> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # Lebesgue Extremal Shape Reconstruction ## Paper 09H — Outward Observable Enclosures and Bridge-Leaf Preconditions ### Reconstructing $A,T,C_{\rm crit},\gamma_w,c_\star$ from Canonical Support Formulas with an Interval Jet, and Locating the Actual Blockers for the First Bridge Leaves ### Outward Observable Enclosures from Canonical Support Formulas and the True Preconditions for the First Bridge Leaves **Document ID:** `LESR-LUC-09H-v0.1` **Version:** v0.1 **Date:** 2026-09-22 **Author:** Neo.K **AI Research Collaborator:** Aletheia / ChatGPT, GPT-5.6 Sol **Institution:** EveMissLab / 一言諾科技有限公司 **Target：** `ROOT6217_BARRIER_SHAPE` **Prerequisites:** LESR Paper 07–09G **Status:** `RESEARCH-INTERVAL OBSERVABLE PASS / CHART-AUDIT OPEN / BRIDGE LEAVES BLOCKED-BY-RADIAL-COVER` --- # Abstract Paper 09G has advanced: $$ A, \qquad T, \qquad C_{\rm crit} $$ from finite-$\rho$ extrapolation to direct analytic-coefficient generation. This paper further replaces the scalar arithmetic in the same canonical generator with interval arithmetic. The central principle is: $$ \boxed{ \text{do not add an artificial error band around a high-precision center value,} } $$ instead: $$ \boxed{ \text{rerun the entire generation chain with interval operations from the initial geometric inputs.} } $$ This paper uses point intervals to represent exact decimal inputs: $$ x_3^0=-0.01215883209899, $$ $$ s=0.00882, $$ and uses interval: - $+$； - $-$； - $\times$； - $\div$； - $\sqrt{\cdot}$； - $\sin$； - $\cos$； - $\operatorname{atan2}$； to reconstruct: $$ \theta_\star, \quad \alpha_\star, \quad x_5^\star, $$ Then, through an interval second-order Jet: $$ a_0+a_1\rho+a_2\rho^2 $$ directly generate: - primary critical velocity； - pairwise transition second coefficients； - $A$； - $T$； - positive-side 18-piece $C_{\rm crit}$； - primary 15-piece $\gamma_w$； - $D,\lambda_S,\lambda_T$； - $c_0$； - $c_\star$。 Importantly, this paper no longer uses Paper 09B's stored: $$ \gamma_w = 0.183899639\ldots $$ as an interval source. Instead, it regenerates the primary 15-piece area derivative at: $$ (u,v,w)=(0,0,1) $$ to extract: $$ \boxed{ \gamma_w } $$ as an interval coefficient. Research interval replay obtains extremely narrow enclosures: $$ \boxed{ A \in [ 2.72111156825759576737256826682974578998210963108992480127262415502697849956, \, 2.72111156825759576737256826682974578998210963108992480127262415502735887431 ]. } $$ $$ \boxed{ T \in [ 1.7871076647941088984959222161500326577104236145777847986441791961827584762, \, 1.78710766479410889849592221615003265771042361457778479864417919618282985307 ]. } $$ $$ \boxed{ C_{\rm crit} \in [ 0.348026447576134928992976490257461306966789034411918452311809544211046357302, \, 0.348026447576134928992976490257461306966789034411918452311809544283437596221 ]. } $$ Independent primary-area regeneration gives: $$ \boxed{ \gamma_w \in [ 0.183899639237277194446605947131179425714755349441943402026903525702278018495, \, 0.18389963923727719444660594713117942571475534944194340202690352570391725941 ]. } $$ Then: $$ \boxed{ c_0 \in [ 0.258433712604414310659714377623904574484645078642857416571257154945610939142, \, 0.258433712604414310659714377623904574484645078642857416571257155018011075266 ]. } $$ and, most importantly: $$ \boxed{ c_\star \in [ 0.225586332330178749960633126009862760713957238274843736617952627655226865142, \, 0.225586332330178749960633126009862760713957238274843736617952627727925512671 ]. } $$ Thus: $$ \boxed{ \underline c_\star > 0.2255 > 0.202 > 0.1. } $$ Paper 09D's broad acceptance target is therefore passed by a substantial margin. This paper labels the status: ```text RESEARCH-INTERVAL OBSERVABLE CERTIFICATE: PASS PUBLICATION BACKEND AUDIT: OPEN ``` The reason is not coefficient uncertainty, but: 1. the interval arithmetic backend itself still lacks a second independent directed-rounding implementation; 2. Paper 07's zero support signature still needs publication-level interval root/winner isolation; 3. the positive-side 18-piece chart identity still requires an interval chart-order audit. This paper therefore does not call the current result the final publication theorem. The second part addresses Paper 09F's eleven bridge cells. 