> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # LESR Root-6217 Positive-Rotation Line ## Local Verification and Series Integration Handoff ### Local Reverification, Tail-Campaign Takeover and Series-Integration Handoff **Document ID:** `LESR-LOCAL-HANDOFF-v0.1` **Date:** 2026-09-24 **Author:** Neo.K **AI Research Collaborator:** Aletheia / ChatGPT, GPT-5.6 Sol **Institution:** EveMissLab / 一言諾科技有限公司 **Status:** `CLOUD RESEARCH FREEZE / LOCAL REPLAY REQUIRED / SERIES INTEGRATION NEXT` --- # 0. This Round's Stopping Point This document formally freezes the current cloud / chat research state of the Root-6217 positive-rotation line. The reason for stopping is neither a mathematical counterexample nor a collapse of the main theory; rather: ```text long-running finite certificate + tool execution timeout + deep persistent DFS / proof-tree traversal ``` has become the main engineering bottleneck. Starting from this node, therefore: $$ \boxed{ \text{stop grinding through the remaining finite tail trees in cloud/chat} } $$ Switch to: $$ \boxed{ \text{long-duration local reverification} + \text{cloud-side series integration / theoretical organization} } $$ --- # 1. Completed Local Positive-Rotation Line ## 1.1 Zero rotation Paper 07 has reconstructed the barrier state at: $$ \rho=0 $$ . zero barrier area： $$ A_0 = 0.8349861105592240998506623565577005921\ldots $$ --- ## 1.2 Micro gate Paper 09U： $$ \boxed{ [0,10^{-4}] } $$ Status: ```text RESEARCH-CLOSED ``` Core: - Paper 09T atlas-agnostic 64-QP lower； - the normalized lower bound is nonincreasing in $\rho$; - chart-free radial secant cover； - 576 / 576 radial cells PASS。 micro boundary inherited lower： $$ \boxed{ 0.16529554824965303>0.1. } $$ --- ## 1.3 Bridge gate Paper 09T completes: $$ \boxed{ [10^{-4},0.003] } $$ The 11 parents: $$ B01,\ldots,B11 $$ are all: ```text RESEARCH-CLOSED ``` Thus: $$ \boxed{ 11/11 } $$ bridge parents closed。 --- ## 1.4 Local seal Paper 09U composes: $$ [0,10^{-4}] + [10^{-4},0.003] $$ to obtain: $$ \boxed{ [0,0.003] } $$ Status: ```text RESEARCH-CLOSED ``` --- # 2. Important Methodological Evolution in the Local Line This series should not be organized as subdivision all the way. The actual evolution is: ```text first-order critical cone -> eight-chart finite atlas -> chartwise Psi2 -> explicit Psi3 -> interval fourth-order remainder -> explicit Psi4 -> explicit Psi5 -> sixth-order interval remainder -> chartwise radial derivative -> rho-only refinement -> chart-free local secant -> finite-chart migration repair -> atlas-agnostic 64-QP -> micro monotone inheritance ``` Along the way it repeatedly shows: $$ \boxed{ \text{the proof compiler must change with the actual obstruction} } $$ The same finite kernel cannot simply be deepened indefinitely. --- # 3. The Important Semantic Correction Exposed by B11 Paper 09T finds: $$ \boxed{ \text{first-order sign chart} \neq \text{finite chart identity} } $$ Chart migration can occur at finite $\rho$. The final boundary certificate therefore changes to: $$ 8 \text{ sign regions} \times 8 \text{ finite chart formulas} = 64 $$ atlas-agnostic quadratic programs. global lower： $$ \boxed{ 0.16529554824965303>0.1. } $$ This correction should be elevated into a formal methodological conclusion in the future integrated account. --- # 4. Current Tail-Gate State tail domain： $$ \boxed{ [0.003,\pi/120]. } $$ target： $$ \boxed{ m(\rho) \ge A_{\rm ref} = 0.8349865. } $$ Paper 08's `AREF-TAIL` is only a certificate target. Paper 08's composition theorem must not be misread as completion of the tail. --- # 5. Corrected parent ledger Paper 08's original target-specific ledger: $$ 200\to195. $$ After BCODR-LUC-06 it is corrected to: $$ \boxed{ 200 \to 192 \to 187 \to 182. } $$ where: ### Permanent NOT-F3 $$ \boxed{ 106,110,127,132,135,162,165,166 } $$ ### Previous $A_{\rm ref}$ closed $$ \boxed{ 28,36,68,144,155 } $$ ### New BCODR-LUC-06 $A_{\rm ref}$ closed $$ \boxed{ 101,103,104,164,167 } $$ --- # 6. Tail-active responsibilities In the corrected ledger, the active parent responsibilities intersecting: $$ [0.003,\pi/120] $$ number: $$ \boxed{ 173 } $$ The cloud tail campaign has established persistent per-parent DFS states. --- # 7. Tail-Campaign Compression Already Completed After the original finite Aref campaign and the lookahead delegated campaign, the hard core had been compressed to: $$ \boxed{ 29 } $$ parents。 