> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # SRBRD-09 ## Lattice-Sparsity / Moment-Polytope Dual Closure ### Finite Occupancy Caps, Polynomial Majorants, Even-Degree Certificates and Layered Moment LP Duality **Parent program:** RG-SCP / Reverse Goldbach Structural Closure Program **Branch:** SRBRD — Symmetric Root-Bounded Relation Domain **Round:** SRBRD-09 **Date:** 2026-09-30 **Version:** v0.1 **Authors:** Neo.K; Aletheia (GPT-5.6 Sol) **Type:** Exact Finite-Moment Closure / LP-Dual Certificate / Lattice-Sparsity Compression **Prerequisites:** SRBRD-00 through SRBRD-08 **Status:** A finite moment certificate family strictly stronger than Bonferroni has been established; uniform Goldbach closure has not yet been obtained --- # 0. This Round's Change of Direction SRBRD-08 has proved that exact danger hypercells can be counted precisely using: - prime-union product； - canonical side assignment； - CRT； - root-valid interval； as displayed above. SRBRD-09 was originally planned to upper-bound cycle-space mass directly. But formal analysis revealed a more general and stronger route: > There is no need to select a particular combinatorial certificate such as a tree, cycle or Bonferroni certificate first. > Using only the **finite occupancy cap** already available in SRBRD, the union upper-bound problem itself can be converted into a finite moment polytope. This produces: $$ \boxed{ \text{Moment-Polytope Closure}. } $$ More importantly, it permits: $$ \boxed{ \text{even-degree rigorous upper certificates}. } $$ Thus Bonferroni's odd-order restriction is no longer a fundamental limitation of SRBRD finite closure. --- # 1. Occupancy Recalled For: $$ d\in\Omega_m, $$ Define the distinct prime-group occupancy: $$ \boxed{ \mu_m(d) = \#\{r:d\in P_{m,r}\}. } $$ Obstruction union indicator: $$ \boxed{ \mathbf1_{\mathcal O_m}(d) = \mathbf1_{\{\mu_m(d)\ge1\}}. } $$ Thus: $$ \boxed{ |\mathcal O_m| = \sum_{d\in\Omega_m} \mathbf1_{\{\mu_m(d)\ge1\}}. } $$ --- # 2. Occupancy Moments For: $$ j\ge1, $$ Define: $$ \boxed{ B_j(m) = \sum_{d\in\Omega_m} \binom{\mu_m(d)}{j}. } $$ SRBRD-02 has proved: $$ B_j(m) = \sum_{\substack{ U\subseteq Q_m\\ |U|=j }} |H_m(U)|. $$ Thus: $$ B_j $$ can be calculated directly from arithmetic hypercells, without needing a Goldbach success label. --- # 3. Local Occupancy Cap SRBRD-03： $$ (m-d)(m+d) = m^2-d^2. $$ If: $$ \mu_m(d)\ge t, $$ then the product of the first: $$ t $$ odd primes: $$ Q_t^{odd} $$ satisfying: $$ Q_t^{odd} \le m^2-d^2. $$ Therefore: $$ \boxed{ \mu_m(d)\le\kappa_m(d), } $$ where: $$ \boxed{ \kappa_m(d) = \max \{t:Q_t^{odd}\le m^2-d^2\}. } $$ --- # 4. Global Occupancy Cap Define: $$ \boxed{ \kappa(m) = \max \{t:Q_t^{odd}\le m^2\}. } $$ then: $$ \boxed{ 0\le\mu_m(d)\le\kappa(m) } $$ for all: $$ d\in\Omega_m. $$ Thus the occupancy indicator needs to be controlled only on the finite integer domain: $$ \boxed{ k=0,1,\ldots,\kappa(m) } $$ as displayed above. This is the central entry point for finite closure in this round. --- # 5. Binomial-Basis Polynomial Consider a degree: $$ s $$ binomial-basis polynomial: $$ \boxed{ P(k) = \sum_{j=1}^{s} \lambda_j \binom{k}{j}. } $$ Automatically: $$ P(0)=0. $$ If it can be proved that: $$ \boxed{ P(k)\ge1 \quad \text{for all } 1\le k\le\kappa(m), } $$ then: $$ P(\mu_m(d)) \ge \mathbf1_{\{\mu_m(d)\ge1\}}. $$ --- # 6. Moment-Polynomial Majorant Theorem ## Theorem SRBRD-09-T01 If: $$ P(k) = \sum_{j=1}^{s} \lambda_j\binom{k}{j} $$ satisfying: $$ P(k)\ge1 $$ for: $$ 1\le k\le\kappa(m), $$ then: $$ \boxed{ |\mathcal O_m| \le \sum_{j=1}^{s} \lambda_j B_j(m). } $$ Proof: For each offset: $$ \mathbf1_{\{\mu\ge1\}} \le P(\mu). $$ Summing: $$ |\mathcal O_m| \le \sum_d \sum_j \lambda_j \binom{\mu(d)}{j} = \sum_j \lambda_jB_j. $$ Note: $$ \lambda_j $$ can be negative. Since: $$ B_j $$ is an exact structural quantity in the theorem. --- # 7. Polynomial Non-Cover Certificate ## Corollary SRBRD-09-C01 If: $$ \boxed{ \sum_{j=1}^{s} \lambda_jB_j(m) < |\Omega_m|, } $$ then: $$ \boxed{ \operatorname{Res}(m)\neq\varnothing. } $$ Therefore: $$ \boxed{ R_G^{-}(2m)>0. } $$ --- # 8. Bonferroni Is a Special Case For odd: $$ q, $$ Take: $$ \boxed{ \lambda_j=(-1)^{j+1}, \qquad 1\le j\le q. } $$ then: $$ P_q(k) = \sum_{j=1}^{q} (-1)^{j+1} \binom{k}{j} \ge \mathbf1_{\{k\ge1\}}. $$ Thus: $$ \boxed{ U_q = B_1-B_2+\cdots+B_q } $$ is merely a special polynomial majorant of T01. --- # 9. Why SRBRD Can Beat Bonferroni Parity If an ordinary Bonferroni partial sum directly uses the coefficients: $$ 1,-1,+1,-1,\ldots $$ as displayed above, only odd truncation gives a union upper bound. But SRBRD knows: $$ \boxed{ \mu\le\kappa(m)<\infty. } $$ so we do not require the polynomial to majorize the indicator for every: $$ k\in\mathbb N $$ as displayed above. It is needed only for: $$ \boxed{ k=0,\ldots,\kappa(m). } $$ Thus we can construct: $$ \boxed{ \text{even-degree upper majorants}. } $$ This is the genuinely new freedom in this round. --- # 10. Degree-2 Cap Majorant For: $$ \kappa\ge2, $$ Define: $$ \boxed{ P_{\kappa}^{(2)}(k) = 1- \frac{ (k-1)(k-\kappa) }{\kappa}. } $$ we have: $$ P_{\kappa}^{(2)}(0)=0. $$ For integers: $$ 1\le k\le\kappa, $$ $$ (k-1)(k-\kappa)\le0. $$ Thus: $$ \boxed{ P_{\kappa}^{(2)}(k)\ge1. } $$ --- # 11. Degree-2 Moment Bound Write: $$ P_{\kappa}^{(2)} $$ in the binomial basis: $$ \boxed{ P_{\kappa}^{(2)}(k) = \binom{k}{1} - \frac{2}{\kappa} \binom{k}{2}. } $$ Therefore: ## Theorem SRBRD-09-T02 $$ \boxed{ |\mathcal O_m| \le B_1(m) - \frac{2}{\kappa(m)} B_2(m). } $$ This is an even-degree rigorous upper bound. --- # 12. Degree-4 Cap-Majorant Family Now choose an integer: $$ a $$ satisfying: $$ \boxed{ 2\le a\le\kappa-2. } $$ Define: $$ \boxed{ P_{\kappa,a}^{(4)}(k) = 1- \frac{ (k-1)(k-a)(k-a-1)(k-\kappa) }{ \kappa a(a+1) }. } $$ --- # 13. Degree-4 Majorant Theorem ## Theorem SRBRD-09-T03 For integers: $$ 0\le k\le\kappa, $$ we have: $$ P_{\kappa,a}^{(4)}(0)=0, $$ and, for: $$ 1\le k\le\kappa, $$ $$ \boxed{ P_{\kappa,a}^{(4)}(k)\ge1. } $$ --- # 14. T03 Sign Proof roots： $$ 1, \quad a, \quad a+1, \quad \kappa. $$ On the integer interval: $$ 1,\ldots,\kappa, $$ product： $$ (k-1)(k-a)(k-a-1)(k-\kappa) $$ is always: $$ \le0. $$ The only open interval where it could become positive: $$ (a,a+1) $$ contains no integer. Thus the polynomial majorizes the occupancy indicator. --- # 15. Degree-4 Binomial Coefficients It can be written as: $$ P_{\kappa,a}^{(4)}(k) = \lambda_1\binom{k}{1} + \lambda_2\binom{k}{2} + \lambda_3\binom{k}{3} + \lambda_4\binom{k}{4}, $$ where: $$ \boxed{ \lambda_1=1, } $$ $$ \boxed{ \lambda_2 = \frac{ -4\kappa a + 2\kappa - 2a^2 + 6a - 4 }{ \kappa