> 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 09D — Paired-Sliver Domination and a Robust Scalar Certificate ### From Analytic Elimination of the Secondary LP to a Finite-Radius Acceptance Box: Compressing the Second-Order Proof to Five Scalars and One Remainder Bound ### Paired-Sliver Domination and a Robust Scalar Certificate: Analytic Elimination of the Secondary LP and a Finite-Radius Acceptance Box **Document ID:** `LESR-LUC-09D-v0.1` **Version:** v0.1 **Date:** 2026-09-21 **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 09A v0.2; Paper 09B; Paper 09C; BCODR-LUC-07 **Status:** `ANALYTIC LP ELIMINATION / ROBUST ACCEPTANCE BOX / CONDITIONAL FINITE-RADIUS CERTIFICATE` --- # Abstract Paper 09C reduced the second epi-derivative of root-6217's local positive-rotation barrier to: $$ \Phi_2 = c_0 + \gamma_w w_2 + \lambda_S \left( |\delta_+| + |\delta_-| \right) + \lambda_T[\delta_T]_+, $$ where the two signed side widths have a mirror-paired affine structure: $$ \delta_+ = A + X(u_2,v_2) + Dw_2, $$ $$ \delta_- = -A + X(u_2,v_2) - Dw_2. $$ Paper 09C consequently formulated the problem as a small convex LP, obtaining the nominal minimum: $$ c_\star \approx 0.22558634 > 0.1. $$ This paper further finds that the LP can be eliminated analytically. Let: $$ H = A+Dw_2. $$ then: $$ \delta_+ = X+H, $$ $$ \delta_- = X-H. $$ The elementary inequality: $$ \boxed{ |X+H| + |X-H| \ge 2|H|, } $$ gives: $$ \Phi_2 \ge c_0 + \gamma_w w_2 + 2\lambda_S|A+Dw_2|. $$ Let: $$ y=A+Dw_2, $$ then: $$ w_2 = \frac{y-A}{D}, $$ Thus: $$ \Phi_2 \ge c_0 - \frac{\gamma_w A}{D} + \frac{\gamma_w}{D}y + 2\lambda_S|y|. $$ If: $$ \boxed{ 0 \le \frac{\gamma_w}{D} < 2\lambda_S, } $$ the right-hand side attains its global minimum at: $$ y=0 $$ . This gives the paper's core theorem: $$ \boxed{ \inf\Phi_2 \ge c_0 - \frac{\gamma_w A}{D}. } $$ For 09C's nominal constants: $$ c_0 \approx 0.258433713, $$ $$ \gamma_w \approx 0.1838996392372776, $$ $$ A \approx 2.721112, $$ $$ D \approx 15.234440, $$ we obtain: $$ \boxed{ c_0 - \frac{\gamma_w A}{D} \approx 0.225586328. } $$ in complete agreement with Paper 09C's numerical LP minimum. This brings two important corrections. First, 09C's: $$ (u_2,v_2,w_2) \approx ( 37.01279, 0.133549, -0.178616 ) $$ is: $$ \boxed{ \text{a canonical boundary representative,} } $$ not the unique minimizer. The actual nominal minimizer face satisfies: $$ A+Dw_2=0, $$ $$ X(u_2,v_2)=0, $$ together with the condition that the top term contributes no additional area. Second, a publication-level second-order lower bound no longer requires solving an interval LP. Interval certification is needed only for: 1. mirror-pair identity； 2. $c_0$； 3. $\gamma_w$； 4. $A$； 5. $D$； 6. $\lambda_S$； 7. omitted / top secondary contributions nonnegative。 The paper further constructs a deliberately loose coarse acceptance box: $$ c_0 \in [0.24,0.28], $$ $$ \gamma_w \in [0.18,0.19], $$ $$ A \in [2.5,3.0], $$ $$ D \in [15.0,15.5], $$ $$ \lambda_S \in [0.0068,0.0076]. $$ Throughout this entire box: $$ \frac{\gamma_w}{D} < 2\lambda_S, $$ and: $$ \boxed{ \Phi_2 \ge 0.202. } $$ Thus even when coefficient enclosures are far coarser than the current high-precision numerical values, the lower bound still exceeds the target: $$ 0.1 $$ by: $$ \boxed{ 0.102 } $$ in second-order coefficient