> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # SRBRD-17 ## Pair-Buchstab Shell / Row-Sieve Quantitative Closure ### Exact Euler-Factor Switching, Dimension-2 Constant Barrier, Prime-Cofactor Row Reduction, and Quantitative Crossing Gate **Parent program:** RG-SCP / Reverse Goldbach Structural Closure Program **Branch:** SRBRD — Symmetric Root-Bounded Relation Domain **Round:** SRBRD-17 **Date:** 2026-09-30 **Version:** v0.1 **Authors:** Neo.K; Aletheia (GPT-5.6 Sol) **Nature:** Quantitative Sieve Interface / Switching Kernel / Constant-Barrier Audit **Prerequisites:** SRBRD-00 through SRBRD-16 **Status:** Establishes a quantitative provider gate, proves a constant-scale no-go for the naive dimension-2 row upper bound, and exactly switches the semiprime row to prime-cofactor × one-residue sieve; Goldbach remains unproved --- # 0. This Round Adds No New Closure Layer SRBRD-16 has established the exact master identity: $$ \boxed{ M_{PP} = |\widehat{\mathcal S}_y| - \sum_{yz_3}n_r + M_{SS}. } $$ where: $$ z_3^3\ge2m-3. $$ Thus, SRBRD-17 adds no new combinatorial object. This round asks only: > If standard dimension-2 lower/upper sieves are inserted directly into these four modules, can the constants cross? Answer: $$ \boxed{ \text{naive separate dimension-2 row upper cannot close.} } $$ The quantitative source actually required is: $$ \boxed{ \text{cofactor switching}. } $$ Moreover, switching is not heuristic: > The local Euler factor of SRBRD itself factors exactly as > $$ > \text{cofactor primality factor} > \times > \text{opposite-endpoint factor}. > $$ --- # 1. Counting Scale Let: $$ \boxed{ X=2m-3. } $$ Ordered odd-endpoint domain: $$ 3\le x\le X. $$ Its cardinality: $$ \boxed{ H=m-2. } $$ Thus: $$ H=\frac{X-1}{2}. $$ --- # 2. Fixed-Cutoff Bulk For a cutoff: $$ z, $$ define the ordered bulk: $$ \boxed{ \widehat\Omega_m^{bulk}(z) = \left\{ x\in\widehat\Omega_m: x\ge z^2, \quad 2m-x\ge z^2 \right\}. } $$ Within the bulk, for every odd prime: $$ p\le z, $$ the root condition holds automatically. Thus: $$ \operatorname{RootRough}_z(x), \quad \operatorname{RootRough}_z(2m-x) $$ reduces completely to residue avoidance. --- # 3. Local Forbidden-Class Count Define: $$ \boxed{ \nu_m(p) = \begin{cases} 1, & p\mid m, \\ 2, & p\nmid m. \end{cases} } $$ For: $$ p\ge3. $$ Reason: - in general, one must avoid: $$ x\equiv0\pmod p, $$ $$ x\equiv2m\pmod p; $$ - if: $$ p\mid m, $$ the two classes collapse. --- # 4. Dimension-2 Sieve Product Define: $$ \boxed{ V_m(z) = \prod_{\substack{ p\in\mathbb P\\ 3\le p\le z }} \left( 1-\frac{\nu_m(p)}p \right). } $$ This is the canonical local-density product for the ordered SRBRD bulk. --- # 5. Generic and Center-Divisor Factors Define the generic product: $$ \boxed{ V_0(z) = \prod_{3\le p\le z} \left( 1-\frac2p \right). } $$ then: ## Theorem SRBRD-17-T01 $$ \boxed{ V_m(z) = V_0(z) \prod_{\substack{ p\mid m\\ 3\le p\le z }} \frac{p-1}{p-2}. } $$ Thus, center divisors can only: $$ \boxed{ \text{increase survivor density}. } $$ The uniform worst-case local