> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # SRBRD-16 ## Buchstab-Row Mass / Upper-Bound Mixed-Flow Closure ### Pair-Buchstab Identity, Cube-Root Semiprime Rows, Dimension-2 Row Sieve, Sifting-Depth Tradeoff and Modular Analytic Closure **Parent program:** RG-SCP / Reverse Goldbach Structural Closure Program **Branch:** SRBRD — Symmetric Root-Bounded Relation Domain **Round:** SRBRD-16 **Date:** 2026-09-30 **Version:** v0.1 **Authors:** Neo.K; Aletheia (GPT-5.6 Sol) **Type:** Exact Buchstab Decomposition / Row-Sieve Kernel / Analytic Interface Audit **Prerequisites:** SRBRD-00 through SRBRD-15 **Status:** Rough-pair mass, Buchstab loss, semiprime rows and transport correction at the P2 boundary have been assembled into a single exact identity; uniform closure remains incomplete --- # 0. The Main Question of This Round SRBRD-15 has obtained: $$ \boxed{ M_{PP} = |\widehat{\mathcal S}_{z_3}| - 2\sum_{r>z_3}n_r + M_{SS}, } $$ where: $$ z_3 = \min\left\{ z\in\mathbb N: z^3\ge2m-3 \right\}. $$ This identity has decomposed the ordered Goldbach count into: 1. Cube-root RootRough pair mass; 2. Semiprime least-factor row mass; 3. Semiprime internal transport correction. But a uniform proof still requires: $$ \boxed{ \text{lower-bound rough-pair mass} } $$ and: $$ \boxed{ \text{upper-bound semiprime-row mass}. } $$ SRBRD-16 further proves: > Both components can be written as standard sieve / Buchstab-type finite arithmetic objects. It also obtains a new exact master identity: $$ \boxed{ M_{PP} = |\widehat{\mathcal S}_y| - \sum_{\substack{ r\in\mathbb P\\ yz_3 }} n_r + M_{SS}, } $$ for any: $$ 2\le y Having no finite nontrivial root divisor. --- # 2. RootRough as Least-Factor Threshold From SRBRD-13: ## Theorem SRBRD-16-T01 $$ \boxed{ \operatorname{RootRough}_z(x) \Longleftrightarrow \ell(x)>z. } $$ for all: $$ x\ge2. $$ This formulation automatically and correctly retains primes with: $$ x\le z $$ as displayed above. --- # 3. Pair Least-Factor Height For an ordered endpoint: $$ x\in\widehat\Omega_m, $$ Let: $$ y_x=2m-x. $$ Define: $$ \boxed{ \rho_m(x) = \min \left( \ell(x), \ell(y_x) \right). } $$ If both endpoints are prime: $$ \rho_m(x)=+\infty. $$ --- # 4. Pair RootRough Characterization ## Theorem SRBRD-16-T02 $$ \boxed{ x\in\widehat{\mathcal S}_z(m) \Longleftrightarrow \rho_m(x)>z. } $$ Thus the survivor sets at distinct cutoffs satisfy: $$ \boxed{ z_1 The smallest prime factor first encountered among composite left/right endpoints lies in: > $$ > (y,z]. > $$ Each state is uniquely assigned by: $$ \rho_m(x) $$ to a shell. Thus there is no inclusion–exclusion multiplicity. --- # 8. Left / Right / Tie Shell Split One can further define: $$ \mathcal B_r^L = \{ x: \ell(x)=r<\ell(2m-x) \}, $$ $$ \mathcal B_r^R = \{ x: \ell(2m-x)=r<\ell(x) \}, $$ $$ \mathcal B_r^T = \{ x: \ell(x)=\ell(2m-x)=r \}. $$ then: ## Theorem SRBRD-16-T04 $$ \boxed{ \mathcal B_r^{pair} = \mathcal B_r^L \bigsqcup \mathcal B_r^R \bigsqcup \mathcal B_r^T. } $$ And: $$ \boxed{ \mathcal B_r^T\neq\varnothing \Longrightarrow r\mid m. } $$ This agrees with the earlier center-divisor shared-factor collapse. --- # 9. Cube-Root Cutoff Let: $$ X_m=2m-3, $$ and: $$ \boxed{ z_3=z_3(m) = \min \left\{ z: z^3\ge X_m \right\}. } $$ SRBRD-13： $$ \widehat{\mathcal S}_{z_3} $$ endpoints in this set can only be: - prime； - semiprime。 --- # 10. Semiprime Row Recalled For: $$ r>z_3 $$ prime, SRBRD-15： $$ \boxed{ \mathcal R_r(m) = \left\{ x\in\widehat{\mathcal S}_{z_3}: r\mid x, \quad x\ge r^2 \right\}. } $$ and: $$ \mathcal R_r $$ is exactly the set of: > Semiprime endpoints whose least prime factor is $r$. Let: $$ \boxed{ n_r=|\mathcal R_r|. } $$ --- # 11. Cofactor Coordinate for a Row If: $$ x\in\mathcal R_r, $$ Write: $$ \boxed{ x=ru. } $$ Since: $$ x\ge r^2, $$ we have: $$ u\ge r. $$ Also: $$ x\le X_m, $$ Thus: $$ \boxed{ u\le \left\lfloor \frac{X_m}{r} \right\rfloor. } $$ --- # 12. Cofactor Primality Is Forced ## Theorem SRBRD-16-T05 If: $$ r>z_3, $$ $$ r\le u, $$ $$ ru\le X_m, $$ and: $$ \operatorname{RootRough}_{z_3}(ru), $$ then: $$ \boxed{ u\in\mathbb P. } $$ --- # 13. T05 Proof Suppose: $$ u $$ is composite. From: $$ ru $$ being RootRough$_{z_3}$, every prime divisor: $$ p\mid u $$ must satisfy: $$ p>z_3. $$ Thus: $$ u>z_3^2. $$ Also: $$ r>z_3, $$ Therefore: $$ ru>z_3^3. $$ However: $$ ru\le X_m\le z_3^3. $$ a contradiction. Thus: $$ u $$ is prime. --- # 14. Exact Row Cofactor Formula From T05: ## Theorem SRBRD-16-T06 $$ \boxed{ n_r = \# \left\{ u: r\le u\le \left\lfloor \frac{X_m}{r} \right\rfloor, \quad \operatorname{RootRough}_{z_3}(ru), \quad \operatorname{RootRough}_{z_3}(2m-ru) \right\}. } $$ This formula need not include: $$ u\in\mathbb P $$ in the definition. It follows automatically from RootRough + the cube-root bound. --- # 15. One-Sided Semiprime Upper Envelope If the complement RootRough condition is discarded, then: $$ u $$ is still forced to be prime by T05. Thus: ## Theorem SRBRD-16-T07 $$ \boxed{ n_r \le A_r^{one}(m) } $$ where: $$ \boxed{ A_r^{one}(m) = \pi \left( \left\lfloor \frac{X_m}{r} \right\rfloor \right) - \pi(r-1). } $$ Therefore: $$ \boxed{ n_{SP} = \sum_{r>z_3}n_r \le U_{SP}^{one} = \sum_{\substack{ r\in\mathbb P\\ z_3 The total number of odd semiprimes in the interval: > $$ > [3,X_m] > $$ > whose least prime factor exceeds $z_3$. It completely ignores the requirement that: $$ 2m-x $$ must also be RootRough$_{z_3}$. Thus it is a valid but usually very loose upper envelope. --- # 17. Finite Audit: One-Sided Envelope selected controls： | $m$ | $z_3$ | exact $n_{SP}$ | $U_{SP}^{one}$ | |---:|---:|---:|---:| | 100 | 6 | 6 | 10 | | 199 | 8 | 8 | 16 | | 496 | 10 | 25 | 63 | | 1000 | 13 | 35 | 86 | | 5000 | 22 | 152 | 489 | | 10000 | 28 | 291 | 1022 | | 100000 | 59 | 2100 | 8856 | Thus: $$ \boxed{ \text{Complement RootRough filtering is a central structure that a row upper bound cannot ignore.