\})\leq1\) | near-Siegel |
| \([0.01,0.10]\) | \(N(3.08)\leq1\) | scalar |
| \([0.10,0.30]\) | \(N(1.58)\leq1\) | scalar |
| \([0.30,0.40]\) | \(N(1.29)\leq1\) | scalar |
| \([0.40,0.60]\) | \(N(0.92)\leq2\) | active \(63\) rows |
| \([0.60,0.62]\) | \(N(0.85)\leq1\) or \(N(0.91)\leq2\) | secondary |
| \([0.62,0.64]\) | \(N(0.85)\leq2\) | secondary |
| \([0.64,0.68]\) | \(N(0.74)\leq2\) | secondary |
| \([0.68,0.702]\) | \(N(0.702)\leq2\) | secondary |
| \([0.702,H]\) | \(\lambda_{i,1}\geq0.857\) for \(i\geq5\) | aligned suffix |
Proof. The zero alternatives are precisely Zhao’s Lemma 2.4, including its three unconditional final assertions. If \(c>H\), the packet is empty and there is nothing to prove. Otherwise, intersect the displayed closed intervals with \([c,H]\) and discard empty intersections; the resulting rows cover \([c,H]\), with harmless overlaps at their endpoints. In the active interval, (6.5) exhausts the possible placements of the at most two low zeros. In every secondary interval the verifier checks each admissible allocation \((m,k)\), where \(m\) is the number of known low zeros and \(k\) the number of classes which contain them. The remaining cases are the four scalar rows and the near-Siegel majorant (7.18). ◻
7. The finite packet theorem
Theorem 19 (Computer-assisted packet theorem at \(A=33/10\)). The coefficient-\(33/10\) packet property \(\mathsf Z(33/10)\) holds. More precisely, every discretized non-near-Siegel branch satisfies \[\mathcal Q_{33/10} <\frac{198479}{200000}=0.992395,\] (7.1) while the near-Siegel branch has a positive gap depending only on the fixed lower zero defect \(c\).
The proof uses Zhao’s inequalities in the forms transcribed in Section 11, the branch exhaustion in Proposition 18, and the exact certificate described below.
Exact arithmetic model
Every terminating decimal used as an input denotes the exact corresponding rational number. All final comparisons are made in \(\mathbb Q\). Displayed packet bounds are rounded upward, while displayed positive separations and slacks are rounded downward. For \(x\geq0\), the verifier encloses \[\mathrm e^x=\sum_{k=0}^{m}\frac{x^k}{k!}+R_m(x)\] using the rational bound \[0\leq R_m(x) \leq 3^{\lceil x\rceil}\frac{x^{m+1}}{(m+1)!}.\] (7.2) Negative arguments are handled by reciprocating a positive enclosure. For the kernel transform, write \[G(s)=\sum_{k\geq0}\frac{(-s)^km_k}{k!},\] where the moments are the exact rationals \[m_k= \frac1{30}\left( \frac{32\,2^{k+1}}{k+1} -\frac{40\,2^{k+3}}{k+3} +\frac{20\,2^{k+4}}{k+4} -\frac{2^{k+6}}{k+6} \right).\] (7.3) The omitted series is bounded by \[\frac89\sum_{j>m}\frac{(2|s|)^j}{j!}.\] (7.4) Floating point performs only coarse scans and one-dimensional searches for detector parameters. The selected decimal rational is accepted only after exact substitution into all support, positivity, and next-integer separation inequalities.
Active branches
The active interval is covered by \[[0.40,0.405],\ [0.405,0.41],\ [0.41,0.42],\ldots,[0.59,0.60].\] (7.5) Each of the \(21\) boxes is checked in all three allocations from (6.5), for \(63\) directed rows.
