> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # SRBRD-00 ## Exact Representation Kernel ### Finiteness, Bidirectional Equivalence, Count-Level Bijection, and Root-Truncated Residue Cells of the Symmetric Root-Bounded Relation Domain **Parent program:** RG-SCP / Reverse Goldbach Structural Closure Program **Branch:** SRBRD — Symmetric Root-Bounded Relation Domain **Round:** SRBRD-00 **Date:** 2026-09-30 **Version:** v0.1 **Authors:** Neo.K; Aletheia (GPT-5.6 Sol) **Nature:** Exact Representation Audit / Formalization Specification **Status:** Not a Goldbach proof; the target is to prove that the new representation and the original reduced Goldbach target are exact in both directions --- # 0. The Sole Task of This Round This round does not attempt to prove: $$ \forall m,\quad \operatorname{Res}(\mathfrak R_m)\neq\varnothing. $$ This round proves only: $$ \boxed{ \operatorname{Res}(\mathfrak R_m)\neq\varnothing \Longleftrightarrow R_G^{-}(2m)>0 } $$ and the stronger count-level statement: $$ \boxed{ |\operatorname{Res}(\mathfrak R_m)| = R_G^{-}(2m) } $$ For: $$ m\ge3 $$ as the domain on which these hold. That is: > SRBRD-00 must first prove itself to be only an exact representation without loopholes. If even count equality cannot hold, all subsequent finite-cover / closure research must stop. --- # 1. Boundary The smallest target of strong Goldbach is: $$ 4=2+2. $$ This pair is not an odd-prime symmetric pair. Thus, the main SRBRD kernel treats: $$ N=4 $$ separately. Main domain: $$ \boxed{ N=2m, \qquad m\ge3. } $$ That is: $$ N\ge6. $$ At this point, no Goldbach representation can contain the prime $2$. For if: $$ 2+(N-2)=N, $$ then: $$ N-2 $$ is an even integer greater than $2$, and hence composite. Therefore: $$ \boxed{ N\ge6,\ N\text{ even} \Longrightarrow \text{all Goldbach pairs are odd prime + odd prime}. } $$ --- # 2. Center Let: $$ \boxed{ N=2m. } $$ Every unordered decomposition: $$ a+b=2m, \qquad a\le b $$ can be written as: $$ \boxed{ a=m-d, \qquad b=m+d, } $$ where: $$ d=m-a\ge0. $$ Thus: $$ m $$ is simultaneously: 1. the compulsory division result of an even integer: $$ 2m\xrightarrow{/2}m; $$ 2. the additive symmetric midpoint. This is the Level-0 origin of SRBRD. --- # 3. Legal Offset Domain For: $$ m\ge3, $$ Define: $$ \boxed{ \Omega_m = \left\{ d\in\mathbb N: d\le m-3, \quad m-d\text{ odd} \right\}. } $$ Here we must allow: $$ d=0. $$ Otherwise we would omit: $$ p+p=2p $$ type Goldbach representations. For example: $$ 10=5+5 $$ corresponds to: $$ m=5, \quad d=0. $$ --- # 4. Endpoint Functions Define: $$ \boxed{ L_m(d)=m-d, } $$ $$ \boxed{ R_m(d)=m+d. } $$ To avoid confusing the relation $R$ with the right endpoint, the suggested formal-code names are: ```text LeftEndpoint RightEndpoint RootObstruction ``` Do not let `R_m` represent two objects at once. --- # 5. Endpoint Bounds ## Theorem SRBRD-00-T01 If: $$ d\in\Omega_m, $$ then: $$ \boxed{ 3\le L_m(d)\le m\le R_m(d)\le2m-3. } $$ proof： From: $$ d\le m-3, $$ we obtain: $$ m-d\ge3. $$ and: $$ d\ge0. $$ Thus: $$ m-d\le m\le m+d. $$ Again, from: $$ d\le m-3 $$ we obtain: $$ m+d\le2m-3. $$ --- # 6. Endpoint Parity ## Theorem SRBRD-00-T02 If: $$ d\in\Omega_m, $$ then: $$ \boxed{ L_m(d)\text{ odd} } $$ and: $$ \boxed{ R_m(d)\text{ odd}. } $$ Since: $$ R_m(d)-L_m(d)=2d $$ is even. Thus, the two endpoints have the same parity. --- # 7. Offset-to-Decomposition Map Define: $$ \boxed{ \Phi_m(d) = \left( m-d, m+d \right). } $$ For: $$ d\in\Omega_m, $$ we have: $$ \boxed{ (m-d)+(m+d)=2m. } $$ and: $$ m-d\le m+d. $$ Thus, each offset produces a unique unordered odd decomposition. --- # 8. Decomposition-to-Offset Map If: $$ a+b=2m, $$ $$ 3\le a\le b, $$ and: $$ a,b $$ odd， Define: $$ \boxed{ d=m-a. } $$ Since: $$ a\le b $$ and: $$ a+b=2m, $$ we have: $$ a\le m. $$ Thus: $$ d\ge0. $$ Also: $$ a\ge3 $$ gives: $$ d=m-a\le m-3. $$ and: $$ m-d=a $$ odd。 Thus: $$ d\in\Omega_m. $$ --- # 9. Odd Decomposition Bijection ## Theorem SRBRD-00-T03 For: $$ m\ge3, $$ there is a bijection between $\Omega_m$ and the unordered odd decompositions: $$ (a,b) $$ satisfying: $$ a+b=2m, $$ $$ 3\le a\le b, $$ $$ a,b\text{ odd} $$ of this form: $$ \boxed{ d \longleftrightarrow (m-d,m+d). } $$ This theorem is the additive-domain completeness of SRBRD. --- # 10. Reduced Goldbach Pair Bijection For: $$ N=2m\ge6, $$ a reduced Goldbach witness from V1-01 satisfies: $$ p\in\mathbb P, $$ $$ 2p\le2m, $$ $$ 2m-p\in\mathbb P. $$ From $N\ge6$: $$ p\neq2. $$ Thus: $$ p\ge3 $$ and is odd. Let: $$ \boxed{ d=m-p. } $$ then: $$ d\in\Omega_m. $$ Conversely, if: $$ d\in\Omega_m $$ and both endpoints are prime, Let: $$ p=m-d. $$ then: $$ 2p\le2m $$ and: $$ 2m-p=m+d $$ prime。 Thus, $p$ is a reduced Goldbach witness. --- # 11. Count-Level Candidate Bijection ## Theorem SRBRD-00-T04 For: $$ m\ge3, $$ map： $$ \boxed{ p \mapsto m-p } $$ In: $$ \operatorname{GoldbachReducedPair}(2m,p) $$ and: $$ \left\{ d\in\Omega_m: m-d,m+d\in\mathbb P \right\} $$ is a bijection between these sets. Inverse function: $$ \boxed{ d \mapsto m-d. } $$ --- # 12. Root-Bounded Division Domain For: $$ x>1, $$ Define: $$ \boxed{ \mathcal D(x) = \left\{ r\in\mathbb P: r^2\le x, \quad r\mid x \right\}. } $$ Note: Here we retain only prime-divisor witnesses directly. There is no need for all divisors satisfying: $$ 2\le d\le\sqrt x. $$ because every composite $x$ has a prime proper factor: $$ r\le\sqrt x. $$ --- # 13. Root-Bounded Primality Equivalence ## Theorem SRBRD-00-T05 If: $$ x\ge2, $$ then: $$ \boxed{ x\in\mathbb P \Longleftrightarrow \mathcal D(x)=\varnothing. } $$ --- # 14. Prime Direction If $x$ is prime, its only positive factors are: $$ 1,x. $$ Any: $$ r\in\mathbb P $$ If: $$ r\mid x, $$ can only satisfy: $$ r=x. $$ However: $$ x^2>x $$ For: $$ x\ge2. $$ Thus: $$ r^2\le x $$ is impossible. Therefore: $$ \mathcal D(x)=\varnothing. $$ --- # 15. Composite Direction If: $$ x $$ is composite and: $$ x\ge2, $$ Take: $$ r=\operatorname{lpf}(x). $$ then: $$ r\in\mathbb P, $$ $$ r\mid x, $$ and: $$ r^2\le x. $$ Thus: $$ \mathcal D(x)\neq\varnothing. $$ --- # 16. Odd Endpoints Remove Factor 2 If: $$ d\in\Omega_m, $$ then both endpoints are odd. Therefore: $$ 2\nmid L_m(d), $$ $$ 2\nmid R_m(d). $$ Thus, every root-bounded witness must satisfy: $$ \boxed{ r\ge3. } $$ This formally realizes the Parity Base-Space note: > The factor $2$ is no longer a residual obstruction prime; it has already been handled globally at the target-generation layer. --- # 17. Global Finite Witness Prime Domain From the endpoint bound: $$ R_m(d)\le2m-3, $$ every root-bounded witness satisfies: $$ r^2\le2m-3. $$ Thus, define: $$ \boxed{ Q_m = \left\{ r\in\mathbb P: 3\le r, \quad r^2\le2m-3 \right\}. } $$ This completely avoids: $$ \sqrt{2m-3} $$ as a source of machine-rounding problems. --- # 18. Side Domain Define the finite side type: $$ \boxed{ \Sigma = \{\mathrm L,\mathrm R\}. } $$ witness universe： $$ \boxed{ W_m = \Sigma\times Q_m. } $$ Therefore: $$ \boxed{ |W_m|<\infty. } $$ --- # 19. Endpoint-by-Side Define: $$ E_m(d,\mathrm L)=m-d, $$ $$ E_m(d,\mathrm R)=m+d. $$ --- # 20. Root Obstruction Relation Define: $$ \boxed{ \mathcal R_m(d,\sigma,r) } $$ if and only if: 1. $d\in\Omega_m$; 2. $r\in Q_m$; 3. $r\mid E_m(d,\sigma)$; 4. $r^2\le E_m(d,\sigma)$. That is: $$ \boxed{ \mathcal R_m \subseteq \Omega_m\times\Sigma\times Q_m. } $$ --- # 21. Relation Finiteness ## Theorem SRBRD-00-T06 For each: $$ m\ge3, $$ we have: $$ \boxed{ |\Omega_m|<\infty, \qquad |Q_m|<\infty, \qquad |\mathcal R_m|<\infty. } $$ Thus: $$ \boxed{ \mathfrak R_m = (\Omega_m,\Sigma\times Q_m,\mathcal R_m) } $$ is a finite relation domain. --- # 22. Sidewise Completeness ## Theorem SRBRD-00-T07 If: $$ d\in\Omega_m, $$ then, for any: $$ \sigma\in\Sigma, $$ $$ \boxed{ E_m(d,\sigma)\in\mathbb P } $$ if and only if there is no: $$ r\in Q_m $$ such that: $$ \mathcal R_m(d,\sigma,r). $$ This is the version of root-bound completeness within the global witness universe. --- # 23. Why Global $Q_m$ Does Not Overreach Although: $$ Q_m $$ uses only the global bound: $$ r^2\le2m-3, $$ the relation still requires the endpoint-specific condition: $$ r^2\le E_m(d,\sigma). $$ Thus, it does not wrongly treat: > A prime small enough for the overall target, but too large for the current endpoint as a proper-factor witness. --- # 24. Obstruction Degree Define: $$ \boxed{ \deg_m(d) = \#\left\{ (\sigma,r)\in\Sigma\times Q_m: \mathcal R_m(d,\sigma,r) \right\}. } $$ --- # 25. Zero Degree iff Prime Pair ## Theorem SRBRD-00-T08 If: $$ d\in\Omega_m, $$ then: $$ \boxed{ \deg_m(d)=0 \Longleftrightarrow m-d\in\mathbb P \land m+d\in\mathbb P. } $$ proof： Degree zero holds if and only if neither side has a witness. By T07: both endpoints are prime. --- # 26. Residual Set Define: $$ \boxed{ \operatorname{Res}(m) = \left\{ d\in\Omega_m: \deg_m(d)=0 \right\}. } $$ Equivalently: $$ \boxed{ \operatorname{Res}(m) = \left\{ d\in\Omega_m: m-d,m+d\in\mathbb P \right\}. } $$ --- # 27. Exact Representation Theorem ## Theorem SRBRD-00-T09 For: $$ m\ge3, $$ $$ \boxed{ \operatorname{Res}(m)\neq\varnothing \Longleftrightarrow R_G^{-}(2m)>0. } $$ That is: $$ \boxed{ 2m \text{ satisfies Goldbach} \Longleftrightarrow \operatorname{Res}(m)\neq\varnothing. } $$ --- # 28. Count-Level Exactness ## Theorem SRBRD-00-T10 For: $$ m\ge3, $$ $$ \boxed{ |\operatorname{Res}(m)| = R_G^{-}(2m). } $$ This is more important than existence equivalence. It means: > Every reduced unordered Goldbach representation corresponds to exactly one residual offset, with neither overcounting nor omissions. --- # 29. Counterexample Exact Form By T09: $$ \boxed{ \operatorname{GoldbachFailure}(2m) \Longleftrightarrow \operatorname{Res}(m)=\varnothing. } $$ For: $$ m\ge3. $$ Note: $$ 2m $$ is automatically an admissible even integer. --- # 30. Obstruction Cells For: $$ \sigma\in\Sigma, \quad r\in Q_m, $$ Define: $$ \boxed{ O_{m,\sigma,r} = \left\{ d\in\Omega_m: \mathcal R_m(d,\sigma,r) \right\}. } $$ then: $$ \boxed{ \operatorname{Res}(m) = \Omega_m \setminus \bigcup_{\sigma,r} O_{m,\sigma,r}. } $$ --- # 31. Failure as Complete Cover ## Theorem SRBRD-00-T11 For: $$ m\ge3, $$ $$ \boxed{ \operatorname{GoldbachFailure}(2m) \Longleftrightarrow \Omega_m = \bigcup_{\sigma,r} O_{m,\sigma,r}. } $$ Thus, a counterexample is exactly equivalent to a finite complete cover. --- # 32. Left Residue Cell Consider: $$ \sigma=\mathrm L. $$ The relation requires: $$ r\mid(m-d). $$ In: $$ d\le m $$ this is equivalent to: $$ \boxed{ d\equiv m\pmod r. } $$ Root bound: $$ r^2\le m-d $$ is equivalent to: $$ \boxed{ d\le m-r^2. } $$ Therefore: $$ \boxed{ O_{m,\mathrm L,r} = \left\{ d\in\Omega_m: d\equiv m\pmod r, \quad d\le m-r^2 \right\}. } $$ If: $$ r^2>m, $$ then: $$ \boxed{ O_{m,\mathrm L,r}=\varnothing. } $$ This is the first very important root-asymmetry fact. --- # 33. Right Residue Cell Consider: $$ \sigma=\mathrm R. $$ The relation requires: $$ r\mid(m+d). $$ Modulo $r$: $$ \boxed{ d\equiv -m\pmod r. } $$ The formal Nat version can be written as: $$ \boxed{ m+d\equiv0\pmod r. } $$ to avoid negative-residue implementation problems. Root bound: $$ r^2\le m+d $$ is equivalent to: $$ \boxed{ d\ge r^2-m. } $$ Adding: $$ d\ge0, $$ we obtain: $$ \boxed{ d\ge\max(0,r^2-m). } $$ Therefore: $$ \boxed{ O_{m,\mathrm R,r} = \left\{ d\in\Omega_m: d\equiv -m\pmod r, \quad d\ge\max(0,r^2-m) \right\}. } $$ --- # 34. Root-Bound Asymmetry The left and right residue classes are formally symmetric modulo $r$: $$ \boxed{ d\equiv\pm m\pmod r. } $$ But the root-bound truncations are completely different: ### Left $$ \boxed{ d\le m-r^2. } $$ ### Right $$ \boxed{ d\ge\max(0,r^2-m). } $$ Thus: $$ \boxed{ \text{residue symmetry} + \text{opposite interval truncation}. } $$ This is the exact geometry that SRBRD adds relative to a simple sieve residue cover. --- # 35. Large Witness Prime Regime If: $$ r^2>m, $$ then: $$ O_{m,\mathrm L,r}=\varnothing. $$ But the right cell may still be nonempty, provided that: $$ r^2\le2m-3. $$ In this case: $$ d\ge r^2-m>0. $$ Thus, all: $$ \sqrt m m - d | Side.right => m + d ``` --- # 46. Root Obstruction ```lean def RootObstruction (m d : Nat) (s : Side) (r : Nat) : Prop := OffsetAdmissible m d ∧ WitnessPrime m r ∧ r ∣ Endpoint m d s ∧ r * r ≤ Endpoint m d s ``` --- # 47. Residual Offset ```lean def ResidualOffset (m d : Nat) : Prop := OffsetAdmissible m d ∧ ∀ s r, WitnessPrime m r → ¬ RootObstruction m d s r ``` In a cleaner version, the relation itself can already contain WitnessPrime, so the quantifier need only be: ```lean ∀ s r, ¬ RootObstruction m d s r ``` But a finite-enumeration theorem will need a witness finset. --- # 48. Formal Theorem Queue ```yaml SRBRD_00: T01: offset_endpoint_bounds T02: offset_endpoint_parity T03: offset_odd_decomposition_bijection T04: reduced_goldbach_pair_offset_bijection T05: prime_iff_no_root_bounded_prime_divisor T06: relation_domain_finite T07: side_endpoint_prime_iff_no_obstruction T08: zero_degree_iff_prime_pair T09: residual_nonempty_iff_goldbach_success T10: residual_card_eq_RGminus T11: failure_iff_complete_cell_cover T12: elimination_finite_termination