# AMRAL × Lebesgue Universal Covering — Round 34 ## Replayable Rational-Hull Certificates and Publication-Candidate Promotion **Document ID:** AMRAL-LUC-FC-R34 **Version:** v0.1 **Date:** 2026-09-20 **Research status:** Finite certificate persistence / Byte-level replay / Depth-30 batch migration **Research initiation and methodology source:** Neo.K **AI collaborating researcher and primary executor:** Aletheia / ChatGPT, GPT-5.6 Sol --- # 0. Round Summary Round 33 achieved cross-backend enclosure dominance, but left behind a genuine proof-object gap: > A `PASS + density + margin` computation ledger is not a concrete finite certificate entity. Round 34 directly fills this gap. This round implements: $$ \boxed{\texttt{RHCERT-v0.1}} $$ binary rational-hull certificate format. cell47 / cell48 both re-emit complete finite rational polygon bytes, and are replayed by an independent verifier that **reads only certificate bytes**. Result: ## cell47 $$ \boxed{9228/9228\text{ leaves PASS}} $$ stored rational vertices: $$ \boxed{2{,}163{,}069} $$ raw certificate: approximately 34.8 MiB. minimum exact rational margin: $$ \boxed{6.75024723\times10^{-9}>0.} $$ ## cell48 $$ \boxed{9285/9285\text{ leaves PASS}} $$ vertices: $$ \boxed{2{,}109{,}700} $$ raw certificate: approximately 34.0 MiB. minimum margin: $$ \boxed{1.03164178\times10^{-7}>0.} $$ Both trees: - membership failures: 0; - convexity failures: 0; - exact-area failures: 0. The Round 33 MPFR enclosure-dominance audit likewise: $$ \boxed{0\text{ failures}} $$ Therefore 47 / 48 can, by: $$ \boxed{ \text{concrete certificate bytes} + \text{byte-only replay} + \text{MPFR dominance} } $$ formally be promoted to: $$ \boxed{\texttt{PUBLICATION-CANDIDATE-SHARD}.} $$ publication-candidate cells become: $$ \boxed{\{47,48,69,70\}} $$ that is: $$ \boxed{4/77\approx5.19\%.} $$ In addition, cell68 completed: $$ \boxed{9255/9255\text{ mpmath/libmp lower PASS}} $$ and marker68's independent upper and MPFR upper both PASS, so it is promoted to: $$ \boxed{\texttt{ARITHMETICALLY-CLOSED-PROTOTYPE}.} $$ Arithmetic-closed-or-better: $$ \boxed{5/77\approx6.49\%.} $$ The global theorem still stands at: $$ \boxed{a_{\mathrm{Leb}}\ge0.835\text{ NOT CERTIFIED}.} $$ --- # 1. Certificate entity principle Round 33 already distinguished: $$ \text{computation summary} \ne \text{finite certificate entity}. $$ Round 34 fixes: > A publication-level shard must leave behind finite bytes that a verifier can replay without trusting the emitter's `PASS` bit. --- # 2. RHCERT-v0.1 global header: ```text magic 8 bytes L34RHC01 version uint16 1 coordinate_exp uint8 18 source_tag_bytes uint8 1 leaf_count uint32 ``` Coordinates: $$ x=X/10^{18},\qquad y=Y/10^{18}, $$ where: $$ X,Y\in\mathbb Z. $$ --- # 3. Leaf record Leaf order is implicitly determined by the preorder of the witness topology. Each leaf: ```text uvarint density uvarint vertex_count repeat vertex_count: zigzag-uvarint delta_x zigzag-uvarint delta_y uint8 source_tag ``` So it does not redundantly store: - leaf path; - witness box; - absolute coordinates. --- # 4. Source tags ```text 0 = diameter-one disk core 1 = B3 common core 2 = B5 common core 3 = B7 common core ``` This lets the verifier verify the source body vertex by vertex. --- # 5. Random-access index `.rhidx` stores: $$ L+1 $$ uint64 byte offsets. So the verifier can: - randomly sample a leaf; - sharded replay; - parallel audit; - selective repair. without needing to scan the entire certificate. --- # 6. Verifier ignores candidate generation The Round 34 verifier performs none of: - support/contact candidate generation; - SciPy hull; - emitter density selection; - emitter pass/fail. It accepts only: $$ \boxed{\text{certificate bytes}.} $$ --- # 7. Quantized inner-core verifier To avoid millions of Fraction membership checks, the Round 34 verifier further shrinks the mpmath directed core into an exact integer core. Take: $$ Q_C=10^{40}. $$ For the center rectangle: - the lower endpoint rounds down; - the upper endpoint rounds up. So the quantized rectangle is larger. For the radius lower bound: - it is quantized downward. So the radius is smaller. Hence: $$ \boxed{ G_{\rm quantized} \subseteq G_{\rm directed}. } $$ Therefore, as long as a vertex passes the quantized core, it is guaranteed to pass the original directed core. --- # 8. Exact membership Certificate vertex: $$ p=(X/10^{18},Y/10^{18}). $$ Exactly lift it to the $10^{40}$ grid. For each defining disk's center rectangle: $$ D^2_{\max} = \max |X-c_x|^2 + \max |Y-c_y|^2. $$ require: $$ \boxed{D^2_{\max}0. } $$ Any non-convex / repeated / reversed vertex sequence: fail closed. --- # 10. Exact area Shoelace: $$ S=\sum_i(X_iY_{i+1}-X_{i+1}Y_i). $$ area: $$ A=\frac{|S|}{2\cdot10^{36}}. $$ Without floating-point conversion, compare directly: $$ \boxed{ 200|S|>167(2\cdot10^{36}). } $$ --- # 11. Cell47 replay leaves: $$ 9228. $$ vertices: $$ 2{,}163{,}069. $$ failures: $$ \boxed{0.