> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # RG-SCP Validation Round 08 ## Phase-I Canonical Synthesis ### Overall Audit of RG-SCP 01–07, Canonical Core Reconstruction, Formalization Priority, and E0–E8 Validation Protocol **Validation series:** RG-SCP Validation **Validation stage:** V0 — Claim / Source Audit **Round:** 08 / overall V0 closure **Corresponding original draft:** RG-SCP 08, Reverse Closure Synthesis: Source-Class Audit, Formalizable Core, and First Experimental Protocol, v0.1 **Date:** 2026-09-28 **Version:** v0.1 **Authors:** Neo.K; Aletheia (GPT-5.6 Sol) **Nature:** Canonical Phase-I Checkpoint / Theorem Ledger / Source Ledger / Validation Protocol **Status:** V0 claim/source audit completed; Phase-II has not yet opened; Formal Core and E0–E8 remain to be executed --- ## Abstract This document is the first overall checkpoint of RG-SCP Validation V0. Rather than adding a new Goldbach route, it formally merges the audit results from V0-R01 through V0-R07, superseding the parts of the original RG-SCP 08 that have been corrected, weakened, or repositioned. The Phase-I canonical architecture no longer uses: $$ \text{obstruction cost} \to \text{matching demand} \to \text{incidence capacity} $$ as its backbone. After seven rounds of article-by-article audit, the cleaner main line should be reconstructed as: $$ \boxed{ \text{Counterexample Universe} } $$ $$ \Downarrow $$ $$ \boxed{ \text{Proof-Admissible Outer Approximation} } $$ $$ \Downarrow $$ $$ \boxed{ \text{Exact Reduced-Domain Obstruction Cover} } $$ $$ \Downarrow $$ $$ \boxed{ \text{Primorial Hard-Sector Geometry} } $$ $$ \Downarrow $$ $$ \boxed{ \text{M6 Fixed-Point / Two-Channel Decomposition} } $$ $$ \Downarrow $$ $$ \boxed{ \text{Canonical Survivor Demand} } $$ $$ \Downarrow $$ $$ \boxed{ \text{Canonical Composite Coverage Capacity} } $$ $$ \Downarrow $$ $$ \boxed{ \text{Minimality Transport Neighborhood} } $$ $$ \Downarrow $$ $$ \boxed{ \text{Lower Bound} > \text{Independent Upper Bound} } $$ $$ \Downarrow $$ $$ \boxed{ \text{Counterexample Branch Exclusion}. } $$ The most important V0 canonical corrections are as follows. --- ### 1. Admissibility No Longer Embeds the Target Predicate in Its Set Definition Original: $$ \mathcal A_k = \{N:F_G(N)=1\land K_1(N)\land\cdots\}. $$ Canonically, change to: $$ \boxed{ \mathcal A_k^{\mathrm P} = \left\{ N\in2\mathbb N_{\ge4}: K_1(N)\land\cdots\land K_k(N) \right\}, } $$ and separately prove: $$ \boxed{ \mathcal C_G \subseteq \mathcal A_k^{\mathrm P}. } $$ Thus: $$ \mathcal A_k^{\mathrm P}=\varnothing $$ only then can one noncircularly infer: $$ \mathcal C_G=\varnothing. $$ --- ### 2. Raw GFOC Is an Exact Characterization on the Reduced Domain For: $$ \mathbb P_N^{-} = \left\{ p\in\mathbb P: 2\le p\le N/2 \right\}, $$ we have: $$ \boxed{ F_G(N)=1 \Longleftrightarrow \mathbb P_N^{-} \subseteq \bigcup_{\substack{ r\in\mathbb P\\ r^2\le N-2 }} \mathcal O_r(N). } $$ Thus, obstruction cover is not merely a necessary condition, but a reduced-domain exact reformulation. --- ### 3. The Primorial Hard Sector Itself Does Not Require a Failure Hypothesis If: $$ M_k\mid N, $$ and: $$ p_k\le N/2, $$ Then: $$ \boxed{ \mathbb H_{p_k}(N) = \left\{ p\in\mathbb P: p_kp_k $$ does one need: $$ F_G(N)=1. $$ --- ### 4. The Effective Wheel Is Removed from the Canonical Exact Core $$ M_k^{\mathrm{eff}}(N) $$ For: $$ p>p_k $$ adds no information beyond the ordinary wheel. At the self-cell: $$ p=r\mid N $$ it may instead wrongly label a genuine small-factor obstruction as hard. Thus: $$ \boxed{ \text{effective wheel} } $$ is no longer listed in the Phase-I exact core. --- ### 5. The M6 Fixed-Point Fiber Multiplier Requires Correction Original: $$ F_p \sim pU_6 $$ is imprecise. Canonical definition: $$ \boxed{ U_p = \left\{ u\ge1: \gcd(u,6)=1, \quad r\mid u,\ r\in\mathbb P \Rightarrow r\ge p \right\}, } $$ Then: $$ \boxed{ F_p = \{pu:u\in U_p\}. } $$ Thus, the exact fiber cone: $$ \boxed{ \mathcal I_{p,q}^{\mathrm{fib}} = \left\{ pu+qv: u\in U_p, \quad v\in U_q \right\}. } $$ --- ### 6. The Positive M6 Shadow Gap Is at Least $20$ For: $$ n\in M6^*, $$ If: $$ e_6(n) = n-T_{\mathrm{top}}(n) >0, $$ Then: $$ \boxed{ e_6(n)\ge20. } $$ Thus: $$ \boxed{ \Delta_6(N)>0 \Longrightarrow \Delta_6(N)\ge20. } $$ and: $$ \boxed{ \max\mathcal S_6(N)\le N-20. } $$ This replaces the weaker statement in the original draft: $$ \Delta_6(N)\ge2. $$ --- ### 7. The Two-Channel Decomposition Can Be Promoted to the Count Level For an even: $$ N\ge10, $$ Define: $$ R_6^{-}(N) = \#\left\{ p\in P^*: p\le N/2, \quad N-p\in P^* \right\}. $$ Then: $$ \boxed{ R_G^{-}(N) = R_6^{-}(N) + \mathbf1_{\mathbb P}(N-3). } $$ Thus, for: $$ 6\mid N, \quad N\ge12, $$ we directly have: $$ \boxed{ R_G^{-}(N) = R_6^{-}(N). } $$ --- ### 8. Discovery Must Include a Truth-Leakage Firewall `exact` does not mean it can legitimately serve as a discovery feature. For example: $$ \Delta_6(N) $$ under: $$ 6\mid N $$ branch already determines failure exactly. whereas: $$ \Delta_6(N) + B_3(N) $$ for a general even: $$ N\ge10 $$ completely encodes failure. Thus, all subsequent Phase-I data must be labeled along both dimensions: ## Epistemic Status ```text exact heuristic mixed ``` ## Data Role ```text proof_input discovery_probe diagnostic truth_label ``` --- ### 9. The Escape Prime Is a Target-Equivalent Observable If: $$ k\ge2, \quad M_k\mid N, \quad N>6, $$ Then: $$ \boxed{ R_G(N)>0 \Longleftrightarrow p_{\mathrm{esc}}(N;k) \text{ exists}. } $$ Thus: - `escape_exists` is a truth label; - `escape_depth` is an exact post-hoc diagnostic; cannot serve as same-target leakage-free features. --- ### 10. Canonical Capacity Is Not $I_N(y)$ The original draft calls: $$ I_N(y) $$ deep obstruction capacity. After V0-R06 reconstruction, the genuinely overlap-free capacity is: $$ \boxed{ C_N^{\mathrm{can}}(y) = \#\left\{ p\in\mathbb H_y(N): N-p\text{ composite} \right\}. } $$ The exact partition: $$ \boxed{ D_N(y) = R_G^{-}(N) + C_N^{\mathrm{can}}(y). } $$ Thus: $$ \boxed{ R_G^{-}(N) = D_N(y) - C_N^{\mathrm{can}}(y). } $$ whereas: $$ I_N(y) $$ should be renamed: $$ \boxed{ \text{deep factor-incidence mass}. } $$ and we have: $$ \boxed{ I_N(y) = C_N^{\mathrm{can}}(y) + \Omega_N(y). } $$ Therefore: $$ R_G^{-} = D+\Omega-I $$ is retained as an audit identity, no longer as the canonical capacity definition. --- ### 11. The Primary Object of Minimality Transport Is the Neighborhood, Not Matching Define: $$ \Gamma_k(N_*) = \left\{ q: \exists h\in\mathcal H_k(N_*), \quad h\sim q \right\}. $$ Then: $$ \boxed{ \Gamma_k(N_*) \subseteq \mathbb H_{p_k}(N_*). } $$ Thus: $$ \boxed{ D_{N_*}(p_k) \ge |\Gamma_k(N_*)|. } $$ under minimal failure: $$ \boxed{ \Gamma_k(N_*) \subseteq \mathbb C_{p_k}(N_*). } $$ Therefore: $$ \boxed{ |\Gamma_k(N_*)| \le C_{N_*}^{\mathrm{can}}(p_k). } $$ Maximum matching: $$ \nu_k $$ remains correct, but is only an optional injective certificate. --- ### 12. Bounded-Degree Transport Demand Does Not Require a Matching Theorem If: $$ \Delta_k^R $$ is the transport graph’s maximum right degree, Then: $$ \boxed{ |\Gamma_k| \ge \left\lceil \frac{ |\mathcal H_k| }{ \Delta_k^R } \right\rceil. } $$ follows directly from edge counting. Hall’s theorem is necessary only when a left-perfect matching is required. --- ### 13. Renaming `primorial-smooth shift` $$ M_k\mid h $$ does not mean that all prime factors of $h$ are: $$ \le p_k. $$ Thus, the canonical name changes to: $$ \boxed{ \text{Primorial-Divisible Shift}. } $$ --- ### 14. The Genuine Phase-I Closure Target It is now unified as: $$ \boxed{ \text{Leakage-Free Demand Lower Bound} > \text{Independent Canonical Coverage Upper Bound}. } $$ General form: $$ \boxed{ L_N(y) > U_N^{\mathrm{can}}(y). } $$ Minimality transport form: $$ \boxed{ L_k^{\mathrm{trans}}(N_*) > U_{N_*}^{\mathrm{can}}(p_k). } $$ This is the only inequality closure that should be treated as the main line after V0. --- Final verdict of this round: $$ \boxed{ \mathrm{V0\ COMPLETE} } $$ but: $$ \boxed{ \mathrm{PHASE\ II\ NOT\ YET\ OPEN}. } $$ The reason is: - exact claim audit is complete; - source audit is complete; - the canonical core has been reconstructed; - but the Formal Core has not yet been machine-checked; - the E0 exact identity suite has not yet been executed; - demand / capacity compression has not yet been measured; - transport incremental gain has not yet been measured; - source novelty has not yet passed the adversarial sufficiency test. Thus, the next stage is not RG-SCP 09. The next stage should be: # V1 — Formal Core then: # V2 — E0 Identity Validation followed, in order, by: $$ E1\to E6. $$ The Phase-II Gate opens only after genuinely observing: $$ \boxed{ \text{nontrivial compression gain} } $$ or: $$ \boxed{ \text{new exact source-separating invariant} } $$ one of these. --- # 1. Has V0 Completed Its Task? The task of V0 is: > Without relying on large-scale computation, check the claim classes, source classes, logical boundaries, data roles, duplicate sources, and obvious correctable points of RG-SCP 01–07 article by article. Now: $$ \boxed{ \text{V0-R01} \to \text{V0-R07} } $$ have all been completed. Thus: $$ \boxed{ \text{V0 Claim / Source Audit = COMPLETE}. } $$ --- # 2. Final Claim Taxonomy Subsequent canonical claims should at least be labeled: ## T — Exact Theorem $$ [\mathrm T] $$ Already has a rigorous proof. ## E — Equivalent Reformulation $$ [\mathrm E] $$ Logically equivalent to the target or an existing theorem, without automatically reducing the proof burden. ## H — Heuristic / Empirical $$ [\mathrm H] $$ Depends on fitting, finite observations, learned patterns, or approximate reconstruction. ## O — Open Target $$ [\mathrm O] $$ Not yet proved, but would reduce counterexample admissibility if true. ## F — Framework / Definition $$ [\mathrm F] $$ Research rules, data structures, definitions, and methodology. ## C — Canonical Correction $$ [\mathrm C] $$ Not a new theorem, but a correction to domain, notation, typing, source ownership, or data role. --- # 3. Final Data-Role Taxonomy Every computable object should additionally be labeled: ## ProofInput Can legitimately enter a rigorous proof chain. ## DiscoveryProbe Does not directly encode the target; used to find patterns. ## Diagnostic May be analyzed post hoc, but should not be used as a same-target leakage-free input. ## TruthLabel Exactly determines the target, directly or in combination. ## HistoryDerived Legitimately derived from minimality / prefix truth, but its target-history provenance must be disclosed. This category may be included as a secondary tag alongside the first four roles. --- # 4. Final Source-Class Ledger ## $\mathcal S_0$ — Elementary arithmetic / logic - parity; - addition; - multiplication; - order; - set / finite counting; - divisibility; - well-ordering. --- ## $\mathcal S_1$ — Factor / least-prime-factor source Including: $$ \operatorname{lpf}(n), $$ $$ T_{\mathrm{top}}(n) $$ under: $$ M6^*, $$ $$ \Phi_N^{\min}(p) $$ on the hard failure branch, and all their deterministic readouts: - least-factor cells; - factor depth; - factor incidence; - lpf entropy; - obstruction alphabet. --- ## $\mathcal S_2$ — Finite quotient / wheel source Including: $$ \bmod6, $$ $$ \bmod30, $$ $$ \bmod210, $$ General: $$ M_k $$ wheels. Adding more finite moduli is usually only source refinement. --- ## $\mathcal S_3$ — Fixed-point relational organization Adds no new primality bit. It reorganizes: - multiplicative fibers; - additive section; - shadow map; - anti-incidence; - zero-lift transport target. --- ## $\mathcal S_4$ — Additive-history / minimality source Derived from: $$ \forall M0. $$ It is a target-internal historical source. Central uses: - lower occupied layers; - primorial-divisible shift; - transport neighborhood; - history-derived lower bound. --- ## $\mathcal S_5$ — External analytic counting Including: - prime count bounds; - primes in AP; - Brun–Titchmarsh; - explicit PNT; - sieve bounds; - future pair-correlation theorems. These do not belong to the RG-SCP elementary core. --- ## $\mathcal S_6$ — Heuristic approximation / skew Including: - fitted $\varepsilon$; - prime