} } $$ --- # 18. Row Bulk / Frontier Split For fixed: $$ r>z_3, $$ Let: $$ x=ru, $$ $$ y=2m-ru. $$ Define the row bulk: $$ \boxed{ \mathcal U_r^{bulk} = \left\{ u: r\le u, \quad ru\le2m-z_3^2 \right\}. } $$ This is equivalent to: $$ \boxed{ y\ge z_3^2. } $$ The row frontier is: $$ \boxed{ \mathcal U_r^{front} = \left\{ u: 2m-z_3^2z_3^2\ge p^2, $$ and: $$ 2m-ru\ge z_3^2\ge p^2, $$ the root conditions for: $$ p $$ are entirely automatic on both sides. --- # 20. Exact Row-Bulk Residue Sieve ## Theorem SRBRD-16-T08 For: $$ u\in\mathcal U_r^{bulk}, $$ we have: $$ \boxed{ ru \text{ RootRough}_{z_3} } $$ if and only if, for every odd prime: $$ p\le z_3, $$ $$ \boxed{ u\not\equiv0\pmod p. } $$ At the same time: $$ \boxed{ 2m-ru \text{ RootRough}_{z_3} } $$ if and only if: $$ \boxed{ u \not\equiv 2m\,r^{-1} \pmod p. } $$ Since: $$ r>z_3, $$ Thus: $$ r^{-1}\pmod p $$ always exists. --- # 21. Local Sieve Dimension For an odd prime: $$ p\le z_3, $$ the number of residue classes excluded by the row bulk is: $$ \boxed{ \nu_m(p) = \begin{cases} 1, & p\mid m, \\ 2, & p\nmid m. \end{cases} } $$ For if: $$ p\mid m, $$ then: $$ 2m r^{-1}\equiv0\pmod p. $$ the two forbidden residues collapse. --- # 22. Row Sieve Dimension Theorem ## Theorem SRBRD-16-T09 Every semiprime row bulk: $$ \mathcal R_r^{bulk} $$ has the same local-dimension profile: $$ \boxed{ \nu_m(p)\in\{1,2\}. } $$ The residue locations vary with: $$ r $$ as displayed above, but, for each: $$ p $$ the forbidden-class cardinality is determined only by: $$ p\mid m $$ as displayed above. Thus, at generic: $$ p\nmid m $$ positions, this is a: # Dimension-2 Sieve --- # 23. Row Frontier Size The row frontier corresponds to: $$ 3\le yz_3, $$ Define any rigorous provider: $$ \boxed{ U_r^{bulk} \ge |\mathcal R_r^{bulk}|. } $$ From T10: $$ \boxed{ n_r \le U_r^{bulk} + F_r^{front}, } $$ where: $$ F_r^{front} = 1+ \left\lfloor \frac{ z_3^2-3 }{ 2r } \right\rfloor. $$ Thus: ## Theorem SRBRD-16-T11 $$ \boxed{ n_{SP} \le \sum_{r>z_3} U_r^{bulk} + F_{SP}^{front}. } $$ --- # 27. Why Upper Sieve Is the Natural Row Tool The row bulk requires only: > An upper bound on integers avoiding a set of local residue classes. This is the direction in which classical upper-bound sieves excel. More importantly: $$ \boxed{ \text{We do not require the row sieve to prove that two primes exist simultaneously.} } $$ It requires only: $$ \boxed{ \text{upper-bound semiprime contamination capacity}. } $$ This differs from a direct binary-Goldbach lower bound. --- # 28. Pair Survivor Lower Bound Is Harder On the other hand, a lower bound on: $$ |\widehat{\mathcal S}_{z_3}| $$ requires a dimension-2 lower sieve. The cube-root cutoff: $$ z_3 \asymp X_m^{1/3} $$ lies precisely in the difficult region for classical lower sieves. --- # 29. Sieve Parameter Classical beta / DHR sieves usually use: $$ \boxed{ s = \frac{\log D}{\log z}, } $$ where: - $z$: Sifting cutoff; - $D$: Available sieve level. For an interval-like sequence, the natural distribution level is at most approximately the scale: $$ D=X_m^{1-o(1)} $$ as displayed above. Thus, at the cutoff: $$ z=X_m^{1/u} $$ we have: $$ \boxed{ s\approx u. } $$ --- # 30. External Dimension-2 Sifting Limit The published sifting limit of the Diamond–Halberstam–Richert dimension-2 sieve is approximately: $$ \boxed{ \beta_2\approx4.266. } $$ That is, for the standard lower-bound sieve