density occurs when small odd center divisors are absent. --- # 6. Multiplicative Residue Count For squarefree odd: $$ d $$ whose prime factors are all: $$ \le z, $$ Define: $$ \boxed{ \nu_m(d) = \prod_{p\mid d}\nu_m(p). } $$ In the odd-endpoint parameter, the bad residue classes modulo: $$ d $$ are exactly multiplicative by CRT. --- # 7. Interval Congruence Remainder If: $$ N $$ is the odd-endpoint count in the bulk, then: ## Theorem SRBRD-17-T02 for squarefree: $$ d, $$ $$ \boxed{ A_d = N\frac{\nu_m(d)}d + R_d } $$ and: $$ \boxed{ |R_d|\le\nu_m(d). } $$ because the count of each forbidden residue class in a finite interval differs from: $$ N/d $$ by at most: $$ 1. $$ This supplies the local model required by a standard finite sieve sequence. --- # 8. Coarse Lower-Sieve Provider Choose: $$ y0 $$ requires: $$ s>\beta_2. $$ Thus, with the natural: $$ D_y=X^{1-o(1)} $$ as the distribution level, choosing: $$ y=X^{1/u} $$ requires: $$ \boxed{ u>\beta_2. } $$ --- # 10. Natural Coarse Choice Therefore: $$ \boxed{ u\approx5 } $$ or: $$ \boxed{ y\approx X^{1/5} } $$ is the first simple classical positive lower-sieve region. This is precisely the SRBRD-16: # lower-cutoff P4 boundary --- # 11. Pair-Buchstab Shell Upper Provider For: $$ yz_3, $$ If one does not use the SRBRD-16 fact that the cofactor is automatically prime, the row bulk can be treated as: $$ u $$ simultaneously avoiding: - $$ u\equiv0\pmod p; $$ - $$ u\equiv2mr^{-1}\pmod p. $$ in a dimension-2 sieve. This is the: # Unswitched Row Sieve --- # 16. Unswitched Row Parameter $$ r=X^t, \qquad t\in[1/3,1/2]. $$ row length： $$ \asymp H/r. $$ cutoff： $$ z_3\asymp X^{1/3}. $$ Thus, the natural parameter is: $$ \boxed{ s_R(t) \approx 3(1-t). } $$ Therefore: $$ \boxed{ s_R(t)\in[3/2,2]. } $$ This is exactly the low-$s$ region where the dimension-2 upper sieve is very loose. --- # 17. External Low-$s$ DHR Upper Function Under the standard DHR normalization, For: $$ 0 0. } $$ This is very important: > **The ideal local-density architecture of SRBRD-16's four modules itself has a positive margin.** Thus, the present failure is not caused by the algebraic identity itself. --- # 22. DHR Unswitched Row Cost Substituting T17: $$ F_2(s) = 8e^{2\gamma}/s^2 $$ into: $$ s_R(t)=3(1-t), $$ the naive DHR row-subtraction coefficient is: $$ 18 \int_{1/3}^{1/2} \frac{ F_2(3(1-t)) }{t} dt. $$ Thus: $$ = 16e^{2\gamma} \int_{1/3}^{1/2} \frac{ dt }{ t(1-t)^2 }. $$ --- # 23. Exact Integral Since: $$ \frac1{t(1-t)^2} = \frac1t + \frac1{1-t} + \frac1{(1-t)^2}, $$ Thus: $$ \int_{1/3}^{1/2} \frac{ dt }{ t(1-t)^2 } = \boxed{ \frac12+\log2. } $$ Therefore: ## Diagnostic SRBRD-17-D03 $$ \boxed{ C_{\mathrm{row}}^{DHR} = 16e^{2\gamma} \left( \frac12+\log2 \right) \approx 60.5588. } $$ --- # 24. Quantitative No-Go for Separate Unswitched DHR Rows The ideal coarse-minus-shell budget leaves at most: $$ 9 $$ at this scale. But the standard unswitched DHR row upper bound alone already consumes: $$ \approx60.56. $$ Thus: $$ \boxed{ \text{Even with the coarse lower and shell upper idealized, the naive DHR row upper bound still cannot cross.