> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # RG-SCP Validation Round 04 ## RG-SCP 04 Claim / Source Audit ### Source Commonality, Two-Channel Decomposition, Fiber Correction, and Shadow-Gap Strengthening in Fixed-Point Fibers, Shadow Gaps, and Goldbach Counterexample Exclusion **Validation series:** RG-SCP Validation **Validation stage:** V0 — Claim / Source Audit **Round:** 04 **Corresponding original draft:** RG-SCP 04, Fixed-Point Fibers, Shadow Gaps, and Goldbach Counterexample Exclusion, v0.1 **Date:** 2026-09-28 **Version:** v0.1 **Authors:** Neo.K; Aletheia (GPT-5.6 Sol) **Nature:** Canonical Audit / Theorem-Source Ledger **Status:** Fourth audit round; no new global Goldbach proof is claimed --- ## Abstract This document performs a claim/source audit of RG-SCP 04, Fixed-Point Fibers, Shadow Gaps, and Goldbach Counterexample Exclusion. This round checks: 1. whether $T_{\mathrm{top}}$ and $\operatorname{lpf}$ share a source; 2. whether the $M6^*$ additive candidate domain is genuinely nonempty for every even $N\ge10$; 3. whether the multiplier representation of M6 multiplicative fibers is correct in both directions; 4. whether the two-channel Goldbach decomposition holds at all boundaries; 5. the exact status of $\Delta_6$, the shadow set, and the shadow gap; 6. whether the naive fixed-point descent no-go holds; 7. whether fixed-point language adds a new arithmetic source; 8. whether the $6\mid N$ branch genuinely makes standard failure completely equivalent to an M6 section hole. This round obtains four important results. First, it confirms: $$ \boxed{ n\in M6^* \Longrightarrow T_{\mathrm{top}}(n) = \operatorname{lpf}(n). } $$ Therefore: $$ T_{\mathrm{top}}, \quad \operatorname{lpf}, \quad \Phi_N^{\min} $$ belong to the same factor source on the M6 hard branch: $$ \boxed{ \mathcal S_1. } $$ Fixed-point language adds no new elementwise primality information. Second, the original draft’s description of fiber multipliers requires a canonical correction. The original draft correctly proves: $$ n\in F_p \Longrightarrow n=pu, \quad u\in U_6. $$ But the converse: $$ u\in U_6 \Longrightarrow pu\in F_p $$ does not hold. For example: $$ 35 = 7\cdot5, $$ but: $$ T_{\mathrm{top}}(35)=5, $$ Thus: $$ 35\notin F_7. $$ The correct fiber multiplier set should be defined as: $$ \boxed{ U_p = \left\{ u\in\mathbb N: \gcd(u,6)=1, \quad \forall r\in\mathbb P, \ r\mid u \Rightarrow r\ge p \right\}. } $$ where $u=1$ is vacuously allowed. Then: $$ \boxed{ F_p = \left\{ pu: u\in U_p \right\}. } $$ Thus, the inflation cone in the original draft: $$ \mathcal I_{p,q} = \left\{ pu+qv: u,v\in U_6 \right\} $$ should be changed to the exact fiber cone: $$ \boxed{ \mathcal I_{p,q}^{\mathrm{fib}} = \left\{ pu+qv: u\in U_p, \quad v\in U_q \right\}. } $$ Third, the two-channel decomposition holds completely and can be strengthened to a reduced representation count identity. Define: $$ R_G^{-}(N) = \#\left\{ p\in\mathbb P: 2\le p\le