> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # SRBRD-03 ## Center–Frontier Occupancy Closure ### Shared-Prime Decomposition, Exact Bonferroni Remainder, Center-Supported Error, and a Cap-Corrected Non-Cover Certificate **Parent program:** RG-SCP / Reverse Goldbach Structural Closure Program **Branch:** SRBRD — Symmetric Root-Bounded Relation Domain **Round:** SRBRD-03 **Date:** 2026-09-30 **Version:** v0.1 **Authors:** Neo.K; Aletheia (GPT-5.6 Sol) **Nature:** Exact Center–Frontier Decomposition / Structural Closure Audit **Prerequisites:** SRBRD-00; SRBRD-01; SRBRD-02 **Status:** Establishes a new rigorous center-correction certificate; uniform Goldbach closure has not yet been obtained --- # 0. Target of This Round SRBRD-02 has established the occupancy moments: $$ B_j(m) = \sum_{d\in\Omega_m} \binom{\mu_m(d)}{j}, $$ and the odd Bonferroni upper hierarchy: $$ U_q(m) = \sum_{j=1}^{q} (-1)^{j+1}B_j(m), \qquad q\text{ odd}. $$ with: $$ |\mathcal O_m| \le U_q(m). $$ The question this round actually seeks to answer is: > Why do low-order $U_q$ fail? Where exactly does their overcount occur? The answer can be written exactly: $$ \boxed{ U_q(m) - |\mathcal O_m| = \sum_{d\in\Omega_m} \binom{\mu_m(d)-1}{q}, } $$ with the convention: $$ \binom{a}{q}=0 \quad \text{when }aq$, and all remaining error is concentrated here. This is precisely the Center–Frontier Occupancy Closure of this round. --- # 1. Sidewise Root-Witness Sets For: $$ d\in\Omega_m, $$ Define: $$ \boxed{ \mathcal P_L(m,d) = \left\{ r\in\mathbb P: r^2\le m-d, \quad r\mid(m-d) \right\}, } $$ $$ \boxed{ \mathcal P_R(m,d) = \left\{ r\in\mathbb P: r^2\le m+d, \quad r\mid(m+d) \right\}. } $$ Because both endpoints are odd, All: $$ r\in\mathcal P_L\cup\mathcal P_R $$ all satisfy: $$ r\ge3. $$ --- # 2. Prime-Group Occupancy The occupancy of SRBRD-02 can be written as: $$ \boxed{ \mu_m(d) = \left| \mathcal P_L(m,d) \cup \mathcal P_R(m,d) \right|. } $$ This directly connects prime-group cell occupancy with endpoint factor structure. --- # 3. Shared Prime Witnesses If: $$ r\in \mathcal P_L(m,d) \cap \mathcal P_R(m,d), $$ then: $$ r\mid(m-d) $$ and: $$ r\mid(m+d). $$ Thus: $$ r\mid2m, $$ and: $$ r\mid2d. $$ Since: $$ r $$ odd， we obtain: $$ r\mid m, $$ $$ r\mid d. $$ Conversely, if: $$ r\mid m, \quad r\mid d, $$ then: $$ r\mid(m-d), $$ $$ r\mid(m+d). $$ it remains only to check the stronger left root bound: $$ r^2\le m-d. $$ The right root bound then holds automatically. --- # 4. Exact Shared-Witness Theorem ## Theorem SRBRD-03-T01 $$ \boxed{ \mathcal P_L(m,d) \cap \mathcal P_R(m,d) = \left\{ r\in\mathbb P: r\mid\gcd(m,d), \quad r^2\le m-d \right\}. } $$ Thus, shared-prime overlap is controlled entirely by: $$ \boxed{ \gcd(m,d) } $$ and the left root frontier. --- # 5. Exact Occupancy Decomposition Define: $$ \nu_L(m,d) = |\mathcal P_L(m,d)|, $$ $$ \nu_R(m,d) = |\mathcal P_R(m,d)|, $$ and: $$ \gamma_m(d) = \left| \mathcal P_L(m,d) \cap \mathcal P_R(m,d) \right|. $$ then: ## Theorem SRBRD-03-T02 $$ \boxed{ \mu_m(d) = \nu_L(m,d) + \nu_R(m,d) - \gamma_m(d). } $$ This is the pointwise exact form of same-prime center collapse. --- # 6. Center-Divisor Interpretation By T01: $$ \gamma_m(d)>0 $$ is possible only