> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # SRBRD-25 ## Structured-Prime Provider / Shift-Free Dispersion Hybrid Closure ### Complement-Exponent Map, Maynard-III Exact Range Audit, $3/8$ Type-II Optimum, $5/8\leftrightarrow3/8$ Weight-Class Duality, and Residual Dispersion Region **Parent program:** RG-SCP / Reverse Goldbach Structural Closure Program **Branch:** SRBRD — Symmetric Root-Bounded Relation Domain **Round:** SRBRD-25 **Date:** 2026-09-30 **Version:** v0.1 **Authors:** Neo.K; Aletheia (GPT-5.6 Sol) **Nature:** Exact External-Range Match / Hybrid Coverage Map / Residual-Band Localization **Prerequisites:** SRBRD-00 through SRBRD-24 **Status:** Aligns Maynard-III uniform large-modulus technology with complement-switched low-row geometry scale by scale; proves that the current linear-sieve weight class cannot let the structured-prime route cover the entire low row; establishes an explicit residual dispersion region; Goldbach remains unproved --- # 0. Purpose of This Round SRBRD-24 established two analytic routes: ## Route P — Structured Prime Provider For the switched modulus: $$ K=kq_2 $$ directly use a uniform large-modulus prime / convolution theorem. ## Route D — Shift-Free Dispersion For the affine lattice: $$ nv+Kq_1=A_X $$ use the shift-free off-diagonal geometry of: $$ n\Delta v+K\Delta q_1=0 $$ as displayed above. The SRBRD-24 strategy is: > Route P first handles as much as it can, > and Route D handles only the remaining region. The sole task of SRBRD-25: $$ \boxed{ \text{Precisely map the genuinely legal / conditional coverage region of Route P.} } $$ This round finds: 1. the complement's magnitude cannot be ignored; 2. Maynard-III Theorem 1.1 can only potentially handle small / medium complementary divisors; 3. after optimization, the arbitrary-convolution lower endpoint of Proposition 5.2 is exactly: $$ 3/8; $$ 4. if the top complement is: $$ W\asymp X, $$ then lowering the complementary divisor to: $$ X^{3/8}, $$ requires precisely the original modulus level: $$ 5/8; $$ 5. this is exactly dual to the triply-well-factorable level of Pascadi 2025: $$ 5/8 $$ as displayed above; 6. but the known level for the current upper linear-sieve weights is: $$ 3/5, $$ not: $$ 5/8. $$ Therefore: $$ \boxed{ \text{5/8 route is a weight-class upgrade route, not a currently legal SRBRD linear-sieve provider.} } $$ --- # 1. Add the Complement Exponent SRBRD-22 / 23 repeatedly use: $$ w=A_X-nv. $$ Now apply a dyadic split to: $$ w $$ as follows: $$ \boxed{ w\sim W=X^\omega. } $$ Original modulus: $$ \boxed{ q\sim Q=X^\theta. } $$ Since: $$ q\mid w, $$ The actual block must satisfy: $$ \boxed{ \omega\ge\theta. } $$ Complementary divisor: $$ \boxed{ k=\frac wq. } $$ Thus: ## Theorem SRBRD-25-T01 $$ \boxed{ k\asymp X^\kappa, \qquad \kappa=\omega-\theta. } $$ up to dyadic constants. --- # 2. Low-Row Rectangle Low prime factor: $$ \boxed{ n\sim N=X^t, \qquad \frac13\le t\le\frac25. } $$ Large prime: $$ \boxed{ v\sim M=X^{1-t}. } $$ If the current original level is: $$ \boxed{ \theta=\frac35, } $$ then: $$ \boxed{ 0\le\kappa\le\frac25. } $$ Thus, the low-row switched geometry lives in: $$ \boxed{ \mathcal R_{\mathrm{low}} = \left\{ (t,\kappa): \frac13\le t\le\frac25, \quad 0\le\kappa\le\frac25 \right\}. } $$ --- # 3. Bare Prime-Side Modulus If no inherited factor is present at all, the prime: $$ v $$ lies in a progression modulo: $$ k. $$ Relative to the prime scale: $$ M=X^{1-t}, $$ the modulus exponent is: ## Theorem SRBRD-25-T02 $$ \boxed{ \rho_0(t,\kappa) = \frac{ \kappa }{ 1-t }. } $$ --- # 4. Ordinary BV Region Ordinary Bombieri–Vinogradov prime-side shape: $$ k\lesssim