predictors; - empirical skew; - anomaly detector; - ML feature; - surrogate scoring. Does not enter the proof chain by default. --- ## Future $\mathcal S_7$ — New nonlocal / pair source Not yet established. If Phase I stalls, one may search for: - pair spectral; - additive convolution invariant; - nonlocal operator; - generating-function positivity certificate; - new source-kernel split. --- # 5. Source de-duplication final rules The following must not be counted repeatedly as independent sources: $$ \boxed{ T_{\mathrm{top}} = \operatorname{lpf} } $$ on: $$ M6^*. $$ and: $$ \boxed{ T_{\mathrm{top}}(N-p) = \Phi_N^{\min}(p) } $$ on the hard failure branch. Thus: ```text fixed point label least factor canonical obstruction factor ``` share, at the element level: $$ \mathcal S_1. $$ --- # 6. Deduplicating Wheel Sources $$ M_2, M_3, M_4,\ldots $$ are not a sequence of independent sources. They belong to the same: $$ \mathcal S_2. $$ Only if a wheel-derived construction produces: > A theorem-level invariant that cannot be deterministically reconstructed from the finite quotient signature can it possibly be promoted to a new source. --- # 7. Effective wheel final status The original RG-SCP 03: $$ M_k^{\mathrm{eff}}(N) $$ from: $$ M_k $$ removes: $$ r\mid N $$ the small primes satisfying this condition. V0 verdict: $$ \boxed{ \text{REMOVE FROM CANONICAL EXACT CORE}. } $$ Reason: - It is identical to the ordinary wheel for $p>p_k$; - it may mislabel the self-cell; - it adds no proof information. It may remain in async heuristic history, but not enter the theorem ledger. --- # 8. Counterexample admissibility final form Define the counterexample universe: $$ \mathcal C_G. $$ Proof-admissible outer approximation: $$ \boxed{ \mathcal A_k^{\mathrm P} = \left\{ N\in2\mathbb N_{\ge4}: K_1(N)\land\cdots\land K_k(N) \right\}. } $$ Every: $$ K_i $$ must have a theorem: $$ F_G(N)=1 \Longrightarrow K_i(N). $$ Therefore: $$ \boxed{ \mathcal C_G \subseteq \mathcal A_k^{\mathrm P}. } $$ --- # 9. Separating the Heuristic Candidate Space If heuristic filters are used: $$ H_j(N), $$ separately define: $$ \mathcal A_m^{\mathrm H}. $$ One must not write: $$ \mathcal C_G \subseteq \mathcal A_m^{\mathrm H} $$ unless every filter has been promoted to a theorem. --- # 10. Goldbach reduced-domain canonical target Use uniformly: $$ \boxed{ \mathbb P_N^{-} = \left\{ p\in\mathbb P: 2\le p\le N/2 \right\}. } $$ and: $$ \boxed{ R_G^{-}(N) = \#\left\{ p\in\mathbb P_N^{-}: N-p\in\mathbb P \right\}. } $$ Then: $$ \boxed{ F_G(N)=1 \Longleftrightarrow R_G^{-}(N)=0. } $$ --- # 11. Canonical GFOC Define: $$ \mathcal O_r(N) = \left\{ p\in\mathbb P_N^{-}: r\mid(N-p) \right\}. $$ Then: $$ \boxed{ F_G(N)=1 \Longleftrightarrow \mathbb P_N^{-} \subseteq \bigcup_{\substack{ r\in\mathbb P\\ r^2\le N-2 }} \mathcal O_r(N). } $$ This is the canonical obstruction cover after the R02 audit. --- # 12. Canonical proper obstruction map Failure is equivalent to the existence of: $$ \Phi_N: \mathbb P_N^{-} \to \mathbb P $$ such that: $$ \Phi_N(p)\mid(N-p), $$ and: $$ \Phi_N(p)^2\le N-p. $$ Canonical choice: $$ \Phi_N^{\min}(p) = \operatorname{lpf}(N-p). $$ Note: > A total proper map is not unique; the least-factor choice is unique. --- # 13. Divisor self-suppression If: $$ r\mid N, $$ and: $$ r\mid(N-p), $$ Then: $$ r\mid p. $$ For a prime: $$ p, $$ Thus: $$ p=r. $$ This is the shared core of R03 primorial geometry and R06 capacity suppression. --- # 14. Primorial hard-sector canonical theorem If: $$ M_k\mid N, $$ and: $$ p_k\le N/2, $$ Then: $$ \boxed{ \mathbb H_{p_k}(N) = \left\{ p\in\mathbb P: p_kp_k. } $$ This is the conditional strengthening of the hard sector under failure. --- # 16. M6 fixed-point canonical theorem For: $$ n\in M6^*, $$ $$ \boxed{ T_{\mathrm{top}}(n) = \operatorname{lpf}(n). } $$ Thus: $$ \boxed{ \operatorname{Fix}(T_{\mathrm{top}}) = P^*. } $$ --- # 17. Canonical M6 fiber For: $$ p\in P^*, $$ Define: $$ U_p = \left\{ u\ge1: \gcd(u,6)=1, \quad r\mid u,\ r\in\mathbb P \Rightarrow r\ge p \right\}. $$ Then: $$ \boxed{ F_p = \{pu:u\in U_p\}. } $$ Old: $$ pU_6 $$ version is superseded. --- # 18. M6 additive candidate nonemptiness For every even: $$ N\ge10, $$ $$ \boxed{ \mathcal A_N^{(6)} \neq\varnothing. } $$ Even the reduced domain: $$ \boxed{ \mathcal A_N^{(6),-} \neq\varnothing. } $$ can be proved elementarily using explicit residue-class witnesses: $$ 5 $$ or: $$ 7 $$ of these values. --- # 19. Two-channel count theorem For: $$ N\ge10 $$ even: $$ \boxed{ R_G^{-}(N) = R_6^{-}(N) + \mathbf1_{\mathbb P}(N-3). } $$ Therefore: $$ F_G(N)=1 $$ is equivalent to: $$ \Delta_6(N)>0 $$ and: $$ N-3\notin\mathbb P. $$ --- # 20. $6\mid N$ clean branch If: $$ 6\mid N, \quad N\ge12, $$ Then: $$ \boxed{ R_G^{-}(N) = R_6^{-}(N), } $$ Thus: $$ \boxed{ F_G(N)=1 \Longleftrightarrow \Delta_6(N)>0. } $$ --- # 21. Canonical shadow identity $$ S_N(a) = T_{\mathrm{top}}(a) + T_{\mathrm{top}}(N-a). $$ $$ E_{6,N}(a) = e_6(a)+e_6(N-a). $$ we have: $$ \boxed{ S_N(a) = N-E_{6,N}(a). } $$ Therefore: $$ \boxed{ \Delta_6(N) = N-\max\mathcal S_6(N). } $$ --- # 22. Shadow positive-gap theorem If: $$ e_6(n)>0, $$ Then: $$ e_6(n)\ge20. $$ Thus: $$ \boxed{ \Delta_6(N) = 0 } $$ or: $$ \boxed{ \Delta_6(N)\ge20. } $$ If there is a section hole: $$ \boxed{ \max\mathcal S_6(N)\le N-20. } $$ --- # 23. Naive shadow descent final status Any: $$ S_N(a) $$ is already a sum of two: $$ P^* $$ primes. Thus: $$ \boxed{ F_G(S_N(a))=0. } $$ If: $$ S_N(a)6, $$ $$ \boxed{ R_G(N)>0 \Longleftrightarrow p_{\mathrm{esc}}(N;k)\text{ exists}. } $$ Thus: ```text escape_exists = TruthLabel escape_depth = PosthocDiagnostic / TruthAdjacent ``` --- # 28. Surrogate world final semantics For: $$ \mathcal Q\subseteq\mathbb N, $$ Define: $$ R_{\mathcal Q}^{-}(N), $$ and: $$ F_{\mathcal Q}(N). $$ Only: $$ \mathcal Q=\mathbb P $$ is this called: $$ F_G. $$ A synthetic world must not directly appropriate real-prime target notation. --- # 29. Canonical survivor demand For: $$ 2\le yy} B_r(N;y). } $$ Therefore: $$ \boxed{ C_N^{\mathrm{can}}(y) = \sum_{r>y} |B_r(N;y)|. } $$ --- # 