} } $$ The gap is approximately: $$ \boxed{ 60.56-9 \approx51.56 } $$ normalized units, before transport is counted. --- # 25. Why Transport Alone Cannot Be Expected to Repair a 60.56 Constant In finite controls: $$ M_{SS} $$ is a meaningful positive correction, but its natural scale has the same order as the actual semiprime mass, rather than being: $$ O(10) $$ times larger than the classical row main term. Thus: $$ \boxed{ \text{The row-upper provider itself must improve.} } $$ All hope cannot be placed in: $$ M_{SS}. $$ --- # 26. Prime-Cofactor Switching SRBRD-16 has proved: If: $$ r>z_3, $$ $$ x=ru\le X, $$ and: $$ x $$ is RootRough$_{z_3}$, then: $$ \boxed{ u\in\mathbb P. } $$ Moreover: $$ r,u>z_3 $$ such that: $$ ru $$ is automatically RootRough$_{z_3}$ itself. Thus, the row condition can be simplified. --- # 27. Exact Switched Row Formula ## Theorem SRBRD-17-T03 For: $$ r>z_3, $$ $$ r\le\sqrt X, $$ we have: $$ \boxed{ n_r = \# \left\{ u\in\mathbb P: r\le u\le X/r, \quad \operatorname{RootRough}_{z_3}(2m-ru) \right\}. } $$ There is no longer any need to check: $$ \operatorname{RootRough}_{z_3}(ru). $$ --- # 28. Switched Local Forbidden Class Fix: $$ p\le z_3 $$ as an odd prime. For a prime cofactor: $$ u>z_3, $$ Thus: $$ u\not\equiv0\pmod p. $$ The opposite-endpoint obstruction: $$ p\mid2m-ru $$ is equivalent to: $$ u \equiv 2mr^{-1} \pmod p. $$ --- # 29. Center-Divisor Collapse After Switching If: $$ p\mid m, $$ then: $$ 2mr^{-1} \equiv0\pmod p. $$ However: $$ u>z_3\ge p $$ and: $$ u $$ prime， Thus: $$ u\not\equiv0\pmod p. $$ Therefore: ## Theorem SRBRD-17-T04 within a switched row, the local forbidden-class counts are: $$ \boxed{ \eta_m(p) = \begin{cases} 0, & p\mid m, \\ 1, & p\nmid m. \end{cases} } $$ This is the: # Dimension-1 Complement Sieve --- # 30. Switched Euler Product Define: $$ \boxed{ P_0(z) = \prod_{3\le p\le z} \left( 1-\frac1p \right), } $$ and: $$ \boxed{ W_m(z) = \prod_{\substack{ 3\le p\le z\\ p\nmid m }} \left( 1-\frac1{p-1} \right). } $$ $P_0$ is the local factor for cofactor primality / roughness, and $W_m$ is the factor for sieving the opposite endpoint within the primes. --- # 31. Exact Euler-Factor Switching Identity ## Theorem SRBRD-17-T05 For each: $$ z, $$ $$ \boxed{ V_m(z) = P_0(z)W_m(z). } $$ --- # 32. T05 Local Proof If: $$ p\nmid m, $$ $$ \left( 1-\frac1p \right) \left( 1-\frac1{p-1} \right) = \frac{p-1}{p} \frac{p-2}{p-1} = 1-\frac2p. $$ If: $$ p\mid m, $$ the second factor is absent, Thus: $$ \left( 1-\frac1p \right)\cdot1 = 1-\frac1p. $$ Multiplying over the primes gives the result. --- # 33. Meaning of T05 This is not a probability heuristic. It is an exact Euler-factor identity: $$ \boxed{ \text{dimension 2 pair density} = \text{cofactor-prime density} \times \text{one-residue opposite density}. } $$ Thus, at the local-density level, cofactor switching: $$ \boxed{ \text{preserves the original sieve density completely}. } $$ It does not replace the problem with another heuristic