Later, the following was introduced: $$ \boxed{ \text{exact eroded-core support lower} } $$ as a new proof tier. --- # 8. Exact-eroded-core proof tier Old kernel: ```text robust erosion core -> K256 / K512 / K1024 inscribed polygon -> hull lower ``` New tier: ```text robust erosion core -> exact support arcs / vertices -> exact convex-hull area lower ``` Core inclusion relation: $$ \boxed{ K_c\ominus\delta B \subseteq K_{\rm actual}. } $$ The analytical eroded-core hull area therefore remains an admissible lower bound. --- # 9. Exact-core tier validation Completed research falsification: ### Against K1024 For 337 hard-frontier boxes: $$ \boxed{ L_{\rm exact-core} \ge L_{\rm K1024}. } $$ The minimum gain is approximately: $$ 1.95\times10^{-6}. $$ ### Against actual placements For 640 random actual placements inside boxes: $$ \boxed{ A_{\rm actual} \ge L_{\rm exact-core}-10^{-9}. } $$ The minimum actual margin is approximately: $$ 1.44\times10^{-4}. $$ --- # 10. Anisotropic robust-core escalation Later, a slower but stronger core was also built: $$ \boxed{ \text{AABB / corner-disk intersection robust core} } $$ Use: - not as the main kernel; - only for near-threshold final leaves. Initial replay: - 31 robust leaves deterministic replay； - replay slack error = 0； - 279 actual placement samples； - minimum： $$ A_{\rm actual} - L_{\rm robust} \approx 3.96\times10^{-5}>0; $$ - 0 containment failures。 This tier should undergo a fresh deterministic exhaustive replay locally. --- # 11. Hard-Core Closure at the Cloud Freeze Among the 29 exact-core hard parents, at least the following proportion are completely CLOSED at the current cloud freeze: $$ \boxed{ 10/29 } $$ : ```text 0 1 13 45 47 95 143 169 178 186 ``` Some final verified slacks are known to be extremely thin, for example: $$ O(10^{-10}) \text{--} O(10^{-9}). $$ Thus: $$ \boxed{ \text{local correctly-rounded / independent replay is important.} } $$ --- # 12. Remaining Hard Core at the Cloud Freeze The remaining: $$ \boxed{ 19 } $$ hard parents still need local continuation / reverification. They are not claimed closed after the cloud freeze. The current status is therefore: ```text TAIL [0.003,pi/120]: OPEN ROOT6217 POSITIVE ROTATION: OPEN GLOBAL LEBESGUE: OPEN ``` --- # 13. Local Replay Priorities ## Priority 0 — Artifact integrity First verify the snapshot: ```text LESR_09V_Tail_Campaign_Snapshot_2026-09-24.zip ``` The existing manifest previously reported: ```text 125 files SHA-256 all match 0 mismatch ``` It should be verified again locally. --- # 14. Priority 1 — Already CLOSED Parents Do not begin by computing the remaining 19. First select: $$ \boxed{ 0,1,13,45,47,95,143,169,178,186 } $$ and replay completely: 1. tree reconstruction； 2. all terminal leaves； 3. `NOT-F3` predicates； 4. exact-core lower； 5. robust-core leaves； 6. safety subtraction； 7. minimum verified slack。 If any replay disagrees: ```text STOP PROMOTION ``` but retain the research evidence. --- # 15. Priority 2 — Exact-core proof-tier independent backend Build at least a second: $$ \boxed{ \text{directed-rounding / interval geometry backend} } $$ to recheck: $$ K_c\ominus\delta B \subseteq K_{\rm actual}. $$ Then recompute the support-arc boundary area. --- # 16. Priority 3 — Robust AABB core Specifically verify: ### $B_3$ The translation rectangle's robust core is built from defining disks at the translation-corner extrema. ### $B_5$ The rotation+translation trajectory must first be rigorously enclosed by a coordinate AABB, then an inner core is built using the corner-disk intersection. This layer cannot retain only random falsification. Publication promotion requires a deterministic geometric inclusion proof. --- # 17. Priority 4 — Remaining 19 hard parents Suggested runtime: ```text persistent parent state 5000-node plain exact-core wave robust direct sweep near-threshold robust split checkpoint repeat ``` Do not again use: ```text one parent -> run to exhaustion ``` because it is unfriendly to long-running execution. --- # 18. Priority 5 — Scheduler Observed: - pure K64 lookahead is usable; - the erosion-radius-only scheduler did not outperform K64; - adaptive child robust lookahead is effective for near-threshold branches, but more costly; - the best strategy is hybrid: - cheap scheduler away from threshold； - expensive robust child lookahead only near closure。 --- # 19. Two-colour boundary locking Cycle 15 fixed-$q$ radial convexity： $$ r\mapsto f_q(r) $$ is convex. Admissible leaf types: ```text SECANT-LOCKED AREF-PRUNED ``` But the cloud probe shows: among 311 coarse unresolved q-projections: $$ 0 $$ can be directly coarse `SECANT-LOCKED`. Some center secants are also negative. Thus: $$ \boxed{ \text{the two-colour route cannot replace the exact-core campaign in one coarse-grained step.} } $$ Any future reactivation requires new 4D boundary subdivision. --- # 20. The Most Important Local State Principle Strictly separate the following three states: ```text RESEARCH-CLOSED LOCAL-REPLAY-PASSED PUBLICATION-CERTIFIED ``` A local replay PASS must not directly promote publication status. --- # 21. The Entire 09A–09U Series Also Needs Replay Not only 09V. A future local implementation should establish: ```text LESR_REPLAY_ALL ``` At least verify, in order: 1. 09E first-order atlas； 2. 09J quadratic KKT； 3. 09K cubic reconstruction； 4. 09O quartic/quintic reconstruction； 5. sixth-order interval Jet； 6. B01–B11 radial certificates； 7. 09T 64-QP； 8. 09U monotone inheritance； 9. 09V exact-core／robust-core tail certificates。 --- # 22. Suggested Integration into Three Artifact Layers ## Layer A — Research narrative The Paper 07–09V series. ## Layer B — Canonical certificate state Machine-readable: ```text ledger/ parents/ charts/ polynomials/ radial/ tail/ checkpoints/ ``` ## Layer C — Replay runtime ```text verify_zero verify_local verify_bridge verify_micro verify_tail verify_root6217 verify_all ``` --- # 23. Do Not Integrate the Series Solely by Paper Number Organization by proof dependency is more appropriate: ```text ZERO BARRIER -> FIRST ORDER -> FINITE ATLAS -> HIGHER ORDER -> RADIAL -> MICRO -> BRIDGE -> TAIL -> ROOT6217 ``` Retain paper numbers as historical lineage. --- # 24. The Next Cloud Work Mode After this document, cloud/chat work is no longer primarily: ```text long finite tree execution ``` Switch to the following workflow (the first item is theory integration; the last is integration with other Lebesgue / AMRAL series). The original workflow block is retained verbatim: ```text 1. 理論統合 2. proof dependency graph 3. local verifier specification 4. artifact normalization 5. publication debt ledger 6. 與其他 Lebesgue / AMRAL 系列整合 ``` Delegate long-running computation to the local environment. --- # 25. Suggested Next Integration Document The next paper need not continue under the name 09W. It is more reasonable to begin with: $$ \boxed{ \textbf{LESR Root-6217 Closure Architecture — Integrated State v1} } $$ Contents: 1. what is currently closed; 2. what is only at research level; 3. what requires local replay; 4. what requires publication audit; 5. tail remaining 19； 6. LinkBack to the AMRAL global Lebesgue line. --- # 26. Final Freeze State ```text ZERO ROTATION: RESEARCH-CLOSED MICRO [0,1e-4]: RESEARCH-CLOSED BRIDGE [1e-4,0.003]: RESEARCH-CLOSED LOCAL [0,0.003]: RESEARCH-CLOSED TAIL [0.003,pi/120]: PARTIAL 173 active tail parents after corrected ledger filtering hard core compressed to 29 at least 10/29 hard parents CLOSED 19 hard parents remain for local replay/completion ROOT6217 POSITIVE ROTATION: OPEN ROOT6217 FULL SHAPE: OPEN GLOBAL LEBESGUE: OPEN PUBLICATION AUTHORITY: OPEN ``` --- # 27. Conclusion This round should not be described as finally stuck. A more precise description is: $$ \boxed{ \text{the cloud finite-computation phase has reached its runtime-efficiency boundary.} } $$ The mathematical research has compressed the problem from: ```text unknown positive-rotation geometry ``` to: ```text 19 persistent tail hard parents + local replay / publication audit ``` The most reasonable action now is therefore to stop cloud brute force, hand the entire certificate stack to long-duration local replay, and simultaneously begin, on the cloud side: $$ \boxed{ \text{series integration and consolidation of the proof architecture.} } $$ --- # Canonical Source Declaration This file is the UTF-8 Markdown canonical source for `LESR-LOCAL-HANDOFF-v0.1`. The only mathematical delimiters used are `$...$` and `$$...$$`. This paper does not claim that the tail is closed, that root6217 positive rotation is closed, or that the global Lebesgue universal covering problem is solved. ===END===