32. Incidence mass final status $$ m_y(N,p) $$ counts deep proper factors. $$ I_N(y) = \sum_p m_y(N,p). $$ It is not canonical coverage capacity. whereas: $$ \Omega_N(y) = \sum_p \max(m_y(N,p)-1,0). $$ we have: $$ \boxed{ I_N(y) = C_N^{\mathrm{can}}(y) + \Omega_N(y). } $$ Thus: $$ R_G^{-} = D+\Omega-I $$ is merely an audit identity. --- # 33. $D>I$ final interpretation $$ \boxed{ D_N(y)>I_N(y) \Longleftrightarrow R_G^{-}(N)>\Omega_N(y). } $$ Thus, $D>I$ is only a sufficient criterion for success. It is no longer the primary Phase-I closure target. --- # 34. AP raw capacity column Define: $$ x_r(N) = \min(N/2,N-r^2), $$ $$ A_r(N) = \#\left\{ p\in\mathbb P: 2\le p\le x_r(N), \quad p\equiv N\pmod r \right\}. $$ Least-factor cell: $$ B_r(N;y) \subseteq A_r(N). $$ Thus: $$ \boxed{ C_N^{\mathrm{can}}(y) \le \sum_{r>y} A_r(N). } $$ --- # 35. Exact divisor-column suppression If: $$ r\mid N, $$ Then: $$ \boxed{ A_r(N) = \mathbf1_{\{N\ge r(r+1)\}}. } $$ This replaces: $$ A_r\le1. $$ --- # 36. Exact small-factor self-cell If: $$ \ell\mid N, $$ and: $$ \ell\le y U_N^{\mathrm{can}}(y) \Longrightarrow R_G^{-}(N)>0. } $$ This is the primary Phase-I closure template. --- # 38. Minimality transport naming final form Old: `primorial-smooth shift` Canonical: $$ \boxed{ \text{primorial-divisible shift} } $$ Namely: $$ M_k\mid h. $$ --- # 39. Transport layer set $$ \boxed{ \mathcal H_k(N_*) = \left\{ h>0: M_k\mid h, \quad N_*-h>2p_k \right\}. } $$ Its size: $$ \boxed{ |\mathcal H_k(N_*)| = \left\lfloor \frac{ N_*-2p_k-2 }{ M_k } \right\rfloor } $$ holds when the numerator is nonnegative. nonempty if and only if: $$ \boxed{ N_* \ge M_k+2p_k+2. } $$ --- # 40. Transport neighborhood $$ \boxed{ \Gamma_k(N_*) = \left\{ q: \exists h\in\mathcal H_k(N_*), \quad q\in\mathcal R^{-}(N_*-h) \right\}. } $$ Then: $$ \boxed{ \Gamma_k(N_*) \subseteq \mathbb H_{p_k}(N_*). } $$ under failure: $$ \boxed{ \Gamma_k(N_*) \subseteq \mathbb C_{p_k}(N_*). } $$ --- # 41. Transport demand Canonical: $$ \boxed{ D_{N_*}(p_k) \ge |\Gamma_k(N_*)|. } $$ Matching: $$ \nu_k $$ is demoted to an optional certificate. --- # 42. Direct bounded-degree lower bound If: $$ \Delta_k^R(N_*) $$ is the maximum right degree, Then: $$ \boxed{ |\Gamma_k(N_*)| \ge \left\lceil \frac{ |\mathcal H_k(N_*)| }{ \Delta_k^R(N_*) } \right\rceil. } $$ No matching theorem is required. --- # 43. Exact right-degree interval For a fixed right endpoint: $$ q, $$ Define: $$ L_{k,q} = \max(q,2p_k-q+1), $$ $$ U_{k,q} = N_*-q-M_k. $$ Then: $$ \boxed{ \deg(q) = \#\left\{ p\in\mathbb P: L_{k,q}\le p\le U_{k,q}, \quad p\equiv N_*-q\pmod{M_k} \right\}. } $$ If: $$ \gcd(N_*-q,M_k)>1, $$ Then: $$ \deg(q)=0. $$ --- # 44. Minimality closure final form If: $$ |\Gamma_k(N_*)| \ge L_k^{\mathrm{trans}}(N_*), $$ and: $$ C_{N_*}^{\mathrm{can}}(p_k) \le U_{N_*}^{\mathrm{can}}(p_k), $$ then a minimal counterexample requires: $$ L_k^{\mathrm{trans}}(N_*) \le U_{N_*}^{\mathrm{can}}(p_k). $$ Thus: $$ \boxed{ L_k^{\mathrm{trans}}(N_*) > U_{N_*}^{\mathrm{can}}(p_k) } $$ excludes a minimal counterexample. --- # 45. The Actual Research Domain of Transport If: $$ M_k\mid N, $$ total demand is already exact: $$ D_N(p_k) = \pi(N/2)-k. $$ Thus, the region where transport is most likely to add quantitative value is: $$ \boxed{ M_k\nmid N, } $$ but there are many: $$ M_k\mid h $$ lower layers of this kind. This is the priority domain for E6. --- # 46. Canonical Theorem Ledger: Core A — Foundation ## C-A01 Emptiness of the counterexample set is equivalent to Goldbach. Class: $$ [\mathrm E]. $$ ## C-A02 A nonempty counterexample set has a minimal counterexample. Class: $$ [\mathrm T]. $$ ## C-A03 minimality implies success for all smaller admissible even integers. Class: $$ [\mathrm T]. $$ ## C-A04 proof-admissible outer approximation soundness. Class: $$ [\mathrm T/F]. $$ --- # 47. Canonical Theorem Ledger: Core B — Obstruction ## C-B01 The reduced prime domain suffices. ## C-B02 Reduced pointwise small-factor characterization. ## C-B03 Reduced GFOC exact characterization. ## C-B04 Total proper obstruction map iff failure. ## C-B05 least-factor canonical partition. ## C-B06 divisor self-suppression. --- # 48. Canonical Theorem Ledger: Core C — Wheel ## C-C01 M6 residue branch classification. ## C-C02 wheel-hard nesting. ## C-C03 primorial-divisible target implies hard-sector equality. ## C-C04 hard-sector cardinality: $$ \pi(N/2)-k. $$ ## C-C05 failure turns the hard sector into a $p_k$-rough composite wall. ## C-C06 graph deficiency: $$ \delta_N = R_G^{-}(N). $$ --- # 49. Canonical Theorem Ledger: Core D — Fixed Point ## C-D01 $$ T_{\mathrm{top}} = \operatorname{lpf} $$ on: $$ M6^*. $$ ## C-D02 exact fiber multiplier: $$ F_p=pU_p. $$ ## C-D03 M6 additive candidate nonempty. ## C-D04 positive fixed-point defect at least: $$ 20. $$ ## C-D05 count-level two-channel decomposition. ## C-D06 shadow identity. ## C-D07 shadow-gap identity. ## C-D08 positive $\Delta_6$ at least: $$ 20. $$ ## C-D09 naive shadow descent no-go. ## C-D10 $6\mid N$ clean branch. --- # 50. Canonical Theorem Ledger: Core E — Heuristic Firewall This group comprises the research-method core, not new number-theoretic theorems. ## C-E01 Two-dimensional Exactness / Data Role labeling. ## C-E02 Truth leakage principle. ## C-E03 world-relative surrogate target: $$ F_{\mathcal Q}. $$ ## C-E04 Discovery Gate / Proof Gate separation. ## C-E05 exact observation collision implies source insufficiency. --- # 51. Canonical Theorem Ledger: Core F — Demand / Capacity ## C-F01 all reduced Goldbach inputs survive: $$ \mathbb G_N^{-} \subseteq \mathbb H_y(N). $$ ## C-F02 canonical partition: $$ H=G\bigsqcup C. $$ ## C-F03 $$ R_G^{-} = D-C^{\mathrm{can}}. $$ ## C-F04 least-factor capacity-cell partition. ## C-F05 $$ I=C^{\mathrm{can}}+\Omega. $$ ## C-F06 incidence decomposition. ## C-F07 raw AP upper envelope. ## C-F08 divisor AP exact indicator. ## C-F09 small-factor self-cell exact theorem. ## C-F10 canonical closure: $$ L>U^{\mathrm{can}} \Rightarrow R>0. $$ --- # 52. Canonical Theorem Ledger: Core G — Minimality Transport ## C-G01 