M^{1/2-o(1)}. $$ Thus: ## Definition SRBRD-25-D01 $$ \boxed{ \mathcal R_{\mathrm{BV}} = \left\{ (t,\kappa): \kappa< \frac{1-t}{2} \right\}. } $$ Selected boundaries: ### $t=1/3$ $$ \kappa<1/3. $$ ### $t=3/8$ $$ \kappa<5/16. $$ ### $t=2/5$ $$ \kappa<3/10. $$ --- # 5. Maynard-III Theorem 1.1 Shape Maynard-III Theorem 1.1: Prime-variable scale: $$ x $$ Modulus: $$ q_1q_2 $$ with: $$ q_1 \lesssim x^{1/10-3\delta}, $$ $$ q_2 \lesssim x^{4/10+4\delta}. $$ Total: $$ q_1q_2 \lesssim x^{1/2+\delta}. $$ The residue is completely uniform. Its error has a weak relative-error form: $$ O \left( \delta\pi(x) + \frac{ x(\log\log x)^2 }{ (\log x)^2 } \right). $$ Thus: $$ \boxed{ \text{shape match} } $$ and: $$ \boxed{ \text{constant-ledger provider} } $$ must be distinguished. --- # 6. Map Theorem 1.1 to the Switched Modulus In SRBRD Route P: - prime variable: $$ v\asymp M; $$ - large switched cofactor: $$ k; $$ - inherited original factor: $$ q_2^{\mathrm{inh}}. $$ The most natural mapping: $$ \boxed{ \text{Maynard small factor} \leftrightarrow q_2^{\mathrm{inh}}, } $$ $$ \boxed{ \text{Maynard large factor} \leftrightarrow k. } $$ Therefore: $$ \boxed{ k \lesssim M^{2/5+4\delta}. } $$ --- # 7. Limiting Uniform-AP Shape Threshold Theorem 1.1 requires: $$ 1/10-3\delta>0 $$ for a nontrivially growing convenient factor. Thus: $$ \boxed{ \delta<1/30. } $$ Let: $$ \delta\to1/30^{-} $$ approach this limit: $$ 2/5+4\delta \to \boxed{ 8/15. } $$ Therefore: ## Theorem-Match SRBRD-25-M01 In the purely shape-level limit, the bare complementary exponent can be pushed by the uniform prime-AP template to: $$ \boxed{ \kappa < \frac{ 8 }{ 15 } (1-t). } $$ provided: - an inherited convenient factor exists; - the factor mapping is legal; - the weak error is acceptable. --- # 8. Shape Region Define: $$ \boxed{ \kappa_{\mathrm{UAP}}(t) = \frac{ 8 }{ 15 } (1-t). } $$ Selected cases: ### $t=1/3$ $$ \boxed{ \kappa_{\mathrm{UAP}} = 16/45 \approx0.35556. } $$ ### $t=3/8$ $$ \boxed{ \kappa_{\mathrm{UAP}} = 1/3. } $$ ### $t=2/5$ $$ \boxed{ \kappa_{\mathrm{UAP}} = 8/25 = 0.32. } $$ Thus, even with the limiting shape of Theorem 1.1, the top complement: $$ \kappa=2/5 $$ remains uncovered. --- # 9. Weak-Error Firewall The main error in Theorem 1.1 contains: $$ C\delta\pi(x). $$ In the published statement, $$ C $$ is a sufficiently large absolute constant. The SRBRD constant ledger has only a small positive margin. Thus: ## Research Firewall SRBRD-25-R01 $$ \boxed{ \mathcal R_{\mathrm{UAP}} \text{ is a shape-coverage region, not yet a constant-admissible SRBRD provider region.} } $$ One cannot directly treat: $$ 8/15 $$ as a usable quantitative crossing constant. --- # 10. Theorem 1.2 Good-Error Regime Maynard-III Theorem 1.2: $$ \delta<1/1000 $$ and gives good error of order: $$ x/(\log x)^A $$ as displayed above, but its total modulus is still: $$ x^{1/2+\delta}. $$ Thus: > The good-error uniform theorem extends only slightly beyond the square root. Therefore: $$ \boxed{ \text{good-error Route P extension beyond ordinary BV is tiny.} } $$ This is: # Shape vs Ledger Strength Separation --- # 11. Ambient-$X$ Type-II Route Another Route P: instead of viewing: $$ v $$ as an isolated prime AP, directly apply the Maynard-III uniform Type-II propositions to: $$ nv\asymp X $$ as displayed above. The ambient scale then returns to: $$ \boxed{ X. } $$ Switched congruence: $$ \boxed{ nv\equiv A_X\pmod K. } $$ --- # 12. Proposition 5.2 Limiting Constraints Maynard-III Proposition 5.2: Let the modulus factors be: $$ Q_1Q_2Q_3=X^{1/2+\delta}. $$ Ignore: $$ \delta,\varepsilon $$ to pass to the limiting exponent layer. Write: $$ Q_i=X^{a_i}. $$ Let: $$ a=a_1, \quad b=a_2, \quad c=a_3. $$ Then: $$ \boxed{ a+b+c=1/2. } $$ Constraints: $$ \boxed{ b\le c, } $$ $$ \boxed{ c\le 1/10 - 3b/5. } $$ And the convolution factor: $$ N=X^t $$ requires: $$ \boxed{ t> \max \left( a, b+c+1/4 \right). } $$ --- # 13. Exact Optimization Let: $$ s=b+c. $$ then: $$ a=1/2-s. $$ From: $$ b\le c=s-b $$ we obtain: $$ b\le s/2. $$ From: $$ c=s-b \le 1/10-3b/5 $$ we obtain: $$ s-1/10 \le 2b/5, $$ That is: $$ \boxed{ b \ge \frac52 \left( s-\frac1{10} \right). } $$ Feasibility requires: $$ \frac52 \left( s-\frac1{10} \right) \le \frac s2. $$ Thus: ## Theorem SRBRD-25-T03 $$ \boxed{ s\le1/8. } $$ Therefore: $$ \boxed{ a\ge3/8. } $$ --- # 14. Proposition 5.2 Optimal Lower Endpoint Lower factor threshold: $$ L(s) = \max \left( 1/2-s, s+1/4 \right). $$ Among feasible values: $$ s\le1/8 $$ as displayed above, the two terms are equal at: $$ s=1/8 $$ with common value: $$ 1/2-1/8 = 1/8+1/4 = 3/8. $$ Thus: ## Theorem SRBRD-25-T04 The smallest arbitrary-convolution factor in the limiting exponent geometry of Maynard-III Proposition 5.2 is: $$ \boxed{ t_{\min}=3/8. } $$ The unique symmetric optimum: $$ \boxed{ (a,b,c) = \left( \frac38, \frac1{16}, \frac1{16} \right). } $$ --- # 15. Meaning of the $3/8$ Barrier This does not mean: > primes cannot be handled at: > $$ > N $$ > as displayed above. The paper itself also contains prime-specific decompositions such as Proposition 5.4. The correct statement: $$ \boxed{ \text{Proposition 5.2 arbitrary bilinear convolution architecture cannot push its small factor below }X^{3/8} \text{ in the limiting exponent model.} } $$ --- # 16. Irreducible Complementary Block In the complement-switched modulus: $$ k $$ has no inherited factorization. If: $$ k>X^{1/10+o(1)}, $$ then, in the factor geometry of Proposition 5.2, it cannot fit into: $$ Q_2 $$ or: $$ Q_3, $$ because, in the optimized / admissible regime, both factors are approximately: $$ \le X^{1/10+o(1)}. $$ Thus: ## Theorem SRBRD-25-T05 The large complementary block: $$ k $$ must be absorbed into: $$ \boxed{ Q_1. } $$ Hence, the lower condition in Proposition 5.2 forces: $$ \boxed{ N > k } $$ up to $X^{o(1)}$ margins. In exponent notation: $$ \boxed{ t>\kappa. } $$ --- # 17. Top Complement Under Current $3/5$ Level If: $$ \theta=3/5, $$ Top complement: $$ W\asymp X $$ gives: $$ \boxed{ \kappa=2/5. } $$ T05 requires: $$ t>2/5. $$ but the low row has: $$ t\le2/5. $$ Thus: ## Corollary SRBRD-25-C01 $$ \boxed{ \text{with current }3/5\text{ original level, Proposition 5.2 cannot cover the top-complement low-row interior.} } $$ This is a factor-geometry obstruction, independent of residue uniformity. --- # 18. General Complement-Level Threshold For the top complement: $$ W\asymp X, $$ we have: $$ \kappa=1-\theta. $$ If Proposition 5.2 is to handle the row: $$ t, $$ it is necessary that: $$ 1-\theta1-t. } $$ This is: # Complementary-Level Threshold --- # 19. Selected Thresholds ### To reach $t=2/5$ need: $$ \boxed{ \theta>3/5. } $$ ### To reach $t=3/8$ need: $$ \boxed{ \theta>5/8. } $$ ### To reach $t=1/3$ necessary from irreducible-$k$ alone: $$ \boxed{ \theta>2/3. } $$ However, Proposition 5.2 itself already has the limiting barrier: $$ t\ge3/8 $$ as displayed above, Thus: $$ 2/3 $$ will not let Proposition 5.2 handle: $$ t=1/3. $$ --- # 20. The $5/8\leftrightarrow3/8$ Duality If: $$ \boxed{ \theta=5/8, } $$ Top complement: $$ W\asymp X $$ gives: $$ \boxed{ k\asymp X^{3/8}. } $$ Meanwhile, the Proposition 5.2 optimum: $$ \boxed{ Q_1\asymp X^{3/8}, } $$ $$ \boxed{ Q_2,Q_3\asymp X^{1/16}. } $$ Thus: ## Theorem-Match SRBRD-25-M02 $$ \boxed{ 5/8 \longleftrightarrow 3/8 } $$ form an exact complement / Type-II scale alignment. This is one of this round's most elegant structural coincidences. --- # 21. Why This Is Not Yet a Legal SRBRD Provider Pascadi 2025: - primes with triply-well-factorable weights: $$ \boxed{ 5/8-\varepsilon; } $$ - well-factorable upper-bound linear sieve weights: $$ \boxed{ 3/5-\varepsilon. } $$ Earlier Maynard 2025: - triply-well-factorable: $$ 3/5-\varepsilon; $$ - upper-bound linear sieve: $$ 7/12-\varepsilon. $$ Therefore: ## Research Firewall SRBRD-25-R02 $$ \boxed{ \text{the }5/8\text{ level does not belong to the same upper linear-sieve weight class currently used by SRBRD row upper bounds.} } $$ One cannot directly insert: $$ 5/8 $$ into: $$ \theta $$ without changing the weight architecture. --- # 22. Weight-Class Upgrade Route This instead produces a new research route: # WCU ## Weight-Class Upgrade Change from: $$ \boxed{ \beta=1 \text{ upper linear-sieve weights} } $$ to an upper / weighted sieve object closer to: $$ \boxed{ \text{triply-well-factorable} } $$ as displayed above, trading off: - sieve-constant efficiency; - factorability; - distribution level. Target: $$ \boxed{ \theta=5/8 } $$ to unlock the top-complement Type-II region at: $$ t\ge3/8 $$ as displayed above. --- # 23. But WCU Has a Constant Cost A more factorable sieve weight need not have an upper-bound constant as good as the linear sieve's: $$ \beta=1 $$ as displayed above. SRBRD-18 has proved: $$ \boxed{ \text{constant quality is theorem gate}. } $$ Thus, WCU cannot consider only: $$ 5/8. $$ The row penalty must be recomputed: $$ \boxed{ C_R } $$ as displayed above. --- # 24. Complement-Aware Coverage Map For the current: $$ \theta=3/5, $$ Define: $$ \boxed{ \kappa=\omega-3/5. } $$ where: $$ 0\le\kappa\le2/5. $$ we can distinguish four regions. --- # 25. Region P0 — Classical Small Complement $$ \boxed{ \kappa< \frac{1-t}{2}. } $$ The prime: $$ v $$ modulo the bare: $$ k $$ lies in the ordinary BV square-root range. This is the cleanest Route P shape. --- # 26. Region P1 — Uniform Large-Modulus Shape Extension Limiting shape of Maynard-III Theorem 1.1: $$ \boxed{ \frac{1-t}{2} \lesssim \kappa < \frac8{15}(1-t). } $$ Requires: - an inherited convenient factor; - theorem-factor mapping; - a weak-error constant audit. Thus: $$ \boxed{ \mathrm{P1=SHAPE\ MATCH,\ NOT\ LEDGER\ PASS}. } $$ --- # 27. Region P2 — Uniform Type-II Top Band Conditional Proposition 5.2 structure: $$ \boxed{ t>3/8, } $$ and: $$ \boxed{ 3/8 \lesssim \kappa < t. } $$ Additionally require: - two small inherited factors; - an appropriate 3-factor modulus decomposition; - prime coefficients / SW; - good error. However: $$ \boxed{ \text{current }\beta=1\text{ linear-sieve weight at }3/5 \text{ does not automatically provide the required triply-factorable geometry}. } $$ Thus: $$ \boxed{ \mathrm{P2=CONDITIONAL}. } $$ --- # 28. Residual Dispersion Region Define: $$ \boxed{ \mathcal R_D = \mathcal R_{\mathrm{low}} \setminus \left( \mathcal R_{P0} \cup \mathcal R_{P1} \cup \mathcal R_{P2} \right). } $$ Note: - P1 does not pass the constant gate; - P2 is not legal for the current weights; so this is: # optimistic structural residual not the final rigorous residual. --- # 29. Selected Residual Slices ### $t=1/3$ $$ \kappa_{\mathrm{P1,max}} = 16/45 \approx0.35556. $$ P2 does not exist. Thus, the optimistic residual is at least: $$ \boxed{ 0.35556 \lesssim \kappa \le0.4. } $$ with width approximately: $$ 0.04444. $$ --- # 30. $t=3/8$ $$ \kappa_{\mathrm{P1,max}} = 1/3. $$ P2 degenerates at the limiting endpoint. Thus, the residual: $$ \boxed{ 1/3 \lesssim \kappa \le2/5. } $$ Width: $$ 1/15 \approx0.06667. $$ --- # 31. $t=2/5$ $$ \kappa_{\mathrm{P1,max}} = 8/25 = 0.32. $$ Conditional P2: $$ \boxed{ 3/8 \lesssim \kappa < 2/5. } $$ Thus, the main optimistic-residual gap: $$ \boxed{ 0.32 \lesssim \kappa < 0.375. } $$ Width: $$ 0.055. $$ Endpoint: $$ \kappa=0.4 $$ is a limiting boundary. --- # 32. Main Hybrid Conclusion Even using: - the limiting Theorem 1.1 shape; - an optimistic Proposition 5.2 factor match; we still have: $$ \boxed{ \mathcal R_D\neq\varnothing } $$ throughout the low-row continuum. Thus: ## Theorem-Match SRBRD-25-M03 $$ \boxed{ \text{first-moment structured-prime technology alone does not eliminate the need for Route D.} } $$ --- # 33. Route D Burden Is Smaller Than Full Rectangle Although: $$ \mathcal R_D $$ is nonempty, its complement-exponent width is only approximately: $$ 0.04\text{--}0.07 $$ in selected slices. Compared with the full: $$ \kappa\in[0,0.4], $$ the analytic burden has been substantially reduced. Thus, the hybrid philosophy of SRBRD-24 still succeeds: $$ \boxed{ \text{do not prove SFLD on the whole low rectangle; only prove it on the narrow residual band.} } $$ --- # 34. Current-Weight Conservative Map If we strictly do not use: - the weak error in Theorem 1.1 as a ledger provider; - triply-factorable P2; then the current conservative Route P retains only: $$ \boxed{ \mathcal R_{P0}. } $$ The residual is then: $$ \boxed{ \kappa \ge \frac{1-t}{2}. } $$ Selected cases: | $t$ | conservative residual begins at | |---:|---:| | $1/3$ | $1/3$ | | $3/8$ | $5/16$ | | $2/5$ | $3/10$ | Thus, SFLD still needs a substantial, but not the entire, complement region. --- # 35. Good-Error Uniform Theorem Does Not Change the Picture Much Maynard-III Theorem 1.2: $$ \delta<1/1000. $$ Hence, the total modulus relative to the prime scale: $$ 1/2+\delta $$ is only slightly above: $$ 1/2. $$ Thus: $$ \boxed{ \text{constant-safe uniform prime-AP technology currently expands P0 only slightly.} } $$ This again shows: > A large headline uniformity range and a usable ledger range are not the same thing. --- # 36. Proposition 5.4 Does Not Directly Solve the Residual Maynard-III itself explicitly states: Propositions 5.1–5.3 fail around: $$ N\approx X^{1/3}, $$ $$ M\approx X^{2/3}. $$ To estimate primes, the paper instead uses Proposition 5.4: - one small arbitrary factor; - three smooth factors; - a ternary-divisor-like decomposition. However, the current SRBRD residual object is: $$ \boxed{ \text{prime}\times\text{prime switched correlation} } $$ with a factorable modulus / lattice coefficient. Thus: ## Research Firewall SRBRD-25-R03 $$ \boxed{ \text{Proposition 5.4 is a technology hint, not a direct DFSPC/SFLD provider.} } $$ If it is used in the future, an explicit Heath-Brown / Buchstab coefficient-decomposition match must be performed first. --- # 37. $5/8$ Technology Reinterpreted The Pascadi 2025 level: $$ 5/8 $$ should not be viewed merely as: > a higher prime-distribution exponent. SRBRD-25 shows that it also has a special complement meaning: $$ \boxed{ 1-\frac58 = \frac38. } $$ And: $$ 3/8 $$ is precisely the limiting arbitrary-convolution endpoint of Maynard-III Proposition 5.2. Thus: # $5/8$ Is a Complement-Matching Threshold This is a structure genuinely worth exploiting later. --- # 38. But Current Upper Linear Weights Stop at $3/5$ Current unconditional summary of Pascadi 2025: $$ \boxed{ \text{triply-WF primes}:5/8-\varepsilon, } $$ $$ \boxed{ \text{WF upper linear-sieve weights}:3/5-\varepsilon. } $$ Thus: $$ \boxed{ 1-\frac35 = \frac25, } $$ rather than: $$ 3/8. $$ This is why the top complement for the current linear-sieve class: $$ k\sim X^{2/5} $$ gets stuck exactly at the uppermost low-row