to enter the positive lower-main-term regime, it requires roughly: $$ \boxed{ s>\beta_2. } $$ This is an external sieve benchmark, not an SRBRD theorem. --- # 31. Cube-Root Boundary vs DHR Limit P2 boundary： $$ z_3\asymp X_m^{1/3}. $$ If: $$ D\asymp X_m, $$ then: $$ \boxed{ s\approx3. } $$ However: $$ 3<4.266. $$ Thus: ## External Barrier Observation At the natural level, the standard dimension-2 DHR lower sieve cannot directly provide its usual positive lower main term for the cube-root P2 boundary. --- # 32. This Is Not an Impossibility Theorem This does not mean: $$ \boxed{ |\widehat{\mathcal S}_{z_3}| } $$ cannot be lower-bounded by other methods. Nor does it mean: $$ \boxed{ \text{SRBRD cannot succeed}. } $$ It means only: > The most direct classical route of treating the P2 boundary as a standard dimension-2 sieve to obtain a lower bound > encounters the known sifting-limit obstacle. --- # 33. Sieve-Depth / Factor-Depth Tradeoff Take the general cutoff: $$ z_h \asymp X_m^{1/(h+1)}. $$ SRBRD-12： $$ \operatorname{RootRough}_{z_h}(x) $$ For: $$ x\le X_m $$ forces: $$ \boxed{ \Omega_{\mathrm{mult}}(x)\le h. } $$ At the same time, the natural sieve parameter is: $$ \boxed{ s\approx h+1. } $$ --- # 34. Tradeoff Table | cutoff scale | max factor count | division depth | sieve parameter | |---|---:|---:|---:| | $X^{1/2}$ | $1$ | $0$ | $\approx2$ | | $X^{1/3}$ | $2$ | $\le1$ | $\approx3$ | | $X^{1/4}$ | $3$ | $\le2$ | $\approx4$ | | $X^{1/5}$ | $4$ | $\le3$ | $\approx5$ | Dimension-2 DHR lower positivity requires approximately: $$ s>4.266. $$ Thus the most natural direct entry into the positive lower-sieve regime is near: $$ \boxed{ X^{1/5} } $$ as displayed above, That is: $$ \boxed{ P_4\text{ boundary}, } $$ not: $$ P_2. $$ --- # 35. Interpretation This explains precisely why: > Moving directly from an ordinary dimension-2 lower sieve to the prime / semiprime boundary is not easy. If the cutoff is lowered to: $$ X^{1/5}, $$ the lower sieve is more comfortable, but a survivor endpoint may then have: $$ 4 $$ prime factors. Thus one must: $$ \boxed{ \text{First obtain mass at a lower cutoff, then use Buchstab / cofactor decomposition to move layer by layer to the cube-root boundary.} } $$ --- # 36. Two-Cutoff Strategy Choose: $$ 2\le y\beta_2. $$ For example, conceptually: $$ u\approx5. $$ First obtain, for: $$ \widehat{\mathcal S}_y $$ a dimension-2 lower bound. Then use T03: $$ |\widehat{\mathcal S}_{z_3}| = |\widehat{\mathcal S}_y| - \sum_{yz_3}n_r + M_{SS}. $$ Substituting T03: ## Theorem SRBRD-16-T12 For any: $$ 2\le yz_3 }} n_r + M_{SS}. } $$ This is the main theorem of this round. --- # 38. Four Modules of T12 T12 decomposes the ordered Goldbach count into: ### Module A — Coarse Rough Mass $$ |\widehat{\mathcal S}_y|. $$ ### Module B — Buchstab Descent Loss $$ \sum_{yz_3}n_r. $$ ### Module D — Internal Transport Rescue $$ M_{SS}. $$ The roles of these four components are clear. --- # 39. Modular Lower-Bound Closure Suppose there are proof-admissible providers: $$ \boxed{ L_y \le |\widehat{\mathcal S}_y|, } $$ $$ \boxed{ |\mathcal