> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # SRBRD-18 ## BV-Switched Row / Weighted Buchstab Constant Closure ### Fixed-Row BV No-Go, Aggregate Bilinear Switching, Near-EH Plain-Sieve Threshold and Constant Ledger **Parent program:** RG-SCP / Reverse Goldbach Structural Closure Program **Branch:** SRBRD — Symmetric Root-Bounded Relation Domain **Round:** SRBRD-18 **Date:** 2026-09-30 **Version:** v0.1 **Authors:** Neo.K; Aletheia (GPT-5.6 Sol) **Type:** Quantitative Constant Audit / Distribution-Level Barrier / Aggregate Bilinear Switching Interface **Prerequisites:** SRBRD-00 through SRBRD-17 **Status:** Fixed-row BV + an ordinary linear sieve has been proved still insufficient at the constant level; an aggregate bilinear switched sequence and near-EH plain-sieve threshold have been established; Goldbach remains unproved --- # 0. This Round's Sole Aim SRBRD-17 has proved: 1. The normalized cost of a naive unswitched dimension-2 row upper bound is approximately: $$ 60.5588; $$ 2. ideal SRBRD ledger： $$ 9-18\log(3/2) \approx1.70163>0; $$ 3. A semiprime row has exact prime-cofactor switching: $$ n_r = \#\left\{ u\in\mathbb P: r\le u\le X/r, \quad 2m-ru\text{ RootRough}_{z_3} \right\}; $$ 4. local Euler product exact factor： $$ V_m(z)=P_0(z)W_m(z). $$ Thus this round must answer: > If Bombieri–Vinogradov and an ordinary linear sieve are connected directly to a switched row, are the constants already sufficient? The answer remains: $$ \boxed{ \text{Insufficient.} } $$ Moreover, the failure can be quantified very clearly. --- # 1. Global Scale Let: $$ \boxed{ X=2m-3. } $$ cube-root cutoff： $$ \boxed{ z_3=X^{1/3} } $$ Integer rounding is ignored in the continuum diagnostic. semiprime least factor： $$ \boxed{ r=X^t, \qquad \frac13\le t\le\frac12. } $$ Scale of the prime-cofactor range: $$ \boxed{ U_r = \frac Xr = X^{1-t}. } $$ --- # 2. Fixed-Row Prime Sequence SRBRD-17's switched row: $$ n_r = \#\left\{ u\in\mathbb P: r\le u\le U_r, \quad 2m-ru \text{ RootRough}_{z_3} \right\}. $$ For: $$ p\le z_3, $$ $$ p\nmid m, $$ one needs to avoid: $$ \boxed{ u \equiv 2m\,r^{-1} \pmod p. } $$ Thus, among prime cofactors, this is a dimension-1 upper sieve. --- # 3. Ordinary Bombieri–Vinogradov Level on One Fixed Row For a prime sequence of size: $$ U_r $$ as displayed above, ordinary Bombieri–Vinogradov gives the natural average modulus level: $$ \boxed{ D_r = U_r^{1/2} (\log U_r)^{-B} } $$ up to logarithmic loss. continuum exponent： $$ \boxed{ D_r \approx X^{(1-t)/2}. } $$ --- # 4. Full Cube-Root Cutoff Is Beyond the Fixed-Row BV Level Since: $$ z_3=X^{1/3}, $$ while, for: $$ t>\frac13, $$ $$ \frac{1-t}{2} < \frac13. $$ Thus: ## Diagnostic SRBRD-18-D01 $$ \boxed{ D_r\frac13. } $$ That is: > Ordinary fixed-row Bombieri–Vinogradov cannot directly sift the linear sieve to the full cube-root cutoff. --- # 5. Partial Linear-Sieve Cutoff Let the linear sieve parameter be: $$ s>1. $$ Take: $$ \boxed{ w_r = D_r^{1/s}. } $$ continuum exponent： $$ \boxed{ w_r \approx X^{(1-t)/(2s)}. } $$ Since: $$ s>1, $$ within the row band: $$ w_r\le z_3. $$ Thus: > We can only sieve to $w_r$ first, > and then treat the missing small-prime