lift-blocking. ## C-G02 primorial-divisible shift suppression. ## C-G03 edge-to-hard-survivor. ## C-G04 transport neighborhood inclusion. ## C-G05 under failure: $$ \Gamma\subseteq C^{\mathrm{can}}. $$ ## C-G06 direct bounded-degree neighborhood lower bound. ## C-G07 exact right-degree interval AP formula. ## C-G08 minimality transport closure: $$ L^{\mathrm{trans}}>U^{\mathrm{can}}. $$ Matching / Hall is no longer listed as primary core. --- # 53. Superseded claim ledger The following old statements are no longer used. ## S-UP01 $$ \mathcal A_k = \{N:F_G(N)=1\land K_i\ldots\}. $$ Superseded by: $$ \mathcal A_k^{\mathrm P} $$ outer approximation. --- ## S-UP02 Raw GFOC is only a necessary condition. Superseded by: $$ \boxed{ \text{reduced GFOC exact characterization}. } $$ --- ## S-UP03 The effective wheel is a new exact obstruction object. Superseded: $$ \boxed{ \text{removed from core}. } $$ --- ## S-UP04 For a fiber multiplier, it suffices to require: $$ u\in U_6. $$ Superseded by: $$ u\in U_p. $$ --- ## S-UP05 A section hole guarantees only: $$ \Delta_6\ge2. $$ Superseded by: $$ \Delta_6\ge20. $$ --- ## S-UP06 $I_N$ serves as canonical capacity. Superseded by: $$ C_N^{\mathrm{can}}. $$ --- ## S-UP07 $D+\Omega-I$ serves as the primary demand-capacity identity. Superseded by: $$ R=D-C^{\mathrm{can}}. $$ --- ## S-UP08 Matching $\nu_k$ serves as canonical transport demand. Superseded by: $$ |\Gamma_k|. $$ --- ## S-UP09 primorial-smooth shift. Superseded terminology: $$ \boxed{ \text{primorial-divisible shift}. } $$ --- # 54. Reordering Formalization Priorities The original RG-SCP 08 recommended comprehensive formalization. After V0, this changes to three tiers. --- # 55. Formalization Tier 0 — Definitions / Safety Prioritize establishing: ```text Basic GoldbachReduced Counterexample ProofAdmissible PrimeCutoff DataRoleMetadata ``` Targets: - unambiguous domain; - unified reduced count; - no hidden embedding of the target; - all cutoffs discretized. --- # 56. Formalization Tier 1 — Elementary Exact Spine Highest priorities: 1. minimal counterexample existence; 2. reduced GFOC exact theorem; 3. proper obstruction map theorem; 4. divisor self-suppression; 5. primorial hard-sector equality; 6. $T_{\mathrm{top}}=\operatorname{lpf}$; 7. exact M6 fiber multiplier; 8. two-channel count theorem; 9. positive defect $\ge20$; 10. shadow identity; 11. canonical survivor partition; 12. $R=D-C^{\mathrm{can}}$; 13. incidence audit identity; 14. AP divisor column exact indicator; 15. transport neighborhood inclusion; 16. bounded-degree neighborhood lower bound; 17. exact right-degree interval formula. --- # 57. Formalization Tier 2 — Closure Templates Do this only after Tier 1 PASS: $$ L>U^{\mathrm{can}} \Rightarrow R>0, $$ and: $$ L^{\mathrm{trans}} > U^{\mathrm{can}} \Rightarrow \text{no minimal counterexample}. $$ These theorems themselves are elementary; the actual difficulty is supplying nontrivial: $$ L, U. $$ --- # 58. External analytic interface Separate: ```text RGSCPExternal/ PrimeCountBounds APUpperBounds BrunTitchmarsh ExplicitPNT SieveBounds PairCorrelationFuture ``` Heuristic modules must not be imported into the proof core. --- # 59. Heuristics Do Not Require Large-Scale Formalization The following are not prioritized for Lean: - $\varepsilon_{\mathrm{full}}$ fitted model; - pressure ridge; - skew anomaly; - ML classifier; - surrogate generator; - cross-AI reconstruction. Only provenance and promotion results need to be retained. --- # 60. V1 Formal Core Gate Only when all the following PASS: ```text [ ] definitions type-check [ ] no sorry [ ] no hidden complement-primality oracle [ ] no target predicate embedded in admissibility sets [ ] lpf/T_top source aliases explicit [ ] old effective-wheel imports removed [ ] fiber multiplier uses U_p [ ] canonical capacity uses C_can [ ] transport primary object uses Gamma [ ] analytic imports isolated ``` may it be labeled: $$ \boxed{ \text{RG-SCP Core-0.1 Formalized}. } $$ --- # 61. Revised E0 — Identity / Invariant Validation Do not run at large scale in the first round. Recommended: $$ N\le10^5. $$ Must verify: ## E0-A $$ R_G^{-} = D-C^{\mathrm{can}}. $$ ## E0-B $$ I = C^{\mathrm{can}}+\Omega. $$ ## E0-C $$ R_G^{-} = D+\Omega-I. $$ ## E0-D $$ \Delta_6 = N-\max\mathcal S_6. $$ ## E0-E $$ \Delta_6=0 $$ or: $$ \Delta_6\ge20. $$ ## E0-F $$ R_G^{-} = R_6^{-} + \mathbf1_{\mathbb P}(N-3). $$ --- # 62. E0 Primorial invariants If: $$ M_k\mid N, $$ Must verify: $$ \boxed{ \mathbb H_{p_k}(N) = \{p:p_k How much exact demand can a leakage-free theorem lower bound capture? --- # 69. E3 — Canonical Capacity Upper-Bound Audit Old E3: $$ U/I. $$ Canonically, change to: $$ \boxed{ \operatorname{CanonicalCapacitySlack} = \frac{ U_N^{\mathrm{can}}(y) }{ C_N^{\mathrm{can}}(y)+1 }. } $$ where: $$ C_N^{\mathrm{can}} $$ is only in the truth/evaluation channel. --- # 70. E3 Bound Ladder Compare, in order: 1. raw AP sum; 2. divisor-suppressed AP; 3. survivor-restricted incidence envelope; 4. Bonferroni; 5. least-factor cell bound; 6. any new cross-$r$ bound. What matters most is not which fits best, instead: $$ \boxed{ U^{\mathrm{can}} \text{can genuinely be compressed below }D\text{ in value}. } $$ --- # 71. E4 — Cutoff Optimization Firewall If one uses: $$ C^{\mathrm{can}}, I,R_G $$ truth values to choose the cutoff and then claims the predictor is effective, this is leakage. Thus: ## Discovery cutoff Only leakage-free inputs may select: $$ y(N). $$ ## Proof cutoff A symbolic theorem may freely specify it, but the selection rule must be written into the theorem. --- # 72. E5 — Primorial Branch For: $$ M_k\mid N, $$ demand is already exact. Thus, E5 need not further study demand prediction. Directly test: $$ \boxed{ \operatorname{PrimorialCapacityRatio} = \frac{ U_N^{\mathrm{can}}(p_k) }{ \pi(N/2)-k }. } $$ If, over the long term, this is: $$ \gg1, $$ the capacity side is the main bottleneck. --- # 73. E5 Escape truth control If: $$ k\ge2, \quad N>6, $$ Verify: $$ R_G(N)>0 \Longleftrightarrow p_{\mathrm{esc}}\text{ exists}. $$ But the escape field serves