T13: elimination_order_independent ``` --- # 49. Exact Count Tests ## $N=6$ $$ m=3. $$ $$ \Omega_3=\{0\}. $$ Endpoints: $$ (3,3). $$ Thus: $$ \operatorname{Res}(3)=\{0\}. $$ $$ R_G^{-}(6)=1. $$ --- # 50. $N=8$ $$ m=4. $$ $$ \Omega_4=\{1\}. $$ Endpoints: $$ (3,5). $$ Thus: $$ |\operatorname{Res}(4)|=1. $$ $$ R_G^{-}(8)=1. $$ --- # 51. $N=10$ $$ m=5. $$ $$ \Omega_5=\{0,2\}. $$ $$ d=0 \Rightarrow (5,5). $$ $$ d=2 \Rightarrow (3,7). $$ Both are residual. Thus: $$ \boxed{ |\operatorname{Res}(5)|=2 = R_G^{-}(10). } $$ This is the regression test for the necessity of retaining $d=0$. --- # 52. $N=12$ $$ m=6. $$ Admissible offsets: $$ \Omega_6=\{1,3\}. $$ $$ d=1 \Rightarrow (5,7) $$ is residual. $$ d=3 \Rightarrow (3,9) $$ is obstructed by the right factor: $$ 3 $$ displayed above. Thus: $$ |\operatorname{Res}(6)|=1 = R_G^{-}(12). $$ --- # 53. $N=14$ $$ m=7. $$ $$ \Omega_7=\{0,2,4\}. $$ $$ d=0 \Rightarrow (7,7) $$ is residual. $$ d=2 \Rightarrow (5,9) $$ is obstructed. $$ d=4 \Rightarrow (3,11) $$ is residual. Thus: $$ |\operatorname{Res}(7)|=2 = R_G^{-}(14). $$ --- # 54. $N=68$ $$ m=34. $$ known reduced Goldbach witnesses include： $$ p=31 \Rightarrow d=3, $$ and: $$ p=7 \Rightarrow d=27. $$ Thus, at least: $$ d=3,27 $$ are residual offsets. This agrees with the V0/V1 statement: $$ R_G^{-}(68)=2 $$ displayed above. --- # 55. Root-Truncation Example Take: $$ m=50, $$ $$ r=7. $$ Since: $$ r^2=49\le50, $$ it belongs to the small-witness regime. The left cell requires: $$ d\le1. $$ The right-side root bound is automatically satisfied for all: $$ d\ge0 $$ of these offsets. Thus, for the same $r$: - the left action is possible only very close to the center; - the right action can extend across the whole offset range, but remains constrained by the residue class. This is precisely the asymmetry. --- # 56. Large Witness Example Again take: $$ m=50. $$ If: $$ r=11, $$ then: $$ r^2=121>2m-3=97. $$ Thus: $$ r\notin Q_m. $$ It cannot act at all. If some $r$ satisfies: $$ 50m, $$ then the left cell must be empty; 5. the witness-prime domain extends only to: $$ r^2\le2m-3. $$ These will be the main audit objects of SRBRD-01. --- # 61. Acceptance Gate SRBRD-00 proceeds to SRBRD-01 only if all the following PASS: 1. The $N=4$ boundary is handled separately; 2. $d=0$ is retained as admissible; 3. the bijection between $\Omega_m$ and unordered odd decompositions; 4. the bijection between reduced Goldbach pairs and residual offsets; 5. root-bounded primality equivalence; 6. finiteness of the witness universe; 7. sidewise obstruction completeness; 8. zero-degree iff prime pair; 9. residual nonempty iff Goldbach success; 10. residual cardinality equals $R_G^{-}(2m)$; 11. failure iff complete cover; 12. the exact form of the left residue cell; 13. the exact form of the right residue cell; 14. the large-witness right-only theorem; 15. finite termination of elimination; 16. order independence of elimination; 17. no unproved prime-distribution assumption. --- # 62. Bug / Falsification Checklist Stop immediately if any of the following occurs: - $d=0$ is excluded; - the residual count for $N=10$ is not $2$; - an odd decomposition has no unique offset; - the relation uses the global root bound but forgets the endpoint-specific root