margin. Finally, if the finite-radius remainder satisfies: $$ |R_3(\rho)| \le 30\rho^3, $$ for: $$ 0\le \rho \le 0.003, $$ then: $$ A(\rho)-A_0 \ge \rho^2 \left( 0.202-30\rho \right) \ge 0.112\rho^2. $$ Thus: $$ \boxed{ A(\rho) \ge A_0 + 0.1\rho^2 } $$ holds throughout the local radius interval. Paper 09D therefore further compresses 09C's: $$ \boxed{ \text{interval LP} + \text{uniform remainder} } $$ to: $$ \boxed{ \text{scalar interval enclosure} + \text{mirror pairing} + \text{very coarse remainder bound}. } $$ This paper still does not claim completion of a publication theorem: an independent interval runtime must actually enclose the canonical coefficients within the acceptance box, and Paper-09B's full neighbouring first-order chart certificate remains incomplete. --- # 0. Recenter Paper 09C has obtained: $$ \boxed{ \Phi_2 = c_0 + \gamma_w w_2 + \lambda_S ( |\delta_+| + |\delta_-| ) + \lambda_T[\delta_T]_+. } $$ nominal constants： $$ c_0 \approx 0.258433713, $$ $$ \gamma_w = 0.1838996392372776, $$ $$ \lambda_S \approx 0.00719758120147, $$ $$ \lambda_T \approx 0.02821424417983. $$ Paper 09C's secondary-width model is: $$ \delta_+ \approx 2.721112 - 0.02682772u_2 + 7.435229v_2 + 15.234440w_2, $$ $$ \delta_- \approx -2.721112 - 0.02682772u_2 + 7.435229v_2 - 15.234440w_2. $$ --- # 1. Mirror-pair normal form Write the common part in: $$ u_2,v_2 $$ as: $$ \boxed{ X = Bu_2 + Cv_2. } $$ Write the constant and horizontal correction as: $$ \boxed{ H = A+Dw_2. } $$ then: $$ \boxed{ \delta_+ = X+H, } $$ $$ \boxed{ \delta_- = X-H. } $$ This is the canonical normal form of the second-scale mirror pair. --- # 2. Why Is This Form More Important Than the Original LP? 09C still treats: $$ u_2, v_2, w_2 $$ as LP variables. The preceding expression shows, however, that: $$ u_2,v_2 $$ enter the paired side penalty only through one scalar combination: $$ X $$ . Thus: $$ \boxed{ \text{side topology sees only one direction in }(u_2,v_2)\text{.} } $$ The LP therefore has a larger hidden degeneracy. --- # 3. Paired Absolute-Value Lemma For all real: $$ X,H, $$ we have: $$ \boxed{ |X+H| + |X-H| \ge 2|H|. } $$ Proof: By the triangle inequality: $$ |X+H| + |X-H| \ge | (X+H)-(X-H) |. $$ the right-hand side is: $$ = |2H| = 2|H|. $$ Q.E.D. --- # 4. Equality condition When: $$ H=0, $$ the left-hand side becomes: $$ 2|X|. $$ If the side penalty must also vanish, we require: $$ \boxed{ X=0. } $$ The zero-side-penalty locus is therefore: $$ \boxed{ A+Dw_2=0, \qquad X(u_2,v_2)=0. } $$ --- # 5. Drop nonnegative top contribution Once it has been proved that: $$ \boxed{ \lambda_T[\delta_T]_+ \ge 0, } $$ the top term can be dropped directly in lower-bound analysis: $$ \Phi_2 \ge c_0 + \gamma_w w_2 + \lambda_S ( |\delta_+| + |\delta_-| ). $$ The top affine coefficients: $$ \delta_T = T+Eu_2+Fv_2 $$ therefore need not enter the arithmetic core of the second-order lower certificate at all. --- # 6. Side domination By the Paired Absolute-Value Lemma: $$ |\delta_+| + |\delta_-| \ge 2|A+Dw_2|. $$ Thus: $$ \boxed{ \Phi_2 \ge c_0 + \gamma_w w_2 + 2\lambda_S |A+Dw_2|. } $$ Note: $$ u_2, \qquad v_2 $$ have disappeared completely. --- # 7. Reduction to One Dimension Let: $$ y = A+Dw_2. $$ Assume: $$ D>0. $$ then: $$ w_2 = \frac{y-A}{D}. $$ Therefore: $$ \Phi_2 \ge c_0 + \frac{\gamma_w}{D} (y-A) + 