> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # Reverse Goldbach Structural Closure Program 06 ## Counterexample Demand–Obstruction Capacity Inequalities ### Reverse Goldbach Structural Closure Program 06 ### Counterexample Demand–Obstruction Capacity Inequalities **Series:** Reverse Goldbach Structural Closure Program **English name:** Reverse Goldbach Structural Closure Program **Abbreviation:** RG-SCP **Paper:** 06 / planned 08–10 **Authors:** Neo.K; Aletheia (GPT-5.6 Sol) **Date:** 2026-09-28 **Version:** v0.1 **Document type:** Structural-inequality draft / counterexample-exclusion framework / interface for subsequent analytic number theory and formalization **Status:** Not a proof of the Goldbach conjecture; this paper establishes exact demand–capacity necessary inequalities for hypothetical Goldbach failure, explicitly marking the capacity-upper-bound problem not yet closed --- ## Abstract The first five papers of the Reverse Goldbach Structural Closure Program have completed the following reconstruction: $$ \text{Goldbach failure} $$ is first rewritten as: $$ \text{complete proper obstruction system}, $$ then projected onto: $$ \text{M6 / Wheel hard branches}, $$ and: $$ \text{fixed-point section hole}. $$ The fifth paper establishes a heuristic discovery layer, but does not allow fitted approximations to enter the proof chain directly. This paper returns to the exact proof channel, for the first time turning the third paper's proposed: $$ \boxed{ \text{obstruction demand} \le \text{obstruction capacity} } $$ into a precise, verifiable inequality architecture. For even $N$, let the reduced prime-input domain be: $$ \mathbb P_N^{-} = \left\{ p\in\mathbb P: 2\le p\le\frac N2 \right\}. $$ Given a prime cutoff: $$ y, $$ define the small-factor survivor set: $$ \mathbb H_y(N) = \left\{ p\in\mathbb P_N^{-}: \gcd(N-p,P(y))=1 \right\}, $$ where: $$ P(y) = \prod_{\substack{ \ell\le y\\ \ell\in\mathbb P }} \ell. $$ Its cardinality: $$ \boxed{ D_N(y) = |\mathbb H_y(N)| } $$ is called **obstruction demand**. Intuitively: > The complements of these prime inputs have avoided every small prime factor not exceeding $y$; if $N$ is a Goldbach failure, each must still be blocked by some proper prime factor greater than $y$. For: $$ p\in\mathbb H_y(N), $$ define the deep obstruction multiplicity: $$ m_y(N,p) = \#\left\{ r\in\mathbb P: y Failure requires every small-factor survivor to remain composite, so every demand unit must consume at least one deep obstruction incidence. Thus: $$ \boxed{ D_N(y)>I_N(y) \Longrightarrow R_G(N)>0. } $$ This is the most basic demand–capacity exclusion criterion of the paper. The paper further decomposes: $$ I_N(y) $$ by obstruction prime $r$: $$ I_N(y) = \sum_{\substack{ r\in\mathbb P\\ r>y }} C_r(N;y), $$ where: $$ C_r(N;y) = \#\left\{ p\in\mathbb H_y(N): p\le N-r^2, \quad p\equiv N\pmod r \right\}. $$ This turns total capacity into a family of arithmetic-progression capacities. If: $$ r\mid N, $$ then by divisor self-suppression: $$ C_r(N;y)\le1. $$ If instead: $$ r\nmid N, $$ then: $$ C_r(N;y) $$ is constrained by the residue class: $$ p\equiv N\pmod r. $$ Thus prime-in-arithmetic-progression upper bounds can control it. The paper also establishes an exact demand formula on the primorial branch. If: $$ M_k = \prod_{i=1}^{k}p_i $$ and: $$ M_k\mid N, $$ then for: $$ y=p_k, $$ every: $$ p>p_k $$ prime input automatically passes the first $k$ small-factor screens, so: $$ \boxed{ D_N(p_k) = \pi\left(\frac N2\right)-k } $$ provided: $$ p_k\le\frac N2. $$ Thus every primorial-divisible hypothetical counterexample must satisfy: $$ \boxed{ \pi\left(\frac N2\right)-k \le I_N(p_k). } $$ Combined with any rigorous capacity upper bound: $$ I_N(p_k) \le U_N(p_k), $$ this yields the exclusion criterion: $$ \boxed{ \pi\left(\frac N2\right)-k > U_N(p_k) \Longrightarrow N \text{is not a Goldbach counterexample}. } $$ The paper also identifies an important limitation: > Simply summing all raw arithmetic-progression capacities usually counts the same inputs repeatedly, so raw first-moment capacity is likely to be much too loose. Thus the genuinely valuable next layer is not more residue classes, but **sieved capacity upper bounds** for canonical least-factor cells, or **overlap-aware compression** of total incidence. Ultimately, this paper compresses the RG-SCP proof target into: $$ \boxed{ \text{Demand Lower Bound} > \text{Capacity Upper Bound}. } $$ If this strict inequality can be established on an infinite branch, an entire family of hypothetical counterexamples can be excluded at once, without verifying Goldbach representations individually. --- # 1. From 'Coverage Cost' to Genuine Inequalities The third paper proposed: $$ \operatorname{Demand}_N \le \operatorname{Cap}_N $$ as a potential contradiction architecture. But at that stage: $$ \operatorname{Demand} $$ and: $$ \operatorname{Cap} $$ were still only concepts. This paper first requires: > Both demand and capacity must be defined by exact finite sets. Only after the exact layer is established can one