only truth/evaluation. --- # 74. Revised E6 — Minimality Transport A genuine minimal counterexample is unknown. Thus, successful-frontier surrogates are used only to test geometry. Retain: ```text k M_k M_k | N ? shift_count |H_k| right_neighborhood_count |Gamma_k| max_right_degree direct_degree_lower_bound matching_optional Hall_optional ordinary_demand_lower_bound transport_demand_lower_bound incremental_gain ``` --- # 75. The Central E6 Metric It is not: $$ \nu_k. $$ instead: $$ \boxed{ \operatorname{TransportIncrement} = \max \left( L_{\mathrm{ordinary}}, L_{\mathrm{trans}} \right) - L_{\mathrm{ordinary}}. } $$ If this remains, over the long term: $$ 0, $$ then $\mathcal S_4$ has no quantitative gain. --- # 76. E6 Priority Domain Priority: $$ \boxed{ M_k\nmid N. } $$ Reason: If: $$ M_k\mid N, $$ total demand: $$ D_N(p_k) $$ is already exactly known. Transport on that branch mainly retains only local-structure uses. --- # 77. E7 — Heuristic Target Selection One can use: - fitted $\varepsilon$; - skew; - pressure; - anomaly detector; to select: - candidate family; - cutoff; - transport scale; - local subset. But the output must be: $$ \boxed{ \text{exact candidate statement}. } $$ If there is only: > A very high model score promotion is not allowed. --- # 78. E8 — Adversarial Source Sufficiency Establish a world-relative: $$ \mathcal Q. $$ Test: - density-matched; - wheel-preserved; - residue-shuffled; - lpf-profile-matched; - pair-target-different surrogate. --- # 79. E8 Formal vs Empirical ## Empirical $$ J_H(X)\approx J_H(Y), $$ with different targets. Only report: ```text empirical_indistinguishability ``` ## Formal $$ J(X)=J(Y) $$ and: $$ T(X)\neq T(Y). $$ One can report: ```text formal_observation_insufficiency_witness ``` and infer: $$ \nexists f, \quad T=f\circ J. $$ --- # 80. The First Formal Research Bottleneck After V0 The situation is no longer simply: > Goldbach has not yet been proved. Instead, there are two more specific bottlenecks. ## Bottleneck A — Demand Can one obtain: $$ L_N(y) $$ sufficiently close to: $$ D_N(y)? $$ ## Bottleneck B — Canonical Capacity Can one obtain: $$ U_N^{\mathrm{can}}(y) $$ sufficiently close to: $$ C_N^{\mathrm{can}}(y)? $$ --- # 81. The Third Bottleneck — History Gain Does the minimality source genuinely increase: $$ L? $$ Namely: $$ \boxed{ L_{\mathrm{trans}} > L_{\mathrm{ordinary}} } $$ Does this hold on any useful family? If not, R07 should stop deepening. --- # 82. The Fourth Bottleneck — Source Sufficiency If: - factor; - wheel; - fixed-point organization; - standard analytic AP; can all produce only loose: $$ L,U, $$ then one should ask: $$ \boxed{ \text{Is the source class itself insufficient?} } $$ rather than continuing to add larger wheels. --- # 83. Final Phase-II Gate Phase II no longer merely asks “is there a beautiful pattern?” At least four gates must be satisfied. --- ## Gate A — Formal Correctness ```text Core-0.1 machine-check PASS no sorry no hidden oracle all corrected domains implemented ``` --- ## Gate B — Compression Gain At least one of: $$ L $$ or: $$ U $$ has nontrivial quantitative improvement over the trivial / raw baseline. --- ## Gate C — Source Usefulness The added source or organization genuinely: - raises $L$; - lowers $U$; - shrinks an admissible family; - or produces a family theorem. --- ## Gate D — Truth-Leakage Clean All discovery claims pass the data-role audit. --- # 84. Phase-II Gate Decision Only: $$ \boxed{ A\land B\land C\land D } $$ opens Phase II. At present: ```text Gate A: PENDING Gate B: PENDING Gate C: PARTIAL-CONCEPTUAL Gate D: DESIGNED, NOT YET TESTED ``` Thus: $$ \boxed{ \text{Phase II = NOT OPEN}. } $$ --- # 85. If Capacity Compression PASS If E3 / E5 show that: $$ U_N^{\mathrm{can}} $$ can be significantly lowered, and there is a theorem candidate, then the next main article may be: # RG-SCP 09 ## Canonical Capacity Compression ### Least-Factor Cells, Survivor Restriction, and Family Exclusion --- # 86. If Transport Gain PASS If: $$ L_{\mathrm{trans}} > L_{\mathrm{ordinary}} $$ appears in a clearly specified family, one can first pursue: # RG-SCP 09 ## Minimality Neighborhood Expansion ### Additive-History Lower Bounds Beyond Ordinary Wheel Demand --- # 87. If Both FAIL If: - capacity upper bounds remain too loose; - transport has no incremental gain; - adversarial source tests show that finite quotient / lpf sources are insufficient; then do not continue: $$ M_k \to M_{k+1} \to M_{k+2}. $$ Directly open: # RG-SCP 09 ## Source-Class Escape ### From Local Factor Obstructions to Pair-Spectral or Nonlocal Additive Structure --- # 88. Pair-Spectral Gate If opening a pair-spectral route, at least answer: > What two-point information does the new operator / transform actually preserve, and is it merely recomputing $R_G$? One cannot merely write: $$ P(z)^2 $$ that its coefficients equal the Goldbach count, because this restates the target. --- # 89. Generating Function Gate $$ P(z) = \sum_{p\in\mathbb P}z^p. $$ $$ [z^N]P(z)^2 $$ directly encodes Goldbach representations. Thus, new proof power must come from: - independent coefficient positivity lower bound; - transform preserving positivity; - analytic restriction producing a new bound; - certificate compression. --- # 90. Nonlocal Operator Gate If one establishes: $$ \mathcal T_N $$ acting on the entire: $$ \{N-p\}_{p\in\mathbb P_N^{-}}, $$ one must check: 1. whether it secretly factors element by element; 2. whether it secretly queries primality element by element; 3. whether it has a bounded-size output; 4. whether it can prove a target-dependent invariant; 5. whether it splits old-source fibers. Only if the last three succeed might it be called source escape. --- # 91. Formal source-novelty test If the old observation: $$ J_{\mathrm{old}} $$ and new observation: $$ J_{\mathrm{new}} $$ satisfying: $$ J_{\mathrm{new}} = f\circ J_{\mathrm{old}}, $$ Then: $$ J_{\mathrm{new}} $$ is not an independent source. Genuine source escape requires at least the existence of: $$ x,y $$ such that: $$ J_{\mathrm{old}}(x) = J_{\mathrm{old}}(y) $$ but: $$ J_{\mathrm{new}}(x) \neq J_{\mathrm{new}}(y). $$ --- # 92. Final Phase-I Proof Compression Criterion For a new result to be labeled: ```text proof_compression_gain: true ``` it must satisfy at least one of: 1. uniformly raise the demand lower bound; 2. uniformly lower the canonical capacity upper bound; 3. eliminate an infinite candidate family; 4. compress many pointwise factor obligations into a bounded-size certificate; 5. produce a cross-$p$ theorem; 6. use minimality history to add quantitative lower-bound power; 7. split a source fiber relevant to the target. --- # 93. What Does Not Count as Proof Compression The following do not count in themselves: - changing notation; - drawing a graph; - adding a wheel; - defining entropy; - defining energy; - more accurate fitting; - verifying larger $N$; - producing more derived metrics from the same lpf source; - renaming target truth as a pressure score. --- # 94. Canonical synchronization policy Synchronized area: ```text 00_SYNC_CORE ``` Contains only: - canonical theorem; - canonical correction; - source ledger; - formal proof; - exact certificate; - Phase Gate; - promoted theorem candidate. --- # 95. Async policy ```text 10_ASYNC_TEXT 20_ASYNC_AUDIT 30_ASYNC_EXPERIMENTS ``` May contain: - narrative summary; - fitted model; - experiment log; - failed candidate; - null-model output; - charts; - raw data; - cross-AI reconstruction. Move into the sync core only after promotion. --- # 96. Canonical Phase-I file set The formal handoff is recommended to contain at least: ```text RG-SCP_Canonical_Definitions_v0.1.md RG-SCP_Canonical_Theorem_Ledger_v0.1.yaml RG-SCP_Canonical_Source_Ledger_v0.1.yaml RG-SCP_Truth_Leakage_Ledger_v0.1.yaml RG-SCP_Formalization_Priority_v0.1.md RG-SCP_E0-E8_Protocol_v0.1.md RG-SCP_PhaseII_Gate_v0.1.yaml ``` These can be separated out from this document before V1. --- # 97. V0 Status Matrix | Round | Topic | Verdict | Main Corrections | |---|---|---|---| | V0-R01 | Counterexample / Admissibility | PASS WITH CORRECTION | remove the $F_G$ circular definition from the outer approximation | | V0-R02 | Obstruction Cover | PASS WITH STRENGTHENING | promote reduced GFOC to exact | | V0-R03 | M6 / Wheel | PASS WITH STRENGTHENING | unconditional primorial hard sector; remove the effective wheel | | V0-R04 | Fixed Point | PASS WITH CORRECTION | $F_p=pU_p$; $\Delta_6>0\Rightarrow\Delta_6\ge20$ | | V0-R05 | Heuristic Layer | PASS WITH FIREWALL | separate TruthLabel / DiscoveryProbe | | V0-R06 | Demand / Capacity | PASS WITH RECONSTRUCTION | $C^{can}$ replaces $I$ as canonical capacity | | V0-R07 | Minimality Transport | PASS WITH REBASE | $\Gamma$ replaces matching as primary transport demand | | V0-R08 | Phase-I Synthesis | V0 COMPLETE | reconstruct canonical core / protocol / gates | --- # 98. Phase-I Current Status ```yaml phase: I conceptual_architecture: status: COMPLETE claim_source_audit: status: COMPLETE canonical_corrections: status: COMPLETE_FOR_V0 formal_core: status: PENDING exact_identity_experiments: status: PENDING demand_compression: status: PENDING capacity_compression: status: PENDING transport_increment: status: PENDING adversarial_source_audit: status: PENDING goldbach_closure: status: OPEN phase_II: status: NOT_OPEN ``` --- # 99. Recommended V1 Execution Order Do not start with the longest theorem. Follow dependencies: ```text V1-01 Basic / Reduced Goldbach V1-02 Counterexample / Admissibility V1-03 Obstruction / GFOC V1-04 Primorial Geometry V1-05 T_top / Exact M6 Fiber V1-06 Two-Channel / Shadow V1-07 Survivor / Canonical Capacity V1-08 Incidence Audit / AP Suppression V1-09 Minimality / Transport Neighborhood V1-10 Closure Templates ``` --- # 100. Minimum Acceptable Formal Validation Without Lean If local resources are temporarily insufficient for full Lean, every theorem at least needs: 1. machine-readable statement; 2. finite-domain sanity test; 3. independent proof rewrite; 4. dependency graph; 5. no-target-oracle audit; 6. counterexample search for the theorem statement. but: $$ \boxed{ \text{This is still not equivalent to formal proof-assistant verification}. } $$ --- # 101. E0 Takes Priority over Large-Scale E1 After the formal core is completed, the first computation task must be: $$ \boxed{ E0. } $$ rather than directly performing: $$ N\le10^8 $$ a branch census. Reason: If: $$ R=D-C^{can} $$ or: $$ \Delta_6 $$ is implemented with even one incorrect domain, the large-scale results are entirely meaningless. --- # 102. E0 Bugs Most Worth Testing First According to the V0 audit: 1. fibers mistaking $U_6$ for $U_p$; 2. effective-wheel self-cell; 3. prime complements wrongly entering lpf obstruction cells; 4. target-derived fields entering discovery inputs; 5. the cutoff: $$ y\ge N/2 $$ boundary; 6. strict / non-strict transport-layer inequalities; 7. AP degree ignoring the lower interval; 8. $I$ being wrongly treated as canonical capacity. All of these should first have regression tests written. --- # 103. The Three Core Phase-I Research Questions Most Worth Retaining If the entire RG-SCP is compressed to only three questions: ## Q1 — Capacity Compression Can one independently prove: $$ C_N^{\mathrm{can}}(y) \le U_N^{\mathrm{can}}(y) $$ whereas: $$ U_N^{\mathrm{can}} $$ to be significantly smaller than the raw AP sum? --- ## Q2 — Demand Compression Can one independently prove: $$ D_N(y)\ge L_N(y) $$ and: $$ L_N $$ to be sufficiently close to exact demand? --- ## Q3 — History Gain Can minimality transport prove: $$ L_{\mathrm{trans}} > L_{\mathrm{ordinary}} $$ on some nontrivial family? --- # 104. Conditions for Genuine Phase-I Success It is not: > Defining many quantities. Instead, at least one must appear: $$ \boxed{ L>U^{\mathrm{can}}. } $$ or: $$ \boxed{ \text{a new cross-input theorem that can uniformly raise }L \text{ or lower }U. } $$ --- # 105. Conditions for Genuine Phase-I Failure If: - $L$ is always too low; - $U^{\mathrm{can}}$ is always too high; - transport increment is approximately $0$; - exact surrogate collisions show the current source