a(a+1) }, } $$ $$ \boxed{ \lambda_3 = \frac{ 6(\kappa+2a-4) }{ \kappa a(a+1) }, } $$ $$ \boxed{ \lambda_4 = -\frac{ 24 }{ \kappa a(a+1) }. } $$ This gives an explicit 4-moment upper bound. --- # 16. General Even-Degree Cap Majorant Choose: $$ s\ge1. $$ Take: $$ s-1 $$ disjoint consecutive root pairs: $$ (a_1,a_1+1), \ldots, (a_{s-1},a_{s-1}+1) $$ satisfying: $$ 2\le a_1, $$ $$ a_i+10. } $$ This is a rigorous continuous-LP certificate. Integrality is not required. --- # 24. Layered Dual Certificate The primal dual gives a particularly concise certificate. Choose: - layer potentials： $$ \alpha_t\in\mathbb Q; $$ - moment coefficients： $$ \lambda_1,\ldots,\lambda_s\in\mathbb Q. $$ Require: $$ \boxed{ \alpha_t + \sum_{j=1}^{s} \lambda_j \binom{k}{j} \ge \mathbf1_{\{k\ge1\}} } $$ for each: $$ t $$ and: $$ 0\le k\le t. $$ --- # 25. Layered Dual Upper Bound ## Theorem SRBRD-09-T06 If all the above finite inequalities hold, then: $$ \boxed{ |\mathcal O_m| \le \sum_t \alpha_tN_t + \sum_{j=1}^{s} \lambda_jB_j. } $$ Proof: For each offset: $$ d\in Z_t, $$ substitute: $$ k=\mu(d)\le t. $$ and sum to obtain the result. --- # 26. Dual Non-Cover Certificate If: $$ \boxed{ \sum_t \alpha_tN_t + \sum_{j=1}^{s} \lambda_jB_j < |\Omega_m|, } $$ then: $$ \boxed{ \operatorname{Res}(m)\neq\varnothing. } $$ Every certificate requires only rational arithmetic for verification. --- # 27. Global Polynomial Is a Special Dual Take: $$ \alpha_t=0 $$ for all： $$ t. $$ this returns to T01: $$ P(k) = \sum_j\lambda_j\binom{k}{j} \ge1 $$ on： $$ 0 49999. $$ But the degree-6 paired-root pattern: $$ \boxed{ (2,3), \quad (5,6) } $$ that is, the roots: $$ 1,2,3,5,6,9, $$ gives: $$ \boxed{ |\mathcal O| \le \frac{441524}{9} \approx 49058.22 < 49999. } $$ Thus the degree-6 certificate succeeds. --- # 35. Computational Control: $m=10^6$ $$ |\Omega|=499999. $$ moments： $$ B_1=1573938, $$ $$ B_2=2303946, $$ $$ B_3=2095573, $$ $$ B_4=1292747, $$ $$ B_5=550290, $$ $$ B_6=159137. $$ global cap： $$ \boxed{ \kappa(10^6)=10. } $$ degree-6 paired roots： $$ \boxed{ 1,3,4,5,6,10 } $$ that is, the pairs: $$ (3,4), \quad (5,6), $$ give the polynomial: $$ \boxed{ P(k) = 1- \frac{ (k-1)(k-3)(k-4)(k-5)(k-6)(k-10) }{ 3600 }. } $$ binomial-basis coefficients： $$ \boxed{ 1, -\frac{71}{75}, \frac{21}{25}, -\frac{17}{25}, \frac{7}{15}, -\frac15. } $$ Thus: $$ \boxed{ |\mathcal O| \le \frac{12476427}{25} = 499057.08 < 499999. } $$ Therefore using only: $$ \boxed{ B_1,\ldots,B_6 } $$ proves non-cover for the fixed: $$ m=10^6 $$ as displayed above. --- # 36. Comparison with Ordinary Bonferroni at $m=10^6$ SRBRD-02 audit： $$ U_5=623108 $$ fails. $$ U_7=492799 $$ succeeds. Thus ordinary Bonferroni requires moments through: $$ B_7. $$ The moment-polytope cap polynomial requires only: $$ \boxed{ B_6. } $$ This is an actual reduction by one in moment order. --- # 37. Degree-6 Exactness at $\kappa=6$ If: $$ \kappa=6, $$ take the roots: $$ 1,2,3,4,5,6. $$ Then, at: $$ k=1,\ldots,6 $$ the polynomial is everywhere: $$ =1. $$ Therefore: $$ \boxed{ P(k)=\mathbf1_{\{k\ge1\}} } $$ on the entire occupancy domain. Thus: $$ \boxed{ |\mathcal O| = \sum_{j=1}^{6} \lambda_jB_j. } $$ This is simply finite binomial interpolation. It shows: > When the degree reaches the occupancy cap, the moment polynomial can recover the union exactly. --- # 38. Exact Finite