N/2, \quad N-p\in\mathbb P \right\}. $$ and: $$ R_6^{-}(N) = \#\left\{ p\in P^*: p\le N/2, \quad N-p\in P^* \right\}. $$ For an even: $$ N\ge10, $$ the prime $2$ channel cannot occur. Thus: $$ \boxed{ R_G^{-}(N) = R_6^{-}(N) + \mathbf1_{\mathbb P}(N-3). } $$ Thus, the positivity decomposition in the original draft: $$ R_G(N)>0 \Longleftrightarrow \Delta_6(N)=0 \lor N-3\in\mathbb P $$ is a correct exact consequence. Fourth, the original draft only points out: $$ \Delta_6(N)>0 \Longrightarrow \Delta_6(N)\ge2. $$ This round can substantially strengthen it. The smallest composite in $M6^*$ is: $$ 25. $$ If: $$ n\in M6^* $$ is composite, Let: $$ p=T_{\mathrm{top}}(n)=\operatorname{lpf}(n). $$ Write: $$ n=pu. $$ Because $p$ is the least prime factor and $n$ is composite: $$ u\ge p\ge5. $$ Thus: $$ e_6(n) = n-p = p(u-1) \ge 5(5-1) = 20. $$ Therefore: $$ \boxed{ e_6(n)>0 \Longrightarrow e_6(n)\ge20. } $$ and hence: $$ \boxed{ \Delta_6(N)>0 \Longrightarrow \Delta_6(N)\ge20. } $$ Thus, the nearest successful shadow of a section hole is not merely at least $2$ lower, but at least: $$ \boxed{ 20. } $$ Namely: $$ \boxed{ \Delta_6(N)>0 \Longrightarrow \max\mathcal S_6(N)\le N-20. } $$ This is the most important exact strengthening of this round. The verdict of this round: $$ \boxed{ \mathrm{PASS\ WITH\ FIBER\ CORRECTION,\ COUNT\ STRENGTHENING,\ AND\ SHADOW\mbox{-}GAP\ STRENGTHENING}. } $$ --- # 1. Audit Scope This round audits only: # RG-SCP 04 ## Fixed-Point Fibers, Shadow Gaps, and Goldbach Counterexample Exclusion It cites the existing element-level fixed-point theorem from M6 v0.2: $$ \operatorname{Fix}(T_{\mathrm{top}}) = P^*. $$ But it does not reaudit: - statistical fixed point; - ergodic fixed point; - LOS; - Gowers; - RH-related interpretation. RG-SCP 04 actually uses only: $$ T_{\mathrm{top}}. $$ --- # 2. Canonical M6 domain Define: $$ \boxed{ M6^* = \left\{ n\in\mathbb N: n>1, \quad \gcd(n,6)=1 \right\}. } $$ is equivalent to: $$ n\equiv1 \quad\text{or}\quad 5 \pmod6. $$ Its prime fixed-point set: $$ \boxed{ P^* = \mathbb P\cap M6^* = \mathbb P_{>3}. } $$ --- # 3. $T_{\mathrm{top}}$ Define: $$ \boxed{ T_{\mathrm{top}}(n) = \min \left\{ d\in M6^*: d\mid n \right\}. } $$ Because: $$ n\in M6^*, $$ the set is nonempty and contains at least: $$ n. $$ --- # 4. The Output of $T_{\mathrm{top}}$ Must Be Prime If: $$ d=T_{\mathrm{top}}(n) $$ is not prime, then there exists: $$ 13 $$ Because: $$ u\in U_6. $$ Thus: $$ r\in M6^* $$ and: $$ r0, $$ Then: $$ \boxed{ e_6(n)\ge20. } $$ --- ## Proof $$ e_6(n)>0 $$ representing: $$ n $$ is not a fixed point, Hence: $$ n $$ is composite. Let: $$ p=T_{\mathrm{top}}(n)=\operatorname{lpf}(n). $$ Write: $$ n=pu. $$ Because $n$ is composite and $p$ is the least prime factor: $$ u\ge p. $$ Also: $$ p\in P^*, $$ Thus: $$ p\ge5. $$ Therefore: $$ e_6(n) = pu-p = p(u-1) \ge p(p-1) \ge 5\cdot4 = 20. $$ Equality is attained at: $$ n=25 $$ this value. This completes the proof. --- # 20. Pair energy