boundary, unable to enter the open interior of Prop 5.2. --- # 39. Weight-Class / Complement-Depth Principle This round obtains a more general design principle: If the original upper weight can reach the level: $$ \theta, $$ then the top-complement irreducible block is: $$ \boxed{ X^{1-\theta}. } $$ If the minimum factor endpoint of the external arbitrary-convolution theorem is: $$ \alpha, $$ then admitting the top complement into that theorem requires: $$ \boxed{ 1-\theta\le\alpha. } $$ That is: ## Theorem SRBRD-25-T07 $$ \boxed{ \theta\ge1-\alpha. } $$ For: $$ \alpha=3/8, $$ we obtain: $$ \boxed{ \theta\ge5/8. } $$ --- # 40. This Is a General Architecture Rule $$ \boxed{ \text{distribution level of original weight} + \text{small-factor depth of switched theorem} \ge1. } $$ may be called: # Complement–Factor Threshold It is not a Goldbach-specific theorem, but a direct consequence of: $$ qk\le X $$ and the switched theorem's small-factor requirement. --- # 41. Shift-Free Dispersion Region Has a New Natural Lower Boundary The region where Route D is genuinely necessary includes at least: $$ \boxed{ \kappa \ge \kappa_{\mathrm{provider}}(t), } $$ where: $$ \kappa_{\mathrm{provider}} $$ is jointly determined by: - the uniform prime-AP range; - Type-II factor geometry; - the weight class; as listed above. Thus, the SFLD of SRBRD-24 should not be proved globally. It should be parameterized: $$ \boxed{ \mathrm{SFLD}(t,\kappa). } $$ --- # 42. Residual Geometry Is Exactly Where Lattice Length Is Shortest SRBRD-24: $$ T\asymp X^{\theta-\sigma-t}. $$ Large: $$ \kappa $$ corresponds to a large complement / difficult switched modulus. Under factor-length conservation, this is also the region with shorter progressions. Thus: $$ \boxed{ \text{The Route D residual is precisely where the elementary lattice sieve is weakest.} } $$ This is not good news, but it pinpoints the true analytic hard core very cleanly. --- # 43. Why This Is Still Progress Previously, the problem was: > the entire binary Goldbach problem. After SRBRD-25, the analytic hard core becomes: > small prime: > $$ > t\in[1/3,2/5], > $$ > complement exponent: > $$ > \kappa > $$ > lying in a narrow residual strip, > with a coefficient-aware second-moment estimate to prove for a shift-free affine lattice. This is a very different target. --- # 44. New Target — RSFLD Define: # RSFLD ## Residual Shift-Free Lattice Dispersion Require only that, for: $$ \boxed{ (t,\kappa)\in\mathcal R_D } $$ one prove: $$ \boxed{ \sum_{h\ne0} \mathcal R_A(n,K;h) \ll \frac{ T^2 }{ \log^2M } \times C(t,\kappa) } $$ and: $$ C(t,\kappa) $$ is sufficient to pass the constant ledger. This is weaker than the full SFLD target of SRBRD-24. --- # 45. RSFLD Can Exploit Boundary Geometry Since: $$ \mathcal R_D $$ is a narrow band, one can use: - stronger dyadic localization; - endpoint-specific Kloosterman estimates; - factor-length conservation; - the exact relation: $$ \rho+\tau=1; $$ - local transport discounts. A universally uniform theorem is not needed. --- # 46. Constant-Ledger Warning Even if the shape residual is narrow, its contribution need not be proportional to geometric width. Sieve-density integrals often carry: $$ 1/t, $$ $$ 1/(1-t), $$ or a large-modulus multiplier. Thus: $$ \boxed{ \text{narrow region} \neq \text{small constant automatically}. } $$ The next round must actually integrate the residual weight. --- # 47. Formal Theorem Queue ```yaml SRBRD_25: T01: complement_exponent: kappa: omega-theta T02: prime_side_relative_modulus_exponent T03: Prop5_2_feasibility_implies_s_le_1_8 T04: Prop5_2_limiting_small_factor: 3/8 T05: irreducible_k_must_enter_large_modulus_factor