The common high-class quantities are \[\begin{split} C_A(\Lambda)&\leq0.066553314396,\\ V_T&\leq6.650837008243,\\ V_R&\leq0.455946922433. \end{split}\] (7.6) The exhaustive cap split in Proposition 15 is recorded in Table 1.
| \([a,b]\) | \(c_A(\Lambda;a,b)\) | nodes | minimum separation |
|---|---|---|---|
| \([0.92,1]\) | \(0.066553314396\) | 203 | \(\geq0.000019922993\) |
| \([1,1.05]\) | \(0.065749401858\) | 195 | \(\geq0.000019546894\) |
| \([1.05,1.10]\) | \(0.065434575306\) | 191 | \(\geq0.000018216157\) |
| \([1.10,1.15]\) | \(0.065122210666\) | 185 | \(\geq0.000055196136\) |
| \([1.15,1.20]\) | \(0.064818914259\) | 182 | \(\geq0.000004433480\) |
| \([1.20,1.25]\) | \(0.064511514566\) | 177 | \(\geq0.000013802128\) |
| \([1.25,1.30]\) | \(0.064214254658\) | 174 | \(\geq0.000007323372\) |
| \([1.30,1.40]\) | \(0.064292723507\) | 170 | \(\geq0.000058651699\) |
| \([1.40,1.466]\) | \(0.063449399440\) | 162 | \(\geq0.000038035052\) |
| \(\mu_i>1.466\) | \(0.065103393638\) | 173 | \(\geq0.000000293236\) |
The active maxima, separated by allocation, are recorded in Table 2.
| allocation | packet upper bound | slack |
|---|---|---|
| two classes/two zeros | \(0.982040541993\) | \(\geq0.017959458007\) |
| one class/one zero | \(0.774022703391\) | \(\geq0.225977296609\) |
| one class/two zeros | \(0.961300315347\) | \(\geq0.038699684653\) |
All three maxima occur in the first half-box.
Secondary branches
If \(m\) known low zeros occupy \(k\) classes, the number of classes is at most \[N_{\mathrm{zeros}}-(m-k),\] (7.7) because \(m-k\) excess zeros cannot begin new classes. The positional count therefore constructs a common ordered \(T\)-cap list. In the same order, the \(R\)-caps have the form \[(\underbrace{a_R,\ldots,a_R}_{k},r_2,r_2,\ldots).\] (7.8) Here \(N_{\mathrm{zeros}}\) is the certified unrestricted count \(N(u)\) at the branch’s separation level \(u\), \(r_2\) is the fixed-class \(R\)-cap when every defect is at least \(u\), and \(a_R\) is the compatible cap for a class containing designated low zeros.
For a secondary lower endpoint \(\ell\), let \(r_{11}^{\mathrm{fixed}}\) be the larger of (A.2) and (A.3) with prescribed lower endpoints \((\ell,\ell)\), and let \(r_{12}^{\mathrm{fixed}}\) be the analogous cap with endpoints \((\ell,u)\). Let \(U_{11}\) and \(U_{12}\) denote the maximum of, respectively, the corresponding unrestricted bound from (A.1) and the direct two-zero guard. The endpoint \(u\) in the \(12\)-bound represents the worst possible next zero when only one low zero is known; an absent next zero only decreases the true sum. For allocations \((m,k)=(1,1),(2,1),(2,2)\), respectively, the compatible fixed cap and unrestricted budget are \[(r_{12}^{\mathrm{fixed}},U_{12}),\quad (r_{11}^{\mathrm{fixed}},U_{11}),\quad (r_{12}^{\mathrm{fixed}},U_{11}).\] (7.9) Applying Theorem 17 gives Table 3.
| branch | \((1,1)\) | \((2,1)\) | \((2,2)\) |
|---|---|---|---|
| \([0.60,0.62]\), \(N(0.91)\leq2\) | \(0.941335795526\) | \(0.979388257544\) | \(0.981239614797\) |
| \([0.60,0.62]\), \(N(0.85)\leq1\) | \(0.967962977997\) | inadmissible | inadmissible |
| \([0.62,0.64]\), \(N(0.85)\leq2\) | \(0.939895062527\) | \(0.967038727055\) | \(0.972356429912\) |
| \([0.64,0.66]\) | \(0.976868013140\) | \(0.987983379260\) | \(0.992394547694\) |
| \([0.66,0.68]\) | \(0.950821675538\) | \(0.957609181221\) | \(0.962392175459\) |
| \([0.68,0.702]\) | \(0.955998283891\) | \(0.956640496475\) | \(0.958101358231\) |
The limiting row is \([0.64,0.66]\), \((m,k)=(2,2)\), with \[\mathcal Q_{33/10}\leq0.992394547694, \qquad 1-\mathcal Q_{33/10}\geq0.007605452306.\] (7.10)
Remark 20 (Necessity of alignment). If the \(T^2\)-energy and the \(R^2+2RT\) terms are maximized separately, the \([0.64,0.66]\) rows \((2,1)\) and \((2,2)\) have negative slacks \(-0.005953317963\) and \(-0.010066017325\). Theorem 17 changes the latter full packet to \(0.992394547694\). Thus simply replacing \(10/3\) by \(33/10\) in Zhao’s scalar table does not prove the result.