Define: $$ \boxed{ E_{6,N}(a) = e_6(a)+e_6(N-a). } $$ Then: $$ E_{6,N}(a)=0 $$ if and only if: $$ a,N-a\in P^*. $$ Claim class: $$ [\mathrm T/E]. $$ --- # 21. Positive pair energy gap By R04-T04: If: $$ E_{6,N}(a)>0, $$ at least one defect term is positive, Thus: $$ \boxed{ E_{6,N}(a)\ge20. } $$ Thus, pair energy has no positive values in: $$ 1,\ldots,19 $$ this range. --- # 22. Section defect Define: $$ \boxed{ \Delta_6(N) = \min_{a\in\mathcal A_N^{(6)}} E_{6,N}(a). } $$ we have: $$ \boxed{ \Delta_6(N)=0 \Longleftrightarrow N\in P^*+P^*. } $$ Claim class: $$ [\mathrm E]. $$ --- # 23. Shadow-gap discreteness strengthening By the pair-energy gap: $$ \boxed{ \Delta_6(N) \in \{0\} \cup [20,\infty)\cap\mathbb N. } $$ More precisely, since all M6 elements are odd, the defect: $$ e_6(n) $$ is even, Thus: $$ \Delta_6(N) $$ is also even. Therefore: $$ \boxed{ \Delta_6(N) \in \{0\} \cup \left( 2\mathbb N\cap[20,\infty) \right). } $$ --- # 24. Two-channel Goldbach count decomposition Define the reduced standard count: $$ \boxed{ R_G^{-}(N) = \#\left\{ p\in\mathbb P: 2\le p\le N/2, \quad N-p\in\mathbb P \right\}. } $$ Define the reduced M6 section count: $$ \boxed{ R_6^{-}(N) = \#\left\{ p\in P^*: p\le N/2, \quad N-p\in P^* \right\}. } $$ --- # 25. $2$ channel absence For an even: $$ N\ge10, $$ If: $$ p=2, $$ Then: $$ N-p=N-2 $$ is even and: $$ N-2>2. $$ Thus: $$ N-2 $$ is not prime. Therefore: $$ p=2 $$ never contributes to: $$ R_G^{-}(N). $$ --- # 26. The Only External Channel Is $3$ Every prime: $$ p>3 $$ belongs to: $$ P^*. $$ Thus, if a standard reduced representation is not in the M6 section, the only possible smaller endpoint is: $$ p=3. $$ --- # 27. Count-level Two-Channel Theorem ## Theorem R04-T05 For every even: $$ N\ge10, $$ we have: $$ \boxed{ R_G^{-}(N) = R_6^{-}(N) + \mathbf1_{\mathbb P}(N-3). } $$ where: $$ \mathbf1_{\mathbb P}(N-3) = \begin{cases} 1,&N-3\in\mathbb P,\\ 0,&\text{otherwise}. \end{cases} $$ This is the count-level strengthening of the original draft’s positivity decomposition. --- # 28. Positivity decomposition By R04-T05: $$ \boxed{ R_G(N)>0 \Longleftrightarrow \left[ R_6^{-}(N)>0 \right] \lor \left[ N-3\in\mathbb P \right]. } $$ Also: $$ R_6^{-}(N)>0 \Longleftrightarrow \Delta_6(N)=0. $$ Thus: $$ \boxed{ R_G(N)>0 \Longleftrightarrow \Delta_6(N)=0 \lor N-3\in\mathbb P. } $$ The original draft’s RG-SCP-04.1: PASS. --- # 29. Failure dual-hole theorem Therefore: $$ \boxed{ F_G(N)=1 \Longleftrightarrow \Delta_6(N)>0 \land N-3\notin\mathbb P. } $$ For an even: $$ N\ge10. $$ PASS. --- # 30. Boundary bit Define: $$ B_3(N) = \mathbf1_{\{N-3\notin\mathbb P\}}. $$ Then: $$ F_G(N) = H_6(N)B_3(N) $$ is exact under Boolean semantics. But this is merely a compact encoding. Claim class: $$ [\mathrm E]. $$ --- # 31. $6\mid N$ clean branch If: $$ 6\mid N, \quad N\ge12, $$ Then: $$ N-3>3 $$ and: $$ 3\mid(N-3). $$ Thus: $$ N-3 $$ is composite. Therefore: $$ \boxed{ F_G(N)=1 \Longleftrightarrow \Delta_6(N)>0. } $$ The original draft’s RG-SCP-04.4: PASS. --- # 32. Count-level clean branch More strongly: For: $$ 6\mid N, \quad N\ge12, $$ we have: $$ \boxed{ R_G^{-}(N) = R_6^{-}(N). } $$ Because the boundary $3$ count is: $$ 0. $$ Thus, on the $6\mathbb N$ branch, the standard Goldbach reduced count is exactly equal to the M6 section count. This is Strengthening S02. --- # 33. Shadow map Define: $$ \boxed{ S_N(a) = T_{\mathrm{top}}(a) + T_{\mathrm{top}}(N-a). } $$ Because: $$ T_{\mathrm{top}} $$ outputs: $$ P^*, $$ Thus: $$ S_N(a)\in P^*+P^*. $$ Thus, every shadow must be a Goldbach-success even integer. Claim class: $$ [\mathrm T]. $$ --- # 34. Shadow identity By definition: $$ e_6(a) = a-T_{\mathrm{top}}(a), $$ $$ e_6(N-a) = N-a-T_{\mathrm{top}}(N-a), $$ Thus: $$ \boxed{ S_N(a) = N-E_{6,N}(a). } $$ Claim class: $$ [\mathrm T]. $$ --- # 35. Shadow set $$ \boxed{ \mathcal S_6(N) = \left\{ S_N(a): a\in\mathcal A_N^{(6)} \right\}. } $$ Because: $$ \mathcal A_N^{(6)} $$ is nonempty and finite, $$ \max\mathcal S_6(N) $$ exists. --- # 36. Shadow-gap theorem $$ \boxed{ \Delta_6(N) = N-\max\mathcal S_6(N). } $$ PASS. Claim class: $$ [\mathrm T/E]. $$ --- # 37. Shadow-Gap Strengthening The original draft only states: $$ \Delta_6(N)>0 \Longrightarrow \Delta_6(N)\ge2. $$ By R04-T04: $$ \boxed{ \Delta_6(N)>0 \Longrightarrow \Delta_6(N)\ge20. } $$ Thus: $$ \boxed{ \max\mathcal S_6(N) \le N-20. } $$ This is Strengthening S03. --- # 38. Nearest-success gap If: $$ \Delta_6(N)>0, $$ Then: $$ N-\Delta_6(N) $$ is a sum of two $>3$ primes. whereas: $$ \Delta_6(N)\ge20. $$ Thus, the section-hole state is separated from the nearest success state generated by its own fixed-point shadow construction by at least: $$ 20. $$ Note: This does not mean that $N$ is separated from “all” Goldbach-success even integers by at least $20$. Only for: $$ \mathcal S_6(N) $$ the construction’s own shadows in this set. This semantic boundary must be retained. --- # 39. Naive descent no-go If: $$ a\in\mathcal A_N^{(6)}, $$ Then: $$ S_N(a) $$ is itself in: $$ P^*+P^*. $$ Thus: $$ \boxed{ F_G(S_N(a))=0. } $$ If: $$ E_{6,N}(a)>0, $$ Then: $$ S_N(a)0, $$ By the positive energy gap: $$ E_{6,N}(a)\ge20. $$ Thus: $$ \boxed{ S_N(a)\le N-20. } $$ Thus, any nontrivial shadow descent drops not merely by at least $2$, but by at least: $$ 20. $$ --- # 41. Shadow Identity After Correction of the Fiber Multiplier Form If: $$ a\in F_p, \quad N-a\in F_q, $$ then, canonically: $$ a=pu, \quad u\in U_p, $$ $$ N-a=qv, \quad v\in U_q. $$ Thus: $$ N=pu+qv. $$ Shadow: $$ S_N(a)=p+q. $$ Gap: $$ \boxed{ N-S_N(a) = p(u-1)+q(v-1). } $$ exact. --- # 42. Zero-inflation condition Because: $$ u,v\ge1, $$ we have: $$ p(u-1)+q(v-1)=0 $$ if and only if: $$ u=v=1. $$ Thus: $$ \boxed{ \Delta_6(N)=0 } $$ is equivalent to the existence of a zero-inflation representative of a fiber pair. PASS. --- # 43. Exact fiber cone Define: $$ \boxed{ \mathcal I_{p,q}^{\mathrm{fib}} = \left\{ pu+qv: u\in U_p, \quad