B_r^{pair}| \le U_r^{B}, } $$ $$ \boxed{ n_r \le U_r^{row}, } $$ $$ \boxed{ L_{SS} \le M_{SS}. } $$ then: ## Theorem SRBRD-16-T13 $$ \boxed{ M_{PP} \ge L_y - \sum_{yz_3}U_r^{row} + L_{SS}. } $$ --- # 40. Buchstab–Row–Flow Closure Criterion ## Corollary SRBRD-16-C01 If: $$ \boxed{ L_y - \sum_{yz_3}U_r^{row} + L_{SS} >0, } $$ then: $$ \boxed{ R_G^{-}(2m)>0. } $$ This is currently the most complete modular analytic closure contract. --- # 41. Provider Assignment The most natural tool assignment: ### $L_y$ A dimension-2 lower sieve. ### $U_r^B$ Buchstab shell upper sieve / least-factor decomposition. ### $U_r^{row}$ A dimension-2 upper sieve on the cofactor row bulk + an explicit frontier. ### $L_{SS}$ SRBRD-15： - row deficiency； - block deficiency； - center-divisor lock； - transport matching。 This deliberately makes it so that: $$ \boxed{ \text{the lower-bound sieve works only at the lower cutoff }y, } $$ while near the cube-root boundary the main tools are: $$ \boxed{ \text{upper bounds + exact transport identities}. } $$ --- # 42. Finite Audit: $m=1000$ $$ X_m=1997. $$ cube-root cutoff： $$ z_3=13. $$ Take the fifth-root-scale integer: $$ y=5. $$ exact： $$ |\widehat{\mathcal S}_y|=268, $$ Buchstab loss： $$ 132, $$ Thus: $$ |\widehat{\mathcal S}_{z_3}|=136. $$ semiprime： $$ n_{SP}=35. $$ internal： $$ M_{SS}=8. $$ Therefore: $$ \boxed{ M_{PP} = 268-132-70+8 = 74. } $$ reduced： $$ R_G^{-}(2000)=37. $$ --- # 43. $m=1000$ Buchstab Shell Distribution For: $$ y=5, \quad z_3=13, $$ exact pair losses： $$ |\mathcal B_7^{pair}|=76, $$ $$ |\mathcal B_{11}^{pair}|=34, $$ $$ |\mathcal B_{13}^{pair}|=22. $$ Sum: $$ \boxed{ 132. } $$ This proves that the descent loss is indeed decomposed disjointly by least-factor shells. --- # 44. Finite Audit: $m=5000$ $$ X_m=9997, $$ $$ z_3=22. $$ Take: $$ y=7. $$ exact： $$ |\widehat{\mathcal S}_y|=954, $$ Buchstab loss： $$ 430, $$ $$ |\widehat{\mathcal S}_{z_3}|=524. $$ semiprime： $$ n_{SP}=152. $$ internal： $$ M_{SS}=34. $$ Thus: $$ \boxed{ M_{PP} = 954-430-304+34 = 254. } $$ That is: $$ R_G^{-}(10000)=127. $$ --- # 45. $m=5000$ Shell Distribution $$ |\mathcal B_{11}^{pair}|=172, $$ $$ |\mathcal B_{13}^{pair}|=120, $$ $$ |\mathcal B_{17}^{pair}|=76, $$ $$ |\mathcal B_{19}^{pair}|=62. $$ Total: $$ \boxed{ 430. } $$ --- # 46. Larger Two-Cutoff Controls ### $m=10000$ $$ z_3=28, $$ fifth-root-scale： $$ y=8. $$ $$ |\widehat{\mathcal S}_y|=1908, $$ $$ |\widehat{\mathcal S}_{z_3}|=954, $$ loss： $$ 954. $$ $$ n_{SP}=291, $$ $$ M_{SS}=90, $$ Thus: $$ \boxed{ M_{PP}=462. } $$ ### $m=100000$ $$ z_3=59, $$ $$ y=12. $$ $$ |\widehat{\mathcal S}_y|=15584, $$ $$ |\widehat{\mathcal S}_{z_3}|=6358, $$ loss： $$ 9226. $$ $$ n_{SP}=2100, $$ $$ M_{SS}=676, $$ Thus: $$ \boxed{ M_{PP}=2834. } $$ --- # 47. One-Sided Row Envelope No-Go If one uses only: $$ U_{SP}^{one}, $$ then the crude lower bound is: $$ M_{PP} \ge |\widehat{\mathcal S}_{z_3}| - 2U_{SP}^{one} + L_{SS}. $$ Even if one temporarily uses the exact: $$ L_{SS}=M_{SS}, $$ selected： ### $m=1000$ $$ 136-2(86)+8=-28. $$ ### $m=5000$ $$ 524-2(489)+34=-420. $$ ### $m=10000$ $$ 954-2(1022)+90=-1000. $$ ### $m=100000$ $$ 6358-2(8856)+676=-10678. $$ Thus: $$ \boxed{ \text{A row upper envelope ignoring opposite-endpoint RootRough is simply insufficient.