> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # SRBRD-14 ## Aggregate Semiprime-Cell Coupling / Diagonal–Off-Diagonal Closure ### Symmetric Transport Matrix, Prime-Majority Criterion, Recursive Diagonal Lower Bound, and Parity-Correlation Identity **Parent program:** RG-SCP / Reverse Goldbach Structural Closure Program **Branch:** SRBRD — Symmetric Root-Bounded Relation Domain **Round:** SRBRD-14 **Date:** 2026-09-30 **Version:** v0.1 **Authors:** Neo.K; Aletheia (GPT-5.6 Sol) **Nature:** Exact Aggregate Coupling Kernel / Transport-Matrix Audit / Recursive Diagonal Closure **Prerequisites:** SRBRD-00 through SRBRD-13 **Status:** Establishes an aggregate exact identity for the semiprime boundary and a recursive diagonal lower bound; Goldbach remains unproved --- # 0. Question of This Round SRBRD-13 has established the cube-root RootRough boundary: $$ \mathcal S_z = PP \bigsqcup SP/P \bigsqcup P/SP \bigsqcup SP/SP. $$ and: $$ R_G^{-}(2m) = |\mathcal S_z| - |C_L| - |C_R| + |\mathcal S_{SS}|. $$ The next question is: > Can we avoid finding each $D_{r,s}$ individually and directly force $PP>0$ from aggregate semiprime coupling? The answer of this round has two parts. First: $$ \boxed{ \text{The entire P/SP boundary can be compressed into a symmetric transport matrix.} } $$ Second: $$ \boxed{ \text{With prime / semiprime marginals alone, the strongest result is the prime-majority criterion.} } $$ To go beyond it, one must add: - forced $SS$ overlap; - especially a same-factor recursive diagonal; - or genuine off-diagonal arithmetic coupling. --- # 1. Ordered Boundary Domain To remove the left/right asymmetry caused by reduced orientation, define the ordered odd-endpoint domain: $$ \boxed{ \widehat\Omega_m = \left\{ x\in\mathbb N: 3\le x\le2m-3, \quad x\text{ odd} \right\}. } $$ For: $$ x\in\widehat\Omega_m, $$ define the complement involution: $$ \boxed{ \tau_m(x) = 2m-x. } $$ Clearly: $$ \boxed{ \tau_m(\tau_m(x))=x. } $$ --- # 2. Ordered RootRough Survivor Set Take an integer: $$ z $$ satisfying: $$ z^3\ge2m-3. $$ Define: $$ \boxed{ \widehat{\mathcal S}_z(m) = \left\{ x\in\widehat\Omega_m: \operatorname{RootRough}_z(x), \quad \operatorname{RootRough}_z(2m-x) \right\}. } $$ By definition: $$ \boxed{ \tau_m : \widehat{\mathcal S}_z \to \widehat{\mathcal S}_z } $$ is an involution. --- # 3. P2 Boundary Label By SRBRD-13, In: $$ z^3\ge2m-3 $$ under this condition, each survivor endpoint can only be: - prime; - semiprime. Define the label: $$ \boxed{ \ell_z(x) = \begin{cases} 0, & x\in\mathbb P, \\ \operatorname{lpf}(x), & x\text{ semiprime}. \end{cases} } $$ Thus: $$ 0 $$ represents prime, and a positive prime label: $$ r>z $$ represents a semiprime's canonical least factor. --- # 4. Label Set Let: $$ \boxed{ \Lambda_z(m) = \{0\} \cup \left\{ r\in\mathbb P: r>z, \quad r^2\le2m-3 \right\}. } $$ All: $$ \ell_z(x) $$ all lie in: $$ \Lambda_z(m). $$ --- # 5. Symmetric Transport Matrix For: $$ a,b\in\Lambda_z(m), $$ Define: $$ \boxed{ M_{a,b}(m,z) = \# \left\{ x\in\widehat{\mathcal S}_z: \ell_z(x)=a, \quad \ell_z(\tau_m(x))=b \right\}. } $$ This is the: # Semiprime-Boundary Transport Matrix --- # 6. Matrix Symmetry ## Theorem SRBRD-14-T01 $$ \boxed{ M_{a,b}=M_{b,a}. } $$ proof： If: $$ x $$ contributes to: $$ M_{a,b}, $$ then: $$ \tau_m(x) $$ contributes to: $$ M_{b,a}. $$ and: $$ \tau_m $$ is a bijection. --- # 7. Row Marginals Define: $$ \boxed{ n_a = \sum_bM_{a,b}. } $$ Thus: $$ n_a $$ is the number of endpoints within: $$ \widehat{\mathcal S}_z $$ with label: $$ a $$ as displayed above. In particular: $$ \boxed{ n_P = n_0, } $$ $$ \boxed{ n_{SP} = \sum_{r>0}n_r. } $$ and: $$ \boxed{ |\widehat{\mathcal S}_z| = n_P+n_{SP}. } $$ --- # 8. Aggregate 2×2 Matrix Collapse all positive semiprime labels into: $$ SP. $$ Define: $$ \boxed{ M_{PP} = M_{0,0}, } $$ $$ \boxed{ M_{PS} = \sum_{s>0}M_{0,s}, } $$ $$ \boxed{ M_{SS} = \sum_{r,s>0}M_{r,s}. } $$ By symmetry: $$ M_{SP}=M_{PS}. $$ Thus, the ordered boundary matrix is: $$ \boxed{ \mathbb M = \begin{pmatrix} M_{PP} & M_{PS}\\ M_{PS} & M_{SS} \end{pmatrix}. } $$ --- # 9. Marginal Identities Directly from the row sums: $$ \boxed{ n_P = M_{PP}+M_{PS}, } $$ $$ \boxed{ n_{SP} = M_{PS}+M_{SS}. } $$ Subtracting: ## Theorem SRBRD-14-T02 $$ \boxed{ M_{PP} = n_P-n_{SP}+M_{SS}. } $$ This is the main aggregate identity of this round. --- # 10. Ordered Goldbach Count $$ M_{PP} $$ is the ordered prime-pair count: $$ \boxed{ M_{PP} = \# \left\{ p\in\mathbb P: 3\le p\le2m-3, \quad 2m-p\in\mathbb P \right\}. } $$ Thus: $$ \boxed{ M_{PP}>0 \Longleftrightarrow R_G^{-}(2m)>0. } $$ --- # 11. Reduced Count Relation If: $$ m $$ prime， center： $$ m+m=2m $$ appears only once in the ordered count. Every other unordered representation appears twice in the ordered count. Thus: ## Theorem SRBRD-14-T03 $$ \boxed{ R_G^{-}(2m) = \frac{ M_{PP} + \mathbf1_{\mathbb P}(m) }{2}. } $$ For: $$ m\ge3. $$ --- # 12. Prime-Majority Criterion By T02: $$ M_{SS}\ge0. $$ Therefore: ## Theorem SRBRD-14-T04 If: $$ \boxed{ n_P>n_{SP}, } $$ then: $$ \boxed{ M_{PP}>0. } $$ Thus: $$ \boxed{ R_G^{-}(2m)>0. } $$ Interpretation: > In the survivor population of the complement involution, > semiprime endpoints can absorb at most the same number of prime endpoints, one to one. > If prime endpoints are more numerous, the excess primes must be paired with primes. --- # 13. Prime-Majority Lower Bound More precisely: $$ \boxed{ M_{PP} \ge n_P-n_{SP}. } $$ Therefore: $$ \boxed{ R_G^{-}(2m) \ge \frac{ n_P-n_{SP} + \mathbf1_{\mathbb P}(m) }{2} } $$ When the right side is positive, an appropriate integer-lower-bound interpretation can also be taken. --- # 14. Marginal-Only Sharpness Consider knowing only: $$ n_P, \quad n_{SP}, $$ but not the arithmetic of pairing. The aggregate symmetric nonnegative matrix: $$ \begin{pmatrix} M_{PP}&M_{PS}\\ M_{PS}&M_{SS} \end{pmatrix} $$ need only satisfy: $$ M_{PP}+M_{PS}=n_P, $$ $$ M_{PS}+M_{SS}=n_{SP}. $$ --- # 15. Marginal Transport No-Go ## Theorem SRBRD-14-T05 In the real nonnegative transport relaxation, given: $$ n_P, n_{SP}, $$ the minimum possible: $$ M_{PP} $$ is: $$ \boxed{ \max \left( 0, n_P-n_{SP} \right). } $$ Thus: $$ \boxed{ \text{With marginals alone, one cannot in general obtain a lower bound stronger than prime majority.