Interpolation Principle More generally: If moments are obtained through: $$ B_1,\ldots,B_{\kappa}, $$ then the indicator: $$ \mathbf1_{\{k\ge1\}} $$ on the finite set: $$ k=0,\ldots,\kappa $$ can be represented exactly in the binomial basis: $$ \boxed{ \mathbf1_{\{k\ge1\}} = \sum_{j=1}^{\kappa} (-1)^{j+1} \binom{k}{j}. } $$ Thus: $$ |\mathcal O| $$ ultimately always has finite exact closure. The actual proof-compression question is: > Can a low-degree certificate with: > $$ > s\ll\kappa > $$ > already yield strict non-cover? --- # 39. Layered LP Can Only Improve The global polynomial uses: $$ \mu\le\kappa(m). $$ The layered dual also knows: $$ d\in Z_t \Longrightarrow \mu(d)\le t. $$ Thus: $$ U_s^{MP} $$ cannot be worse than the best global degree-$s$ majorant. Therefore: $$ \boxed{ \text{Global Cap Polynomial} \subseteq \text{Layered Moment Polytope}. } $$ --- # 40. LP Dual Is a Formal Certificate, Not an Oracle A solver can search for: $$ \alpha_t, \lambda_j. $$ But the formal proof does not need to trust the solver. It needs only to output rational coefficients, and then machine-check: 1. All the inequalities: $$ \alpha_t+\sum_j\lambda_j\binom{k}{j} \ge\mathbf1_{\{k\ge1\}}; $$ 2. moment values / rigorous moment bounds； 3. The final rational sum: $$ <|\Omega|. $$ Thus the solver is only a certificate searcher. --- # 41. Local-Cap Dual Interpretation For each layer: $$ t, $$ function： $$ \boxed{ P_t(k) = \alpha_t + \sum_j\lambda_j\binom{k}{j} } $$ needs only to majorize the indicator on: $$ 0\le k\le t. $$ frontier layers： $$ t $$ is small, so a global polynomial can be allowed to be insufficiently strong at high $k$, and then use: $$ \alpha_t $$ to repair it locally. This formally connects SRBRD-03 center–frontier geometry to the LP dual. --- # 42. Moment Bounds Can Replace Exact Moments T01 / T06 currently use the exact: $$ B_j. $$ But if: $$ \lambda_j>0, $$ only a rigorous upper bound is needed: $$ B_j\le U_j. $$ If: $$ \lambda_j<0, $$ a rigorous lower bound is needed: $$ L_j\le B_j. $$ Therefore: $$ \boxed{ \sum_{\lambda_j>0} \lambda_jU_j + \sum_{\lambda_j<0} \lambda_jL_j } $$ remains a valid union upper bound. This is the key interface for moving toward a uniform theorem. --- # 43. Signed Moment Provider Contract Define: $$ \boxed{ \operatorname{MomentEnvelope}_j = \begin{cases} U_j, & \lambda_j>0, \\ L_j, & \lambda_j<0. \end{cases} } $$ Then, as long as: $$ P $$ is an occupancy majorant, we obtain: $$ \boxed{ |\mathcal O_m| \le \sum_j \lambda_j \operatorname{MomentEnvelope}_j. } $$ This directly connects uniform moment bounds to a finite polynomial certificate. --- # 44. Why Signs Matter For example, a degree-4 polynomial typically has: $$ \lambda_1>0, $$ $$ \lambda_2<0, $$ $$ \lambda_3>0, $$ $$ \lambda_4<0. $$ Thus, to upper-bound the union uniformly: one needs: - An upper bound for $B_1$; - A lower bound for $B_2$; - An upper bound for $B_3$; - A lower bound for $B_4$. This agrees entirely with SRBRD-03's warning about Bonferroni polarity. --- # 45. Relation to Forest / Cycle Certificates Forest methods produce a: $$ E_q $$ lower bound. The moment polynomial instead directly majorizes: $$ \mathbf1_{\{\mu\ge1\}}. $$ The two are not mutually exclusive. One can: 1. Use a forest / danger complex to provide certain negative-sign moment lower bounds; 2. Use