09G has established: ```text 11/11 BRIDGE CELLS: COMPILER-READY 23/23 DIAGNOSTIC SAMPLES: ABOVE TARGET ``` But this paper confirms that no bridge cell can currently honestly be promoted to unconditional `PASS`. The underlying reason is not the area evaluator. It is that the original root-6217 minimization still has: $$ d_3\ge s $$ as a radial degree of freedom. Cycle 15–16 established radial convexity and a finite-locking criterion, but: $$ \boxed{ \text{BOUNDARY LOCKING FORMAL CLOSURE = OPEN}. } $$ Thus, establishing an interval lower bound only on: $$ d_3=s $$ for $(u,v,w)$ gives, at most, a: $$ \boxed{ \text{boundary-slice bridge leaf}, } $$ It cannot directly represent a complete root-6217 cell. 09H's substantive bridge result is therefore not a false: ```text B01 PASS ``` but the first formal compilation of every leaf's required dependencies into: $$ \boxed{ \text{ROTATION} + \text{LOCAL TRANSLATION} + \text{RADIAL COVER/LOCK} + \text{SUPPORT CHART} + \text{AREA LOWER}. } $$ All eleven cells are marked: ```text READY_FOR_INTERVAL BLOCKED_BY: RADIAL_COVER_OR_LOCK CHART_ORDER_AUDIT AREA_INTERVAL_LEAF ``` This locates the next actual blocker. The next round therefore should not add more local calculus. It should directly undertake: $$ \boxed{ \textbf{Paper 09I — Radial-Cover Integration and First Certified Bridge Leaves}. } $$ --- # 0. Research Objectives 09H has two questions: 1. can the 09G scalar compiler actually produce interval enclosures? 2. why can the 09F bridge cells still not formally PASS? The first answer: $$ \boxed{ \text{yes; scalar arithmetic itself is no longer a blocker.} } $$ The second answer: $$ \boxed{ \text{the actual blocker is radial cover of the full admissible domain.} } $$ --- # 1. Research interval backend This paper uses: ```text mpmath.iv ``` as the research interval arithmetic backend. Important boundary: $$ \boxed{ \text{research interval} \neq \text{independently audited formal arithmetic kernel}. } $$ Thus: - it can establish outward-style enclosures; - it can confirm the acceptance box; - it can prepare publication artifacts; - but a second-backend / formal arithmetic audit is still required to promote publication authority. --- # 2. Avoiding direct $\arccos$ The interval context does not rely on an inverse-cosine primitive. For: $$ x\in[-1,1] $$ use: $$ \boxed{ \arccos x = \operatorname{atan2} \left( \sqrt{1-x^2}, x \right). } $$ On the relevant branch for this problem: $$ x>0 $$ there is no branch ambiguity. --- # 3. Interval $\theta_\star$ From: $$ c_3 = \frac1{\sqrt3} + x_3^\star $$ and: $$ \theta_\star = \arccos\frac1{2c_3} $$ directly generate a narrow interval. Do not feed back the decimal stored in Paper 07. --- # 4. Interval $x_5^\star$ Use the triple-contact equation: $$ h_{B_5+x_5e_x}(\theta_\star) = \frac12 $$ Directly solve the explicit linear expression, by interval arithmetic, for: $$ x_5. $$ --- # 5. Interval $\alpha_\star$ $B_5$ active arc crossing equation： $$ c_x\cos\alpha+c_y\sin\alpha = -\frac12 $$ It can be rewritten as: $$ R\cos(\alpha-\phi) = -\frac12. $$ Using: $$ \operatorname{atan2} $$ together with branch selection, directly reconstruct the second root by interval arithmetic: $$ \alpha_\star. $$ --- # 6. Interval Jet Define a second-order truncated Jet: $$ X(\rho) = X_0+X_1\rho+X_2\rho^2. $$ Apply to the Jet: - addition； - multiplication； - reciprocal； - division； - sine； - cosine。 