} } $$ --- # 16. T05 Construction If: $$ n_P\le n_{SP}, $$ Take: $$ M_{PP}=0, $$ $$ M_{PS}=n_P, $$ $$ M_{SS}=n_{SP}-n_P. $$ If: $$ n_P>n_{SP}, $$ Take: $$ M_{PS}=n_{SP}, $$ $$ M_{SS}=0, $$ $$ M_{PP}=n_P-n_{SP}. $$ Thus, the bound is sharp. --- # 17. Consequence Going beyond: $$ n_P>n_{SP}, $$ requires a: $$ \boxed{ M_{SS} } $$ positive lower bound on this quantity. This returns precisely to the SRBRD-13: $$ SP/SP $$ overlap. --- # 18. Semiprime Matrix Diagonal Define: $$ \boxed{ D_{\mathrm{diag}} = \sum_{r>0}M_{r,r}. } $$ Clearly: $$ \boxed{ M_{SS} \ge D_{\mathrm{diag}}. } $$ Thus, T02 immediately gives: ## Theorem SRBRD-14-T06 $$ \boxed{ M_{PP} \ge n_P-n_{SP}+D_{\mathrm{diag}}. } $$ --- # 19. Recursive Diagonal Closure Criterion ## Corollary SRBRD-14-C01 If: $$ \boxed{ n_P-n_{SP}+D_{\mathrm{diag}}>0, } $$ then: $$ \boxed{ R_G^{-}(2m)>0. } $$ This allows: > Even when semiprime endpoints are slightly more numerous, > sufficient forced same-factor semiprime pairing > can still force a prime-prime pair. --- # 20. Same-Factor Entry SRBRD-13 has proved: If: $$ M_{r,r}>0, $$ then: $$ r\mid m. $$ Since: $$ rp+rq=2m $$ and: $$ r $$ odd。 Thus: $$ \boxed{ D_{\mathrm{diag}} = \sum_{\substack{ r>z\\ r\mid m }} M_{r,r}. } $$ --- # 21. Ordered Smaller-Goldbach Count Define: $$ \boxed{ \widetilde R_{G,\ge r}(N) } $$ as the ordered prime-representation count: $$ \#\left\{ p\in\mathbb P: p\ge r, \quad N-p\in\mathbb P, \quad N-p\ge r \right\}. $$ There is no requirement that: $$ p\le N-p. $$ --- # 22. Diagonal Recursive Bijection ## Theorem SRBRD-14-T07 For: $$ r>z, $$ $$ r\mid m, $$ we have: $$ \boxed{ M_{r,r}(m,z) = \widetilde R_{G,\ge r} \left( \frac{2m}{r} \right). } $$ proof： $$ x=rp, $$ $$ 2m-x=rq $$ if and only if: $$ p+q=2m/r, $$ and: $$ p,q\ge r $$ ensures that: $$ r $$ is the canonical least factor at both endpoints. --- # 23. Recursive Diagonal Formula Therefore: ## Corollary SRBRD-14-C02 $$ \boxed{ D_{\mathrm{diag}} = \sum_{\substack{ r\in\mathbb P\\ r>z\\ r\mid m }} \widetilde R_{G,\ge r} \left( \frac{2m}{r} \right). } $$ --- # 24. Recursive Aggregate Closure Combining with T06: ## Theorem SRBRD-14-T08 $$ \boxed{ M_{PP} \ge n_P-n_{SP} + \sum_{\substack{ r>z\\ r\mid m }} \widetilde R_{G,\ge r} \left( \frac{2m}{r} \right). } $$ Thus, if the right side is positive, Goldbach success holds. --- # 25. Minimal-Counterexample Interface Suppose: $$ 2m_* $$ is the minimal counterexample. Then every admissible smaller even integer: $$ \frac{2m_*}{r} < 2m_* $$ has at least one Goldbach representation. But T08 requires: $$ \boxed{ p,q\ge r. } $$ Thus, minimality guarantees only: $$ \widetilde