The final scalar and near-Siegel branches
For \(\lambda_{1,1}\geq0.702\), Zhao’s class separation allows at most four classes before the later level \(0.857\). Use aligned cap lists \[t(v)=\text{the certified fixed-class \(T\)-cap under }\mu_i\geq v, \qquad r(v)=\text{the corresponding fixed-class \(R\)-cap},\] after replacing each raw list, together with its repeated tail, by the coordinatewise suffix-maximum majorant. Thus \[\begin{split} \mathbf t&=(t(0.702),t(0.702),t(0.702),t(0.702),t(0.857),\ldots),\\ \mathbf r&=(r(0.702),r(0.702),r(0.702),r(0.702),r(0.857),\ldots). \end{split}\] (7.11) The old separated scalar is \(1.002462657342\); the aligned packet is \[\mathcal Q_{33/10}\leq0.981677779591.\] (7.12) The remaining distinguished-zero branches are listed in Table 4.
| branch | packet upper bound | slack |
|---|---|---|
| \([0.702,H]\) (aligned) | \(0.981677779591\) | \(\geq0.018322220409\) |
| \([0.01,0.10]\) | \(0.946741143646\) | \(\geq0.053258856354\) |
| \([0.10,0.30]\) | \(0.884075525138\) | \(\geq0.115924474862\) |
| \([0.30,0.40]\) | \(0.974798896903\) | \(\geq0.025201103097\) |
For the near-Siegel branch, set \[b_A=2.22-\left(\frac{10}{3}-A\right).\] Write \[L=\Lambda(\lambda) =\max\left\{5.68,\ 1.09\log\frac1\lambda\right\}, \qquad \mathcal T_A^{\mathrm{all}}(L)=\sum_i T_i(L).\] If \(c\leq\lambda=\lambda_{1,1}\leq0.01\), Zhao’s first-zero alternative gives \[N(L)\leq1.\] Indeed, \[L\leq \max\left\{5.68,\ 1.09\log\frac1c\right\} =H_c\leq H,\] so this cutoff lies inside the packet rectangle. Consequently, in the decomposition (3.1), \[R_1=\mathrm e^{-A\lambda}-\mathrm e^{-AL}, \qquad R_i=0\quad(i\geq2).\] (7.13) It follows that \[\begin{split} \mathcal Q_A &=R_1^2+2R_1T_1(L)+\sum_iT_i(L)^2\\ &\leq \mathrm e^{-2A\lambda} +\left(\max_iT_i(L)\right) \left(2+\sum_iT_i(L)\right). \end{split}\] (7.14)
For the fixed-class detector \[x=\frac{29}{1000},\qquad y=\frac{83}{1000},\qquad z=\frac{52}{1000},\qquad k=\frac{833}{750},\] the exact rational enclosure used by the verifier gives \[\mathcal C(x,y,z,5.2,0) <99.729.\] (7.15) For \(\lambda_0=0\), the source kernel simplifies to \[\mathcal B(t,L,0) =\frac{\phi}{2}\frac{1-\mathrm e^{-2Lt}}L +\left(\frac{1-\mathrm e^{-Lt}}L\right)^2.\] Both terms are nonincreasing in \(L>0\), since \[\frac{1-\mathrm e^{-aL}}L=\int_0^a\mathrm e^{-Lu}\,du\] is nonincreasing. Thus \(\mathcal C(x,y,z,L,0)\) is nonincreasing in its cutoff. Moreover, \[(1+10^{-3})\,99.729=99.828729<100, \qquad A-k-b_A=\frac1{375}>0.\] Hence, for \(0<\varepsilon_z\leq10^{-3}\), Zhao’s fixed-class estimate and the coefficient shift of Lemma 11 give \[\max_iT_i(L)\leq100\mathrm e^{-b_AL}.\] (7.16) The unrestricted exact enclosure, with the same source-error allowance, gives \[\mathcal T_A^{\mathrm{all}}(L) \leq\mathcal T_A^{\mathrm{all}}(5.68) <0.016308621213<\frac1{50}.\] (7.17) Substitution in (7.14) proves \[\mathcal Q_A \leq \mathrm e^{-2A\lambda} +202\mathrm e^{-b_A\Lambda(\lambda)}.\] (7.18)