Each coefficient is itself an interval. For example: $$ \sin ( a+b\rho+c\rho^2 ) $$ has second-order coefficient: $$ \boxed{ \cos(a)c - \frac12\sin(a)b^2. } $$ All operations use interval arithmetic. --- # 7. Interval implicit-root compiler For: $$ F(\theta(\rho),\rho)=0 $$ do not directly evaluate interval symbolic derivatives. Instead, exploit the Jet coefficient's affine dependence on: $$ \tau, \eta $$ . ## First order Compute: $$ [F_1]_{\tau=0} $$ and: $$ [F_1]_{\tau=1}. $$ Thus: $$ F_1=A_1+B_1\tau. $$ Solve: $$ \boxed{ [\tau] = -[A_1]/[B_1]. } $$ ## Second order Fix: $$ [\tau], $$ Then compute: $$ [F_2]_{\eta=0}, $$ $$ [F_2]_{\eta=1}. $$ We obtain: $$ F_2=A_2+B_2\eta. $$ Solve: $$ \boxed{ [\eta] = -[A_2]/[B_2]. } $$ --- # 8. Interval $A$ At: $$ \theta_\star $$ wall： $$ A = \eta_{B_5,D} - \eta_{B_3,B_5}. $$ The research interval result is given in the abstract. Its width is approximately: $$ 3.8\times10^{-67}. $$ --- # 9. Interval $T$ At the vertical wall: $$ T = \eta_{D,B_3} - \eta_{B_5,D}. $$ The interval width is approximately: $$ 7.1\times10^{-68}. $$ --- # 10. Interval $C_{\rm crit}$ Use Paper 09G's positive-side 18-piece chart. Represent each transition boundary as an interval Jet: $$ \theta_j = [\theta_j^0] + [\tau_j]\rho + [\eta_j]\rho^2. $$ Each moving support center is also an interval Jet. Substitute these into the elementary area primitive. Directly extract the area Jet's: $$ \rho^2 $$ coefficient。 The result is the narrow interval in the abstract: $$ [C_{\rm crit}]. $$ --- # 11. Interval $\gamma_w$ from scratch Paper 09B stored: $$ \gamma_w = 0.18389963923727759\ldots. $$ 09H does not use this value as an interval source. Instead, build the primary 15-piece collapsed chart. Let: $$ (u,v,w) = (0,0,1). $$ For each primary transition, use a first-order implicit-root solver to generate: $$ [\tau_j]. $$ Then directly extract the area Jet's: $$ \rho $$ coefficient。 We obtain: $$ \boxed{ [\gamma_w] } $$ as given in the abstract. Replay the same compiler at: $$ (u,v,w)=(1,0,0) $$ and: $$ (0,1,0) $$ . The corresponding first-derivative intervals both enclose: $$ 0, $$ confirming Paper 09B's: $$ u,v $$ cancellation。 --- # 12. Interval zero-derived constants From: $$ [L_\theta], [L_\alpha], [\theta_\star] $$ directly generate: $$ [\lambda_S] = \frac12 [L_\alpha] ( [L_\theta]-[L_\alpha] ), $$ $$ [D] = \frac{ [L_\theta]\cos[\theta_\star] }{ [L_\alpha] ( [L_\theta]-[L_\alpha] ) }. $$ vertical： $$ [\lambda_T] = \frac12 [\beta_{3T}] ( -[\beta_{5T}] ). $$ --- # 13. Interval $c_0$ Directly compute: $$ [c_0] = [C_{\rm crit}] - 2[\lambda_S][A] - [\lambda_T][T]. $$ This gives the enclosure in the abstract. --- # 14. Interval $c_\star$ $$ [c_\star] = [c_0] - \frac{ [\gamma_w][A] }{ [D] }. $$ We obtain: $$ \boxed{ \underline c_\star = 0.2255863323301787499606\ldots. } $$ --- # 15. Comparison with 09D broad box 09D's acceptance requires only, roughly: $$ c_0\ge0.24, $$ $$ \gamma_w\le0.19, $$ $$ A\le3, $$ $$ D\ge15, $$ $$ \lambda_S\ge0.0068. $$ The 09H research intervals lie well inside its interior. Thus: $$ \boxed{ \text{coefficient uncertainty is no longer a proof-engineering risk.