} } $$ --- # 48. Exact Location of the Needed Gain If a uniform proof follows T13, what actually needs strengthening is not: $$ \boxed{ \text{one-sided semiprime counting}. } $$ but: $$ \boxed{ \text{two-form row sieve} } $$ that is, simultaneously requiring: $$ ru $$ and: $$ 2m-ru $$ to avoid small root divisors. In the row bulk, this is exactly T08's dimension-2 residue system. --- # 49. Why This Is Better Localized Than Original Goldbach Original Goldbach: $$ x, \quad 2m-x $$ both must be prime simultaneously. Row problem: $$ ru, \quad 2m-ru $$ requires only: > An upper bound on cofactor candidates that pass the RootRough sieve. This is still a two-form sieve, but we need only an: $$ \boxed{ \text{upper bound}. } $$ There is no need to lower-bound simultaneous primes directly. This is an important methodological asymmetry. --- # 50. Pair-Buchstab Loss Is Also an Upper-Bound Problem In T13: $$ |\mathcal B_r^{pair}| $$ is also a quantity being subtracted. Thus it needs only an: $$ \boxed{ \text{upper bound}. } $$ Therefore, throughout the cube-root descent stage: - Buchstab shells: Upper bounds; - Semiprime rows: Upper bounds; - Transport overlap: Lower bounds from exact combinatorics. The only hard classical lower sieve is moved to: $$ \boxed{ \text{the lower cutoff }y. } $$ --- # 51. This Is the Main Architectural Gain The old problem: > Applying a dimension-2 lower sieve directly at: > $$ > z=X^{1/3} > $$ > as displayed above. SRBRD-16 changes this to: 1. At: $$ y\approx X^{1/5} $$ Obtain lower mass; 2. Raise the cutoff using upper bounds on Buchstab losses; 3. Subtract semiprime rows at the P2 boundary; 4. Correct overlap using the transport correction. Thus: $$ \boxed{ \text{The hardest lower-bound work is moved to a cutoff more comfortable for the sieve.} } $$ This is still not a proof, but it genuinely changes the placement of the analytic burden. --- # 52. Relation to Weighted / Chen-Type Sieve Philosophy One of the central ideas of external Chen-type methods is also: > Not to require the ordinary sieve to complete all the work directly at the prime boundary; > Instead, weighted / switching / almost-prime decompositions are added to cross the parity-sensitive region. SRBRD-16's: $$ \boxed{ \text{lower cutoff} \to \text{Buchstab descent} \to P_2 \to \text{transport correction} } $$ is compatible with this philosophy at a high structural level. But this document has not reproduced Chen's theorem, nor has it thereby obtained a Goldbach proof. --- # 53. Potential New Source Compared with an ordinary weighted sieve, SRBRD additionally retains: 1. symmetric center coordinate； 2. exact pair least-factor shell； 3. RootRough dynamic boundary； 4. semiprime least-factor rows； 5. center-divisor transport lock； 6. $SS$ matching / flow correction。 