when: $$ \gcd(m,d)>1. $$ Thus: > The odd prime divisors of the center $m$ produce left/right shared occupancy only when the offset $d$ also lies on the corresponding divisor lattice. This promotes the SRBRD-01 statement: $$ r\mid m \Longrightarrow O_{L,r}\subseteq O_{R,r} $$ to a pointwise occupancy decomposition. --- # 7. Odd Primorial Sequence Let: $$ q_1=3, \quad q_2=5, \quad q_3=7, \ldots $$ be the odd primes. Define: $$ \boxed{ Q_t = \prod_{i=1}^{t}q_i, } $$ and: $$ Q_0=1. $$ --- # 8. Local Product Budget For: $$ d\in\Omega_m, $$ we have: $$ \boxed{ (m-d)(m+d) = m^2-d^2. } $$ If: $$ \mu_m(d)\ge t, $$ then there are at least: $$ t $$ distinct odd obstruction primes, whose product divides: $$ (m-d)(m+d). $$ Therefore: ## Theorem SRBRD-03-T03 $$ \boxed{ \mu_m(d)\ge t \Longrightarrow Q_t \le m^2-d^2. } $$ This is the local form of SRBRD-02's global occupancy bound. --- # 9. Local Occupancy Cap Define: $$ \boxed{ \kappa_m(d) = \max \left\{ t: Q_t\le m^2-d^2 \right\}. } $$ then: $$ \boxed{ \mu_m(d)\le\kappa_m(d). } $$ --- # 10. Frontier Monotonicity If: $$ 0\le d_1\le d_2, $$ then: $$ m^2-d_2^2 \le m^2-d_1^2. $$ Thus: ## Theorem SRBRD-03-T04 $$ \boxed{ \kappa_m(d_2) \le \kappa_m(d_1). } $$ That is: > The closer one is to the frontier, the maximum possible obstruction multiplicity decreases monotonically. --- # 11. Center Support Zones For: $$ t\ge1, $$ Define: $$ \boxed{ \mathcal C_t(m) = \left\{ d\in\Omega_m: d^2+Q_t\le m^2 \right\}. } $$ Call this: # $t$-Occupancy Center Zone By T03: ## Theorem SRBRD-03-T05 If: $$ \mu_m(d)\ge t, $$ then: $$ \boxed{ d\in\mathcal C_t(m). } $$ --- # 12. Nested Center Zones Since: $$ Q_{t+1}>Q_t, $$ we have: $$ \boxed{ \mathcal C_{t+1}(m) \subseteq \mathcal C_t(m). } $$ Thus, higher-order overlap is confined to increasingly smaller nested center regions. --- # 13. Frontier Zones Define: $$ \boxed{ \mathcal F_t(m) = \Omega_m \setminus \mathcal C_t(m). } $$ If: $$ d\in\mathcal F_t(m), $$ then: $$ Q_t>m^2-d^2. $$ By the contrapositive of T03: $$ \boxed{ \mu_m(d)\le t-1. } $$ --- # 14. Bonferroni Point Polynomial For: $$ q\ge1, $$ Define: $$ \boxed{ P_q(k) = \sum_{j=1}^{q} (-1)^{j+1} \binom{k}{j}. } $$ For a covered offset: $$ k\ge1. $$ Use the standard binomial identity: $$ \sum_{j=0}^{q} (-1)^j \binom{k}{j} = (-1)^q \binom{k-1}{q}. $$ --- # 15. Odd-Order Pointwise Exact Remainder If: $$ q $$ odd， then: ## Theorem SRBRD-03-T06 For: $$ k\ge1, $$ $$ \boxed{ P_q(k) = 1+ \binom{k-1}{q}. } $$ If: $$ k=0, $$ then: $$ P_q(0)=0. $$ Thus: $$ \boxed{ P_q(k) = \mathbf1_{\{k\ge1\}} + \binom{k-1}{q}. } $$ with the convention: $$ \binom{k-1}{q}=0 $$ when： $$ k\le q. $$ --- # 16. Global Exact Bonferroni Remainder For odd: $$ q, $$ SRBRD-02: $$ U_q(m) = \sum_{d\in\Omega_m} P_q(\mu_m(d)). $$ By T06: ## Theorem SRBRD-03-T07 $$ \boxed{ U_q(m) = |\mathcal O_m| + E_q(m), } $$ where: $$ \boxed{ E_q(m) = \sum_{d\in\Omega_m} \binom{\mu_m(d)-1}{q}. } $$ Therefore: $$ \boxed{ |\mathcal O_m| = U_q(m)-E_q(m). } $$ --- # 17. Exact Residual Formula with Center Correction Since: $$ |\operatorname{Res}(m)| = |\Omega_m|-|\mathcal O_m|, $$ Thus: ## Corollary SRBRD-03-C01 $$ \boxed{ |\operatorname{Res}(m)| = |\Omega_m| - U_q(m) + E_q(m). } $$ This is a more precise