The exact verifier establishes \[\begin{gathered} b_A=\frac{164}{75},\qquad 1.09b_A=\frac{4469}{1875}>2,\\ d_0:=2A-\frac{2A^2+202}{200} =\frac{54811}{10000}>0,\\ \mathrm e^{-2A/200}+202\mathrm e^{-5.68b_A} \leq0.968353823265<1. \end{gathered}\] (7.19) For \(0<\lambda\leq1/200\), \(L\geq1.09\log(1/\lambda)\), so \[\mathrm e^{-2A\lambda} \leq1-2A\lambda+2A^2\lambda^2, \qquad \lambda^{1.09b_A}\leq\lambda^2,\] give \[\mathcal Q_A\leq1-d_0\lambda.\] (7.20) On \(1/200\leq\lambda\leq1/100\), no monotonicity assertion is needed: termwise, \[\mathrm e^{-2A\lambda}\leq\mathrm e^{-2A/200}, \qquad \mathrm e^{-b_AL}\leq\mathrm e^{-5.68b_A},\] and the last line of (7.19) applies. Thus the explicit fixed-\(c\) gap \[\kappa_{\mathrm{NS}}(33/10,c) := \min\left\{ \frac{54811}{10000}\min\left(c,\frac1{200}\right), \ \frac{6329235347}{200000000000} \right\} >0\] (7.21) is valid. No uniformity as \(c\downarrow0\) is claimed; the exceptional-real-zero alternative is removed in Theorem 9 before this estimate is used.
Absorbing the auxiliary error
The certificate evaluates the limiting \(\varepsilon_z=0\) formulas only after proving a finite collection of strict separations: \[\begin{gathered} A-k>0,\qquad \Delta^2-\xi>0,\\ (B+1)(\Delta^2-\xi)+2\Delta D-(1-\xi)>0,\\ (B+1)\Delta^2+2\Delta D-1>0,\\ 1-\mathcal Q_A^{\mathrm{upper}}>0. \end{gathered}\] (7.22) Every bound is continuous in \(\varepsilon_z\), with denominators separated from zero. Since the list is finite, one may choose a common \(\varepsilon_z^*>0\) for which every non-near-Siegel row remains valid. Choose once and for all \[0<\varepsilon_z\leq\min\left\{\varepsilon_z^*,10^{-3}\right\},\] and only then take the maximum of the finitely many modulus thresholds. This same positive source-error parameter is used in the near-Siegel bounds (7.16) and (7.17). Since \[\frac1{250}=0.004<0.007605452306,\] the non-near-Siegel branches retain gap \(1/250\). Together with the near-Siegel gap \(\kappa_{\mathrm{NS}}(A,c)>0\), this gives \[\mathcal Q_{33/10} \leq1-\min\left\{\frac1{250}, \kappa_{\mathrm{NS}}(33/10,c)\right\}.\] (7.23) The finite detector levels and the tail estimate (A.6) are independent of the outer rectangle height \(H\). Hence (7.23) is a common zero-source gap for every \(H\geq H_c\), and the strict separations in (7.22) provide the common perturbation tolerance required by Proposition 5. Applying that proposition for an arbitrary fixed \(C\geq1\) moves all conductor dependence into \(Q_0(33/10,c,H,C)\). This is precisely the quantifier order in Definition 4, and proves Theorem 19.