lattice sparsity to provide positive-sign moment upper bounds; 3. Assemble them through the polynomial dual at the end. Thus SRBRD-09 begins to unify the earlier distinct branches into a: $$ \boxed{ \text{signed finite moment certificate}. } $$ --- # 46. Moment-Polytope Barrier Even if the finite LP can be solved for every fixed: $$ m $$ as displayed above, this is still not a uniform Goldbach proof. Because: - moments： $$ B_j(m) $$ Vary with $m$; - occupancy cap： $$ \kappa(m) $$ Grows slowly; - The optimal polynomial / dual coefficients may also change. Thus the actual next step is: $$ \boxed{ \text{uniformize moment envelopes and dual polynomials}. } $$ --- # 47. New Research Target We can now compress the uniform Goldbach closure target into: finding a degree: $$ s(m) $$ and rigorous moment envelopes such that a dual certificate exists: $$ \boxed{ \sum_t\alpha_tN_t + \sum_j \lambda_j \operatorname{MomentEnvelope}_j < |\Omega_m| } $$ for all： $$ m\ge3. $$ Ideally: $$ s(m) $$ is constant. The next-best case: $$ s(m) $$ needs to grow only extremely slowly. --- # 48. Why This Is Potentially Stronger Than Raising Bonferroni Order Bonferroni permits only the fixed coefficients: $$ +1,-1,+1,-1,\ldots $$ The moment polytope permits: $$ \boxed{ \text{any rational coefficients} } $$ provided the majorant holds on the finite occupancy domain. Thus a more effective signed combination can be sought for: - $\kappa(m)$； - local cap layers； - moment sizes； as displayed above. This is a genuinely larger proof-certificate class. --- # 49. Formal Theorem Queue ```yaml SRBRD_09: T01: finite_occupancy_polynomial_majorant_union_bound T02: degree2_cap_majorant T03: degree4_paired_root_majorant T04: general_even_degree_paired_root_majorant T05: moment_polytope_primal_upper_bound T06: layered_LP_dual_certificate T07: global_polynomial_as_dual_special_case T08: exact_interpolation_at_degree_kappa T09: signed_moment_envelope_substitution ``` --- # 50. Acceptance Gate Before SRBRD-09 can become a formal core, it requires: 1. global occupancy cap theorem； 2. polynomial majorant theorem； 3. rational signed summation proof； 4. degree-2 sign proof； 5. general paired-root sign proof； 6. degree-4 binomial coefficient derivation； 7. cap-layer partition； 8. moment-polytope primal； 9. An LP dual weak-duality theorem or a direct finite summation proof; 10. exact interpolation； 11. signed moment envelope polarity； 12. Reverification of finite numerical certificates using exact rationals; 13. no solver-trust assumption； 14. No uniform Goldbach claim. --- # 51. Verdict This round can formally give: $$ \boxed{ \mathrm{SRBRD\mbox{-}09\ MOMENT\mbox{-}POLYTOPE\ DUAL\ PASS}. } $$ The reason: SRBRD's finite occupancy cap makes it so that: $$ \boxed{ \text{the union upper bound is no longer restricted to Bonferroni's odd-order form.} } $$ We obtain: - degree-2 even certificate； - general degree-$2s$ paired-root majorants； - layered moment-polytope LP； - rational dual certificate； - signed moment-envelope interface。 