closure identity than the Bonferroni upper bound alone. --- # 18. Error Support Is Center-Only If: $$ \binom{\mu_m(d)-1}{q}>0, $$ then: $$ \mu_m(d)\ge q+1. $$ By T05: $$ d\in\mathcal C_{q+1}(m). $$ Thus: ## Theorem SRBRD-03-T08 $$ \boxed{ E_q(m) = \sum_{d\in\mathcal C_{q+1}(m)} \binom{\mu_m(d)-1}{q}. } $$ All Bonferroni overcount is completely absent from: $$ \boxed{ \mathcal F_{q+1}(m). } $$ --- # 19. Frontier Exactness Theorem If: $$ d\in\mathcal F_{q+1}(m), $$ then: $$ \mu_m(d)\le q. $$ Thus: $$ \binom{\mu_m(d)-1}{q}=0. $$ Therefore: ## Theorem SRBRD-03-T09 For odd: $$ q, $$ on the frontier: $$ \boxed{ P_q(\mu_m(d)) = \mathbf1_{\{\mu_m(d)\ge1\}}. } $$ That is: > Order-$q$ Bonferroni is not an approximation on the frontier, but is pointwise exact. This is the most important center–frontier theorem of this round. --- # 20. Interpretation Failure of low-order closure is not because the whole offset domain needs higher-order correction. The actual reason can only be: $$ \boxed{ \mathcal C_{q+1}(m) } $$ containing: $$ \mu\ge q+1 $$ high-occupancy states. Thus, higher-order research can focus entirely on the center zone. --- # 21. Global Occupancy Cap Define: $$ \boxed{ \kappa(m) = \max \left\{ t: Q_t\le m^2 \right\}. } $$ From: $$ m^2-d^2\le m^2, $$ we have: $$ \boxed{ \mu_m(d)\le\kappa(m) } $$ for all: $$ d. $$ --- # 22. Error to Next Even Moment For: $$ \mu\ge q+1, $$ we have: $$ \binom{\mu}{q+1} = \frac{\mu}{q+1} \binom{\mu-1}{q}. $$ Thus: $$ \boxed{ \binom{\mu-1}{q} = \frac{q+1}{\mu} \binom{\mu}{q+1}. } $$ Also: $$ \mu\le\kappa(m). $$ Therefore: $$ \boxed{ \binom{\mu-1}{q} \ge \frac{q+1}{\kappa(m)} \binom{\mu}{q+1}. } $$ --- # 23. Cap-Corrected Error Lower Bound After summation: ## Theorem SRBRD-03-T10 $$ \boxed{ E_q(m) \ge \frac{q+1}{\kappa(m)} B_{q+1}(m). } $$ Since the left side is an integer, this can be strengthened to: $$ \boxed{ E_q(m) \ge \left\lceil \frac{(q+1)B_{q+1}(m)} {\kappa(m)} \right\rceil. } $$ --- # 24. Cap-Corrected Union Upper Bound From: $$ |\mathcal O_m| = U_q-E_q, $$ we obtain: ## Theorem SRBRD-03-T11 $$ \boxed{ |\mathcal O_m| \le U_q(m) - \left\lceil \frac{(q+1)B_{q+1}(m)} {\kappa(m)} \right\rceil. } $$ This uses: - odd moments up to $q$; - the next even moment $B_{q+1}$; - a global structural occupancy cap. It does not require the next positive odd moment: $$ B_{q+2}. $$ --- # 25. Cap-Corrected Non-Cover Criterion ## Theorem SRBRD-03-T12 If: $$ \boxed{ U_q(m) - \left\lceil \frac{(q+1)B_{q+1}(m)} {\kappa(m)} \right\rceil < |\Omega_m|, } $$ then: $$ \boxed{ \operatorname{Res}(m)\neq\varnothing. } $$ Therefore: $$ \boxed{ R_G^{-}(2m)>0. } $$ This is the first new rigorous structural certificate of this round. --- # 26. Layered Local-Cap Correction global： $$ \kappa(m) $$ may be too loose. Thus, stratify by the local cap. Define: $$ \boxed{ Z_t(m) = \left\{ d\in\Omega_m: \kappa_m(d)=t \right\}. } $$ and the restricted even moment: $$ \boxed{ B_{q+1}^{[t]}(m) = \sum_{d\in Z_t(m)} \binom{\mu_m(d)}{q+1}. } $$ --- # 27. Layerwise Error Lower Bound In: $$ Z_t, $$ we have: $$ \mu\le t. $$ Thus: $$ \binom{\mu-1}{q} \ge \frac{q+1}{t} \binom{\mu}{q+1}. $$ Therefore: ## Theorem SRBRD-03-T13 $$ \boxed{ E_q(m) \ge \sum_{t\ge q+1} \left\lceil \frac{(q+1)B_{q+1}^{[t]}(m)} {t} \right\rceil. } $$ This is no weaker than the global cap correction. --- # 28. Layered Center–Frontier Closure Define: $$ \boxed{ C_q^{\mathrm{layer}}(m) = \sum_{t\ge q+1} \left\lceil \frac{(q+1)B_{q+1}^{[t]}(m)} {t} \right\rceil. } $$ then: $$ \boxed{ |\mathcal O_m| \le U_q(m)-C_q^{\mathrm{layer}}(m). } $$ If: $$ \boxed{ U_q(m)-C_q^{\mathrm{layer}}(m) < |\Omega_m|, } $$ then Goldbach success holds. --- # 29. Why This Is a Genuine Center–Frontier Architecture frontier： $$ \kappa_m(d)\le q $$ when this holds: $$ B_{q+1} $$ its contribution is automatically zero, and order $q$ is already exact. Thus: $$ C_q^{\mathrm{layer}} $$ needs to handle only the center layers: $$ t\ge q+1. $$ This is not a heuristic domain split, but is forced by the exact error identity. --- # 30. Center Correction Criterion By C01: $$ |\operatorname{Res}| = |\Omega|-U_q+E_q. $$ If: $$ U_q\ge|\Omega|, $$ this still does not mean the low-order route has failed. All that is actually required is: $$ \boxed{ E_q > U_q-|\Omega|. } $$ Therefore: > Whatever excess low-order Bonferroni has over the domain is precisely the amount of forced redundancy that must be proved from the center. This shifts the proof burden from the global union to: $$ \boxed{ \text{central high-multiplicity correction}. } $$ --- # 31. Computational Control: $m=100$ SRBRD-02： $$ |\Omega|=49, $$ $$ U_1=B_1=59, $$ $$ B_2=21. $$ global： $$ \kappa(100)=4 $$ Since: $$ Q_4=1155\le10000, $$ $$ Q_5=15015>10000. $$ cap correction： $$ \left\lceil \frac{2\cdot21}{4} \right\rceil = 11. $$ Thus: $$ |\mathcal O| \le 59-11 = 48 < 49. $$ Thus, using only: $$ \boxed{ B_1+B_2+\kappa } $$ already suffices to prove, for the fixed target: $$ 2m=200 $$ that it has a Goldbach representation. actual union： $$ 41. $$ This bound is not exact, but is sufficient for non-cover. --- # 32. Computational Control: $m=199$ We already have: $$ |\Omega|=99, $$ $$ U_3=94, $$ Thus, the original bound already directly gives: $$ 94<99. $$ Additionally: $$ B_4=2, $$ $$ \kappa(199)=5. $$ The cap correction further gives: $$ \left\lceil \frac{4\cdot2}{5} \right\rceil = 2. $$ Thus: $$ |\mathcal O| \le92. $$ This is exactly equal to the exact union. --- # 33. Computational Control: $m=1000$ SRBRD-02： $$ |\Omega|=499, $$ $$ U_3=511, $$ Thus, third-order Bonferroni fails by itself: $$ 511>499. $$ However: $$ B_4=53, $$ and: $$ \kappa(1000)=6, $$ Since: $$ Q_6=255255\le10^6, $$ $$ Q_7=4849845>10^6. $$ Thus, the cap correction: $$ \left\lceil \frac{4\cdot53}{6} \right\rceil = 36. $$ we obtain: $$ \boxed{ |\mathcal O| \le 511-36 = 475 < 499. } $$ Therefore: $$ \boxed{ q=3 + \text{one even moment }B_4 } $$ already suffices to structurally certify $N=2000$. There is no need to go all the way to: $$ U_5. $$ --- # 34. Computational Control: $m=5000$ $$ |\Omega|=2499, $$ $$ U_5=2390<2499. $$ The original fifth-order bound already succeeds. Additionally: $$ B_6=18, $$ $$ \kappa(5000)=7. $$ cap correction： $$ \left\lceil \frac{6\cdot18}{7} \right\rceil = 16. $$ gives: $$ |\mathcal O| \le2374. $$ exact union： $$ 2372. $$ --- # 35. Computational Control: $m=100000$ SRBRD-02： $$ |\Omega|=49999, $$ $$ U_5=53157, $$ Thus, fifth order fails by itself. But the