band: > $$ > (w_r,z_3] > $$ > as a loss. --- # 6. Exact Missing-Band Penalty SRBRD-17's opposite-endpoint factor: $$ W_m(z) = \prod_{\substack{ 3\le p\le z\\ p\nmid m }} \left( 1-\frac1{p-1} \right). $$ Define the exact partial/full ratio: $$ \boxed{ \Psi_m(w,z_3) = \frac{ W_m(w) }{ W_m(z_3) } } $$ $$ = \prod_{\substack{ w1). } $$ --- # 9. Optimally Tuned Fixed-Row BV Multiplier partial sieve upper relative to full ideal density： $$ \boxed{ R_{\mathrm{fixBV}}(t,s) = F_1(s) \Psi_0(w_r,z_3). } $$ From D02: $$ R_{\mathrm{fixBV}} \sim F_1(s) \frac{2s}{3(1-t)}. $$ Also: $$ sF_1(s)\ge2e^\gamma. $$ Thus: ## Diagnostic SRBRD-18-D03 $$ \boxed{ R_{\mathrm{fixBV}}(t,s) \ge \frac{ 4e^\gamma }{ 3(1-t) }. } $$ For: $$ 19. } $$ But it remains impossible to complete the ideal ledger crossing using only: > fixed-row BV + ordinary linear upper sieve as displayed above. --- # 13. Fixed-Row BV No-Go Even if: - coarse lower； - Buchstab shell upper； are both idealized to the best possible local-density constants, they leave only: $$ \boxed{ 9. } $$ But the fixed-row BV row already costs at least: $$ \boxed{ 29.63. } $$ Thus, with zero transport, the ledger is at most: $$ \boxed{ 9-29.63 \approx -20.63. } $$ This is the: # Fixed-Row BV Constant No-Go It is not a theorem that Goldbach is impossible, but only a quantitative rejection of this specific provider architecture. --- # 14. Full-Cutoff Aggregate Distribution Model Now consider the stronger hypothetical / provider interface: > Instead of using BV row by row, > obtain a distribution exponent directly for the entire switched bilinear sequence: > $$ > \theta. > $$ That is, the aggregate bilinear sequence can be controlled for moduli up to: $$ \boxed{ D=X^\theta } $$ as displayed above. At the full cutoff: $$ z_3=X^{1/3}, $$ linear sieve parameter： $$ \boxed{ s=3\theta. } $$ --- # 15. Plain Linear-Sieve Aggregate Multiplier If: $$ 19. } $$ Even if the aggregate sequence can fully use the BV level: $$ 1/2 $$ to sift to: $$ z_3, $$ the plain linear upper bound is still insufficient. --- # 18. Plain-Sieve Distribution Threshold With zero transport and ideal coarse/shell contributions, one needs: $$ C_{\mathrm{row}}^{agg}(\theta)<9. $$ That is: $$ \frac{ 12e^\gamma\log(3/2) }{ \theta } <9. $$ Thus: ## Diagnostic SRBRD-18-D06 $$ \boxed{ \theta > \theta_* = \frac{ 12e^\gamma\log(3/2) }{9} } $$ $$ \boxed{ \theta_* \approx 0.9628836273. } $$ That is: > **Using only a plain linear upper sieve + the distribution level, > a level almost equal to 1 is needed.** --- # 19. Near Elliott–Halberstam Interpretation Bombieri–Vinogradov： $$ \theta=\frac12. $$ The ideal direction of the Elliott–Halberstam conjecture: $$ \theta<1. $$ T18 shows: $$ \boxed{ \theta_*\approx0.963. } $$ Thus the plain-sieve crossing threshold is very close to: # Elliott–Halberstam Level This again proves: $$ \boxed{ \text{Increasing the level of distribution alone is not a practical unconditional solution.