v\in U_q \right\}. } $$ This is the actual image of: $$ F_p\times F_q $$ under addition. --- # 44. Fiber-cone global cover Because: $$ M6^* = \bigsqcup_{p\in P^*} F_p $$ and every even: $$ N\ge10 $$ can be written as the sum of two: $$ M6^* $$ elements, we have: $$ \boxed{ 2\mathbb N_{\ge10} = \bigcup_{p,q\in P^*} \mathcal I_{p,q}^{\mathrm{fib}}. } $$ This is exact. The union, however, generally overlaps heavily, and is not a partition. --- # 45. Status of the Original Inflation Cone The original draft: $$ \mathcal I_{p,q} = \left\{ pu+qv: u,v\in U_6 \right\} $$ is: $$ \mathcal I_{p,q}^{\mathrm{fib}} $$ an over-approximation of this exact cone. It can still serve as a coarse generating cone, but cannot be called: $$ \boxed{ \text{exact fiber-pair image}. } $$ This is Canonical Correction C03. --- # 46. Hard-branch obstruction identity If: $$ p\in P^* $$ and: $$ N-p\in M6^*, $$ Then: $$ T_{\mathrm{top}}(N-p) = \operatorname{lpf}(N-p). $$ If there is also: $$ F_G(N)=1, $$ Then: $$ N-p $$ is composite, Thus: $$ \boxed{ T_{\mathrm{top}}(N-p) = \Phi_N^{\min}(p). } $$ PASS. --- # 47. Source de-duplication Thus, the following three elementwise objects: $$ T_{\mathrm{top}}(N-p), $$ $$ \operatorname{lpf}(N-p), $$ $$ \Phi_N^{\min}(p) $$ have exactly the same values on the hard failure branch. Thus: $$ \boxed{ \text{must not be counted as three independent sources}. } $$ Shared source: $$ \mathcal S_1. $$ --- # 48. What the Fixed-Point Relational Layer Genuinely Adds Although the elementwise source is the same, fixed-point language adds: - multiplicative fiber partition; - additive candidate fiber; - section incidence; - zero set; - shadow map; - pair energy; - transport questions. Thus, it belongs to: $$ \boxed{ \mathcal S_3 = \text{relational organization layer}. } $$ but: $$ \mathcal S_3 $$ is not a new primality oracle. --- # 49. Does $\Delta_6$ Reduce the Proof Burden? On its own: $$ \Delta_6(N)=0 $$ is merely: $$ N\in P^*+P^*. $$ an equivalent rewriting of this condition. Thus: $$ \Delta_6 $$ has not itself completed proof compression. Genuine proof value must come from: - nontrivial bound; - transport; - family theorem; - capacity incompatibility. --- # 50. Global $T_{\min}$ baseline The original draft also defines: $$ T_{\min}(n) = \min \left\{ d>1: d\mid n \right\}. $$ we have: $$ \operatorname{Fix}(T_{\min}) = \mathbb P. $$ Then it defines the global energy: $$ E_N(a). $$ Then: $$ R_G(N)>0 \Longleftrightarrow \min E_N=0. $$ This is entirely: $$ [\mathrm E]. $$ It may be retained as an audit baseline, but not listed as new theorem novelty. --- # 51. Zero-lift criterion The original draft proposes: If: $$ N'0 \Longrightarrow \Delta_6(N')>0. $$ This is an exact elementary implication. Claim class: $$ [\mathrm T]. $$ But at present: $$ \Psi $$ is only a hypothetical transport target. Thus: ```text criterion: theorem existence of useful Psi: open ``` --- # 52. Standard Failure Descent Still Requires a Boundary Condition