8. Proof of the main theorem
Proof of Theorem 1. By Theorem 19, the packet property \(\mathsf Z(33/10)\) holds. Apply Theorem 9: \[E(X)\ll_\varepsilon X^{1-10/33+\varepsilon} =X^{23/33+\varepsilon}.\] (8.1) Let \[\gamma=\frac{69697}{100000}.\] The exact margins are \[\gamma-\frac{23}{33}=\frac1{3300000}>0, \qquad \frac1{1-\gamma}-\frac{33}{10} =\frac1{303030}>0.\] (8.2) This is Theorem 1. Choose the \(\varepsilon\) in (8.1) smaller than \(1/3300000\). Then \(E(X)\ll X^\gamma\), proving Corollary 2. ◻
9. Reproducibility and source correspondence
Verifier architecture
The release separates parameter search, exact verification, witness export, and witness checking. Table 5 records the proof-to-code correspondence.
| Analytic object | Implementation |
|---|---|
| release integrity and orchestration | verify_release.py |
| active \(63\)-row packet | zhao_069697_full_active_audit.py |
| independent active/scalar optimization check | export_active_scalar_witness.py,
check_active_scalar_witness.py |
| secondary aligned construction | zhao_secondary_aligned_trial.py |
| independent secondary optimization check | export_secondary_aligned_witness.py,
check_secondary_aligned_witness.py |
| final scalar rows | zhao_secondary_scalar_audit.py |
| near-Siegel branch | zhao_near_siegel_exact.py |
| early-or-late cap | zhao_high_class_positional_trial.py |
| aligned majorization and tail fill | zhao_aligned_rt_staircase.py |
| same-class positional caps | zhao_positional_fixed_density.py |
| outward rational kernel arithmetic | zhao_packet_certificate.py |
Commands
From the project root, the complete manifest-driven release check is
python3 -u \
preprints/goldbach-exception-069697/verify_release.pyA fast integrity-only pass is
python3 -u \
preprints/goldbach-exception-069697/verify_release.py --hash-onlyThe three standard-library witness checks can also be run directly:
python3 -u experiments/check_active_scalar_witness.py \
preprints/goldbach-exception-069697/active-scalar-witness.json
python3 -u experiments/check_secondary_aligned_witness.py \
preprints/goldbach-exception-069697/secondary-aligned-witness.json
python3 -u experiments/zhao_near_siegel_exact.py \
--witness preprints/goldbach-exception-069697/\
near-siegel-witness.jsonThe exact primary command-line arguments are stored in
certificate-manifest.json, rather than duplicated in the
paper. The entry points refuse to run when
sys.flags.optimize is nonzero: the verifier must not be
invoked with python -O, because Python optimization removes
assertions used as proof guards.
Frozen release identifiers
The machine-readable file certificate-manifest.json
records the exact constants, the complete source dependency closure,
every theorem-critical command and argument, expected row counts,
success markers, and SHA-256 digests of the exact standard output. The
companion MANIFEST.sha256 identifies the manuscript and all
release artifacts. Keeping these identifiers in machine-readable files
avoids an unaudited discrepancy between a printed hash table and the
executable release.
The cold runs were performed on macOS arm64 with Python 3.9.6 and NumPy 2.0.2. NumPy is used for detector search, but every accepted detector is substituted into outward rational inequalities. The active/scalar, secondary, and near-Siegel witness checkers use only the Python standard library.
Verification boundary
The finite witnesses contain the exact ordered class coordinates, raw \(R/T\) caps, suffix majorants, mass budgets, greedy fills, objectives, and slacks for all \(16\) rows in Table 3, all \(63\) active rows, and all four scalar rows. Their standard-library checkers import none of the detector, staircase, interval-kernel, or optimization modules. They first verify that the witness has exactly the theorem row cover, then recompute the class-count recurrences, majorants, fills, and objectives from the exported boundary data. The active/scalar checker also executes an adversarial upward-tail test; both finite-row checkers reject a missing row. The near-Siegel checker independently recomputes the rational enclosures and the gap in (7.21).