} } $$ --- # 20. Current Structured-Weight Exponents as a Diagnostic The following is only an external comparison; the weight hypotheses differ between theorems, so they cannot be directly inserted into the SRBRD sequence without checking the hypotheses. ### Bombieri–Friedlander–Iwaniec style well-factorable： $$ \theta=\frac47. $$ ### Maynard well-factorable The published result can reach: $$ \theta=\frac35-\varepsilon. $$ ### stronger structured-weight literature Certain more specialized weights can reach: $$ \theta=\frac58 $$ or nearby or higher levels. --- # 21. Plain Linear-Sieve Costs at Selected Exponents Use D05 solely as a constant diagnostic: | $\theta$ | $s=3\theta$ | $F_1(s)$ | row cost | |---:|---:|---:|---:| | $1/2$ | $1.5$ | $2.37476$ | $17.3319$ | | $4/7$ | $12/7$ | $2.07792$ | $15.1654$ | | $3/5$ | $1.8$ | $1.97897$ | $14.4433$ | | $5/8$ | $1.875$ | $1.89981$ | $13.8655$ | | $1$ | $3$ | $1.18738$ | $8.66595$ | Even with: $$ \theta=\frac58, $$ the plain row cost remains: $$ \boxed{ 13.87>9. } $$ --- # 22. Consequence Thus SRBRD-18 reaches a strong architectural conclusion: $$ \boxed{ \text{We cannot pursue only a higher distribution exponent.} } $$ One must also use: - weighted Buchstab subtraction； - switching identities； - transport overlap； - possibly well-factorable / bilinear structure； to change the ledger itself. --- # 23. Required Effective Row Multiplier Write the aggregate row cost as: $$ \boxed{ 18\log\frac32 \cdot \overline R. } $$ With zero transport and ideal coarse/shell contributions, one needs: $$ 18\log\frac32 \cdot \overline R <9. $$ Thus: ## Diagnostic SRBRD-18-D07 $$ \boxed{ \overline R < \frac{ 9 }{ 18\log(3/2) } } $$ $$ \boxed{ \overline R < 1.2331517312. } $$ This is currently the most intuitive row-constant target. --- # 24. Transport-Adjusted Threshold If the normalized transport gain is: $$ \Phi\ge0, $$ and the available coarse-shell budget is: $$ B_{CS}, $$ then the row multiplier needs only: $$ \boxed{ \overline R < \frac{ B_{CS}+\Phi }{ 18\log(3/2) }. } $$ ideal： $$ B_{CS}=9. $$ Thus transport relaxes the: $$ 1.23315 $$ threshold. But finite evidence shows that: $$ \Phi $$ is usually an order-one correction, not a huge compensation sufficient to absorb: $$ 29.63 $$ or: $$ 60.56 $$ as displayed above. --- # 25. Finite Effective-Row Calibration SRBRD-17's finite calibration: $$ U_{SP}^{one} W_m(z_3) $$ is the ideal switched main term. Define the observed multiplier: $$ \boxed{ R_{\mathrm{obs}} = \frac{ n_{SP} }{ U_{SP}^{one} W_m(z_3) }. } $$ selected： | $m$ | $R_{\mathrm{obs}}$ | |---:|---:| | $1000$ | $1.18393$ | | $5000$ | $1.02128$ | | $10000$ | $0.98007$ | | $100000$ | $1.00355$ | All are below the zero-transport target: $$ 1.23315. $$ This is only computational evidence, but indicates: $$ \boxed{ \text{The actual arithmetic row constant seems close to }1, } $$ rather than the classical fixed-row sieve's: $$ 3\text{--}5 $$ or even larger values. --- # 26. Aggregate Bilinear Switched Sequence Now define the object that should actually be estimated as a whole. The row bulk requires: $$ 2m-ru\ge z_3^2. $$ Define the prime-pair parameter set: $$ \boxed{ \mathcal P_m = \left\{ (r,u): r,u\in\mathbb P, \quad z_3z_3, $$ Thus: $$ r,u $$ are both units modulo: $$ p $$ as displayed above. bad condition： $$ p\mid2m-ru. $$ --- # 29. Center Divisor If: $$ p\mid m, $$ then the bad congruence