Even if: $$ \Delta_6(N')>0, $$ standard failure still requires: $$ N'-3\notin\mathbb P. $$ Thus, a genuine descent certificate requires: $$ \boxed{ N'0, \quad N'-3\notin\mathbb P. } $$ PASS. --- # 53. Source Simplification of the $6\mid N$ Branch For: $$ 6\mid N, \quad N\ge12, $$ the boundary bit is automatically: $$ B_3(N)=1. $$ Thus: $$ F_G(N)=1 \Longleftrightarrow \Delta_6(N)>0. $$ The value of this branch lies in: $$ \boxed{ \text{eliminating standard failure’s boundary exception}. } $$ not in adding a primality source. --- # 54. Fixed-to-nonfixed complement map For: $$ 6\mid N, $$ Define: $$ C_N(p)=N-p, $$ where: $$ p\in P^*, \quad p\le N/2. $$ Then: $$ C_N(p)\in M6^*. $$ Goldbach success in this branch is equivalent to: $$ \boxed{ C_N(P_N^*) \cap P^* \neq\varnothing. } $$ failure is equivalent to: $$ \boxed{ C_N(P_N^*) \subseteq M6^*\setminus P^*. } $$ exact. This is a very clean incidence formulation. --- # 55. Source status of anti-incidence The anti-incidence set can still be determined: $$ \operatorname{lpf} $$ element by element. Thus, elementwise, it is still: $$ \mathcal S_1. $$ But the whole-image condition: $$ C_N(P_N^*) \cap P^*=\varnothing $$ is a cross-$p$ relational target. This is: $$ \mathcal S_3 $$ an object genuinely worth retaining. --- # 56. Canonical theorem ledger — RG-SCP 04 | ID | Claim | Class | Source | Status | |---|---|---|---|---| | R04-T01 | $T_{\mathrm{top}}=\operatorname{lpf}$ on $M6^*$ | T | S1 | PASS | | R04-T02 | exact fiber multiplier characterization $F_p=\{pu:u\in U_p\}$ | T | S1 | **AUDIT CORRECTION / PASS** | | R04-T03 | $\mathcal A_N^{(6)}\neq\varnothing$ for even $N\ge10$ | T | S0+S2 | PASS with explicit witnesses | | R04-T04 | positive M6 defect satisfies $e_6(n)\ge20$ | T | S1 | **AUDIT STRENGTHENING / PASS** | | R04-T05 | count-level two-channel decomposition | T | S0+S1+S2 | **AUDIT STRENGTHENING / PASS** | | R04-T06 | failure iff dual hole | T/E | S1+S2+S3 | PASS | | R04-T07 | shadow identity | T | S1 relational | PASS | | R04-T08 | shadow-gap identity | T/E | S1+S3 | PASS | | R04-T09 | $\Delta_6>0\Rightarrow\Delta_6\ge20$ | T | S1 | **AUDIT STRENGTHENING / PASS** | | R04-T10 | naive shadow descent always lands on Goldbach-success | T | S1+S3 | PASS | | R04-T11 | hard-branch $T_{\mathrm{top}}=\Phi_N^{\min}$ | T | S1 | PASS / source duplicate | | R04-T12 | $6\mid N,\ N\ge12$: failure iff $\Delta_6>0$ | T/E | S1+S2+S3 | PASS | | R04-T13 | zero-lift criterion | T | S3 | PASS as conditional criterion | | R04-E01 | global $T_{\min}$ energy reformulation | E | S1 | PASS | | R04-O01 | useful zero-lift morphism exists | O | future | OPEN | | R04-O02 | fixed-point anti-incidence yields capacity contradiction | O | future | OPEN | --- # 57. Canonical corrections ledger ## C01 — Fiber multiplier correction Old: $$ F_p \leadsto pU_6. $$ Canonical: $$ \boxed{ F_p = pU_p. } $$ --- ## C02 — Reduced additive candidate domain Computing: $$ \Delta_6 $$ can use only: $$ \mathcal