finite audit gives: $$ B_6=5170. $$ and: $$ \kappa(100000)=9. $$ Thus: $$ \left\lceil \frac{6\cdot5170}{9} \right\rceil = 3447. $$ This yields: $$ \boxed{ |\mathcal O| \le 53157-3447 = 49710 < 49999. } $$ Therefore: $$ \boxed{ q=5+B_6+\kappa } $$ is already sufficient, without using the complete seventh-order positive term. This is an important finite control showing that this round's certificate genuinely increases proof compression. --- # 36. Structural Meaning of $B_{q+1}$ When: $$ q $$ odd， the next: $$ B_{q+1} $$ is even-order hyperintersection mass. It does not merely add more positive terms. What it supplies is: $$ \boxed{ \text{how much central redundancy must exist in the low-order Bonferroni overcount} } $$ as direct evidence. This understanding differs from merely increasing the odd order. --- # 37. Exact Remainder as Occupancy Excess T07 can be read as: $$ \boxed{ E_q = \sum_{\text{covered }d} \binom{\mu(d)-1}{q}. } $$ Thus: - $\mu\le q$: no error; - $\mu=q+1$: error per point: $$ 1; $$ - $\mu=q+2$: error per point: $$ q+1; $$ - the higher the occupancy, the faster the error grows. Thus, the high-overlap center is not a nuisance, but: $$ \boxed{ \text{an asset for subtracting the Bonferroni overestimate}. } $$ --- # 38. Improved Research Target The question in SRBRD-02 was: > How many orders of moments must be counted? After SRBRD-03, the question can become: > Can we obtain only a lower odd order $U_q$, and then prove that the central even moment $B_{q+1}$ is sufficiently large? That is: $$ \boxed{ B_{q+1} > \frac{\kappa(m)} {q+1} \left( U_q-|\Omega| \right) } $$ suffices for closure, ignoring the ceiling. --- # 39. Layered Version Is Better More precisely, we should not use only: $$ \kappa(m). $$ Different radii of the center have different: $$ \kappa_m(d). $$ Thus, the actual target is: $$ \boxed{ \sum_t \frac{q+1}{t} B_{q+1}^{[t]} > U_q-|\Omega|. } $$ This is the most natural inequality for Center–Frontier Occupancy Closure. --- # 40. Moment Support Radius From: $$ \mu(d)\ge q+1 $$ necessarily implies: $$ Q_{q+1}\le m^2-d^2. $$ Thus: $$ \boxed{ d^2+Q_{q+1}\le m^2. } $$ That is, higher-order correction is confined to a centered radius. Formalization should retain the squared form, and not use: $$ \sqrt{m^2-Q_{q+1}}. $$ --- # 41. Center Size Interface Define: $$ \boxed{ C_t^{\#}(m) = |\mathcal C_t(m)|. } $$ This quantity is determined entirely by: - $m$; - the odd primorial; - parity; as listed above. It does not read any endpoint primality / factorization truth. Thus, it can serve as: $$ \boxed{ \text{proof-admissible geometry input}. } $$ --- # 42. Center Size Alone Still Insufficient Knowing only: $$ |\mathcal C_{q+1}| $$ cannot lower-bound: $$ E_q. $$ because a center zone means only: > High occupancy can occur. It does not mean: > High occupancy must occur. Therefore: $$ \boxed{ \text{support localization} \neq \text{forced overlap}. } $$ Genuine closure still needs a lower bound on: $$ B_{q+1} $$ or the equivalent central multiplicity. --- # 43. Shared-Prime Term May Help Lower-Bound Even Moments T02： $$ \mu = \nu_L+\nu_R-\gamma. $$ where: $$ \gamma $$ is controlled by: $$ \gcd(m,d) $$ as displayed above. Future work can use: - center