bound; - a prime endpoint still has a relation edge; - a composite endpoint has no relation edge; - the factor $2$ appears as a residual obstruction for an odd endpoint; - a left cell remains nonempty when $r^2>m$; - the negative residue of the right congruence is implemented incorrectly in Nat; - residual cardinality differs from $R_G^-$; - the finite-elimination order changes the final residual; - exact equivalence is wrongly called a Goldbach proof. --- # 63. Verdict The target verdict of this document is: $$ \boxed{ \mathrm{SRBRD\mbox{-}00\ EXACT\ REPRESENTATION\ SPECIFICATION\ READY} } $$ But the machine status remains: $$ \boxed{ \text{not yet Lean-verified}. } $$ --- # 64. Next Round: SRBRD-01 If the formal / computational audit of SRBRD-00 PASS, the next round should immediately study: # **SRBRD-01 — Structural Compression Audit** Core question: $$ \boxed{ \text{Is SRBRD merely a finite recoding of Goldbach, or does root-truncated residue geometry genuinely reduce the degrees of freedom of a complete cover?} } $$ Priority checks: 1. small / large witness regimes; 2. left/right cell-cardinality formulas; 3. overlap / disjointness of $O_{L,r}$ and $O_{R,r}$; 4. for the same $r$, the collision condition of the $\pm m$ residues; 5. CRT overlap for distinct $r$; 6. exact union capacity; 7. whether a residual lower bound exists without reading primality truth; 8. whether one can construct: $$ |\cup O_w|<|\Omega_m|. $$ Only if this round produces genuinely nontrivial compression does SRBRD merit promotion to a new main proof line. --- # Appendix A: Machine Contract ```yaml module: SRBRD.ExactRepresentation version: 0.1 domain: m_min: 3 N: 2*m offset: d_min: 0 d_max: m-3 parity: left_endpoint_odd: true right_endpoint_odd: derived witness: side: - left - right prime_min: 3 global_bound: r*r <= 2*m-3 endpoint_bound: r*r <= endpoint divides_endpoint: true residual: definition: no_root_obstruction_on_either_side: true exact_targets: - residual_nonempty_iff_RGminus_positive - residual_card_equals_RGminus - failure_iff_complete_cover truth_role: residual_nonempty: TruthLabel residual_cardinality: TruthLabel ``` --- # Appendix B: Formal Dependency Graph ```text Parity Base Layer ↓ N = 2m ↓ Offset Domain Ω_m ↓ Odd-Decomposition Bijection ↓ Reduced Goldbach Pair Bijection ``` Parallel: ```text Root-Bounded Prime-Divisor Theorem ↓ Finite Witness Prime Domain Q_m ↓ Sidewise Root Obstruction ↓ Zero Degree ↓ Residual Set ``` Join: ```text Reduced Goldbach Pair Bijection + Zero-Degree Prime-Pair Theorem ↓ |Res(m)| = RGminus(2m) ↓ Failure iff Complete Cover ``` --- # Appendix C: Canonical Cell Geometry ```yaml left_cell: congruence: d == m mod r root_truncation: d <= m-r*r empty_if: r*r > m right_cell: congruence: m+d == 0 mod r conceptual: d == -m mod r root_truncation: d >= max(0,r*r-m) global: d <= m-3 r*r <= 2*m-3 ``` --- # Appendix D: Commit Decision ```yaml research_round: SRBRD-00 decision: EXACT_REPRESENTATION_SPECIFICATION_READY claims: goldbach_proved: false exact_reformulation_targeted: true count_level_bijection_targeted: true finite_relation_targeted: true new_candidate_structure: - opposite_root_truncation - large_witness_right_only_regime - finite_residue_cells next: SRBRD-01_Structural_Compression_Audit ```