A_N^{(6),-}. $$ --- ## C03 — Exact fiber cone Old: $$ \mathcal I_{p,q} = pU_6+qU_6. $$ Canonical exact: $$ \boxed{ \mathcal I_{p,q}^{\mathrm{fib}} = pU_p+qU_q. } $$ --- ## C04 — Shadow-gap lower bound Old: $$ \Delta_6>0 \Rightarrow \Delta_6\ge2. $$ Canonical: $$ \boxed{ \Delta_6>0 \Rightarrow \Delta_6\ge20. } $$ --- ## C05 — Count decomposition The positivity theorem is retained, and the following canonical statement is added: $$ \boxed{ R_G^{-} = R_6^{-} + \mathbf1_{\mathbb P}(N-3). } $$ --- ## C06 — Source de-duplication $$ T_{\mathrm{top}}, \operatorname{lpf}, \Phi_N^{\min} $$ share: $$ \mathcal S_1. $$ --- # 58. Formalization priority Recommended priorities: ## F04-01 $T_{\mathrm{top}}=\operatorname{lpf}$. ## F04-02 exact fiber multiplier theorem. ## F04-03 M6 additive candidate nonemptiness with explicit residue witnesses. ## F04-04 positive defect lower bound: $$ e_6>0\Rightarrow e_6\ge20. $$ ## F04-05 pair energy zero iff both endpoints fixed. ## F04-06 count-level two-channel decomposition. ## F04-07 shadow identity. ## F04-08 shadow gap and $20$-gap strengthening. ## F04-09 naive descent no-go. ## F04-10 $6\mid N$ clean branch. ## F04-11 zero-lift criterion. --- # 59. Not Prioritized for Formalization Not prioritized for now: - counterexample cost vectors; - qualitative shadow pressure; - AI search tasks; - heuristic gap statistics; - speculative cross-$p$ invariant. --- # 60. Implementation control If the program constructs a fiber: $$ F_p, $$ it must check: ```text n in F_p iff minFac(n) = p ``` It cannot merely check: ```text n % p == 0 and gcd(n,6)==1 ``` Otherwise: $$ 35 $$ would wrongly enter both: $$ F_5 $$ and: $$ F_7. $$ This is an important implementation control for the first experimental stage. --- # 61. Shadow implementation control For each: $$ a\in\mathcal A_N^{(6)}, $$ one must verify: $$ \boxed{ N-S_N(a)=E_{6,N}(a). } $$ and: $$ E_{6,N}(a) $$ can only be: $$ 0 $$ or: $$ \ge20. $$ If the program produces: $$ 2,4,\ldots,18, $$ the fixed-point / minFac implementation has a bug. --- # 62. Two-channel implementation control For an even: $$ N\ge10, $$ one should verify: $$ \boxed{ R_G^{-}(N) - R_6^{-}(N) \in \{0,1\} } $$ and the difference is exactly: $$ \mathbf1_{\mathbb P}(N-3). $$ This is a very strong E0 identity test. --- # 63. Source novelty verdict RG-SCP 04 adds no new elementwise arithmetic source. Its main sources: $$ \boxed{ \mathcal S_1 + \mathcal S_2. } $$ What is added is: $$ \boxed{ \mathcal S_3 = \text{fixed-point / fiber relational organization}. } $$ Its proof value still awaits: - transport; - incidence bound; - capacity contradiction; to realize it. --- # 64. Proof-compression verdict RG-SCP 04 has two compressions genuinely worth retaining. First: $$ \boxed{ R_G^{-}(N) = R_6^{-}(N) + \mathbf1_{\mathbb P}(N-3). } $$ splits standard Goldbach into the M6 section and a single boundary channel. Second, for: $$ 6\mid N, $$ the boundary channel closes automatically: $$ \boxed{ R_G^{-}(N) = R_6^{-}(N). } $$ This is a clean branch reduction. --- # 65. Validation verdict RG-SCP 04: $$ \boxed{ \mathrm{PASS\ WITH\ FIBER\ CORRECTION,\ COUNT\ STRENGTHENING,\ AND\ SHADOW\mbox{-}GAP\ STRENGTHENING} } $$ The most important correction: $$ \boxed{ F_p\neq pU_6 } $$ instead: $$ \boxed{ F_p=pU_p. } $$ The most important strengthening: $$ \boxed{ \Delta_6(N)>0 \Longrightarrow \Delta_6(N)\ge20. } $$ The most important source conclusion: $$ \boxed{ T_{\mathrm{top}}, \operatorname{lpf}, \Phi_N^{\min} \text{ are one factor source}. } $$ --- # 66. Next Round V0 Round 05 audits only: # RG-SCP 05 ## Prime Approximation, Skew Field, and Counterexample Pressure Probing Focus: 1. which parts of $\varepsilon_{\mathrm{full}}$ are exact and which are fitted; 2. separation of the exact residuals of $\pi-\operatorname{Li}$ and $\psi-x$ from heuristic interpretation; 3. whether the complement pressure field has any implicit probability semantics; 4. the exact status of escape prime / escape depth; 5. whether null models / surrogate contradictory fibers have clear falsification semantics; 6. whether the proof-promotion gate sufficiently prevents heuristic leakage; 7. which objects in 05 merit retention in async discovery, and which should not enter the sync proof core. --- # Appendix A: Canonical Fiber Multiplier ```text P*: primes > 3 For p in P*: U_p: u >= 1 gcd(u,6)=1 every prime divisor r of u satisfies r >= p Then: F_p = {p*u : u in U_p} ``` Example: ```text 35 = 7*5 5 < 7 => 5 not allowed in U_7 => 35 not in F_7 ``` --- # Appendix B: Canonical Two-Channel Count $$ R_G^{-}(N) = R_6^{-}(N) + \mathbf1_{\mathbb P}(N-3), \quad N\ge10 \text{ even}. $$ For: $$ 6\mid N, \quad N\ge12, $$ $$ R_G^{-}(N) = R_6^{-}(N). $$ --- # Appendix C: Canonical Shadow Gap For: $$ n\in M6^*, $$ $$ e_6(n) = n-\operatorname{lpf}(n). $$ Then: $$ e_6(n)=0 \iff n\text{ prime}, $$ and: $$ e_6(n)>0 \Rightarrow e_6(n)\ge20. $$ Therefore: $$ \Delta_6(N)=0 $$ or: $$ \Delta_6(N)\ge20. $$ --- # Appendix D: Source Signature ```yaml RG-SCP-04: S0_elementary: true S1_factor_lpf: primary S2_M6_quotient: primary S3_fixed_point_relational: organizational S4_minimality: interface_only S5_analytic: false S6_heuristic: diagnostics_only source_identity: T_top: S1 lpf: S1 Phi_min_on_hard_branch: S1 novelty: new_prime_oracle: false new_relational_coordinates: true clean_two_channel_reduction: true exact_shadow_gap_strengthening: true ``` --- # Appendix E: Commit Decision ```yaml validation_round: V0-R04 decision: PASS_WITH_FIBER_CORRECTION_COUNT_STRENGTHENING_AND_SHADOW_GAP_STRENGTHENING strengthening: - count_level_two_channel_decomposition - positive_defect_minimum_20 - delta_positive_implies_delta_at_least_20 - exact_fiber_cone - reduced_candidate_domain canonical_corrections: - F_p_equals_pU_p_not_pU6 - inflation_cone_uses_U_p_U_q - source_deduplication - reduced_delta_domain commit_to_sync_core: true promote_new_goldbach_theorem: false next_round: V0-R05 ```