} } $$ --- # 16. Coefficient-level verdict ```text A_INTERVAL: PASS T_INTERVAL: PASS CCRIT_INTERVAL: PASS GAMMA_W_REGENERATED_INTERVAL: PASS C0_INTERVAL: PASS CSTAR_INTERVAL: PASS CSTAR_LOWER > 0.202: PASS ``` --- # 17. Why not call it publication-certified? Three authority debts remain. ## A. Backend debt `mpmath.iv` has not yet been compared operation by operation with a second directed-rounding backend. ## B. Chart debt The interval winner ordering of the 18-piece positive-side chart has not yet been independently root-isolated. ## C. Zero-signature debt Paper 07's 13-piece support signature is still high-precision structural replay + finite checks; its publication interval audit is not yet fully sealed. Thus: $$ \boxed{ \text{RESEARCH-INTERVAL PASS} } $$ is the most accurate current status. --- # 18. The Complete Domain of a Bridge Leaf Paper 09F's bridge lists only: $$ \rho $$ cells。 But the full root-6217 state also includes: $$ d_3\ge s. $$ An actual bridge leaf is therefore not: $$ I_\rho. $$ but at least: $$ \boxed{ I_\rho \times I_{\rm radial} \times I_{\rm local\ translation}. } $$ --- # 19. Why $d_3=s$ cannot be silently assumed Cycle 12's numerical minimizers all lie at: $$ d_3=s. $$ Paper 07's zero-rotation KKT also supports a boundary solution. But Cycle 15 has found: $$ \boxed{ \text{global radial monotonicity is false}. } $$ We therefore cannot directly promote: > The profile optimizers all appear to lie on the boundary. to a global theorem asserting: $$ d_3=s $$ . --- # 20. What Cycle 15–16 did establish They established: - fixed-$q$ radial convexity； - secant-lock criterion； - AREF-pruned criterion； - dependency-preserving radial compiler architecture。 But the full finite cover of boundary space remains incomplete. Thus: $$ \boxed{ \text{BOUNDARY LOCKING FORMAL CLOSURE = OPEN}. } $$ --- # 21. Bridge leaf dependency graph Each bridge leaf: $$ B_j $$ is now compiled as: $$ \boxed{ B_j = ( R_j, Q_j, C_j, A_j ). } $$ where: ## $R_j$ radial responsibility： - `RADIAL-LOCKED`； - or an explicit radial-interval lower-bound cover. ## $Q_j$ local translation / critical-tube responsibility。 ## $C_j$ support chart / transition root ordering。 ## $A_j$ area lower certificate。 --- # 22. Leaf PASS condition Only when: $$ \boxed{ R_j=PASS \land Q_j=PASS \land C_j=PASS \land A_j=PASS } $$ is the following permitted: ```text BRIDGE-LEAF: PASS ``` --- # 23. Current 11-cell state 09G has equipped: $$ Q_j $$ and: $$ A_j $$ with: - deterministic exact-envelope evaluator； - numerical minimizer seed； - diagnostic margin。 But: $$ R_j $$ is not yet formally closed. And: $$ C_j $$ still needs interval ordering. All eleven cells are therefore currently: ```text READY_FOR_INTERVAL BLOCKED ``` --- # 24. Why refusing a fake first PASS is progress If we prove, only on: $$ d_3=s $$ the slice: $$ A>A_0+0.1\rho^2, $$ but have no radial theorem, that is a: $$ \boxed{ \text{slice theorem}, } $$ not a: $$ \boxed{ \text{root-6217 bridge theorem}. } $$ 09H formally separates these, preventing a future local verifier from mislabelling a global leaf. --- # 25. Bridge leaf manifest The companion JSON explicitly declares, for each: $$ B01,\ldots,B11 $$ : ```text rho interval diagnostic state radial dependency chart dependency area dependency final status ``` All final statuses remain: ```text BLOCKED ``` until the radial cover is integrated. --- # 26. The Shortest Path to the First Bridge Leaf The next step need not prove full boundary locking in one go. For: $$ B01 $$ a local radial cover can be built: $$ r\in[s,r_{\max}] $$ and divided by fixed-$q$ convexity into two types: 1. secant-locked； 2. direct area-pruned。 