is: $$ ru\equiv0\pmod p. $$ However: $$ r,u $$ are units. Thus: ## Theorem SRBRD-18-T02 $$ \boxed{ p\mid m \Longrightarrow \text{aggregate bad local density}=0. } $$ --- # 30. Noncenter Prime If: $$ p\nmid m, $$ For each nonzero: $$ r\bmod p, $$ there exists a unique: $$ u \equiv 2m\,r^{-1} \pmod p. $$ In: $$ (p-1)^2 $$ unit pairs, the number of bad pairs is: $$ p-1. $$ Thus: ## Theorem SRBRD-18-T03 $$ \boxed{ g_m(p) = \frac1{p-1} } $$ for： $$ p\nmid m. $$ This agrees exactly with SRBRD-17's: $$ W_m $$ as displayed above. --- # 31. Squarefree Aggregate Density For squarefree odd: $$ d $$ whose prime factors are all: $$ \le z_3, $$ If: $$ (d,m)=1, $$ CRT gives: $$ \boxed{ g_m(d) = \frac1{\varphi(d)}. } $$ If: $$ (d,m)>1, $$ then: $$ \boxed{ g_m(d)=0 } $$ for the simultaneous divisibility event containing a center-divisor prime. This is the exact local-density law of the aggregate bilinear sequence. --- # 32. Aggregate Distribution Remainder Let: $$ A_d^{bil} = \# \left\{ (r,u)\in\mathcal P_m: d\mid2m-ru \right\}. $$ Define: $$ \boxed{ R_d^{bil} = A_d^{bil} - g_m(d) |\mathcal P_m|. } $$ The actual analytic question is: $$ \boxed{ \text{For how large a }D \text{ can the average of } R_d^{bil} \text{ be controlled?} } $$ rather than controlling the prime row by row: $$ u. $$ --- # 33. Bilinear Distribution Provider It is recommended to define formally: $$ \boxed{ \operatorname{BilDist}(\theta,A) } $$ means: for appropriate sieve weights: $$ \lambda_d, $$ support： $$ d\le X^\theta, $$ we have: $$ \boxed{ \sum_{d\le X^\theta} \lambda_d R_d^{bil} \ll \frac{ |\mathcal P_m| }{ (\log X)^A }. } $$ The exact admissible weight class must be specified in a future formal theorem. --- # 34. Ordinary BV Does Not Automatically Give BilDist Bombieri–Vinogradov directly controls: > Primes in one arithmetic-progression variable. Here: $$ A_d^{bil} $$ is the: $$ \boxed{ \text{prime} \times \text{prime} } $$ multiplicative-convolution congruence: $$ ru\equiv2m\pmod d. $$ Thus: $$ \boxed{ \text{The ordinary BV theorem itself is not BilDist}. } $$ One needs: - bilinear form estimates； - convolution-type BV； - dispersion； - well-factorable weights； - or a switching decomposition。 --- # 35. Current Literature Interface Modern distribution literature does include: - well-factorable prime AP estimates beyond exponent $1/2$； - convolution-type Bombieri–Vinogradov estimates； - bilinear / trilinear structured moduli results。 But each theorem has very specific: - coefficient factorability； - support ranges； - modulus factorization； - variable-length hypotheses。 Thus: $$ \boxed{ \text{One cannot insert an exponent directly into SRBRD merely after seeing it}. } $$ SRBRD-19 must perform a hypothesis match item by item. --- # 36. Even Full BV-Scale BilDist Is Not Enough with Plain Linear Sieve Suppose, very optimistically, that: $$ \operatorname{BilDist}(\theta=1/2) $$ holds, and that a plain linear upper sieve can be used directly at the full cutoff: $$ z_3 $$ as displayed above. T17 has proved: $$ C_{\mathrm{row}} \approx17.33>9. $$ Thus: $$ \boxed{ \text{BilDist}(1/2) + \text{plain linear sieve} } $$ is still insufficient. --- # 37. Even Known