R_G(2m_*/r)>0, $$ It does not guarantee: $$ \widetilde R_{G,\ge r}(2m_*/r)>0. $$ This threshold remains the gap in diagonal recursion. --- # 26. Threshold-by-Representation-Count Lemma If: $$ N $$ has all its reduced Goldbach representations with smaller primes: $$ \pi(r-1)-1, $$ then: $$ \boxed{ \widetilde R_{G,\ge r}(N)>0. } $$ For: $$ N\ge6 $$ as the odd-prime-pair setting. --- # 27. Interpretation of T09 Thus, the diagonal term can be activated by representation abundance at smaller targets: $$ \boxed{ \text{smaller Goldbach count} > \text{small-prime slots} } $$ $$ \Longrightarrow $$ $$ \boxed{ \text{same-factor semiprime overlap at the larger target}. } $$ This is not a proof of abundance. But it establishes an exact recursive capacity interface. --- # 28. Off-Diagonal Semiprime Mass Define: $$ \boxed{ D_{\mathrm{cross}} = \sum_{\substack{ r,s>0\\ r\neq s }} M_{r,s}. } $$ then: $$ \boxed{ M_{SS} = D_{\mathrm{diag}} + D_{\mathrm{cross}}. } $$ By symmetry: $$ M_{r,s}=M_{s,r}. $$ Thus: $$ \boxed{ D_{\mathrm{cross}} = 2 \sum_{r0}^{\text{disjoint accounting}} L_{r,s}. } $$ provided that there is no double counting, we have: $$ \boxed{ M_{PP} \ge n_P-n_{SP}+L_{SS}. } $$ Thus, the diagonal is only the most natural first provider. --- # 31. Prime/Semiprime Sign On the cube-root P2 boundary, define: $$ \boxed{ \varepsilon_z(x) = \begin{cases} -1, & x\text{ prime}, \\ +1, & x\text{ semiprime}. \end{cases} } $$ Since: - prime: $$ \Omega_{\mathrm{mult}}=1; $$ - semiprime: $$ \Omega_{\mathrm{mult}}=2; $$ Thus: $$ \boxed{ \varepsilon_z(x) = \lambda(x) } $$ where: $$ \lambda $$ is the Liouville sign: $$ (-1)^{\Omega_{\mathrm{mult}}(x)}. $$ --- # 32. First Boundary Parity Moment Define: $$ \boxed{ A_z(m) = \sum_{x\in\widehat{\mathcal S}_z} \varepsilon_z(x). } $$ then: $$ \boxed{ A_z = n_{SP}-n_P. } $$ Thus, prime majority: $$ n_P>n_{SP} $$ is equivalent to: $$ \boxed{ A_z<0. } $$ --- # 33. Complement Parity Correlation Define: $$ \boxed{ C_z(m) = \sum_{x\in\widehat{\mathcal S}_z} \varepsilon_z(x) \varepsilon_z(2m-x). } $$ then: - PP contributes: $$ +1; $$ - SS contributes: $$ +1; $$ - mixed contributes: $$ -1. $$ --- # 34. Exact Parity-Correlation Identity indicator： $$ \mathbf1_{\{x\text{ prime}\}} = \frac{1-\varepsilon_z(x)}{2} $$ holds on the boundary. Thus: ## Theorem SRBRD-14-T11 $$ \boxed{ M_{PP} = \frac{ |\widehat{\mathcal S}_z| - 2A_z + C_z }{4}. } $$ --- # 35. Exact Correlation Closure Criterion By T11: $$ \boxed{ M_{PP}>0 } $$ if and only if: $$ \boxed{ C_z > 2A_z - |\widehat{\mathcal S}_z|. } $$ Since: $$ M_{PP} $$ is a nonnegative integer, failure gives precisely the equality: $$ \boxed{ C_z = 2A_z - |\widehat{\mathcal S}_z|. } $$ --- # 36. Why This Is the Parity Barrier in One Equation SRBRD-13 found: fixed semiprime cells re-create two-affine-prime problems. SRBRD-14 now compresses the aggregate version into: $$ \boxed{ A_z } $$ and: $$ \boxed{ C_z. } $$ That is: - one-point Liouville imbalance; - complement