T06: top_complement_level_threshold: theta > 1-t T07: complement_factor_threshold: theta >= 1-alpha ``` --- # 48. External-Match Queue ```yaml MaynardIII_Theorem1_1: residue: complete_uniformity: true convenient_factor: exponent_max: 1/10-3delta large_factor: exponent_max: 4/10+4delta total: 1/2+delta error: weak_relative_delta: true SRBRD: shape_only_until_constant_audit: true MaynardIII_Prop5_2: arbitrary_convolution: true limiting_min_factor: 3/8 optimum_modulus_exponents: - 3/8 - 1/16 - 1/16 Pascadi_2025: triply_well_factorable_primes: level: 5/8-o(1) upper_linear_sieve_weights: level: 3/5-o(1) SRBRD: five_eighths_not_same_weight_class: true ``` --- # 49. Acceptance Gate Before SRBRD-25 publication / formal core, the following are required: 1. introduce the complement exponent $\omega$; 2. derive $\kappa=\omega-\theta$; 3. ordinary BV relative scale; 4. Maynard-III Theorem 1.1 factor ranges; 5. weak-error firewall; 6. Proposition 5.2 limiting linear program; 7. exact $3/8,1/16,1/16$ optimum; 8. irreducible-$k$ mapping; 9. distinguish $5/8\leftrightarrow3/8$ from applicability; 10. Pascadi weight-class distinction; 11. no use of $5/8$ for current upper linear weights; 12. label the residual region optimistic where appropriate; 13. do not treat Proposition 5.4 as a direct provider; 14. no constant-crossing claim; 15. no Goldbach-proof claim. --- # 50. Verdict This round can formally give: $$ \boxed{ \mathrm{SRBRD\mbox{-}25\ HYBRID\ COVERAGE\ MAP\ PASS}. } $$ The most important external-range result: $$ \boxed{ \text{Maynard-III Prop 5.2 arbitrary-convolution floor} = 3/8. } $$ The most important complement result: $$ \boxed{ \theta=5/8 \iff 1-\theta=3/8. } $$ Thus: $$ \boxed{ 5/8 } $$ is precisely the threshold at which the top complementary divisor meets the optimal small-factor endpoint of Prop 5.2. However, the current upper linear-sieve weight class has only: $$ \boxed{ 3/5, } $$ Hence, the top complement remains: $$ \boxed{ k\sim X^{2/5}, } $$ and gets stuck at the low-row boundary. Thus: $$ \boxed{ \text{current Route P does not close the whole low row}. } $$ Even with optimistic external shapes, there remains a nonempty: $$ \boxed{ \mathcal R_D. } $$ Therefore: $$ \boxed{ \mathrm{RSFLD\ REMAINS\ NECESSARY}. } $$ However, the analytic target has shrunk from the full low rectangle to: $$ \boxed{ \text{complement-aware narrow residual band}. } $$ Currently: $$ \boxed{ \mathrm{RSFLD\ NOT\ YET\ PROVED}. } $$ $$ \boxed{ \mathrm{CONSTANT\ CROSSING\ NOT\ YET\ PROVED}. } $$ and: $$ \boxed{ \mathrm{GOLDBACH\ STILL\ OPEN}. } $$ --- # 51. Recommendation for the Next Round # SRBRD-26 ## Residual-Band Constant Integral / Shift-Free Dispersion Budget Closure The next round should no longer expand the external theorem list. Directly put: $$ \mathcal R_D $$ into the constant ledger. ### Part A — Residual Weight Measure Derive from the original semiprime-row integral: $$ d\mu(t,\kappa) $$ rather than using geometric area. ### Part B — Conservative vs Optimistic Maps Compute separately: 1. conservative: $$ \mathcal R_D^{cons}; $$ 2. Theorem-1.1 shape: $$ \mathcal R_D^{shape}; $$ 3. hypothetical WCU: $$ \theta=5/8 $$ maps. ### Part C — Required RSFLD Multiplier Find: $$ \boxed{ R_D^{\max} } $$ such that: $$ C_L-C_B-C_{P}-C_D+C_T>0. $$ This will tell us: > how close to ideal the residual dispersion actually needs to be. ### Part D — Decide the Next Theorem If the required multiplier is very loose: > an ordinary coefficient-aware large sieve may suffice. If it is close to: $$ 1, $$ > a genuinely sharp new theorem is needed. This will be the next most important quantitative decision