Beyond-$1/2$ Structured Exponents Are Not Sufficient by Themselves pure constant diagnostic： - $$ \theta=4/7 $$ row cost： $$ \approx15.17; $$ - $$ \theta=3/5 $$ row cost： $$ \approx14.44; $$ - $$ \theta=5/8 $$ row cost： $$ \approx13.87. $$ All are: $$ \boxed{ >9. } $$ Thus the actual breakthrough must change the: $$ \boxed{ \text{weighted ledger} } $$ rather than merely increasing: $$ \theta. $$ --- # 38. Weighted Buchstab Is Therefore Mandatory The SRBRD-16 master identity currently penalizes semiprime rows by: $$ -2n_{SP} $$ as displayed above. But the cost of all semiprime states to failure coverage need not have weight: $$ 2. $$ Since: - $SS$ states later have the following added back: $$ M_{SS}; $$ - Center rows have forced internal flow; - Distinct factor ranges may have different contamination efficiencies; - Buchstab shells and rows have structural overlap. Thus one should introduce a: # Weighted Semiprime Penalty --- # 39. Abstract Weighted Row Functional Choose a weight function: $$ \boxed{ 0\le w(r)\le2. } $$ Decompose: $$ 2\sum_r n_r $$ into: $$ \boxed{ \sum_r w(r)n_r + \mathcal R_w, } $$ where the residual: $$ \mathcal R_w $$ is compensated by a transport / overlap certificate. Target: > Apply the analytic row upper bound only to the lower-weight part, > then pay the remaining penalty with arithmetic transport. This is a natural version of Chen's weighted-sieve philosophy in the language of SRBRD transport. --- # 40. Exact Transport Reweighting Interface From: $$ M_{PP} = |\widehat{\mathcal S}_{z_3}| - 2n_{SP} + M_{SS}, $$ If one can prove: $$ \boxed{ M_{SS} \ge \sum_r (2-w(r))n_r - E_w, } $$ then: $$ \boxed{ M_{PP} \ge |\widehat{\mathcal S}_{z_3}| - \sum_rw(r)n_r - E_w. } $$ then the weighted row penalty is not an arbitrary heuristic. It can be rigorously justified by a forced transport lower bound. --- # 41. Weighted Buchstab / Transport Closure Combining the coarse cutoff: $$ |\widehat{\mathcal S}_{z_3}| = |\widehat{\mathcal S}_y| - \sum_{yz_3} (2-w(r))n_r - E_w, $$ then: $$ \boxed{ M_{PP} \ge |\widehat{\mathcal S}_y| - \sum_{yz_3} w(r)n_r - E_w. } $$ This is the: # Weighted Buchstab–Transport Identity --- # 42. Why T04 Matters The plain ledger requires the effective row multiplier: $$ <1.233. $$ If transport can validly lower the average row weight: $$ 2 $$ as displayed above, the required quality of the analytic row upper bound is greatly relaxed. Thus: $$ \boxed{ \text{Transport is not only an additive bonus, but can also redefine the contamination weight.} } $$ This may be better suited to quantitative optimization than merely: $$ +L_{SS} $$ as displayed above. --- # 43. Transport Weight from Center-Divisor Rows SRBRD-15： If: $$ r\mid m, $$ then: $$ M_{r,0}\le1. $$ Thus, for row: $$ r $$ most of its mass is internal. For large: $$ n_r, $$ the effective contamination weight is close to: $$ \boxed{ 1 } $$ rather than: $$ 2. $$ For an $SS$ ordered pair causes, in: $$ 2n_{SP}-M_{SS} $$ a net penalty of only: $$ 1 $$ per semiprime endpoint-pair contribution. This provides an explicit source of weight reduction. --- # 44. Cross-Row Transport Can Lower Weight Too SRBRD-15 row/block deficiency： $$ M_{SS} \ge L_{SS}^{row/block}. $$ Any such lower bound can be converted to: $$ w(r) $$ or block weights. Thus future constant optimization can become a joint linear program of: $$ \boxed{ \text{analytic upper cost} + \text{transport discount} } $$ as displayed above. --- # 45. Quantitative Constant Ledger Define the normalized quantities: $$ \boxed{ C_L } $$ = Coarse lower contribution, $$ \boxed{ C_B } $$ = Buchstab band upper loss, $$ \boxed{ C_R } $$ = Weighted switched row upper cost, $$ \boxed{ C_T } $$ = Transport correction. global gate： $$ \boxed{ C_L-C_B-C_R+C_T>0. } $$ --- # 46. Ledger Levels Currently: ### Ideal local density $$ C_L-C_B=9. $$ ### Ideal unweighted row $$ C_R = 18\log(3/2) \approx7.298. $$ ### Ideal zero-transport margin $$ \boxed{ 1.7016. } $$ ### Fixed-row BV switched row $$ C_R \ge29.629. $$ ### Aggregate BV plain linear $$ C_R \approx17.332. $$ ### Needed target roughly： $$ \boxed{ C_R-C_T<9. } $$ --- # 47. Transport Calibration in Normalized Units This is only a finite heuristic calibration. Use: $$ H V_m(z_3) $$ as the cube-boundary density scale, and the corresponding dimension-2 continuum factor is approximately: $$ 9. $$ Define: $$ \boxed{ \Phi_{\mathrm{obs}} = 9 \frac{ M_{SS} }{ H V_m(z_3) }. } $$ selected： - $m=1000$: Approximately: $$ 0.55; $$ - $m=5000$: Approximately: $$ 0.59; $$ - $m=10000$: Approximately: $$ 0.85; $$ - $m=100000$: Approximately: $$ 0.96. $$ This is still far from sufficient, by itself, to repair: $$ 17.33 $$ or: $$ 29.63. $$ Thus: $$ \boxed{ \text{Weighted shell/row constant reduction is still necessary.} } $$ --- # 48. Distribution Exponent with Transport If only the plain linear row multiplier is used, the normalized transport gain: $$ \Phi $$ changes the threshold to: $$ \boxed{ \theta > \frac{ 12e^\gamma\log(3/2) }{ 9+\Phi } } $$ In: $$ \Phi\approx1 $$ the scale remains approximately: $$ \boxed{ 0.8666 } $$ as displayed above. This remains far higher than standard BV and current general structured-weight results. Thus transport alone also cannot replace a weighted sieve. --- # 49. Three Different Mean-Value Problems In subsequent work, the following must not be confused: ## Problem A — Prime AP ordinary BV： $$ p\equiv a\pmod q. $$ ## Problem B — Bilinear Product AP SRBRD aggregate rows： $$ ru\equiv2m\pmod q. $$ ## Problem C — Weighted Buchstab Bilinear AP Add: - $r$ range weights； - Buchstab coefficients； - transport weights。 What SRBRD-19 actually needs is: $$ \boxed{ \text{Problem C}. } $$ not merely citing Problem A's exponent. --- # 50. Current 2026 Convolution Literature New convolution-type Bombieri–Vinogradov / well-factorable-weight results already exist in 2026. These types of theorems and SRBRD Problem B/C are conceptually closer to ordinary BV. However: $$ \boxed{ \text{Hypothesis-to-hypothesis matching has not yet been done, so this document does not treat them as providers.} } $$ The next round should check, theorem by theorem: - variables support； - coefficient bounds； - well-factorability； - modulus range； - fixed residue dependence； - endpoint truncation。 --- # 51. Formal Theorem Queue ```yaml SRBRD_18: T01: bulk_semiprime_mass_as_sifted_bilinear_prime_pair_sequence T02: center_prime_gives_zero_local_bad_density T03: noncenter_prime_aggregate_local_density_one_over_p_minus_one T04: weighted_Buchstab_transport_identity ``` --- # 52. Quantitative Diagnostic Queue ```yaml SRBRD_18_Diagnostics: D01: fixed_row_BV_level_below_cube_cutoff D02: generic_missing_band_ratio D03: fixed_row_effective_multiplier_lower_envelope D04: fixed_row_BV_aggregate_cost: 24*exp(gamma)*log(2) D05: aggregate_plain_sieve_row_cost: 12*exp(gamma)*log(3/2)/theta D06: zero_transport_distribution_threshold: theta_star: approx: 0.9628836273 D07: zero_transport_effective_row_multiplier_target: approx: 1.2331517312 ``` --- # 53. Acceptance Gate Before SRBRD-18 can enter a formal / publication core, it requires: 1. fixed-row BV level derivation； 2. partial cutoff definition； 3. exact missing-band ratio； 4. Mertens-scale ratio clearly marked asymptotic； 5. linear sieve $F_1$ formula sourced； 6. $sF(s)$ monotonic theorem sourced； 7. fixed-row cost integral； 8. aggregate $\theta$-model clearly marked provider diagnostic； 9. distribution exponent table marked non-applicative comparison； 10. aggregate bilinear local-density theorem； 11. bilinear remainder definition； 12. ordinary BV ≠ BilDist firewall； 13. weighted transport identity； 14. finite calibration separated from rigorous bounds； 15. no current-literature theorem silently assumed applicable； 16. No Goldbach proof claim. --- # 54. Verdict This round can formally give: $$ \boxed{ \mathrm{SRBRD\mbox{-}18\ DISTRIBUTION\mbox{-}LEVEL\ BARRIER\ PASS}. } $$ The most important negative theorem-level diagnosis: $$ \boxed{ \text{fixed-row BV + ordinary linear sieve cannot provide the required row constant.} } $$ continuum lower envelope： $$ \boxed{ C_{\mathrm{row}}^{fixBV} \approx29.63>9. } $$ More strongly, even assuming that the aggregate plain linear sieve directly has a distribution exponent: $$ \theta, $$ zero-transport crossing still requires: $$ \boxed{ \theta>0.96288\ldots } $$ Thus: $$ \boxed{ \text{distribution level alone is not the missing ingredient.} } $$ The actual remaining target is: $$ \boxed{ \text{weighted Buchstab} + \text{aggregate bilinear distribution} + \text{transport discount}. } $$ The exact positive structural result is: $$ \boxed{ g_m(p) = \begin{cases} 0,&p\mid m,\\ 1/(p-1),&p\nmid m, \end{cases} } $$ for aggregate prime-pair switched rows， showing that the aggregate switched sequence is genuinely a dimension-1 bilinear sieve object. However: $$ \boxed{ \mathrm{UNIFORM\ CONSTANT\ CROSSING\ STILL\ OPEN}. } $$ and: $$ \boxed{ \mathrm{GOLDBACH\ STILL\ OPEN}. } $$ --- # 55. Recommendations for the Next Round # SRBRD-19 ## Convolution-BV Hypothesis Match / Weighted Transport Ledger The next round no longer invents new objects itself. Directly perform external theorem matching: 1. 2025 Maynard well-factorable AP： $$ 3/5-\varepsilon; $$ 2. BFI： $$ 4/7-\varepsilon; $$ 3. newer bilinear / trilinear / convolution-type BV results； 4. Chen linear-sieve switching framework。 Check item by item whether the SRBRD weighted bilinear sequence satisfies: - coefficient factorability； - dyadic support； - fixed-shift condition； - modulus factorization； - well-factorable sieve weights； - allowable exponent。 