2\lambda_S|y|. $$ Rearrange: $$ \boxed{ \Phi_2 \ge c_0 - \frac{\gamma_w A}{D} + \frac{\gamma_w}{D}y + 2\lambda_S|y|. } $$ --- # 8. Kink function Define: $$ g(y) = \frac{\gamma_w}{D}y + 2\lambda_S|y|. $$ When: $$ y>0, $$ the slope is: $$ g_+'(y) = \frac{\gamma_w}{D} + 2\lambda_S. $$ When: $$ y<0, $$ the slope is: $$ g_-'(y) = \frac{\gamma_w}{D} - 2\lambda_S. $$ --- # 9. Scalar criticality condition If: $$ \boxed{ 0 \le \frac{\gamma_w}{D} < 2\lambda_S, } $$ then: $$ g_-'(y)<0, $$ while: $$ g_+'(y)>0. $$ Thus: $$ \boxed{ y=0 } $$ is the unique scalar minimizer. --- # 10. Paired-Sliver Domination Theorem Under the following conditions: 1. $D>0$； 2. $\lambda_S>0$； 3. the side pair has the exact form: $$ \delta_\pm = X \pm (A+Dw_2); $$ 4. omitted / top contributions nonnegative； 5. $$ 0 \le \frac{\gamma_w}{D} < 2\lambda_S; $$ then: $$ \boxed{ \Phi_2 \ge c_0 - \frac{\gamma_w A}{D}. } $$ --- # 11. Equality structure To attain the theorem's lower bound, we require at least: $$ \boxed{ A+Dw_2=0, } $$ Thus: $$ \boxed{ w_2^\star = -\frac{A}{D}. } $$ The paired side penalty must simultaneously vanish: $$ \boxed{ X(u_2,v_2)=0. } $$ The top / omitted contribution must also vanish. --- # 12. Correcting 09C's Minimizer Uniqueness Paper 09C gives the representative: $$ (u_2,v_2,w_2) \approx ( 37.01279, 0.133549, -0.178616 ) $$ and selects: $$ \delta_T=0. $$ Paper 09D corrects this: $$ \boxed{ \text{this is a boundary representative on the minimizer face, not the unique minimizer.} } $$ In the nominal LP, whenever: $$ X=0, $$ $$ w_2=-A/D, $$ and the top penalty is zero, the same objective value is attained. --- # 13. Why Does 09C's Numerical Minimum Remain Correct? 09C's representative simultaneously satisfies: $$ X=0, $$ $$ A+Dw_2=0, $$ and: $$ \delta_T=0. $$ It therefore does belong to the minimizer set. The correction concerns: $$ \boxed{ \text{uniqueness interpretation}, } $$ not: $$ \boxed{ \text{minimum value}. } $$ --- # 14. Nominal scalar replay Take: $$ A = 2.721112, $$ $$ D = 15.234440, $$ then: $$ -\frac{A}{D} \approx -0.17861582. $$ in agreement with 09C's: $$ w_2^\star \approx -0.178616 $$ . --- # 15. Nominal slope condition Compute: $$ \frac{\gamma_w}{D} \approx 0.01207130943. $$ On the other hand: $$ 2\lambda_S \approx 0.01439516240. $$ Therefore: $$ \boxed{ \frac{\gamma_w}{D} < 2\lambda_S. } $$ The margin is approximately: $$ \boxed{ 0.00232385. } $$ --- # 16. Nominal lower bound Directly evaluating: $$ c_0 - \frac{\gamma_w A}{D} $$ gives: $$ \boxed{ 0.225586328\ldots } $$ in agreement with the 09C LP solver's: $$ 0.22558634 $$ . --- # 17. The LP Has Been Eliminated Analytically The publication certificate therefore no longer requires: $$ \boxed{ \text{general-purpose interval LP solver}. } $$ Interval arithmetic need only verify: $$ \boxed{ c_0 - \frac{\gamma_w A}{D} > 0.1 } $$ and: $$ \boxed{ \frac{\gamma_w}{D} < 2\lambda_S. } $$ --- # 18. Coarse Acceptance Box This paper deliberately avoids a narrow box close to the nominal value. Instead, use: $$ \boxed{ c_0 \in [0.24,0.28], } $$ $$ \boxed{ \gamma_w \in [0.18,0.19], } $$ $$ \boxed{ A \in [2.5,3.0], } $$ $$ \boxed{ D \in [15.0,15.5], } $$ $$ \boxed{ \lambda_S \in [0.0068,0.0076]. } $$ All nominal constants lie in the box interior. --- # 19. Robust slope inequality Over the entire box, the largest