Liouville correlation. Therefore: $$ \boxed{ \text{If a uniform method does not control parity sign / correlation at all, it cannot force, from this identity, }M_{PP}>0. } $$ This is the exact parity audit of the semiprime boundary. --- # 37. Marginal No-Go Reinterpreted Knowing only: $$ n_P, n_{SP} $$ is equivalent to knowing only: $$ A_z. $$ But if: $$ A_z\ge0, $$ prime majority disappears. One then also needs: $$ C_z $$ or, equivalently: $$ M_{SS} $$ as lower-bound information. Thus: $$ \boxed{ \text{When one-point parity information is insufficient, two-point coupling information must be obtained.} } $$ --- # 38. Diagonal Recursion as Correlation Source Same-factor entries: $$ M_{r,r} $$ all belong to: $$ SS, $$ and therefore supply, for: $$ C_z $$ positive same-sign states. And: $$ M_{r,r} $$ can also be represented by smaller Goldbach targets. Thus, diagonal recursion is: $$ \boxed{ \text{smaller-target additive structure} \to \text{larger-target positive parity coupling} } $$ as an exact bridge. --- # 39. Finite Audit: $m=100$ Take: $$ z=6. $$ Ordered survivors: $$ |\widehat{\mathcal S}_z|=28. $$ Endpoint marginals: $$ n_P=22, $$ $$ n_{SP}=6. $$ Matrix aggregate: $$ M_{PP}=16, $$ $$ M_{SS}=0. $$ Thus: $$ \boxed{ 16=22-6+0. } $$ Prime majority alone already achieves closure. --- # 40. Finite Audit: $m=199$ $$ z=8. $$ Ordered survivors: $$ 29. $$ $$ n_P=21, $$ $$ n_{SP}=8. $$ $$ M_{PP}=13, $$ $$ M_{SS}=0. $$ Thus: $$ \boxed{ 13=21-8. } $$ The center: $$ m=199 $$ prime， Therefore: $$ R_G^{-}(398) = (13+1)/2 = 7. $$ --- # 41. Finite Audit: $m=1000$ $$ z=13. $$ $$ |\widehat{\mathcal S}_z|=136. $$ $$ n_P=101, $$ $$ n_{SP}=35. $$ $$ M_{SS}=8, $$ where: $$ D_{\mathrm{diag}}=0. $$ Thus: $$ \boxed{ M_{PP} = 101-35+8 = 74. } $$ Therefore: $$ R_G^{-}(2000)=37. $$ The prime-majority lower bound itself gives: $$ M_{PP}\ge66. $$ Cross-semiprime overlap adds: $$ 8. $$ --- # 42. Finite Audit: $m=5000$ $$ z=22. $$ $$ |\widehat{\mathcal S}_z|=524. $$ $$ n_P=372, $$ $$ n_{SP}=152. $$ $$ M_{SS}=34, $$ $$ D_{\mathrm{diag}}=0. $$ Thus: $$ \boxed{ M_{PP} = 372-152+34 = 254. } $$ Therefore: $$ R_G^{-}(10000)=127. $$ --- # 43. Finite Audit: Diagonal-Rich $m=105$ $$ z=6. $$ $$ |\widehat{\mathcal S}_z|=54. $$ $$ n_P=43, $$ $$ n_{SP}=11. $$ $$ M_{SS}=6, $$ and: $$ \boxed{ D_{\mathrm{diag}}=6. } $$ All of it comes from: $$ r=7. $$ Thus: $$ \boxed{ M_{PP} = 43-11+6 = 38. } $$ reduced： $$ R_G^{-}(210)=19. $$ diagonal： $$ M_{7,7}=6 $$ is exactly the ordered Goldbach count for the smaller target: $$ 30 $$ with: $$ p,q\ge7 $$ as the threshold on both primes. --- # 44. Finite Scan: $3\le m\le5000$ Computational audit: For each: $$ 3\le m\le5000 $$ take the least integer: $$ z $$ satisfying: $$ z^3\ge2m-3. $$ Result: $$ \boxed{ n_P>n_{SP} } $$ In: $$ 4998/4998 $$ of the cases, holding in all of them. No prime-majority failure was found. --- # 45. Scan Margin In: $$ 3\le m\le5000 $$ within this range: - the minimum absolute margin: $$ n_P-n_{SP}=1 $$ occurs in a very small boundary case; - for: $$ n_{SP}>0 $$ as the set of cases, the minimum observed ratio: $$ \boxed{ n_P/n_{SP}=1.8 } $$ occurs at: $$ m=496. $$ This is only computational evidence. One cannot claim: $$ \boxed{ n_P>n_{SP} \text{ universally}. } $$ --- # 46. Larger Selected Controls Selected finite audits: ### $m=10000$ $$ n_P=663, $$ $$ n_{SP}=291. $$ ### $m=20000$ $$ n_P=1146, $$ $$ n_{SP}=516. $$ ### $m=50000$ $$ n_P=2379, $$ $$ n_{SP}=1053. $$ ### $m=100000$ $$ n_P=4258, $$ $$ n_{SP}=2100. $$ Prime majority still holds. This, too, is only evidence. --- # 47. Why the Majority Pattern Is Interesting Cube-root RootRough composite survivors must be: $$ rp, $$ where: $$ zn_{SP} $$ is not entirely unexpected. But the paired condition: $$ 2m-x \text{ also RootRough} $$ means that this is not simply a comparison of prime and semiprime counts over the whole interval. --- # 48. Why Majority Is Still a Parity-Sensitive Target $$ n_P-n_{SP} = -A_z. $$ And: $$ A_z $$ is the Liouville sign sum over the RootRough pair population. Thus, proving: $$ \boxed{ n_P>n_{SP} } $$ is equivalent to proving: $$ \boxed{ A_z<0. } $$ This is not an ordinary unsigned sieve-density statement. It directly requires: $$ \boxed{ \text{parity imbalance}. } $$ Thus, however strong the finite evidence, it cannot be promoted directly to a theorem. --- # 49. Aggregate Closure Hierarchy Semiprime-boundary closure can now be divided into: ### Level 1 — Prime Majority $$ n_P>n_{SP}. $$ ### Level 2 — Recursive Diagonal Rescue $$ n_P-n_{SP} + D_{\mathrm{diag}} >0. $$ ### Level 3 — Selected Cross-Cell Rescue $$ n_P-n_{SP} + D_{\mathrm{diag}} + L_{\mathrm{cross}} >0. $$ ### Level 4 — Exact $$ M_{PP} = n_P-n_{SP} + M_{SS}. $$ --- # 50. Transport-Polytope Interpretation matrix： $$ M_{a,b} $$ is a: - symmetric; - nonnegative; - integer; transport object, row sums： $$ n_a $$ are fixed. Arithmetic determines which: $$ (a,b) $$ cells are: - possible; - impossible; - forced; - recursively lower-bounded. Thus, SRBRD-14 converts the problem into: $$ \boxed{ \text{symmetric arithmetic transport polytope}. } $$ The Goldbach cell is: $$ \boxed{ M_{0,0}. } $$ --- # 51. Fine-Label Capacity For a semiprime label: $$ r, $$ the row: $$ n_r $$ is decomposed as: $$ \boxed{ n_r = M_{r,0} + M_{r,r} + \sum_{s\neq r,\ s>0}M_{r,s}. } $$ If: $$ M_{r,r} $$ or selected cross terms are forced to be positive, this can directly reduce the maximum capacity of row: $$ r $$ for pairing with the prime label: $$ 0 $$ displayed above. This is a transport mechanism finer than majority. --- # 52. General Row-Capacity Closure If, for each semiprime label: $$ r $$ one can prove the internal semiprime-coupling lower bound: $$ \boxed{ I_r \le M_{r,r} + \sum_{s>0,s\neq r}M_{r,s}, } $$ then: $$ M_{r,0} \le n_r-I_r. $$ Summation: $$ M_{PS} \le n_{SP} - \sum_r I_r $$ provided that the accounting is not duplicated. Therefore: $$ \boxed{ M_{PP} = n_P-M_{PS} \ge