gate. --- # Appendix A: Complement-Aware Coordinates ```yaml coordinates: row: t: 1/3_to_2/5 complement: omega: theta_to_1 original_modulus: theta complementary_divisor: kappa: omega-theta current_linear_weight: theta: 3/5 rectangle: t: 1/3_to_2/5 kappa: 0_to_2/5 ``` --- # Appendix B: Provider Regions ```yaml P0_BV: condition: kappa < (1-t)/2 P1_uniform_prime_shape: condition: kappa < (8/15)*(1-t) status: shape_only P2_uniform_TypeII: conditions: t > 3/8 3/8 < kappa < t additional: inherited_three_factor_geometry: required status: conditional Residual: low_rectangle_minus_provider_regions ``` --- # Appendix C: Prop 5.2 Optimization ```yaml delta_epsilon: limiting_zero modulus_exponents: a_plus_b_plus_c: 1/2 constraints: b <= c c <= 1/10 - 3*b/5 factor_lower_bound: t: > max( a, b+c+1/4 ) optimum: a: 3/8 b: 1/16 c: 1/16 t_min: 3/8 ``` --- # Appendix D: Weight-Class Threshold ```yaml external_small_factor_floor: alpha: 3/8 required_original_level: theta: >= 1-alpha therefore: theta: >= 5/8 current_upper_linear_sieve: theta: 3/5 gap: 5/8 - 3/5: 1/40 ``` --- # Appendix E: Selected Residual Slices ```yaml t_1_3: P1_max_kappa: 16/45 residual_to: 2/5 t_3_8: P1_max_kappa: 1/3 residual_to: 2/5 t_2_5: P1_max_kappa: 8/25 conditional_P2: 3/8_to_2/5 middle_residual: 8/25_to_3/8 ``` --- # Appendix F: Commit Decision ```yaml research_round: SRBRD-25 decision: complement_exponent_map: true MaynardIII_exact_range_audit: true Prop5_2_three_eighth_floor: true five_eighth_three_eighth_duality: true current_linear_weight_five_eighth_use: false hybrid_residual_region: true RSFLD_proved: false constant_crossing: false goldbach_proved: false next: SRBRD-26_ResidualBand_ConstantIntegral_ShiftFreeDispersion_Budget_Closure ``` --- # References / Audited External Results 1. James Maynard, **Primes in arithmetic progressions to large moduli III: Uniform residue classes**, *Memoirs of the American Mathematical Society* 306 (2025), no. 1544. Audited facts used: - Theorem 1.1: complete residue uniformity for moduli with a factor up to roughly $x^{1/10-3\delta}$ and complementary factor up to roughly $x^{4/10+4\delta}$; error includes an $O(\delta\pi(x))$-scale term. - Theorem 1.2: good logarithmic error with $\delta<1/1000$ and a three-factor modulus. - Proposition 5.2: arbitrary Type-II convolution near $x^{2/5}$ with explicit $Q_1,Q_2,Q_3$ constraints. - the paper explicitly states Propositions 5.1–5.3 do not handle the $N\approx x^{1/3}$, $M\approx x^{2/3}$ regime. - Proposition 5.4 is a different prime-specific multi-factor decomposition, not a direct arbitrary prime×prime Type-II proposition. 2. Alexandru Pascadi, **On the exponents of distribution of primes and smooth numbers**, arXiv:2505.00653 (2025). Audited facts used: - primes reach $x^{5/8-o(1)}$ with triply-well-factorable weights; - upper-bound linear sieve weights have an improved $x^{3/5-o(1)}$ level; - these are different weight classes and cannot be interchanged silently. 3. James Maynard, **Primes in arithmetic progressions to large moduli II: Well-factorable estimates**, *Memoirs of the American Mathematical Society* 306 (2025), no. 1543. Relevant comparison: - triply-well-factorable weights reach $3/5-\varepsilon$; - the factorable upper linear sieve weights in that paper reach $7/12-\varepsilon$; - the distinction between triply-well-factorable and upper linear-sieve weights is structurally important. 4. Jared Duker Lichtman, **Primes in arithmetic progressions to large moduli, and Goldbach beyond the square-root barrier**, arXiv:2309.08522. Used only as a distribution-level benchmark; no direct SRBRD provider is claimed from the headline exponent alone.