Then establish an actual: # Constant Ledger v1 If, after all known unconditional providers have been inserted, we still have: $$ \boxed{ C_L-C_B-C_R+C_T\le0, } $$ then formally establish: > The size of the gap in existing sieve technology. If, unexpectedly: $$ >0, $$ only then enter a complete rigorous proof audit. --- # Appendix A: Fixed-Row BV Constant Ledger ```yaml row_band: t: min: 1/3 max: 1/2 cofactor_size: U_r: X^(1-t) ordinary_BV_level: D_r: approximately: U_r^(1/2) partial_linear_cutoff: w_r: D_r^(1/s) generic_missing_band_penalty: approximately: 2s/(3*(1-t)) linear_sieve: sF_s: >= 2*exp(gamma) effective_multiplier: >=: 4*exp(gamma)/(3*(1-t)) aggregate_double_row_cost: exact_diagnostic: 24*exp(gamma)*log(2) decimal: 29.6290877977 ``` --- # Appendix B: Aggregate Plain-Sieve Threshold ```yaml full_cube_cutoff: z3: X^(1/3) distribution_exponent: D: X^theta linear_sieve_parameter: s: 3*theta plain_row_cost: 12*exp(gamma)*log(3/2)/theta zero_transport_crossing: requires: theta: > 0.9628836273 ``` --- # Appendix C: Selected Exponent Diagnostic | exponent $\theta$ | provenance class | plain row cost | below ideal budget 9? | |---:|---|---:|:---| | $1/2$ | Bombieri–Vinogradov | $17.3319$ | no | | $4/7$ | BFI well-factorable | $15.1654$ | no | | $3/5$ | Maynard well-factorable | $14.4433$ | no | | $5/8$ | stronger structured-weight scale | $13.8655$ | no | | $1$ | EH-level ideal endpoint | $8.66595$ | yes | **Note:** Different distribution theorems have different weight hypotheses. This table compares only the constant effect of exponents in the plain linear-sieve model, and does not mean the corresponding theorems can be applied directly to SRBRD. --- # Appendix D: Aggregate Bilinear Sequence Contract ```yaml prime_pair_parameters: r: prime r > z3 u: prime u >= r product: r*u <= 2m-z3^2 residual: a_r_u: 2m-r*u bulk_semiprime_mass: sift: a_r_u: RootRough_z3 local_bad_density: p_divides_m: 0 p_not_divides_m: 1/(p-1) squarefree_density: if_gcd(d,m)==1: 1/phi(d) ``` --- # Appendix E: Finite Row-Multiplier Calibration | $m$ | exact $n_{SP}$ | ideal switched main term | observed multiplier | |---:|---:|---:|---:| | $1000$ | $35$ | $29.5625$ | $1.18393$ | | $5000$ | $152$ | $148.8330$ | $1.02128$ | | $10000$ | $291$ | $296.9190$ | $0.98007$ | | $100000$ | $2100$ | $2092.5710$ | $1.00355$ | zero-transport target： $$ \boxed{ \overline R<1.2331517312. } $$ The selected exact data all fall within the target. --- # Appendix F: Commit Decision ```yaml research_round: SRBRD-18 decision: fixed_row_BV_no_go: true aggregate_plain_sieve_near_EH_threshold: true bilinear_switched_local_density: true weighted_transport_identity: true current_distribution_exponents_alone_insufficient: true uniform_constant_crossing: false goldbach_proved: false main_gap: weighted_bilinear_distribution_plus_transport_constant next: SRBRD-19_ConvolutionBV_HypothesisMatch_WeightedTransport_Ledger ``` --- # References 1. T. Tao, “254A, Supplement 5: The linear sieve and Chen’s theorem,” 2015. Linear-sieve functions satisfy: $$ F(s)=2e^\gamma/s $$ for: $$ 1