introduce: - analytic bounds; - sieve bounds; - asymptotics; - heuristic prioritization. --- # 2. Reduced prime-input domain For even: $$ N\ge4, $$ Define: $$ \boxed{ \mathbb P_N^{-} = \left\{ p\in\mathbb P: 2\le p\le\frac N2 \right\}. } $$ If there exists: $$ p\in\mathbb P_N^{-} $$ such that: $$ N-p\in\mathbb P, $$ then: $$ N=p+(N-p) $$ is a Goldbach representation. Thus only the reduced domain needs to be studied. --- # 3. Unordered representation count Define: $$ \boxed{ R_G^{-}(N) = \#\left\{ p\in\mathbb P_N^{-}: N-p\in\mathbb P \right\}. } $$ then: $$ R_G^{-}(N)>0 $$ if and only if: $$ R_G(N)>0. $$ Thus: $$ \boxed{ F_G(N)=1 \Longleftrightarrow R_G^{-}(N)=0. } $$ --- # 4. Prime cutoff product For real: $$ y\ge2, $$ Define: $$ \boxed{ P(y) = \prod_{\substack{ \ell\in\mathbb P\\ \ell\le y }} \ell. } $$ This is a primorial-like product at cutoff $y$. If: $$ y=p_k, $$ then: $$ P(y)=M_k. $$ --- # 5. Small-factor survivor set Define: $$ \boxed{ \mathbb H_y(N) = \left\{ p\in\mathbb P_N^{-}: \gcd(N-p,P(y))=1 \right\}. } $$ This means: > For these prime inputs, the complement $N-p$ has avoided every prime divisor not exceeding $y$. --- # 6. Demand Define: $$ \boxed{ D_N(y) = |\mathbb H_y(N)|. } $$ called: # Obstruction Demand Because, if: $$ F_G(N)=1, $$ then every complement associated with: $$ p\in\mathbb H_y(N) $$ the prime inputs displayed above: $$ N-p $$ must still be composite. They must therefore be handled by a deeper factor source. --- # 7. Admissible Cutoffs This paper mainly considers: $$ 2\le y<\frac N2. $$ The reason is: If: $$ p\le\frac N2, $$ then: $$ N-p\ge\frac N2. $$ Thus, if: $$ N-p $$ is itself prime, and: $$ y<\frac N2, $$ then necessarily: $$ N-p>y. $$ A prime complement is therefore not wrongly excluded from the survivor set merely because it equals a cutoff prime. --- # 8. Deep obstruction multiplicity For: $$ p\in\mathbb H_y(N), $$ Define: $$ \boxed{ m_y(N,p) = \#\left\{ r\in\mathbb P: yy $$ such that: $$ r\le\sqrt{N-p} $$ and: $$ r\mid(N-p). $$ Therefore: $$ m_y(N,p)\ge1. $$ --- # 9. Total deep obstruction incidence Define: $$ \boxed{ I_N(y) = \sum_{p\in\mathbb H_y(N)} m_y(N,p). } $$ It counts: > The total number of deep prime-factor incidences of survivor complements that can serve as proper compositeness witnesses. This is the first exact capacity quantity of the paper. --- # 10. Demand–Incidence Necessary Inequality ## Theorem RG-SCP-06.1 If: $$ F_G(N)=1, $$ then for any: $$ 2\le y<\frac N2 $$ we have: $$ \boxed{ D_N(y) \le I_N(y). } $$ --- ## Proof Suppose: $$ F_G(N)=1. $$ Then for every: $$ p\in\mathbb P_N^{-}, $$ the complement: $$ N-p $$ is composite. In particular, for: $$ p\in\mathbb H_y(N), $$ the complement is not only composite, but avoids every: $$ r\le y. $$ Thus its least prime factor: $$ \operatorname{lpf}(N-p) $$ satisfying: $$ \operatorname{lpf}(N-p)>y $$ and: $$ \operatorname{lpf}(N-p) \le \sqrt{N-p}. $$ Thus: $$ m_y(N,p)\ge1. $$ for all: $$ p\in\mathbb H_y(N) $$ sum over all such inputs: $$ I_N(y) = \sum_{p\in\mathbb H_y(N)} m_y(N,p) \ge \sum_{p\in\mathbb H_y(N)}1 = D_N(y). $$ This completes the proof. --- # 11. The First Exclusion Criterion Theorem RG-SCP-06.1 immediately gives: ## Corollary RG-SCP-06.1a If there exists: $$ y $$ such that: $$ \boxed{ D_N(y)>I_N(y), } $$ then: $$ R_G^{-}(N)>0. $$ Thus: $$ \boxed{ N \text{is not a Goldbach counterexample}. } $$ This is a purely pigeonhole-type exclusion criterion. --- # 12. Demand Units and Capacity Slots The inequality has a very intuitive interpretation. Each: $$ p\in\mathbb H_y(N) $$ is a **demand unit**. If $N$ is a failure, it requires at least one: $$ r>y $$ to prove: $$ N-p $$ remains composite. Every pair: $$ (p,r) $$ where: $$ r\mid(N-p) $$ is a **capacity slot**. Failure requires: $$ \boxed{ \text{every demand unit to be assigned at least one slot}. } $$ Therefore: $$ D_N(y)\le I_N(y). $$ --- # 13. A Prime Complement Has Zero Slots If: $$ N-p $$ is prime, then: $$ m_y(N,p)=0. $$ Thus every Goldbach representation in the demand–capacity picture is: $$ \boxed{ \text{a demand unit with no proper-factor slot}. } $$ Goldbach success can therefore be interpreted as an obstruction-capacity deficiency. --- # 14. Decomposing Deep Incidence by Prime Factor For: $$ r>y, $$ Define: $$ \boxed{ C_r(N;y) = \#\left\{ p\in\mathbb H_y(N): r\le\sqrt{N-p}, \quad r\mid(N-p) \right\}. } $$ Interchanging the order of summation gives: $$ \boxed{ I_N(y) = \sum_{\substack{ r\in\mathbb P\\ r>y }} C_r(N;y). } $$ In fact, only: $$ r\le\sqrt{N-2} $$ can contribute. --- # 15. Proper-factor geometry The condition: $$ r\le\sqrt{N-p} $$ is equivalent to: $$ p\le N-r^2. $$ And: $$ r\mid(N-p) $$ is equivalent to: $$ p\equiv N\pmod r. $$ Thus: $$ \boxed{ C_r(N;y) = \#\left\{ p\in\mathbb H_y(N): p\le x_r(N), \quad p\equiv N\pmod r \right\}, } $$ where: $$ \boxed{ x_r(N) = \min \left( \frac N2, N-r^2 \right). } $$ If: $$ x_r(N)<2, $$ then: $$ C_r(N;y)=0. $$ --- # 16. Raw arithmetic-progression capacity Remove the survivor restriction: $$ p\in\mathbb H_y(N) $$ and define the raw capacity: $$ \boxed{ A_r(N) = \#\left\{ p\in\mathbb P: 2\le p\le x_r(N), \quad p\equiv N\pmod r \right\}. } $$ Clearly: $$ \boxed{ C_r(N;y)\le A_r(N). } $$ Therefore: $$ \boxed{ I_N(y) \le \sum_{\substack{ r\in\mathbb P\\ r>y }} A_r(N). } $$ --- # 17. Divisor Self-Suppression Reappears If: $$ r\mid N, $$ and also: $$ p\equiv N\pmod r, $$ then: $$ p\equiv0\pmod r. $$ Since $p$ is prime, the only possibility is: $$ p=r. $$ Thus: $$ \boxed{ r\mid N \Longrightarrow A_r(N)\le1. } $$ Moreover, if: $$ r>x_r(N), $$ then in fact: $$ A_r(N)=0. $$ --- # 18. Non-divisor obstruction prime If: $$ r\nmid N, $$ then: $$ \gcd(N,r)=1. $$ Thus the residue: $$ N\bmod r $$ is a reduced residue class. In this case: $$ A_r(N) $$ is precisely the prime-in-arithmetic-progression count: $$ \boxed{ A_r(N) = \pi \left( x_r(N); r, N\bmod r \right), } $$ using the standard: $$ \pi(x;q,a) $$ notation. --- # 19. Exact Capacity Upper Envelope We therefore define: $$ \boxed{ U_r^{\mathrm{AP}}(N) = \begin{cases} 1, & r\mid N,\\ \pi(x_r(N);r,N\bmod r), & r\nmid N. \end{cases} } $$ Interpret the case: $$ x_r(N)<2 $$ as: $$ U_r^{\mathrm{AP}}(N)=0. $$ then: $$ \boxed{ C_r(N;y) \le U_r^{\mathrm{AP}}(N). } $$ Thus: $$ \boxed{ I_N(y) \le U_N^{\mathrm{AP}}(y) } $$ where: $$ \boxed{ U_N^{\mathrm{AP}}(y) = \sum_{\substack{ r\in\mathbb P\\ y U_N^{\mathrm{AP}}(y), } $$ then: $$ F_G(N)=0. $$ This is already a wholly exact finite-$N$ exclusion template. The remaining question is: > Without computing every $C_r$ individually, how can demand be lower-bounded and capacity upper-bounded? --- # 22. The Exact Complement Form of Demand Define the small obstruction sets: $$ S_\ell(N) = \left\{ p\in\mathbb P_N^{-}: \ell\mid(N-p) \right\} $$ For: $$ \ell\le y. $$ then: $$ \mathbb H_y(N) = \mathbb P_N^{-} \setminus \bigcup_{\substack{ \ell\le y\\ \ell\in\mathbb P }} S_\ell(N). $$ Thus: $$ \boxed{ D_N(y) = |\mathbb P_N^{-}| - \left| \bigcup_{\ell\le y} S_\ell(N) \right|. } $$ --- # 23. A First-Order Demand Lower Bound By the union bound: $$ \left| \bigcup_{\ell\le y} S_\ell(N) \right| \le \sum_{\ell\le y} |S_\ell(N)|. $$ Thus: $$ \boxed{ D_N(y) \ge |\mathbb P_N^{-}| - \sum_{\ell\le y} |S_\ell(N)|. } $$ Also: $$ |\mathbb P_N^{-}| = \pi\left(\frac N2\right). $$ Therefore: $$ \boxed{ D_N(y) \ge \pi\left(\frac N2\right) - \sum_{\ell\le y} |S_\ell(N)|. } $$ If the right-hand side is negative, take: $$ 0 $$ as a trivial lower bound. --- # 24. Self-Suppression of a Small-Factor Class If: $$ \ell\mid N, $$ then: $$ S_\ell(N) \subseteq \{\ell\}. $$ Thus: $$ |S_\ell(N)|\le1. $$ If: $$ \ell\nmid N, $$ then: $$ S_\ell(N) $$ is a prime arithmetic-progression slice: $$ p\equiv N\pmod\ell. $$ Thus it can also be controlled by: $$ \pi \left( \frac N2; \ell, N\bmod\ell \right) $$ the count displayed above. --- # 25. Demand Lower Envelope Let: $$ V_\ell(N) $$ be any rigorous upper bound for: $$ |S_\ell(N)| $$ the cardinality displayed above. Define: $$ \boxed{ L_N(y) = \max \left\{ 0, \pi\left(\frac N2\right) - \sum_{\substack{ \ell\in\mathbb P\\ \ell\le y }} V_\ell(N) \right\}. } $$ then: $$ \boxed{ D_N(y)\ge L_N(y). } $$ --- # 26. General Demand–Capacity Exclusion Theorem ## Theorem RG-SCP-06.3 If, for some: $$ y<\frac N2 $$ one establishes rigorous bounds: $$ D_N(y)\ge L_N(y) $$ and: $$ I_N(y)\le U_N(y), $$ and: $$ \boxed{ L_N(y)>U_N(y), } $$ then: $$ \boxed{ R_G(N)>0. } $$ --- ## Proof Suppose instead: $$ F_G(N)=1, $$ then by RG-SCP-06.1: $$ D_N(y)\le I_N(y). $$ Therefore: $$ L_N(y) \le D_N(y) \le I_N(y) \le U_N(y), $$ and: $$ L_N(y)>U_N(y) $$ a contradiction. Hence: $$ F_G(N)=0. $$ This completes the proof. --- # 27. This Is the First Genuine Inequality-Closure Template of RG-SCP The first three papers describe necessary structures of hypothetical failure. This paper obtains for the first time: $$ \boxed{ \text{if lower demand exceeds upper capacity, the counterexample is directly excluded.} } $$ This is an interface into which external number-theoretic theorems can genuinely be inserted. --- # 28. Classical AP bound interface If: $$ r\nmid N $$ and: $$ x_r(N)>r, $$ then any rigorous upper bound for primes in progressions can be inserted into: $$ \pi(x_r;r,N\bmod r). $$ For example, in its standard range of applicability, a classical Brun–Titchmarsh-type upper bound can be used: $$ \pi(x;q,a) \le \frac{ 2x }{ \varphi(q) \log(x/q) } $$ For: $$ \gcd(a,q)=1, \quad qr $$ the displayed strict condition holds. --- # 29. Small-range fallback If: $$ x_r(N)\le r, $$ a single residue class contains at most one integer representative in: $$ [1,x_r(N)] $$ the interval displayed above. Thus a trivial bound may be used: $$ A_r(N)\le1. $$ The practical upper envelope can therefore be piecewise. This paper does not present this piecewise form as a new number-theory theorem, but only as a capacity interface. --- # 30. The Fatal Weakness of Raw AP Capacity Even if: $$ A_r(N) $$ has a very good upper bound term by term, summing directly: $$ \sum_r A_r(N) $$ may still be extremely loose. The reason is that the same: $$ p $$ may simultaneously satisfy: $$ r_1\mid(N-p), $$ $$ r_2\mid(N-p), $$ $$ r_3\mid(N-p). $$ Thus the same demand unit is counted repeatedly in several raw capacities. This is: $$ \boxed{ \text{capacity overlap}. } $$ --- # 31. Incidence multiplicity identity For: $$ p\in\mathbb H_y(N), $$ we already have: $$ m_y(N,p). $$ Thus: $$ I_N(y) = \sum_{p\in\mathbb H_y(N)} m_y(N,p). $$ If a composite complement has many different deep prime divisors, it contributes several capacity slots, but remains only one demand unit. Therefore: $$ I_N(y) $$ is a **capacity with multiplicity**. --- # 32. Zero-slot count Define: $$ \boxed{ Z_N(y) = \#\left\{ p\in\mathbb H_y(N): m_y(N,p)=0 \right\}. } $$ In: $$ y<\frac N2 $$ when this condition holds, $$ m_y(N,p)=0 $$ for a survivor $p$ is equivalent to: $$ N-p $$ being prime. Therefore: $$ \boxed{ Z_N(y) = R_G^{-}(N). } $$ --- # 33. Overlap surplus Define: $$ \boxed{ \Omega_N(y) = \sum_{p\in\mathbb H_y(N)} \max \left( m_y(N,p)-1, 0 \right). } $$ It measures: > How much raw capacity is overcounted when composite survivors have more than one deep small-prime witness. --- # 34. Exact Demand–Incidence Identity ## Theorem RG-SCP-06.4 For: $$ 2\le y<\frac N2, $$ we have: $$ \boxed{ R_G^{-}(N) = D_N(y) + \Omega_N(y) - I_N(y). } $$ --- ## Proof Divide: $$ \mathbb H_y(N) $$ into: ### A. Zero Slots $$ m_y(N,p)=0. $$ Their count is: $$ R_G^{-}(N). $$ ### B. At Least One Slot $$ m_y(N,p)\ge1. $$ Their count is: $$ D_N(y)-R_G^{-}(N). $$ For class B, each element first contributes one base incidence, then adds: $$ m_y(N,p)-1 $$ surplus incidences. Thus: $$ I_N(y) = D_N(y) - R_G^{-}(N) + \Omega_N(y). $$ Rearranging gives: $$ R_G^{-}(N) = D_N(y) + \Omega_N(y) - I_N(y). $$ This completes the proof. --- # 35. This Identity Is More Complete than Mere $D\le I$ Failure: $$ R_G^{-}(N)=0 $$ is equivalent to: $$ \boxed{ I_N(y) = D_N(y) + \Omega_N(y). } $$ Thus a counterexample requires not merely: $$ I_N(y)\ge D_N(y). $$ Instead, it requires: > Total incidence must exactly fill all demand, together with all unavoidable overlap surplus. --- # 36. Overlap-aware exclusion By the exact identity: if one can prove: $$ \Omega_N(y)\ge\Omega_N^{\mathrm{LB}}(y), $$ and: $$ I_N(y)\le U_N(y), $$ then: $$ R_G^{-}(N) \ge D_N(y) + \Omega_N^{\mathrm{LB}}(y) - U_N(y). $$ Therefore, if: $$ \boxed{ D_N(y) + \Omega_N^{\mathrm{LB}}(y) > U_N(y), } $$ then: $$ R_G^{-}(N)>0. $$ This is stronger than the first-order criterion. --- # 37. But an Overlap Lower Bound Is Difficult Note: $$ \Omega_N(y) $$ is not necessarily large. For example: $$ N-p $$ may be: $$ r^2 $$ or: $$ rs $$ with only one prime factor not exceeding the square root. Then: $$ m_y(N,p)=1. $$ Thus: $$ \Omega_N(y)=0 $$ may perfectly well occur for many complements. Thus the paper does not assume overlap surplus automatically supplies closure. --- # 38. Canonical least-factor cells Another way to avoid overlap is to use the canonical least factor directly. For: $$ r>y, $$ Define: $$ \boxed{ B_r(N;y) = \left\{ p\in\mathbb H_y(N): \operatorname{lpf}(N-p)=r \right\}. } $$ Cells for different: $$ r $$ are disjoint: $$ B_r(N;y) \cap B_s(N;y) = \varnothing $$ If: $$ r\neq s. $$ --- # 39. The Canonical Partition Under Failure If: $$ F_G(N)=1, $$ then every survivor complement is composite. Thus: $$ \boxed{ \mathbb H_y(N) = \bigsqcup_{\substack{ r\in\mathbb P\\ r>y }} B_r(N;y). } $$ Therefore: $$ \boxed{ D_N(y) = \sum_{r>y} |B_r(N;y)|. } $$ This is a genuine capacity decomposition without overlap. --- # 40. Sieved capacity Each: $$ B_r(N;y) $$ requires not only: $$ p\equiv N\pmod r, $$ but also: $$ N-p $$ not to be divisible by any prime: $$ sy} U_r^{\mathrm{sieve}}(N;y) < D_N(y), $$ failure is directly excluded. --- # 42. Where the Parity Problem Appears To control: $$ B_r(N;y) $$ one must distinguish: > The least prime factor of a complement is exactly $r$. This is much more refined than: $$ r\mid(N-p) $$ the divisibility relation displayed above. It requires excluding every smaller prime factor. This is precisely where difficulties such as sieve parity / almost-prime distinctions can easily enter. Thus this paper does not claim: $$ U_r^{\mathrm{sieve}} $$ is already sufficiently strong. --- # 43. The Demand Side Can Also Be Refined by Sieving Likewise, $$ D_N(y) $$ is itself a sifted prime-input count: $$ p\in\mathbb P, \quad \gcd(N-p,P(y))=1. $$ Thus one can seek: $$ \boxed{ L_N^{\mathrm{sieve}}(y) \le D_N(y) } $$ as a rigorous lower bound. Genuine closure needs: $$ \boxed{ L_N^{\mathrm{sieve}}(y) > U_N^{\mathrm{sieve}}(y). } $$ --- # 44. Asymmetry of Demand and Capacity Note: - Demand is **how many prime inputs remain unhandled by small factors**; - Capacity is **how many obstruction slots the remaining large factors can supply**. These are not the same sieve problem. This asymmetry may be one source of RG-SCP's value. Because: $$ \boxed{ \text{survivor lower bound} } $$ and: $$ \boxed{ \text{obstruction upper bound} } $$ can use different mathematical tools. --- # 45. $6\mid N$ clean branch The fourth paper proved: For: $$ N\ge12, \quad 6\mid N, $$ standard Goldbach failure is equivalent to: $$ \Delta_6(N)>0. $$ In the demand language, for every: $$ p>3, $$ we have: $$ \gcd(N-p,6)=1. $$ Thus: $$ \boxed{ \mathbb H_3(N) = \left\{ p\in\mathbb P: 5\le p\le\frac N2 \right\}. } $$ Therefore: $$ \boxed{ D_N(3) = \pi\left(\frac N2\right)-2. } $$ provided: $$ N/2\ge5. $$ --- # 46. M6 Demand–Capacity Inequality Thus every: $$ 6\mid N, \quad N\ge12 $$ hypothetical failure in this domain must satisfy: $$ \boxed{ \pi\left(\frac N2\right)-2 \le I_N(3). } $$ This is the cleanest first-layer demand–capacity inequality. It says: > Almost every prime not exceeding $N/2$ must consume, at the complement endpoint, at least one factor incidence with $r\ge5$. --- # 47. Primorial branch Let: $$ M_k = \prod_{i=1}^{k}p_i. $$ Suppose: $$ M_k\mid N. $$ Take: $$ y=p_k. $$ For any: $$ p>p_k, $$ Since: $$ p $$ is not divisible by the first $k$ primes, and: $$ N\equiv0\pmod{p_i}, $$ Thus: $$ N-p \equiv -p \not\equiv0 \pmod{p_i} $$ for all: $$ i\le k. $$ Therefore: $$ \gcd(N-p,M_k)=1. $$ --- # 48. Primorial Demand Formula ## Theorem RG-SCP-06.5 If: $$ M_k\mid N $$ and: $$ p_k\le\frac N2, $$ then: $$ \boxed{ \mathbb H_{p_k}(N) = \left\{ p\in\mathbb P: p_kp_k $$ is prime, then: $$ p $$ does not equal any: $$ p_i $$ and: $$ p_i\nmid p $$ For: $$ i\le k. $$ Since: $$ p_i\mid N, $$ Thus: $$ N-p \equiv-p \not\equiv0 \pmod{p_i}. $$ Hence: $$ \gcd(N-p,M_k)=1. $$ Conversely, If: $$ p=p_i $$ for some: $$ i\le k, $$ then: $$ p_i\mid N-p_i, $$ so it is not in the survivor set. Thus the survivor set is precisely: $$ p>p_k $$ prime inputs in this range. This completes the proof. --- # 49. Primorial Demand–Capacity Necessary Inequality By RG-SCP-06.1: if: $$ M_k\mid N $$ and: $$ F_G(N)=1, $$ then: $$ \boxed{ \pi\left(\frac N2\right)-k \le I_N(p_k). } $$ This makes the demand of hypothetical failure completely explicit. --- # 50. Primorial raw capacity Also: $$ I_N(p_k) \le \sum_{\substack{ r\in\mathbb P\\ r>p_k }} A_r(N). $$ Thus failure must satisfy: $$ \boxed{ \pi\left(\frac N2\right)-k \le \sum_{\substack{ r\in\mathbb P\\ p_kp_k $$ but: $$ r\mid N, $$ then: $$ A_r(N)\le1. $$ Thus, if $N$ has additional prime divisors beyond: $$ M_k $$ the primorial displayed above, deep raw capacity is suppressed further. Therefore: $$ \boxed{ \text{more prime divisors of }N \to \text{more suppressed capacity columns}. } $$ This formally incorporates the third paper's self-suppression into the capacity inequality. --- # 52. Primorial exclusion template If, for some: $$ k $$ and every: $$ N $$ satisfying: $$ M_k\mid N $$ one can prove: $$ \boxed{ \sum_{r>p_k} U_r(N) < \pi\left(\frac N2\right)-k, } $$ where: $$ U_r(N) $$ are rigorous upper bounds, then the whole branch cannot contain a Goldbach counterexample. This would be a: $$ \boxed{ \text{family-level exclusion theorem}. } $$ --- # 53. This Is More Important than Verifying $N$ Individually Finite computation can prove: $$ N $$ is not a counterexample. A family inequality, in contrast, might prove at once: $$ \forall N\in\mathcal F. $$ The genuine goal of RG-SCP is the latter. --- # 54. Capacity density heuristic Temporarily ignoring self-suppression and the survivor restriction, For: $$ r\nmid N $$ the rough density scale for raw AP capacity of this form resembles: $$ \frac{ x_r(N) }{ (r-1)\log x_r(N) }. $$ Thus total raw capacity roughly involves: $$ \sum_{r>y} \frac1{r-1}. $$ Such sums decline very slowly. Thus first-order raw AP capacity alone may be far larger than demand. --- # 55. Raw-capacity no-go warning This paper therefore expects: $$ \boxed{ \text{using only a raw AP union bound is very likely insufficient for global closure}. } $$ This is a methodological warning, not a proved impossibility theorem. The reason is that raw capacity: - does not deduct smaller-factor avoidance; - does not deduct overlap; - does not exploit least-factor uniqueness; - does not exploit cross-$r$ consistency; - does not exploit fixed-point section geometry. --- # 56. The Genuine Target of Capacity Compression Thus subsequent work needs to compress: $$ A_r(N) $$ into: $$ B_r(N;y) $$ or another smaller canonical capacity. The ideal form: $$ \boxed{ \sum_r U_r^{\mathrm{canonical}}(N;y) < D_N(y). } $$ Only this has a chance of producing a contradiction. --- # 57. Cross-$r$ compatibility Raw capacity treats different: $$ r $$ as independent columns. But they are actually generated by the same: $$ N $$ target displayed above. Thus the residue classes: $$ p\equiv N\pmod r $$ are not chosen arbitrarily. Every column shares the same: $$ N. $$ This may produce cross-$r$ compatibility constraints. --- # 58. Chinese-remainder coupling If some: $$ p $$ is simultaneously obstructed by: $$ r $$ and: $$ s $$ the two primes displayed above, then: $$ p\equiv N\pmod r $$ and: $$ p\equiv N\pmod s. $$ Thus: $$ p\equiv N\pmod{rs} $$ for distinct primes: $$ r,s. $$ Thus overlap is controlled by the deeper modulus: $$ rs $$ displayed above. This supplies an exact overlap structure. --- # 59. Pair-overlap set Define: $$ A_{r,s}(N) = \left\{ p\in\mathbb P_N^{-}: r\mid(N-p), \quad s\mid(N-p) \right\}. $$ If: $$ r\neq s, $$ then: $$ rs\mid(N-p). $$ Thus: $$ p\equiv N\pmod{rs}. $$ Therefore: $$ A_{r,s}(N) $$ can be controlled by an AP count with modulus: $$ rs $$ as displayed above. --- # 60. Higher-order overlap hierarchy Likewise, If: $$ r_1,\ldots,r_j $$ all divide: $$ N-p, $$ then: $$ p\equiv N \pmod{ r_1\cdots r_j }. $$ Thus capacity overlap has a natural inclusion–exclusion hierarchy: $$ \text{single columns} \to \text{pair intersections} \to \text{triple intersections} \to \cdots. $$ --- # 61. Bonferroni capacity refinement If the raw cover is used: $$ \bigcup_r A_r(N), $$ the first-order upper bound is: $$ \left| \bigcup_r A_r \right| \le \sum_r|A_r|. $$ At higher orders, Bonferroni truncation can improve the upper bound. But higher-order intersections cause: $$ r_1r_2\cdots r_j $$ to grow rapidly. Thus this route may soon encounter combinatorial explosion. --- # 62. Canonical Cells Are Cleaner than Inclusion–Exclusion Least-factor cells: $$ B_r(N;y) $$ are intrinsically disjoint. Thus, if one can upper-bound directly: $$ |B_r|, $$ there is no need to handle an infinite overlap hierarchy. Therefore: $$ \boxed{ \text{canonical least-factor capacity} } $$ is the subsequent direction preferred by this paper. --- # 63. Capacity as matching problem One can also build a bipartite graph: Left side: $$ \mathbb H_y(N). $$ Right side: $$ \mathcal R_y(N) = \left\{ r\in\mathbb P: y Demand–capacity inequalities need not choose between obstruction theory and fixed-point theory; at the canonical-capacity level they are actually the same structure. --- # 71. A Source Audit Is Again Necessary But this also means: $$ \operatorname{FCap}_r $$ and: $$ |B_r| $$ cannot be treated as two independent sources. They are only: $$ \operatorname{lpf} $$ two representations of this structure. A genuinely new source must supply: - a cross-fiber constraint; - a nonlocal invariant; - a global capacity upper bound; - a transport theorem; - spectral coupling. --- # 72. Exact capacity gap Define: $$ \boxed{ G_N(y) = D_N(y)-I_N(y). } $$ If: $$ G_N(y)>0, $$ then Goldbach success holds. If: $$ F_G(N)=1, $$ then: $$ G_N(y)\le0. $$ However: $$ G_N(y)\le0 $$ does not imply failure. The overlap surplus may make: $$ I_N(y) $$ large. --- # 73. Overlap-corrected gap From the exact identity: $$ R_G^{-}(N) = D_N(y) + \Omega_N(y) - I_N(y), $$ Define: $$ \boxed{ G_N^{\mathrm{corr}}(y) = D_N(y) + \Omega_N(y) - I_N(y). } $$ then: $$ \boxed{ G_N^{\mathrm{corr}}(y) = R_G^{-}(N). } $$ This is exact, but computationally it is almost equivalent to already knowing the answer. It therefore serves mainly as a theoretical calibrator. --- # 74. Proof Value Comes from Bounds, Not from the Identity Itself If we directly compute: $$ D, \Omega,I, $$ we can already determine: $$ R_G^{-}. $$ This has no advantage over finding Goldbach pairs directly. Genuine proof value comes from: $$ D\ge L, $$ $$ \Omega\ge O, $$ $$ I\le U $$ as provable bounds. Then: $$ L+O-U>0. $$ --- # 75. Three-bound closure template The most general closure form is therefore: $$ \boxed{ L_N(y) + O_N(y) > U_N(y) } $$ where: $$ L_N(y)\le D_N(y), $$ $$ O_N(y)\le\Omega_N(y), $$ $$ I_N(y)\le U_N(y). $$ then: $$ R_G^{-}(N)>0. $$ This is the most complete inequality architecture currently given in this paper. --- # 76. Branch-level version If, for some family: $$ \mathcal F $$ we can prove: $$ L_N(y_N) + O_N(y_N) > U_N(y_N) $$ for all: $$ N\in\mathcal F, $$ then: $$ \boxed{ \mathcal F \cap \mathcal C_G = \varnothing. } $$ This is the branch elimination pursued by RG-SCP. --- # 77. Dynamic cutoff There is no need to fix: $$ y. $$ We can set: $$ y=y(N). $$ For example: $$ y=N^\theta, $$ or: $$ y=(\log N)^A, $$ or: $$ y=p_k $$ where $k$ grows with $N$. This allows demand and capacity to be compared at the most favorable scale. --- # 78. Cutoff optimization Define: $$ \mathcal G_N(y) = L_N(y)-U_N(y). $$ We can study: $$ \max_y \mathcal G_N(y). $$ If there exists: $$ y $$ such that: $$ \mathcal G_N(y)>0, $$ then exclusion is achieved. The fifth paper's heuristic can predict which: $$ y $$ is most worth attacking first, but the theorem must hold rigorously for the selected cutoff. --- # 79. Cutoff trade-off Increasing: $$ y $$ will: ### Reduce Demand because more complements are covered first by small factors. But at the same time it will: ### Reduce Deep Capacity because the available: $$ r>y $$ become fewer. Thus: $$ y $$ is a natural trade-off parameter. The optimal cutoff may lie at an intermediate scale rather than at: $$ y=3 $$ or: $$ y\approx\sqrt N. $$ --- # 80. Capacity phase diagram We can use: $$ y $$ as the horizontal axis, and plot: $$ D_N(y), $$ $$ I_N(y), $$ $$ L_N(y), $$ $$ U_N(y). $$ If at some point: $$ L_N(y)>U_N(y), $$ that is a rigorous exclusion zone. Even without a crossing, the shapes of the curves can tell us which side lacks a bound. --- # 81. A Heuristic Crossing Must Not Be Passed Off as a Proof If the fitted curves: $$ \widehat L_N(y) $$ and: $$ \widehat U_N(y) $$ cross, this can only produce a candidate cutoff. The actual conclusion must come from the rigorous inequality: $$ L_N(y)>U_N(y). $$ This directly inherits the fifth paper's proof-promotion gate. --- # 82. Minimal counterexample version Suppose: $$ N_* = \min\mathcal C_G. $$ Then for: $$ N_* $$ all demand–capacity inequalities must hold. If we also use the information that: $$ M All success states below a minimal counterexample may constrain the capacity geometry of the larger counterexample. This is another route, distinct from failure-preserving descent. --- # 84. Success-background constraint We can investigate whether: $$ \boxed{ \forall M0 } $$ implies that: $$ I_{N_*}(y) $$ cannot be too large, or: $$ D_{N_*}(y) $$ cannot be too small. If this can be done, minimality becomes an inequality source. This is a direction worth studying in the next stage. --- # 85. Capacity certificate For fixed $N$ and $y$, a finite capacity certificate can contain: ```text N y survivor prime-input set H_y(N) for each deep prime r: admissible p-range x_r(N) residue class N mod r exact or certified upper capacity suppressed divisors r | N total capacity upper bound demand lower bound comparison ``` If: $$ L_N(y)>U_N(y), $$ the certificate verifier need only verify the bounds and inequalities. --- # 86. Family certificate A higher-level family certificate should contain symbolic: $$ L(N) $$ and: $$ U(N) $$ and prove: $$ L(N)>U(N) $$ For: $$ N\in\mathcal F. $$ This is the most valuable proof artifact. --- # 87. Formalization targets This paper is particularly suitable for first formalizing the elementary core in Lean / Coq. Suggested targets: ### L1 Define: $$ \mathbb H_y(N), D_N(y), m_y(N,p), I_N(y). $$ ### L2 Prove: $$ F_G(N)=1 \Rightarrow D_N(y)\le I_N(y). $$ ### L3 Prove the incidence decomposition: $$ I_N(y)=\sum_r C_r(N;y). $$ ### L4 Prove: $$ C_r(N;y)\le A_r(N). $$ ### L5 Prove self-suppression: $$ r\mid N \Rightarrow A_r(N)\le1. $$ ### L6 Prove the exact identity: $$ R_G^{-} = D+\Omega-I. $$ ### L7 Prove the primorial demand formula: $$ D_N(p_k) = \pi(N/2)-k. $$ --- # 88. External Analytic Theorems Should Be Isolated Brun–Titchmarsh, explicit PNT bounds, sieve estimates, and the like should not be mixed directly into the elementary core. A layered formalization is suggested: ```text Core/ Demand.lean Capacity.lean Incidence.lean PrimorialDemand.lean Interfaces/ PrimeAPUpperBound.lean PrimeCountLowerBound.lean SieveCapacityUpperBound.lean External/ BrunTitchmarshInterface.lean ExplicitPNTInterface.lean ``` This makes it clear which external theorems each conclusion consumes. --- # 89. Source provenance The main source graph of this paper is: ```text Goldbach failure ↓ small-factor survivor demand ↓ deep proper-factor necessity ↓ factor-incidence capacity ↓ arithmetic progression columns ↓ capacity upper bounds ``` and: ```text M_k | N ↓ small-prime self-suppression ↓ explicit survivor demand ↓ primorial demand-capacity inequality ``` --- # 90. This Paper Has Not Broken the Parity Barrier It must be made explicit: This paper establishes: $$ D\le I $$ and: $$ D\le U $$ but this does not mean that the sieve parity problem has been overcome. On the contrary, compressing raw AP capacity sufficiently far is very likely to encounter parity-sensitive difficulty. The position of this paper is therefore: $$ \boxed{ \text{place parity-type difficulty in an explicit capacity term}. } $$ --- # 91. This Is Still Progress Even though closure has not yet been achieved, we can now ask clearly: > Is the missing ingredient a demand lower bound that is not strong enough, or a capacity upper bound that is too loose? instead of saying vaguely: > Goldbach is difficult. This kind of failure localization itself has research value. --- # 92. Capacity Bottleneck Classification If an inequality does not cross in subsequent work, the bottlenecks can be classified as follows: ### B1 Demand collapse: $$ L_N(y) $$ is too small. ### B2 Raw capacity inflation: $$ U_N(y) $$ is too large. ### B3 Overlap inflation. ### B4 Self-suppression is not fully exploited. ### B5 The least-factor sieve is too weak. ### B6 The wrong cutoff is selected. ### B7 The source class itself is insufficient. Each type of bottleneck corresponds to a different next step. --- # 93. AI Search Tasks AI can be asked to: 1. Search for the most effective cutoff $y(N)$; 2. Search for the strongest matched family; 3. Search for capacity overlap patterns; 4. Conjecture a sieved capacity inequality; 5. Find extremal $N$ on the primorial branch; 6. Search for local demand subsets $S$; 7. Find a cross-$r$ invariant; 8. Translate an empirical relation into an exact statement. But a numerical crossing cannot be treated directly as a theorem. --- # 94. Candidate theorem queue High-value candidate forms include: $$ D_N(y) \ge L(N,y), $$ $$ I_N(y) \le U(N,y), $$ $$ \Omega_N(y) \ge O(N,y), $$ $$ |B_r(N;y)| \le U_r(N,y), $$ and: $$ \sum_r U_r(N,y) < L(N,y). $$ These are theorem shapes that can genuinely advance closure. --- # 95. The First Branch Most Worth Attacking This paper suggests prioritizing: $$ \boxed{ M_k\mid N } $$ as a primorial-divisible family. Reasons: 