What actually needs verification is: > Whether these additional couplings suffice for the T13 crossing. This has not yet been completed. --- # 54. Analytic Provider API It is recommended to define formally in subsequent work: ```yaml RoughPairLower: cutoff: y output: L_y PairBuchstabUpper: prime_r: y < r <= z3 output: U_B_r SemiprimeRowUpper: prime_r: r > z3 output: U_row_r InternalFlowLower: output: L_SS ``` closure： ```yaml if: L_y - sum(U_B_r) - 2*sum(U_row_r) + L_SS > 0 then: Goldbach_success: true ``` --- # 55. Formal Theorem Queue ```yaml SRBRD_16: T01: RootRough_iff_extended_lpf_gt_cutoff T02: pair_RootRough_iff_pair_lpf_height_gt_cutoff T03: exact_pair_Buchstab_identity T04: left_right_tie_shell_partition T05: cube_root_row_cofactor_forced_prime T06: exact_row_cofactor_formula T07: one_sided_semiprime_upper T08: row_bulk_exact_residue_sieve T09: row_bulk_dimension_profile T10: row_frontier_capacity T11: abstract_row_upper_provider T12: exact_Buchstab_Row_Flow_identity T13: modular_provider_lower_bound ``` --- # 56. Acceptance Gate Before SRBRD-16 can become a formal core, it requires: 1. extended lpf with prime infinity semantics； 2. RootRough equivalence； 3. pair minimum-factor shell； 4. Buchstab shell disjointness； 5. cube-root cofactor primality theorem； 6. exact row cofactor formula； 7. row bulk / frontier split； 8. exact forbidden residues； 9. center-divisor residue collapse； 10. frontier bound； 11. one-sided upper no-go audit； 12. DHR sifting-limit statement clearly marked external； 13. no claim that DHR barrier is impossibility； 14. modular closure theorem； 15. finite audit separated from proof； 16. No Goldbach proof claim. --- # 57. Verdict This round can give: $$ \boxed{ \mathrm{SRBRD\mbox{-}16\ BUCHSTAB\mbox{-}ROW\mbox{-}FLOW\ KERNEL\ PASS}. } $$ The most important new exact identity: $$ \boxed{ M_{PP} = |\widehat{\mathcal S}_y| - \sum_{yz_3}n_r + M_{SS}. } $$ This brings: - coarse sieve mass； - cutoff descent； - semiprime contamination； - transport rescue； fully into a single formula for the first time. The external sieve audit also indicates that: $$ \boxed{ z=X^{1/3} } $$ is too close to the classical sifting-limit barrier for the dimension-2 lower sieve, while: $$ \boxed{ y\approx X^{1/5} } $$ is a more natural starting point for the lower sieve. Therefore: $$ \boxed{ \text{lower sieve at low cutoff} + \text{upper-bound descent} + \text{transport correction} } $$ becomes the most reasonable analytic architecture currently available. However: $$ \boxed{ \mathrm{UNIFORM\ CROSSING\ NOT\ YET\ PROVED}. } $$ and: $$ \boxed{ \mathrm{GOLDBACH\ STILL\ OPEN}. } $$ --- # 58. Recommendations for the Next Round # SRBRD-17 ## Pair-Buchstab Shell / Row-Sieve Quantitative Closure The next round should begin making T13 genuinely quantitative. Priorities: ### A. Lower cutoff Choose: $$ y=X^{1/u}, \qquad u>\beta_2, $$ Establish an explicit: $$ L_y. $$ ### B. Buchstab shells For: $$ yz_3, $$ Use a dimension-2 upper sieve in the row bulk, and T10's elementary bound in the frontier. ### D. Transport Insert: - center-divisor exact lock； - row / block deficiency； - selected matching lower bound。 