model. --- # 34. Switched Row as a Linear Sieve over Primes T03 + T04 convert: $$ n_r $$ into: > Among prime $u$, avoid, for each: > $$ > p\le z_3,\quad p\nmid m > $$ > one nonzero residue class. Thus: $$ \boxed{ \text{row sieve dimension} : 2 \longrightarrow 1. } $$ The cost is: > The base sequence is no longer all integers, > but primes. This requires prime distribution in arithmetic progressions. --- # 35. External Distribution Interface Typical unconditional interface: # Bombieri–Vinogradov supplies prime distribution, averaged over moduli, at approximately: $$ X^{1/2} $$ as the level of distribution. Combined with a linear upper sieve, this can give a switched row a: $$ \boxed{ \text{dimension-1 upper bound}. } $$ This agrees with the philosophy of Chen switching / weighted sieves. This round does not claim to have completed the corresponding error analysis. --- # 36. Switched Row Provider Contract Define an abstract: $$ \boxed{ U_r^{sw} } $$ such that: $$ n_r\le U_r^{sw}. $$ Its intended analytic construction: 1. prime-cofactor range: $$ r\le u\le X/r; $$ 2. base prime count / prime distribution; 3. local weights: $$ \eta_m(p)=0\text{ or }1; $$ 4. linear upper sieve with product: $$ W_m(z_3); $$ 5. an explicit endpoint frontier. --- # 37. Switched Quantitative Closure In SRBRD-16 T13, replace: $$ U_r^{row} $$ with: $$ U_r^{sw}. $$ ## Theorem SRBRD-17-T06 If: $$ \boxed{ L_y - \sum_{yz_3}U_r^{sw} + L_{SS} >0, } $$ then: $$ \boxed{ R_G^{-}(2m)>0. } $$ T06 is only a provider-composition theorem; the actual analytic work is to prove each provider. --- # 38. Finite Product Calibration The following is computational evidence, not a bound. Define the one-sided semiprime count: $$ \boxed{ U_{SP}^{one} = \sum_{r>z_3} \left[ \pi(X/r)-\pi(r-1) \right]. } $$ This counts only possible prime cofactors, entirely ignoring the opposite RootRough condition. Multiplying by: $$ W_m(z_3), $$ gives: # Switched Independence Main Term --- # 39. Calibration: $m=1000$ $$ z_3=13. $$ $$ U_{SP}^{one}=86. $$ $$ W_m(13)=0.34375. $$ Thus: $$ U_{SP}^{one}W_m = 29.5625. $$ actual： $$ n_{SP}=35. $$ --- # 40. Calibration: $m=5000$ $$ U_{SP}^{one}=489, $$ $$ W_m(22) = 0.304361979\ldots $$ product： $$ \boxed{ 148.8330\ldots } $$ actual： $$ \boxed{ 152. } $$ --- # 41. Calibration: $m=10000$ $$ U_{SP}^{one}=1022, $$ $$ W_m(28) = 0.29052734375. $$ product： $$ \boxed{ 296.9190\ldots } $$ actual： $$ 291. $$ --- # 42. Calibration: $m=100000$ $$ U_{SP}^{one}=8856, $$ $$ W_m(59) = 0.236288503\ldots $$ product： $$ \boxed{ 2092.57099\ldots } $$ actual： $$ 2100. $$ --- # 43. Interpretation of the Calibration For the larger selected controls: $$ U_{SP}^{one}W_m(z_3) $$ is very close to the exact: $$ n_{SP} $$ as displayed above. This supports the interpretation that: $$ \boxed{ W_m(z_3) } $$ does capture the main density of opposite-endpoint RootRough survivors. However: $$ \boxed{ U_{SP}^{one}W_m } $$ is not a rigorous upper bound. For example: $$ m=1000 $$ it is below the actual: $$ 35. $$ Thus, one still needs: - a sieve