1. Demand has an exact formula; 2. Small-prime capacities are extensively self-suppressed; 3. $6\mid N$ automatically eliminates boundary-$3$; 4. A fixed-point section hole is equivalent to standard failure; 5. Heuristic pressure is also most concentrated here. This is currently the structurally cleanest subdomain of RG-SCP. --- # 96. Primorial family target theorem The ideal first branch theorem could be: > There exists some $k_0$ such that, for all $k\ge k_0$ and all sufficiently large $N$, if $M_k\mid N$, then $N$ has a Goldbach representation. In RG-SCP language, it suffices to prove: $$ \boxed{ L_N(p_k)>U_N(p_k). } $$ This paper does not currently contain this proof. It is a clear target. --- # 97. A Weaker but Valuable Version We could first prove the result for some fixed: $$ k. $$ For example: $$ M_k\mid N $$ has no counterexample throughout the entire family. Even excluding just one infinite branch is genuine theorem-level progress. --- # 98. One Level Weaker: A Finite-Scale Theorem If we can only prove: $$ N\le B $$ as the scale of branch exclusion, it can still be a machine-checked finite theorem. But it must be labeled: $$ \text{finite}. $$ It must not be promoted to: $$ \forall N. $$ --- # 99. Division of Labor with the Fifth Paper The fifth paper: $$ \text{find which }y,\mathcal F,S \text{ are worth attacking}. $$ The sixth paper: $$ \text{require an actual proof of } L>U. $$ Thus the interface between the two papers is: $$ \boxed{ \text{heuristic target selection} \to \text{rigorous inequality closure}. } $$ --- # 100. Core Exact Results of This Paper The exact framework actually established so far in this paper includes: ## T1 $$ F_G(N)=1 \Rightarrow D_N(y)\le I_N(y). $$ ## T2 $$ I_N(y) = \sum_{r>y} C_r(N;y). $$ ## T3 $$ C_r(N;y) \le A_r(N). $$ ## T4 $$ r\mid N \Rightarrow A_r(N)\le1. $$ ## T5 $$ R_G^{-}(N) = D_N(y) + \Omega_N(y) - I_N(y). $$ ## T6 If: $$ M_k\mid N, $$ then: $$ D_N(p_k) = \pi(N/2)-k. $$ ## T7 If: $$ L_N(y)>U_N(y), $$ and: $$ D_N(y)\ge L_N(y), \quad I_N(y)\le U_N(y), $$ then: $$ R_G(N)>0. $$ --- # 101. What This Paper Has Not Proved This paper has not yet proved: $$ L_N(y)>U_N(y) $$ for all even numbers. Nor has it proved: $$ L_N^{\mathrm{sieve}} > U_N^{\mathrm{sieve}} $$ on any complete infinite branch. Thus: $$ \boxed{ \text{Demand–Capacity Framework} \neq \text{Goldbach proof}. } $$ --- # 102. Interface to the Next Paper The sixth paper has reduced the proof target to upper and lower bounds. The most natural next paper is: # RG-SCP 07 ## Minimal Counterexamples, Transport Operators, and Demand–Capacity Closure ### Minimal Counterexamples, Transport Operators, and Demand–Capacity Closure Core question: > Can the fact that all states below a minimal counterexample are successful be turned into additional constraints on $D_N(y)$, $I_N(y)$, or canonical capacities? That is, rewrite the unsuccessful failure-preserving descent of the fourth paper as: $$ \boxed{ \text{minimality} \to \text{capacity restriction}. } $$ This may be more natural than directly seeking: $$ N_* \to N'I_N(y) \Longrightarrow \text{Goldbach success}. } $$ Decomposing: $$ I_N(y) $$ by obstruction primes: $$ I_N(y) = \sum_{r>y} C_r(N;y), $$ then turns the proof burden into: $$ \text{prime counts in constrained residue classes}. $$ On the primorial branch: $$ M_k\mid N, $$ the demand becomes fully explicit: $$ \boxed{ D_N(p_k) = \pi\left(\frac N2\right)-k. } $$ Thus a hypothetical failure must satisfy: $$ \boxed{ \pi\left(\frac N2\right)-k \le I_N(p_k). } $$ This is the first genuine counterexample-admissibility inequality sought by this paper. Only one question remains: the hardest, but also the clearest: $$ \boxed{ \text{Can we prove that the demand lower bound eventually exceeds the admissible obstruction capacity upper bound?} } $$ If the answer is affirmative on some branch, that branch is permanently removed from the Goldbach counterexample universe. --- # 104. The Shortest Version If only one sentence is retained: > **If a Goldbach counterexample exists, every prime complement that avoids small prime factors must still consume at least one deeper proper-factor obstruction slot; thus a counterexample must satisfy demand not exceeding capacity, and any scale at which the demand lower bound can be proved greater than the capacity upper bound immediately excludes that counterexample.** Core formulas: $$ \boxed{ F_G(N)=1 \Longrightarrow D_N(y)\le I_N(y). } $$ Exclusion formula: $$ \boxed{ L_N(y)>U_N(y) \Longrightarrow R_G(N)>0. } $$ Primorial branch: $$ \boxed{ M_k\mid N \Longrightarrow D_N(p_k) = \pi\left(\frac N2\right)-k. } $$ Thus the real next step is not to compute a few more Goldbach pairs, but to: $$ \boxed{ \text{compress capacity below this demand.} } $$ --- # Appendix A: Canonical Definitions $$ \mathbb P_N^{-} = \left\{ p\in\mathbb P: 2\le p\le N/2 \right\}. $$ $$ P(y) = \prod_{\ell\le y,\ \ell\in\mathbb P}\ell. $$ $$ \mathbb H_y(N) = \left\{ p\in\mathbb P_N^{-}: \gcd(N-p,P(y))=1 \right\}. $$ $$ D_N(y) = |\mathbb H_y(N)|. $$ $$ m_y(N,p) = \#\left\{ r\in\mathbb P: y