divisors; - the offset divisor lattice; - left/right factor counts; Study: $$ B_{q+1} $$ as the quantity for which to study a forced lower bound. This may be better than directly enumerating CRT hypercells. --- # 44. Center–Frontier Two-Zone Proof Pattern A future proof could take the form: ### Frontier Prove: $$ \mu\le q. $$ Thus, order-$q$ closure is exact. ### Center There is no need to solve the union completely. It suffices to prove: $$ E_q > U_q-|\Omega|. $$ or use: $$ B_{q+1} $$ with the occupancy cap to derive sufficient correction. This is a very clean two-zone proof architecture. --- # 45. New Canonical Closure Form This round suggests upgrading the SRBRD closure target to: $$ \boxed{ \mathcal C_{\mathrm{SRBRD}}(m,q) : \Longleftrightarrow U_q(m) - L_{E,q}(m) < |\Omega_m|, } $$ where: $$ L_{E,q}(m) \le E_q(m) $$ is any proof-admissible error lower bound. Whenever: $$ \mathcal C_{\mathrm{SRBRD}}(m,q) $$ holds, That is: $$ R_G^{-}(2m)>0. $$ --- # 46. Cap Provider The most basic provider: $$ \boxed{ L_{E,q}^{\mathrm{cap}}(m) = \left\lceil \frac{(q+1)B_{q+1}(m)} {\kappa(m)} \right\rceil. } $$ A stronger provider: $$ \boxed{ L_{E,q}^{\mathrm{layer}}(m) = \sum_t \left\lceil \frac{(q+1)B_{q+1}^{[t]}(m)} {t} \right\rceil. } $$ This establishes a pluggable center-correction provider. --- # 47. Data-Role Audit ### $\Omega_m$ ProofInput. ### $U_q$ Constructed from root-obstruction cells / hyperintersections. A proof-admissible structural quantity. ### $B_{q+1}$ Likewise structural hyperintersection mass. Not a prime-pair truth label. ### Exact $E_q$ Interconvertible with the exact union, and classified as a truth-adjacent diagnostic. ### Lower bound $L_{E,q}$ If derived from: - the occupancy cap; - CRT; - gcd; - cofactor constraints; then it is a legitimate ProofInput. --- # 48. Important Distinction Although a complete moment hierarchy can reconstruct the exact residual, an individual: $$ B_j $$ is not itself a Goldbach label. Formal research may use: $$ B_j $$ as a proof quantity, provided that it is not obtained by: > First labeling prime pairs / residuals, and then inferring the moment backward. The canonical implementation should calculate directly from relation edges / CRT hypercells. --- # 49. Formal Theorem Queue ```yaml SRBRD_03: T01: shared_root_primes_eq_center_offset_divisors T02: occupancy_left_plus_right_minus_shared T03: local_odd_primorial_product_bound T04: local_occupancy_cap_monotone_in_offset T05: high_occupancy_implies_center_zone T06: odd_bonferroni_pointwise_remainder_identity T07: global_exact_bonferroni_remainder T08: error_supported_only_in_center T09: frontier_pointwise_exactness T10: cap_corrected_error_lower_bound T11: cap_corrected_union_upper_bound T12: cap_corrected_noncover T13: layered_local_cap_error_lower_bound ``` --- # 50. Acceptance Gate Before SRBRD-03 is promoted to the formal core, the following are needed: 1. shared-witness set equality; 2. occupancy decomposition; 3. the local product bound; 4. local cap monotonicity; 5. nested center support; 6. the pointwise binomial remainder identity; 7. the global exact remainder; 8. frontier exactness; 9. the $E_q$ / $B_{q+1}$ ratio theorem; 10. integer ceiling correction; 11. the layered cap theorem; 12. finite controls independent of theorem proofs; 13. exact $E_q$ not treated as a leakage-free provider; 14. no claim of uniform closure. --- # 51. Verdict This round can formally give: $$ \boxed{ \mathrm{SRBRD\mbox{-}03\ CENTER\mbox{-}FRONTIER\ CLOSURE\ PASS} } $$ The reason is not that Goldbach has been proved, but that we obtain a new exact statement: $$ \boxed{ \text{All low-order Bonferroni error occurs only in the central high-occupancy zone.