multiplier; - distribution error; - an endpoint correction. --- # 44. Coarse Survivor Product Calibration Selected finite data also show: $$ |\widehat{\mathcal S}_y| $$ is close to: $$ H V_m(y). $$ For example, the previous audit: | $m$ | $y$ | $V_m(y)$ | $|\widehat{\mathcal S}_y|/(H V_m(y))$ | |---:|---:|---:|---:| | 1000 | 5 | 0.266667 | 1.0070 | | 5000 | 7 | 0.190476 | 1.0021 | | 10000 | 8 | 0.190476 | 1.0019 | | 100000 | 12 | 0.155844 | 1.0000 | This, too, is only finite calibration. --- # 45. Cube-Root Survivor Product Calibration | $m$ | $z_3$ | $V_m(z_3)$ | $|\widehat{\mathcal S}_{z_3}|/(H V_m(z_3))$ | |---:|---:|---:|---:| | 1000 | 13 | 0.131868 | 1.0334 | | 5000 | 22 | 0.104106 | 1.0071 | | 10000 | 28 | 0.095054 | 1.0038 | | 100000 | 59 | 0.063222 | 1.0057 | This explains why the ideal local-density diagnostic fits finite behavior fairly well. --- # 46. Quantitative Constant Gate Let the abstract normalized multipliers be: - coarse lower: $$ L_2(u); $$ - shell upper: $$ B_2(t); $$ - switched row upper: $$ R_1(t); $$ - normalized transport gain: $$ \Phi. $$ Then the continuum diagnostic gate is: $$ \boxed{ \mathcal K(u) = u^2L_2(u) - 2 \int_{1/u}^{1/3} \frac{ B_2(t) }{t^3} dt - 18 \int_{1/3}^{1/2} \frac{ R_1(t) }{t} dt + \Phi. } $$ If: $$ \mathcal K(u)>0 $$ and all asymptotic / remainder providers have been rigorously established, only then is large-$m$ uniform crossing possible. --- # 47. Ideal Gate $$ L_2=B_2=R_1=1, $$ $$ \Phi=0 $$ in this idealized case: $$ \boxed{ \mathcal K_{\mathrm{ideal}} = 9 - 18\log\frac32 \approx1.70163. } $$ Thus, the target constant margin is: $$ \boxed{ \text{small but strictly positive}. } $$ --- # 48. Consequence for Provider Quality Since the ideal margin is only: $$ \approx1.70, $$ a quantitative proof cannot tolerate: - too much loss in the coarse lower bound; - too much inflation in the shell upper bound; - a row-upper constant several times the ideal value. Thus, after SRBRD-17: $$ \boxed{ \text{constant quality itself is a theorem gate.} } $$ A correct big-$O$ order alone is not enough. --- # 49. Why Standard DHR Row Upper Fails the Gate Unswitched DHR row contribution: $$ \approx60.56. $$ ideal row： $$ \approx7.30. $$ ratio： $$ \boxed{ \approx8.30. } $$ Thus, the standard dimension-2 row upper bound is far from the required constant regime. This quantitatively proves: $$ \boxed{ \text{Switching is not an optimization; it is a necessary architectural change.} } $$ --- # 50. Shell Provider Is the Next Constant Risk Even if row switching succeeds, the pair-Buchstab shell still uses: $$ s_B(t)\in[2,u-1]. $$ Near: $$ t=1/3 $$ as the endpoint of the band: $$ s_B\to2, $$ the dimension-2 upper factor also becomes looser. Thus, subsequent work may also need: - shell switching; - weighted Buchstab decomposition; - dyadic treatment; - exact center-divisor gains. --- # 51. Three Quantitative Regimes After SRBRD-17, the analytic work can be divided into: ## Regime A — Low Cutoff $$ p\le y\approx X^{1/5}. $$ Use a dimension-2 lower sieve. ## Regime B — Descent Band $$ y The semiprime row of SRBRD naturally requires the same type of switching, rather than incidentally borrowing an external technique. --- # 56. What Would Count as Real Progress Next Only completion of at least one of the following is a quantitative advance: ### Gate A Establish an explicit switched row upper bound: $$ \boxed{ n_r \le U_r^{sw} } $$ with an aggregate constant close to the: $$ W_m(z_3) $$ main term. ### Gate B For the pair-Buchstab shell, find a weighted / switched upper bound better than raw DHR: $$ F_2(s_B) $$ as displayed above. ### Gate C Prove that transport: $$ L_{SS} $$ supplies a fixed positive constant at the normalized scale. ### Gate D Insert A+B+C into T06, and prove: $$ \boxed{ \mathcal K(u)>0 } $$ for some： $$ u>\beta_2. $$ At present: $$ \boxed{ A,B,C,D \text{ have all not yet been completed in full.} } $$ --- # 57. Formal Theorem Queue ```yaml SRBRD_17: T01: center_divisor_factorization_of_Vm T02: finite_interval_local_remainder T03: exact_prime_cofactor_switched_row T04: switched_local_dimension_profile T05: exact_Euler_factor_switching_V_equals_P0_times_W T06: switched_provider_composition_closure ``` --- # 58. External Quantitative Audit Queue ```yaml ExternalAudit: DHR: beta2: approx: 4.26645 lower_function: f2 upper_function: F2 low_s_formula: F2_s_le_2: 8*exp(2*gamma)/s^2 linear_sieve: switched_row: dimension: 1 BombieriVinogradov: use: prime_cofactor_distribution_average BrunTitchmarsh: use: isolated_prime_progression_upper ``` --- # 59. Acceptance Gate Before SRBRD-17 enters the formal core, the following are needed: 1. exact local factor: $$ \nu_m(p); $$ 2. exact: $$ V_m=V_0\times\text{center bonus}; $$ 3. an exact interval remainder; 4. the coarse bulk restriction; 5. shell provider clearly conditional on an external sieve theorem; 6. the exact switched-row theorem; 7. the switched forbidden-class theorem; 8. exact Euler factorization: $$ V_m=P_0W_m; $$ 9. continuum calculations marked as diagnostic; 10. the DHR low-$s$ row no-go not confused with an impossibility theorem; 11. finite calibration not treated as a bound; 12. no Goldbach proof claim. --- # 60. Verdict This round can give: $$ \boxed{ \mathrm{SRBRD\mbox{-}17\ QUANTITATIVE\ SWITCHING\ KERNEL\ PASS}. } $$ The two most important results: ## Exact Structural Result $$ \boxed{ V_m(z) = P_0(z)W_m(z). } $$ That is: $$ \boxed{ \text{dimension-2 pair density} = \text{cofactor-prime factor} \times \text{opposite-endpoint one-residue factor}. } $$ ## Quantitative Barrier Result The standard unswitched dimension-2 DHR row upper bound gives, in the natural semiprime band, a normalized cost: $$ \boxed{ \approx60.56, } $$ whereas the total available margin of the ideal architecture is only: $$ \boxed{ \approx1.70. } $$ Thus: $$ \boxed{ \text{The naive DHR row provider cannot close; prime-cofactor switching is necessary.