} $$ minimum margin: $$ \boxed{6.75024723\times10^{-9}.} $$ raw bytes: $$ 36{,}506{,}484. $$ compressed certificate package, approximately: $$ 8.1\text{ MiB}. $$ --- # 12. Cell48 replay leaves: $$ 9285. $$ vertices: $$ 2{,}109{,}700. $$ failures: $$ \boxed{0.} $$ minimum margin: $$ \boxed{1.03164178\times10^{-7}.} $$ raw: $$ 35{,}657{,}092\text{ bytes}. $$ compressed, approximately: $$ 7.6\text{ MiB}. $$ --- # 13. Cross-backend transfer becomes concrete The Round 33 theorem: If: $$ R^{\rm MPFR}_j\subseteq R^{\rm mpmath}_j $$ and: $$ \rho^{\rm MPFR}_{\rm lo} \ge \rho^{\rm mpmath}_{\rm lo}, $$ then all mpmath-certified rational vertices automatically transfer to the MPFR core. Round 33 audit: ## cell47 unique core states: $$ 1564, $$ failures: $$ 0. $$ ## cell48 unique: $$ 1574, $$ failures: $$ 0. $$ Round 34 now has **concrete vertex bytes**, so the theorem no longer acts on the abstract "if a polygon exists," but on content-hashed finite proof objects. --- # 14. Promotion of cells 47 and 48 Both now simultaneously satisfy: 1. finite complete geometry tree; 2. explicit topology / split axes; 3. marker upper independent replay PASS; 4. marker MPFR PASS; 5. concrete rational lower certificate bytes; 6. bytes-only verifier PASS all leaves; 7. MPFR enclosure dominance PASS; 8. all inequalities strict. So: $$ \boxed{ 47,48 \to \texttt{PUBLICATION-CANDIDATE-SHARD}. } $$ --- # 15. Certificate size economics Full rational vertices did not produce an unacceptable certificate explosion. cell47: $$ \sim16.88\text{ raw bytes/vertex}. $$ cell48: $$ \sim16.90\text{ raw bytes/vertex}. $$ After delta-varint + DEFLATE: it drops to approximately: $$ 3.5\text{--}3.9\text{ bytes/vertex}. $$ So the concrete finite lower certificate for a tree of about 9000 leaves needs only about 8 MiB compressed. This means: $$ \boxed{ \text{full certificate persistence is practical}. } $$ There is no need to fall back to summary-only evidence just to save a few dozen MB. --- # 16. Cell68 starts the next batch cell68 topology: $$ 18509\text{ nodes}, \qquad 9255\text{ leaves}. $$ Round 34 mpmath/libmp lower: $$ \boxed{9255/9255\text{ PASS}.} $$ minimum rational margin: $$ \boxed{2.23292\times10^{-8}.} $$ marker68: - independent exact-polar upper: PASS; - MPFR upper: PASS. So: $$ \boxed{ 68\to\texttt{ARITHMETICALLY-CLOSED-PROTOTYPE}. } $$ The next step only needs to emit RHCERT / dominance / promotion. --- # 17. Stage coverage after Round 34 Geometry complete: $$ \boxed{66/77}. $$ Arithmetic closed or better: $$ \boxed{ \{47,48,68,69,70\} } $$ that is: $$ \boxed{5/77\approx6.49\%.} $$ Publication candidate: $$ \boxed{ \{47,48,69,70\} } $$ that is: $$ \boxed{4/77\approx5.19\%.} $$ --- # 18. Geometry scope Round 34 compute is mainly invested in the certificate ABI and the arithmetic batch. No new global common-depth geometry wave was completed. Therefore geometry strictly remains at: $$ \boxed{66/77\text{ known complete}} $$ and the latest exact common-budget: $$ \boxed{51/77\text{ at }d=36.} $$ --- # 19. Round 35 ## Concrete-Certificate Bulk Migration and Geometry Residual Wave Next round: ### Arithmetic 1. cell68 directly emits RHCERT; 2. 75 / 76 complete mpmath lower; 3. 47/48/68/69/70 build a unified publication-candidate manifest index; 4. begin certificate dedup / parent-state reuse, further compressing bytes. ### Geometry Resume the 11-cell residual wave, aiming for: - new strict closures; - or a new common-budget checkpoint. --- # 20. Shortest Handoff The Round 33 blocker: $$ \boxed{ \text{certificate summary} \ne \text{certificate entity} } $$ has now genuinely been closed in Round 34. 47 / 48 are now not "the computation says they pass," but rather: > finite rational proof bytes that can be saved, hashed, random-accessed, independently replayed, and cross-backend transferred. So publication-candidate coverage officially doubles: $$ \boxed{2/77\to4/77.} $$ Globally, still: $$ \boxed{ a_{\mathrm{Leb}}\ge0.835 \text{ NOT CERTIFIED}. } $$