possible: $$ \frac{\gamma_w}{D} $$ is: $$ \frac{0.19}{15} = 0.0126666666\ldots $$ the smallest possible: $$ 2\lambda_S $$ is: $$ 2(0.0068) = 0.0136. $$ Thus: $$ \boxed{ \frac{\gamma_w}{D} \le 0.0126667 < 0.0136 \le 2\lambda_S. } $$ robust slope margin： $$ \boxed{ 0.00093333\ldots } $$ --- # 20. Robust $\Phi_2$ lower bound Over the entire box, evaluate the worst case of: $$ c_0 - \frac{\gamma_w A}{D} $$ using: $$ c_0=0.24, $$ $$ \gamma_w=0.19, $$ $$ A=3.0, $$ $$ D=15.0. $$ to obtain: $$ \boxed{ \underline{\Phi_2} = 0.24 - \frac{0.19\cdot3}{15} = 0.202. } $$ --- # 21. How Wide Is This Box? Relative to the nominal: $$ c_0 \approx0.2584, $$ the downward-error allowance is: $$ >0.018. $$ $$ \gamma_w \approx0.1839 $$ may rise to: $$ 0.19. $$ $$ A \approx2.721 $$ may rise to: $$ 3.0. $$ $$ D \approx15.234 $$ may fall to: $$ 15. $$ $$ \lambda_S \approx0.007198 $$ may fall to: $$ 0.0068. $$ This is not a tiny floating-point tolerance. It is a deliberately loose: $$ \boxed{ \textbf{publication acceptance region}. } $$ --- # 22. Conditional Interval Certificate Thus, if the local interval runtime obtains: $$ [c_0] \subset [0.24,0.28], $$ $$ [\gamma_w] \subset [0.18,0.19], $$ $$ [A] \subset [2.5,3.0], $$ $$ [D] \subset [15.0,15.5], $$ $$ [\lambda_S] \subset [0.0068,0.0076], $$ and proves the exact mirror-pair normal form and nonnegativity of omitted terms, we obtain directly: $$ \boxed{ \Phi_2 \ge 0.202 > 0.1. } $$ No interval LP needs to be optimized further. --- # 23. Top-Wall Responsibility Drops Another Level Paper 09C included: $$ \lambda_T, \qquad \delta_T $$ in the LP data. The lower certificate now requires only proving: $$ \boxed{ \lambda_T[\delta_T]_+ \ge0. } $$ It does not even require exact knowledge of the affine coefficients of: $$ \delta_T $$ . --- # 24. Is Five-Sliver Exhaustion Still Necessary? For the exact 09C formula, we must still ensure that no unmodelled chart has been omitted which generates: $$ \boxed{ \text{negative second-order correction} } $$ . Either of the following certificates therefore suffices: ## Route A Complete five-sliver exhaustion. ## Route B Prove that any additional second-scale winner interval adds only a nonnegative correction to the full area lower bound. Route A is currently more direct. --- # 25. Separate the Robust Coefficient Theorem from the Topology Theorem 09D divides responsibility into: $$ \boxed{ \text{Topology completeness} } $$ and: $$ \boxed{ \text{Arithmetic positivity}. } $$ Arithmetic positivity has now been almost completed analytically. The actual remaining difficulty is more about topology / interval enclosure. --- # 26. Finite-radius remainder The second epi-derivative gives only: $$ \rho\downarrow0. $$ To obtain the local gate on: $$ 0\le\rho\le0.003 $$ : write: $$ \boxed{ A(\rho) = A_0 + \rho^2\Phi_2 + R_3(\rho). } $$ Assume: $$ \boxed{ |R_3(\rho)| \le C_3\rho^3. } $$ --- # 27. Remainder budget theorem If we know: $$ \Phi_2 \ge L_2, $$ then: $$ A(\rho)-A_0 \ge \rho^2 ( L_2-C_3\rho ). $$ To obtain: $$ A(\rho)-A_0 \ge 0.1\rho^2, $$ it suffices that: $$ \boxed{ C_3\rho_{\max} \le L_2-0.1. } $$ --- # 28. Using Robust $L_2=0.202$ Take: $$ L_2 = 0.202, $$ $$ \rho_{\max} = 0.003. $$ The largest admissible bound is: $$ C_3 \le \frac{0.202-0.1}{0.003} = 34. $$ Therefore: $$ \boxed{ C_3<34 } $$ already suffices. --- # 29. Choose a More Conservative Acceptance Bound This paper chooses: $$ \boxed{ C_3 \le 30. } $$ then: $$ L_2-C_3\rho_{\max} \ge 0.202 - 30(0.003). $$ Thus: $$ \boxed{ L_2-C_3\rho_{\max} \ge 0.112. } $$ Therefore: $$ \boxed{ A(\rho) \ge A_0 + 0.112\rho^2 } $$ for: $$ 0\le\rho\le0.003. $$ --- # 30. Finite-Radius Margin Relative to the Target We actually need only: $$ 0.1\rho^2. $$ The conditional certificate gives: $$ 0.112\rho^2. $$ leaving: $$ \boxed{ 0.012\rho^2 } $$ in finite-radius safety margin. --- # 31. 09C's Finite-$\rho$ Diagnostic and $C_3$ 09C numerical replay： $$ \frac{A(\rho)-A_0}{\rho^2} \rightarrow 0.2255863\ldots $$ and over: $$ \rho = 10^{-4} \text{ to } 10^{-5} $$ the deviation is only approximately of order: $$ 10^{-6} $$ . The corresponding empirical cubic coefficient is approximately of order: $$ 10^{-2} $$ . The acceptance bound: $$ C_3\le30 $$ is therefore thousands of times looser than the current numerical indication. Status： `DIAGNOSTIC ONLY / NOT A REMAINDER PROOF` --- # 32. This Changes the Difficulty of the Remainder Task The task initially appeared to require very sharp third-order analysis. In fact, because the margin: $$ \boxed{ 0.202-0.1=0.102 } $$ is large, only a very coarse uniform third-order bound needs to be proved. This makes the remainder better suited to: - interval automatic differentiation； - Taylor model； - directed rounding； - finite chart subdivision。 --- # 33. Conditional Local Radius Theorem If all of the following hold: 1. Paper-09B's first-order critical chart is correct; 2. 09D mirror-pair normal form certified； 3. the scalar intervals lie within the acceptance box; 4. omitted secondary corrections nonnegative / five-sliver atlas exhaustive； 5. uniform remainder： $$ |R_3(\rho)| \le 30\rho^3 $$ for: $$ 0\le\rho\le0.003; $$ then: $$ \boxed{ A(\rho) \ge A_0 + 0.112\rho^2 } $$ and, in particular: $$ \boxed{ A(\rho) \ge A_0 + 0.1\rho^2. } $$ --- # 34. This Is Still Not the Complete Root-6217 Theorem Paper-09B's following certificate is still required: $$ \boxed{ \text{full neighboring first-order chart nonnegativity}. } $$ If a first-order chart has not been enumerated, it may bypass the 09D critical cone. 09D therefore closes only: $$ \boxed{ \text{second-order critical-cone arithmetic}. } $$ --- # 35. First-order off-cone responsibility For: $$ (u,v,w) \notin \mathcal C_1, $$ prove: $$ \Phi_1(u,v,w)>0 $$ or enter an explicitly declared neighbouring chart. Once a uniform first-order margin exists, it dominates second-order effects for sufficiently small: $$ \rho $$ . --- # 36. Secondary Sequences Are Fully Included For: $$ (u,v,w) = (u_\star,v_\star,0) + \rho (u_2,v_2,w_2), $$ 09D's scalar lower bound permits arbitrary finite: $$ u_2, v_2, w_2. $$ It therefore does not verify only one numerical optimizer path. --- # 37. Geometric Meaning of the Minimizer Face The nominal minimum is no longer an isolated second-order point. It is a secondary face constrained by: $$ X=0, $$ $$ A+Dw_2=0 $$ . Nonnegative omitted terms then select an admissible subset. The second-order shape may therefore have: $$ \boxed{ \text{flat direction / secondary moduli}. } $$ --- # 38. This Does Not Mean Global Nonuniqueness Secondary moduli are only: $$ \boxed{ \text{second-order local degeneracy}. } $$ Higher orders: $$ \rho^3, \rho^4,\ldots $$ may select a unique path again. 