} } $$ and a new rigorous certificate: $$ \boxed{ U_q - \left\lceil \frac{(q+1)B_{q+1}} {\kappa(m)} \right\rceil < |\Omega| \Longrightarrow R_G^{-}(2m)>0. } $$ Compared with simply increasing to: $$ U_{q+2} $$ this supplies a different proof-compression route. However: $$ \boxed{ \mathrm{UNIFORM\ GOLDBACH\ CLOSURE\ STILL\ OPEN}. } $$ --- # 52. Suggestion for the Next Round # SRBRD-04 ## Forced Central Redundancy / Even-Moment Lower Bound The most worthwhile target is now very clear: > Instead of computing exact $B_{q+1}$, lower-bound it from center geometry itself. Priority sources: 1. the $\gcd(m,d)$ shared-prime term; 2. the center divisor lattice; 3. cofactor hyperclosure; 4. cell-mass pigeonhole; 5. CRT forced intersections; 6. center-zone incidence density; 7. even-moment convexity; 8. partition-by-$\kappa_m(d)$. The actual target: $$ \boxed{ B_{q+1}(m) \ge L_{q+1}^{\mathrm{struct}}(m) } $$ and: $$ \boxed{ U_q - \frac{q+1}{\kappa} L_{q+1}^{\mathrm{struct}} < |\Omega|. } $$ If this can hold for a uniform $q$ or a slowly growing $q(m)$, only then does this line genuinely start approaching a global non-cover theorem. --- # Appendix A: Canonical Center–Frontier Contract ```yaml occupancy: mu_m_d: distinct_prime_group_count: true local_cap: kappa_m_d: defined_by: odd_primorial <= m^2-d^2 monotone: decreases_with_d: true center: C_t: condition: d*d + Q_t <= m*m frontier: F_t: complement_of: C_t odd_bonferroni: U_q: q: odd error: E_q: exact: sum_choose_mu_minus_1_q: true support: C_q_plus_1: true ``` --- # Appendix B: Cap-Corrected Certificate ```yaml global_cap_provider: L_E: ceil((q+1)*B_(q+1)/kappa(m)) closure: if: U_q - L_E < card(Omega_m) then: residual_nonempty: true RGminus_positive: true ``` --- # Appendix C: Selected Numerical Audit | $m$ | $|\Omega|$ | $q$ | $U_q$ | $B_{q+1}$ | $\kappa(m)$ | Cap correction | Corrected upper | Exact union | |---:|---:|---:|---:|---:|---:|---:|---:|---:| | 100 | 49 | 1 | 59 | 21 | 4 | 11 | 48 | 41 | | 199 | 99 | 3 | 94 | 2 | 5 | 2 | 92 | 92 | | 1000 | 499 | 3 | 511 | 53 | 6 | 36 | 475 | 462 | | 5000 | 2499 | 5 | 2390 | 18 | 7 | 16 | 2374 | 2372 | | 100000 | 49999 | 5 | 53157 | 5170 | 9 | 3447 | 49710 | 48582 | **Note:** The exact union serves only the audit truth channel. --- # Appendix D: Commit Decision ```yaml research_round: SRBRD-03 decision: center_frontier_exact_decomposition: true bonferroni_error_localized: true cap_corrected_certificate: true uniform_goldbach_closure: false new_exact_structures: - shared_prime_gcd_decomposition - nested_center_support - pointwise_bonferroni_remainder - global_error_identity - frontier_exactness - cap_corrected_union_bound - layered_local_cap_correction next: SRBRD-04_Forced_Central_Redundancy_EvenMoment_LowerBound ```