The finite-row checkers deliberately treat the analytic derivation of
the raw caps, integer count inequalities, and mass budgets from Zhao’s
estimates as their trusted boundary. That derivation is specified in Section 3, Section 4, Section
6, and Section 11 and checked by the primary
rational verifier; it is not made independent merely by orchestration
through verify_release.py. Likewise, the computer
certificate does not replace the source-level argument in Proposition 5 and Theorem 9. These boundaries are recorded
so that an independent specialist audit can target the remaining
analytic correspondence rather than repeat the elementary polytope
calculation.
10. Concluding remarks
The gain from \(7/10\) to \(0.69697\) is not obtained from a new density theorem. It comes from preserving dependence which is lost when character classes or the \(R/T\) decomposition are decoupled. This distinction matters: the old separated estimates actually cross one in two branches at \(A=33/10\), whereas the aligned common-order polytope remains below one with visible room.
The next substantial improvement is unlikely to come from a still finer scalar mesh alone. It would require either a stronger within-class two-zero density statement, a new restriction on cross-class concentration, or a different use of the common class geometry. Eliminating the exceptional set altogether remains a qualitatively different problem.
11. Analytic formulas used by the certificate
This appendix records the limiting forms of the Zhao inequalities which are evaluated by the certificate. It fixes the direction of every endpoint substitution and makes the dependence on \(A\) explicit.
Fix \(x>0\), \(0\leq\lambda_0<\Lambda\), and set \[F_x(s)=G(s/x),\qquad \psi(v)=\frac{F_x(v-\lambda_0)}{F_x(-\lambda_0)}, \qquad \xi=\frac{\phi xg(0)}{2F_x(-\lambda_0)},\quad \phi=\frac13.\] Assume \[\Delta=\psi(\Lambda)-\xi>0.\] Let \[w_A(v)=\mathrm e^{-Av}-\mathrm e^{-A\Lambda}, \qquad 0\leq v\leq\Lambda,\] and suppose \(m\in\{1,2\}\) zeros have prescribed lower endpoints \(\lambda_1,\ldots,\lambda_m\leq\lambda^*<\Lambda\), while every later zero satisfies \(\lambda_j\geq\lambda^*\).
The unrestricted \(R\)-bound used in the certificate is \[\begin{split} R_A(\Lambda)\leq& \sum_{j=1}^m w_A(\lambda_j)\\ &+ \frac{w_A(\lambda^*)} {\psi(\lambda^*)-\psi(\Lambda)} \left\{ \frac{1-\xi}{2\Delta} -\sum_{j=1}^{m} \bigl(\psi(\lambda_j)-\psi(\Lambda)\bigr) \right\}. \end{split}\] (A.1) Inside one fixed class, choose an integer \(N^*\geq2\). In the alternative \(N\leq N^*\), \[R_A(\Lambda) \leq \sum_{j=1}^m w_A(\lambda_j) +(N^*-m)w_A(\lambda^*).\] (A.2) In the alternative \(N\geq N^*+1\), \[\begin{split} R_A(\Lambda)\leq& \sum_{j=1}^m w_A(\lambda_j)\\ &+ \frac{w_A(\lambda^*)} {\psi(\lambda^*)-\psi(\Lambda)} \left\{ \frac{1-(N^*+1)\Delta^2}{2\Delta} -\sum_{j=1}^{m} \bigl(\psi(\lambda_j)-\psi(\Lambda)\bigr) \right\}. \end{split}\] (A.3) At positive \(\varepsilon_z\), the numerator in the final fraction is \[1-(N^*+1)(\Delta^2-\varepsilon_z).\] (A.4) The fixed-class cap is the larger of (A.2) and (A.3); the unrestricted mass uses (A.1) and the direct small-\(N\) guard.
Write \[N=N(\Lambda)=\#\{j:\lambda_j\leq\Lambda\}, \qquad D=D(\Lambda)= \sum_{\lambda_j\leq\Lambda} \bigl(\psi(\lambda_j)-\psi(\Lambda)\bigr).\] The zero-count inequalities are, respectively, \[\begin{aligned} (\Delta^2-\xi-\varepsilon_z)N+2\Delta D&\leq1-\xi, &&\text{unrestricted},\\ (\Delta^2-\varepsilon_z)N+2\Delta D&\leq1, &&\text{one fixed class}. \end{aligned}\] (A.5) The positive \(D\)-term is retained only when the corresponding zeros are known to lie in that same class. Global known positions are used only in the unrestricted line.