insufficient; then one should: $$ \boxed{ \text{Freeze factor/wheel branch}. } $$ This is not research failure. It is: $$ \boxed{ \text{source-class falsification}. } $$ --- # 106. Final V0 Source Audit Principle If: $$ J_2 = f\circ J_1, $$ Then: $$ J_2 $$ is not an independent source. Thus: - fixed-point label; - lpf; - canonical factor obstruction; cannot be treated as multiple pieces of evidence. --- # 107. Final V0 Truth-Leakage Principle If: $$ T = f(X_1,\ldots,X_m) $$ holds exactly, then, in a discovery model with: $$ T $$ as its target, using the complete set: $$ X_1,\ldots,X_m $$ is target leakage. Thus: $$ \boxed{ \text{Exact} \neq \text{Leakage-Free}. } $$ --- # 108. Final V0 Closure Principle Genuine closure accepts only: $$ \boxed{ \text{rigorous lower bound} > \text{rigorous independent upper bound}. } $$ For RG-SCP Phase I: $$ \boxed{ L_N(y) > U_N^{\mathrm{can}}(y) } $$ is the primary form. Minimality: $$ \boxed{ L_k^{\mathrm{trans}}(N_*) > U_{N_*}^{\mathrm{can}}(p_k) } $$ is the history-enhanced form. --- # 109. Final V0 Conclusion After eight V0 audit rounds, RG-SCP Phase I may formally be retained as: $$ \boxed{ \text{a counterexample-admissibility and proof-compression research program}. } $$ but cannot be described as: $$ \boxed{ \text{a Goldbach proof}. } $$ What it has genuinely completed at present is: 1. counterexample state normalization; 2. source de-duplication; 3. exact obstruction characterization; 4. primorial hard-sector compression; 5. fixed-point relational normalization; 6. truth-leakage firewall; 7. canonical demand / coverage identity; 8. minimality neighborhood architecture; 9. formalization-ready dependency order; 10. falsifiable Phase-II gate. --- # 110. Next Step Not directly RG-SCP 09. The formal next stage: # RG-SCP Validation V1 ## Formal Core The first article is recommended to be: # V1-01 ## Basic / Reduced Goldbach / Counterexample Kernel Its content does only: - reduced domain; - $R_G^{-}$; - $F_G$; - $\mathcal C_G$; - minimal counterexample existence; - proof-admissible outer approximation schema. Only after completion proceed to: $$ \boxed{ \text{V1-02 Obstruction Core}. } $$ --- # 111. Shortest Version If only one sentence is retained: > **After V0 completion, the main line of RG-SCP has converged from “many obstruction indicators” to a clean proof-compression problem: prove, without peeking at Goldbach truth, a survivor demand lower bound greater than a canonical composite-coverage upper bound; minimality transport has value only when it can additionally raise this demand lower bound.** Phase-I canonical closure: $$ \boxed{ L_N(y)>U_N^{\mathrm{can}}(y) } $$ Minimality-enhanced closure: $$ \boxed{ L_k^{\mathrm{trans}}(N_*) > U_{N_*}^{\mathrm{can}}(p_k). } $$ Phase-II gate: $$ \boxed{ \text{Formal Correctness} \land \text{Compression Gain} \land \text{Source Usefulness} \land \text{Truth-Leakage Clean}. } $$ At present: $$ \boxed{ \text{V0 COMPLETE}, \qquad \text{PHASE II NOT OPEN}. } $$ --- # Appendix A: Canonical Object Ledger ```yaml counterexample: C_G: N_star: admissibility: A_proof: A_heuristic: obstruction: reduced_domain: raw_GFOC: proper_map: canonical_lpf: wheel: H_y: D_N_y: primorial_hard_sector: fixed_point: T_top: U_p: F_p: delta_6: shadow_set: capacity: C_can_N_y: B_r: I_N_y: Omega_N_y: A_r: transport: primorial_divisible_shift: H_k_shift: Gamma_k: Delta_R: matching_optional: heuristic: epsilon_full: skew: pressure: truth: R_G_minus: delta_6: escape_exists: ``` --- # Appendix B: Canonical Source Ledger ```yaml S0: name: elementary_arithmetic_logic S1: name: factor_lpf aliases: - T_top_on_M6 - Phi_min_on_hard_failure S2: name: finite_quotient_wheel S3: name: fixed_point_relational S4: name: additive_history_minimality type: target_internal_history_source S5: name: external_analytic_counting S6: name: heuristic_approximation_skew S7: name: future_nonlocal_pair_source status: not_established ``` --- # Appendix C: Truth-Leakage Ledger ```yaml safe_exact_probes: - N_mod_Mk - Mk_divides_N - omega_N - eta_Mk - small_factor_screening - standard_arithmetic_residuals truth_or_truth_adjacent: - R_G_minus - delta_6 - boundary_3 - escape_exists - escape_depth - C_can - exact_I - exact_Omega history_derived: - lower_Goldbach_pairs - transport_Gamma - transport_Delta_R ``` --- # Appendix D: Superseded Objects ```yaml remove_from_core: - effective_wheel rename: primorial_smooth_shift: to: primorial_divisible_shift replace: admissibility_with_FG_inside: by: proof_admissible_outer_approximation fiber_pU6: by: fiber_pUp canonical_capacity_I: by: canonical_capacity_C_can transport_demand_matching: by: transport_demand_neighborhood_Gamma ``` --- # Appendix E: E0 Kill-Switch Tests ```text [ ] R = D - C_can [ ] I = C_can + Omega [ ] R = D + Omega - I [ ] delta_6 = N - max(shadow) [ ] delta_6 == 0 or delta_6 >= 20 [ ] R = R6 + indicator(N-3 prime) [ ] primorial hard-sector equality [ ] D(N,p_k) = pi(N/2)-k [ ] A_r exact divisor-column indicator [ ] S_l = {l} for suppressed small factor [ ] T_top = lpf on M6* [ ] fiber membership uses U_p [ ] transport layer boundary uses >= [ ] exact right-degree interval count ``` --- # Appendix F: Phase-II Gate ```yaml formal_correctness: status: pending compression_gain: status: pending source_usefulness: status: partial_conceptual truth_leakage_clean: status: protocol_defined_not_executed phase_II: open: false next: - V1_Formal_Core - E0_Identity_Validation - E1_to_E6 ``` --- # Appendix G: Commit Decision ```yaml validation_round: V0-R08 decision: V0_COMPLETE_PHASE_I_CANONICALIZED major_rebases: - proof_admissible_outer_approximation - reduced_GFOC_exact - primorial_hard_sector_unconditional - effective_wheel_removed - exact_M6_fiber_uses_U_p - shadow_gap_minimum_20 - truth_leakage_firewall - escape_as_truth_observable - canonical_capacity_C_can - incidence_demoted_to_audit_mass - transport_neighborhood_Gamma_primary - matching_and_Hall_demoted - primorial_divisible_shift_terminology phase_II_open: false next_stage: V1-01_Basic_Reduced_Goldbach_Counterexample_Kernel commit_to_sync_core: true ```