} } $$ However: $$ \boxed{ \mathrm{SWITCHED\ UNIFORM\ CONSTANT\ CROSSING\ NOT\ YET\ PROVED}. } $$ and: $$ \boxed{ \mathrm{GOLDBACH\ STILL\ OPEN}. } $$ --- # 61. Suggestion for the Next Round # SRBRD-18 ## BV-Switched Row / Weighted Buchstab Constant Closure The next round should stop expanding the algebra. Work only on quantitative providers: ### Part A — Switched Row For: $$ n_r = \#\left\{ u\in\mathbb P: r\le u\le X/r, \quad 2m-ru\text{ RootRough}_{z_3} \right\} $$ Use: - Bombieri–Vinogradov; - the linear upper sieve; - dyadic $r$ blocks; to establish an explicit: $$ U_r^{sw}. $$ ### Part B — Buchstab Band For: $$ y0? } $$ This would be the round currently closest to a genuine analytic crossing test. --- # Appendix A: Exact Local Factorization ```yaml local_pair_classes: if_p_divides_m: nu_m_p: 1 else: nu_m_p: 2 dimension2_product: V_m_z: product: 1 - nu_m_p/p prime_cofactor_factor: P0_z: product: 1 - 1/p opposite_prime_sieve_factor: W_m_z: for_p_not_dividing_m: product: 1 - 1/(p-1) identity: V_m_z: P0_z * W_m_z ``` --- # Appendix B: Ideal Constant Ledger ```yaml coarse_minus_shell: normalized: 9 double_semiprime_row_cost: normalized: 18*log(3/2) decimal: 7.2983719459 zero_transport_margin: normalized: 9 - 18*log(3/2) decimal: 1.7016280541 ``` --- # Appendix C: Naive DHR Row Barrier ```yaml row_parameter: t: 1/3 <= t <= 1/2 s_R: 3*(1-t) DHR_dimension2_upper: for_s_le_2: F2(s): 8*exp(2*gamma)/s^2 normalized_double_row_cost: exact: 16*exp(2*gamma)*(1/2+log(2)) decimal: 60.5587856961 diagnosis: naive_unswitched_DHR_row: quantitatively_too_loose: true ``` --- # Appendix D: Switched Row Contract ```yaml row_r: r: prime z3 < r <= sqrt(X) exact: n_r: count_prime_u: r <= u <= X/r RootRough_z3(2m-r*u) local_opposite_classes: if_p_divides_m: forbidden_nonzero_classes: 0 else: forbidden_nonzero_classes: 1 sieve_dimension_after_switch: 1 ``` --- # Appendix E: Finite Switching Calibration | $m$ | $z_3$ | one-sided prime-cofactor count | $W_m(z_3)$ | product | exact $n_{SP}$ | |---:|---:|---:|---:|---:|---:| | 1000 | 13 | 86 | 0.343750 | 29.5625 | 35 | | 5000 | 22 | 489 | 0.304362 | 148.8330 | 152 | | 10000 | 28 | 1022 | 0.290527 | 296.9190 | 291 | | 100000 | 59 | 8856 | 0.236289 | 2092.5710 | 2100 | **Note:** `one-sided count × W_m` is only density calibration, not a rigorous upper/lower bound. --- # Appendix F: Commit Decision ```yaml research_round: SRBRD-17 decision: quantitative_constant_gate: true naive_dimension2_row_no_go: true exact_prime_cofactor_switching: true exact_local_Euler_factor_switching: true finite_switching_calibration: true switched_uniform_crossing: false goldbach_proved: false next: SRBRD-18_BVSwitchedRow_WeightedBuchstab_Constant_Closure ``` --- # References - H. G. Diamond and H. Halberstam, *A Higher-Dimensional Sieve Method*, Cambridge Tracts in Mathematics 177, Cambridge University Press, 2008. - C. S. Franze, “Sifting limits for the $\Lambda^2\Lambda^-$ sieve,” *Journal of Number Theory* 131 (2011), 1962–1982. - A. R. Booker and T. D. Browning, “Square-free values of reducible polynomials,” *Discrete Analysis* 2016:8. - T. Tao, “254A, Supplement 5: The linear sieve and Chen’s theorem,” 2015. - Encyclopedia of Mathematics, “Brun–Titchmarsh theorem.”