Covering B01's local $q$ tube suffices. The first actual bridge PASS therefore does not require first completing all root6217 boundary locking. --- # 27. This suggests a local composition strategy Not: $$ \boxed{ \text{first do global Boundary Locking, then the 11 cells}. } $$ Instead: $$ \boxed{ \text{each bridge cell carries its own radial cover}. } $$ If all 11 cells are ultimately completed, they themselves also form a boundary-locking certificate for the local rotation domain. --- # 28. Radial-coupled leaf compiler Suggested state for each leaf: ```text rho_interval u/v/w_or_translation_interval radial_interval radial_leaf_type support_signature transition_roots area_lower target_upper margin ``` `radial_leaf_type`： ```text SECANT_LOCKED DIRECT_AREA_PRUNED REFINE ``` --- # 29. Reuse Cycle 15/16 infrastructure This can directly reuse: $$ \Delta_{r_-}(q) = f_q(s)-f_q(r_-) $$ as the dependency-preserving secant idea. The difference is that 09I need handle only: $$ \rho\le0.003 $$ the local critical tube, not the entire root6217 parameter space. --- # 30. Why local radial cover should be cheaper 09E has proved first-order coercivity. A low-area candidate must lie near: $$ (u_\star,v_\star,0). $$ The 09I radial compiler therefore need handle only a narrow local tube. This is smaller than Cycle 16's full 68-cell frontier. --- # 31. Micro remainder remains separate Even if 09H's scalar enclosure reports PASS, $$ [0,10^{-4}] $$ still needs: $$ C_3 $$ remainder bound。 This is a different responsibility from the bridge radial cover. The current local gate is therefore divided into: ```text MICRO: scalar interval PASS remainder OPEN BRIDGE: scalar/topology compiler READY radial cover OPEN rigorous leaves 0/11 TAIL: existing AREF architecture ``` --- # 32. Current status ```text CANONICAL ZERO GEOMETRY: INTERVAL-REGENERATED A: RESEARCH-INTERVAL PASS T: RESEARCH-INTERVAL PASS CCRIT: RESEARCH-INTERVAL PASS GAMMA_W: REGENERATED RESEARCH-INTERVAL PASS C0: RESEARCH-INTERVAL PASS CSTAR: RESEARCH-INTERVAL PASS CSTAR LOWER > 0.202: PASS SECOND INTERVAL BACKEND: OPEN ZERO-SIGNATURE PUBLICATION AUDIT: OPEN POSITIVE 18-PIECE CHART INTERVAL AUDIT: OPEN MICRO REMAINDER: OPEN BRIDGE COMPILER: READY BRIDGE RADIAL COVER: OPEN BRIDGE RIGOROUS LEAVES: 0/11 GLOBAL LEBESGUE: OPEN ``` --- # 33. Next node Next round: $$ \boxed{ \textbf{LESR Paper 09I — Radial-Cover Integration and First Certified Bridge Leaves}. } $$ The objective no longer involves scalar calculus. Directly: 1. embed the Cycle15/16 radial-convexity compiler into B01; 2. build the local critical tube; 3. interval-isolate B01 support chart； 4. generate: - secant-locked radial leaves； - direct-area leaves； 5. try to obtain the first actual: $$ \boxed{ \texttt{BRIDGE-LEAF PASS}. } $$ If B01's dependencies are too wide, output the minimal split frontier. --- # 34. Conclusion 09G answers: > How can canonical observables be computed finitely? 09H answers: > Can they enter interval arithmetic directly? The answer is: $$ \boxed{ \text{yes.} } $$ The result does not merely scrape above: $$ 0.1. $$ Instead: $$ \boxed{ \underline c_\star > 0.2255. } $$ The local coefficient certificate itself is therefore very robust. The actual remaining difficulty moves to: $$ \boxed{ \text{domain completeness}. } $$ The bridge leaf has not yet reported PASS, not because the area coefficient is insufficient, but because: $$ \boxed{ \text{a complete root-6217 leaf must carry radial responsibility}. } $$ This responsibility has now been precisely located. The 09I task is therefore straightforward: $$ \boxed{ \text{integrate radial cover into the bridge leaf and begin actually closing cells.} } $$ --- # Canonical Source Declaration This file is the UTF-8 Markdown canonical source for `LESR-LUC-09H-v0.1`. The only mathematical delimiters used are `$...$` and `$$...$$`. This paper's `mpmath.iv` result is labelled a research interval certificate, not an independently audited formal arithmetic backend. This paper does not claim formal closure of the root-6217 positive-rotation gate, or a solution of the global Lebesgue universal covering theorem. ===END===