and finite controls have already shown a real reduction in moment order: - $m=5000$: Degree $4$ succeeds, whereas ordinary Bonferroni requires $5$; - $m=10^6$: Degree $6$ succeeds, whereas ordinary Bonferroni requires $7$. However: $$ \boxed{ \mathrm{UNIFORM\ GOLDBACH\ CLOSURE\ STILL\ OPEN}. } $$ --- # 52. Recommendations for the Next Round # SRBRD-10 ## Uniform Moment Envelope / Dual-Polynomial Closure What is most worth attacking now is not to calculate more exact moments of higher order. Instead: 1. Find uniform upper bounds for positive-coefficient moments; 2. Find uniform lower bounds for negative-coefficient moments; 3. Use CRT / root intervals / cofactor load to bound: $$ B_j; $$ 4. Establish symbolic inequalities for degree-4 and degree-6 polynomials; 5. Study how to choose paired-root patterns when: $$ \kappa(m) $$ grows slowly with $m$; 6. Attempt to make: $$ \text{dual upper}<|\Omega| $$ hold for the entire family. This is the first time so far that: $$ \boxed{ \text{Goldbach non-cover} } $$ has been compressed into: $$ \boxed{ \text{finite-degree signed moment inequalities}. } $$ --- # Appendix A: Degree-4 Contract ```yaml inputs: kappa: integer >= 4 a: 2 <= a <= kappa-2 polynomial: P(k): 1 - ((k-1)(k-a)(k-a-1)(k-kappa))/(kappa*a*(a+1)) integer_majorant: P(0): 0 P(k)>=1: for: 1..kappa binomial_coefficients: lambda1: 1 lambda2: (-4*kappa*a + 2*kappa - 2*a^2 + 6*a - 4)/(kappa*a*(a+1)) lambda3: 6*(kappa+2*a-4)/(kappa*a*(a+1)) lambda4: -24/(kappa*a*(a+1)) ``` --- # Appendix B: General Even-Degree Contract ```yaml roots: - 1 - paired_consecutive_integer_roots: - [a1,a1+1] - [a2,a2+1] - ... - kappa normalization: P(0): 0 property: product_term <= 0 on every integer k in [1,kappa] therefore: P(k) >= 1 on occupied integer states ``` --- # Appendix C: Selected Exact-Rational Certificates ### $m=1094$ ```yaml Omega: 546 kappa: 6 degree: 4 paired_root_a: 3 moments: B1: 1114 B2: 921 B3: 415 B4: 108 coefficients: - 1 - -8/9 - 2/3 - -1/3 union_upper: 536 noncover: true ``` ### $m=5000$ ```yaml Omega: 2499 kappa: 7 degree: 4 paired_root_a: 3 moments: B1: 5495 B2: 5206 B3: 2870 B4: 944 union_upper: exact: 52165/21 decimal: 2484.047619... noncover: true ``` ### $m=100000$ ```yaml Omega: 49999 kappa: 9 degree: 6 paired_roots: - [2,3] - [5,6] moments: B1: 139168 B2: 176344 B3: 136402 B4: 69743 B5: 23674 B6: 5170 union_upper: exact: 441524/9 decimal: 49058.222222... noncover: true ``` ### $m=1000000$ ```yaml Omega: 499999 kappa: 10 degree: 6 paired_roots: - [3,4] - [5,6] moments: B1: 1573938 B2: 2303946 B3: 2095573 B4: 1292747 B5: 550290 B6: 159137 coefficients: - 1 - -71/75 - 21/25 - -17/25 - 7/15 - -1/5 union_upper: exact: 12476427/25 decimal: 499057.08 noncover: true ``` --- # Appendix D: Moment-Polytope Dual Contract ```yaml cap_layers: Z_t: local_occupancy_cap_equals: t N_t: card(Z_t) dual_variables: alpha_t: rational: true lambda_j: rational: true finite_constraints: for_each_t: for_k_in_0_to_t: alpha_t + sum(lambda_j*choose(k,j)) >= indicator(k>=1) union_upper: sum(alpha_t*N_t) + sum(lambda_j*B_j) certificate: if_union_upper_lt_card_Omega: residual_nonempty: true ``` --- # Appendix E: Commit Decision ```yaml research_round: SRBRD-09 decision: finite_occupancy_polynomial_majorants: true even_degree_upper_certificates: true moment_polytope_duality: true moment_order_reduction_observed: true uniform_goldbach_closure: false new_exact_structures: - degree2_cap_majorant - paired_root_even_degree_majorants - layered_moment_polytope - rational_dual_certificate - signed_moment_envelope_interface next: SRBRD-10_Uniform_MomentEnvelope_DualPolynomial_Closure ```