Finally, check directly: $$ \boxed{ L_y - \sum U_r^B - 2\sum U_r^{row} + L_{SS} >0. } $$ This will be the first round to put classical sieve constants and SRBRD transport constants into the same numerical inequality. --- # Appendix A: Pair-Buchstab Contract ```yaml extended_lpf: prime: infinity composite: lpf pair_height: rho_m_x: min( extended_lpf(x), extended_lpf(2m-x) ) survivor_cutoff_z: iff: rho_m_x > z shell_r: rho_m_x == r identity: S_y: S_z disjoint_union shell_r_for_y_less_r_le_z ``` --- # Appendix B: Semiprime Row Cofactor Contract ```yaml cube_cutoff: z3^3 >= X X: 2m-3 row_r: r: prime r > z3 endpoint: x: r*u range: r <= u <= floor(X/r) conditions: RootRough_z3(r*u) RootRough_z3(2m-r*u) theorem: u_prime: forced: true ``` --- # Appendix C: Row Bulk Sieve ```yaml bulk: r*u <= 2m-z3^2 for_each_odd_prime_p_le_z3: forbidden_residues: - u == 0 mod p - u == 2m*r_inverse mod p count: if_p_divides_m: nu_p: 1 else: nu_p: 2 sieve_dimension: generic: 2 ``` --- # Appendix D: Sieve-Depth Tradeoff ```yaml cutoff: z: X^(1/(h+1)) RootRough_endpoint: max_prime_factor_count: h natural_sieve_parameter: s: approximately_h_plus_1 external_dimension2_DHR: beta2: approximately_4.266 implication: cube_root_P2: s_approximately: 3 direct_DHR_lower_positive_main_term: unavailable_in_standard_regime: true fifth_root_P4: s_approximately: 5 standard_lower_sieve_regime: plausible: true ``` --- # Appendix E: Selected Exact Two-Cutoff Audit | $m$ | $y$ | $z_3$ | $|\widehat{\mathcal S}_y|$ | Buchstab loss | $|\widehat{\mathcal S}_{z_3}|$ | $n_{SP}$ | $M_{SS}$ | $M_{PP}$ | |---:|---:|---:|---:|---:|---:|---:|---:|---:| | 1000 | 5 | 13 | 268 | 132 | 136 | 35 | 8 | 74 | | 5000 | 7 | 22 | 954 | 430 | 524 | 152 | 34 | 254 | | 10000 | 8 | 28 | 1908 | 954 | 954 | 291 | 90 | 462 | | 100000 | 12 | 59 | 15584 | 9226 | 6358 | 2100 | 676 | 2834 | --- # Appendix F: One-Sided Row No-Go ```yaml m_1000: exact_nSP: 35 one_sided_upper: 86 resulting_PP_lower_even_with_exact_SS: -28 m_5000: exact_nSP: 152 one_sided_upper: 489 resulting_PP_lower_even_with_exact_SS: -420 m_10000: exact_nSP: 291 one_sided_upper: 1022 resulting_PP_lower_even_with_exact_SS: -1000 m_100000: exact_nSP: 2100 one_sided_upper: 8856 resulting_PP_lower_even_with_exact_SS: -10678 diagnosis: opposite_endpoint_RootRough_filter: essential: true ``` --- # Appendix G: Commit Decision ```yaml research_round: SRBRD-16 decision: exact_pair_Buchstab_identity: true row_cofactor_prime_forced: true row_bulk_dimension2_sieve: true frontier_elementary_bound: true sieve_depth_tradeoff_identified: true modular_analytic_closure: true uniform_crossing: false goldbach_proved: false next: SRBRD-17_PairBuchstabShell_RowSieve_Quantitative_Closure ``` --- # References - C. S. Franze, “Sifting Limits for the $\Lambda^2\Lambda^-$ Sieve,” *Journal of Number Theory* 131 (2011), 1707–1725. - H. G. Diamond and H. Halberstam, *A Higher-Dimensional Sieve Method*, Cambridge Tracts in Mathematics 177, Cambridge University Press, 2008, with computational procedures by W. F. Galway. - T. Tao, “254A, Notes 4: Some sieve theory,” 2015. - T. Tao, “254A, Supplement 5: The linear sieve and Chen’s theorem,” 2015.