For completeness, define Zhao’s auxiliary function (with continuous extension at removable singularities) by \[\begin{split} \mathcal B(t,\lambda,\lambda_0) :=\;&\frac{\phi}{2} \frac{1-\mathrm e^{-2(\lambda+\lambda_0)t}}{\lambda+\lambda_0}\\ &+\frac{1-\mathrm e^{-\lambda t}}{\lambda(\lambda+\lambda_0)} +\frac{\mathrm e^{-2(\lambda+\lambda_0)t}-\mathrm e^{-\lambda t}} {(\lambda+\lambda_0)(\lambda+2\lambda_0)}, \end{split}\] and \[\mathcal C(x,y,z,\Lambda,\lambda_0) = \frac1{xz}\left(\frac12+\frac{\phi+x}{y}\right) \sqrt{ \mathcal B(\phi+x+y,\Lambda,\lambda_0) \mathcal B(z,\Lambda,\lambda_0)}.\] Finally, Zhao’s Lemma 3.1 supplies \[\sum_j\mathrm e^{-k\max(\lambda_j,\Lambda)} \leq(1+\varepsilon_z)\mathcal C(x,y,z,\Lambda,\lambda_0),\] (A.6) with \[k= \begin{cases} 2(\phi+3x+y+z),&\text{fixed class},\\ 2(2\phi+3x+y+z),&\text{unrestricted union}. \end{cases}\] (A.7) The smaller coefficient is never used for an unrestricted mass. Lemma 11 converts (A.6) to coefficient \(A\).
12. Tail filling and branch inventory
Suppose the explicit cap prefix is followed by constant tail caps \(\alpha,\beta>0\), and the residual masses after the prefix are \[R=q\alpha+\rho,\qquad T=p\beta+\sigma, \qquad 0\leq\rho<\alpha,\quad0\leq\sigma<\beta.\] The exact tail contribution to the aligned cross term is \[C_{\mathrm{tail}}= \begin{cases} q\alpha\beta+\rho\sigma,&q=p,\\ q\alpha\beta+\rho\beta,&q<p,\\ p\alpha\beta+\alpha\sigma,&p<q. \end{cases}\] (B.1) This is obtained by writing out the two left-greedy fills. The same construction simultaneously gives the two square terms.
For raw listed caps \(c_1,\ldots,c_K\) and repeated tail \(d\), the decreasing coordinatewise majorant is formed as \[(\widetilde c_1,\ldots,\widetilde c_K,\widetilde d) = \operatorname{suffixmax}(c_1,\ldots,c_K,d).\] (B.2) The tail must be inserted before this operation; replacing it by the minimum of the last listed cap and \(d\) can lower a valid raw cap.
The complete branch inventory checked by the manifest-driven release is: \[\begin{array}{lr} \text{active directed rows}&63,\\ \text{secondary aligned rows}&16,\\ \text{scalar rows}&4,\\ \text{near-Siegel analytic branch}&1. \end{array}\] (B.3) The active driver asserts the \(21\)-box cover and all three low-zero allocations. The secondary theorem command must explicitly request the full scope, as recorded in Section 9.
References
H. Davenport, Multiplicative Number Theory, 3rd ed., revised by H. L. Montgomery, Graduate Texts in Mathematics 74, Springer, 2000.
J. Pintz, A new explicit formula in the additive theory of primes with applications II. The exceptional set in Goldbach’s problem, arXiv:1804.09084v2 (2018). https://arxiv.org/abs/1804.09084v2.
J. Pintz, A new explicit formula in the additive theory of primes with applications I. The explicit formula for the Goldbach problem and the generalized twin prime problem, Acta Arith. 210 (2023), no. 1, 53–94. https://doi.org/10.4064/aa220728-31-3.
G. Zhao, The exceptional set of Goldbach problem and Linnik’s constant, arXiv:2511.05631v2 (2026). https://arxiv.org/abs/2511.05631v2.