n_P-n_{SP} +\sum_rI_r. } $$ This is general transport-capacity closure. --- # 53. Double-Counting Warning If a cross cell: $$ M_{r,s} $$ is counted in both: $$ I_r $$ and: $$ I_s $$ as displayed above, it will be double-counted. Thus, a rigorous provider must: - orient cross cells; - or use half-weights; - or directly construct a matching / flow certificate. This is the graph layer that SRBRD-15 can handle. --- # 54. Parity-Correlation vs Transport The two languages are exactly equivalent: ## Transport $$ M_{PP} = n_P-n_{SP}+M_{SS}. $$ ## Parity $$ M_{PP} = \frac{ |\widehat{\mathcal S}_z| - 2A_z + C_z }{4}. $$ The transport language suits: - row capacities; - diagonal recursion; - cell graphs. The parity language suits: - barrier comparisons; - correlation estimates; - analytic number-theory interfaces. --- # 55. Main No-Go of This Round This round has found no: $$ \boxed{ \text{purely combinatorial marginal theorem} } $$ that can, when: $$ n_P\le n_{SP} $$ holds, automatically force: $$ M_{PP}>0. $$ T05 proves this impossible. Thus, genuinely new information must come from: $$ \boxed{ \text{arithmetic forced semiprime–semiprime coupling}. } $$ --- # 56. Formal Theorem Queue ```yaml SRBRD_14: T01: transport_matrix_symmetric T02: M_PP_eq_nP_minus_nSP_plus_MSS T03: reduced_Goldbach_from_ordered_PP T04: prime_majority_implies_Goldbach T05: marginal_only_transport_bound_is_sharp T06: diagonal_recursive_lower_bound T07: diagonal_entry_eq_smaller_ordered_Goldbach_count T08: aggregate_recursive_diagonal_closure T09: smaller_representation_abundance_implies_thresholded_diagonal T10: diagonal_parity_center_rule T11: exact_boundary_parity_correlation_identity ``` --- # 57. Acceptance Gate Before SRBRD-14 enters the formal core, the following are needed: 1. the ordered boundary domain; 2. the complement involution; 3. the canonical P/SP label; 4. matrix symmetry; 5. row-marginal identities; 6. the ordered/reduced Goldbach bridge; 7. the prime-majority theorem; 8. the marginal-sharpness no-go; 9. same-factor diagonal recursion; 10. the thresholded smaller-Goldbach interface; 11. cross-cell parity; 12. the Liouville sign identity; 13. the finite scan separated from the theorem; 14. no universal prime-majority claim; 15. no Goldbach proof claim. --- # 58. Verdict This round can formally give: $$ \boxed{ \mathrm{SRBRD\mbox{-}14\ AGGREGATE\ TRANSPORT\ KERNEL\ PASS}. } $$ Core exact identity: $$ \boxed{ M_{PP} = n_P-n_{SP}+M_{SS}. } $$ The first very concise closure: $$ \boxed{ n_P>n_{SP} \Longrightarrow \text{Goldbach success}. } $$ A second, stronger recursive closure: $$ \boxed{ n_P-n_{SP} + \sum_{\substack{ r>z\\ r\mid m }} \widetilde R_{G,\ge r} \left( \frac{2m}{r} \right) >0 } $$ $$ \Longrightarrow $$ $$ \boxed{ \text{Goldbach success}. } $$ However: $$ \boxed{ \mathrm{UNIFORM\ PRIME\mbox{-}MAJORITY\ NOT\ PROVED}. } $$ Its sign form: $$ A_z<0 $$ already clearly exposes parity sensitivity. Thus: $$ \boxed{ \mathrm{GOLDBACH\ STILL\ OPEN}. } $$ --- # 59. Suggestion for the Next Round # SRBRD-15 ## Boundary Parity-Imbalance / Semiprime Transport Capacity Closure Two routes are worth attacking simultaneously: ### Route A — Prime-Majority Study: $$ \boxed{ A_z = n_{SP}-n_P } $$ Can one prove uniformly that: $$ A_z<0. $$ Finite evidence is currently very strong, but this route directly encounters the parity sign. ### Route B — Transport Rescue Even if: $$ A_z\ge0, $$ investigate whether the fine semiprime labels: $$ r,s $$ have forced internal coupling between them, such that: $$ M_{SS}>A_z. $$ Since: $$ \boxed{ M_{PP} = M_{SS}-A_z. } $$ The most worthwhile first tasks: 1. the row-capacity graph; 2. same-factor recursive diagonal; 3. off-diagonal cell-support restrictions; 4. a matching / flow lower bound; 5. selected cross-cell lower certificates; 6. turn cell coupling into a transport lower bound without double counting. If this layer succeeds, one may be able, without directly proving: $$ A_z<0 $$ as displayed above, still to use: $$ \boxed{ M_{SS}>A_z } $$ to achieve parity rescue. --- # Appendix A: Ordered Transport Contract ```yaml ordered_survivor: x: odd: true 3 <= x <= 2m-3 complement: 2m-x both_RootRough: true label: 0: prime r_positive: semiprime_lpf matrix: M_a_b: count: label(x)=a label(2m-x)=b symmetry: M_a_b = M_b_a ``` --- # Appendix B: Aggregate Identity ```yaml marginals: nP: M_PP + M_PS nSP: M_PS + M_SS identity: M_PP: nP - nSP + M_SS majority: if: nP > nSP then: M_PP > 0 ``` --- # Appendix C: Recursive Diagonal ```yaml D_diag: sum: M_r_r support: only_if: r divides m r > z identity: M_r_r: ordered_Goldbach_count: target: 2m/r both_primes_ge: r lower: M_PP >= nP - nSP + D_diag ``` --- # Appendix D: Parity Correlation ```yaml epsilon: prime: -1 semiprime: +1 A_z: sum_epsilon: nSP - nP C_z: sum: epsilon(x)*epsilon(2m-x) identity: M_PP: (card(S_hat)-2*A_z+C_z)/4 failure_equality: C_z: 2*A_z-card(S_hat) ``` --- # Appendix E: Selected Audit | $m$ | $z$ | $|\widehat{\mathcal S}_z|$ | $n_P$ | $n_{SP}$ | $M_{PP}$ | $M_{SS}$ | $D_{\mathrm{diag}}$ | |---:|---:|---:|---:|---:|---:|---:|---:| | 100 | 6 | 28 | 22 | 6 | 16 | 0 | 0 | | 199 | 8 | 29 | 21 | 8 | 13 | 0 | 0 | | 1000 | 13 | 136 | 101 | 35 | 74 | 8 | 0 | | 5000 | 22 | 524 | 372 | 152 | 254 | 34 | 0 | | 105 | 6 | 54 | 43 | 11 | 38 | 6 | 6 | --- # Appendix F: Prime-Majority Scan ```yaml range: m: min: 3 max: 5000 cutoff: z: least_integer_with: z^3 >= 2m-3 result: nP_greater_than_nSP: passed: 4998 total: 4998 minimum_observed_ratio_with_nSP_positive: nP_over_nSP: 1.8 m: 496 status: computational_evidence_only: true ``` --- # Appendix G: Commit Decision ```yaml research_round: SRBRD-14 decision: symmetric_transport_matrix: true prime_majority_closure: true marginal_only_no_go: true recursive_diagonal_provider: true parity_correlation_identity: true universal_prime_majority: false goldbach_proved: false next: SRBRD-15_BoundaryParityImbalance_SemiprimeTransportCapacity_Closure ```