09D therefore does not directly imply: $$ \boxed{ \text{global minimizer nonunique}. } $$ --- # 39. Formal Supersession of Paper 09C Paper 09D supersedes two levels of description in 09C: ## Superseded A “An interval solve of the 6-variable LP is required to prove $\Phi_2>0.1$.” Replaced by: $$ \boxed{ \text{paired-sliver scalar certificate suffices}. } $$ ## Superseded B “The 09C numerical LP point is the unique secondary minimizer.” Replaced by: $$ \boxed{ \text{it is a canonical minimizer-face representative}. } $$ 09C's: - second-scale reduction； - nominal minimum value； - five potential slivers； - affine width model； are all retained. --- # 40. Only a Few Publication Arithmetic Obligations Remain The core interval arithmetic needs only: $$ [c_0], $$ $$ [\gamma_w], $$ $$ [A], $$ $$ [D], $$ $$ [\lambda_S]. $$ Even: $$ B, C, E,F,T $$ no longer enter the final scalar lower bound. --- # 41. Minimal Local Work Package The local independent verifier should prioritize: 1. exact / interval mirror pairing； 2. interval $c_0$； 3. interval $\gamma_w$； 4. interval $A$； 5. interval $D$； 6. interval $\lambda_S$； 7. check interval boxes fit 09D acceptance box； 8. interval / Taylor-model remainder $C_3\le30$； 9. topology exhaustion； 10. independent first-order chart audit。 --- # 42. Acceptance Box Machine Contract This paper's companion JSON fixes: ```text c0 : [0.24, 0.28] gamma_w : [0.18, 0.19] A : [2.5, 3.0] D : [15.0, 15.5] lambda_S : [0.0068, 0.0076] rho_max : 0.003 C3_max : 30 target : 0.1 ``` This is not a measured interval. It is an: $$ \boxed{ \text{acceptance region that automatically gives PASS whenever actual rigorous intervals lie inside}. } $$ --- # 43. Machine-checkable robust arithmetic The companion verifier uses exact decimal arithmetic to check: $$ \frac{\gamma_{\max}}{D_{\min}} < 2\lambda_{S,\min}, $$ and: $$ L_2 = c_{0,\min} - \frac{ \gamma_{\max}A_{\max} }{ D_{\min} }. $$ We obtain: $$ \boxed{ L_2=0.202. } $$ Then check: $$ L_2 - C_{3,\max}\rho_{\max} = 0.112 > 0.1. $$ --- # 44. Why Is This Verifier Valuable? It does not prove: > The actual coefficients lie within the box. It proves: > Once an independent interval runtime encloses the coefficients inside the box, the conclusion no longer requires human interpretation. This is: $$ \boxed{ \text{proof obligation separation}. } $$ --- # 45. Local result template The companion package also includes: `LESR_Paper09D_Local_Interval_Result_Template_v0.1.json` Its fields initially all contain: ```text PENDING ``` The local runtime must supply: - rigorous interval endpoints； - pairing certificate； - topology status； - remainder constant； - backend identity。 Nominal numbers must not automatically be converted into PASS. --- # 46. Secondary verification Ideally, the second independent backend should: 1. not directly reuse the primary numerical fitter; 2. reconstruct from the canonical support formulas; 3. derive the interval constants again; 4. then let the 09D acceptance verifier perform the final gate. --- # 47. Parallel Relationship with BCODR-LUC-07 09D handles: $$ \boxed{ \text{local positive-rotation second-order barrier}. } $$ Local BCODR-LUC-07 work handles: $$ \boxed{ \text{Cycle-17 candidate / germ / mirror alignment}. } $$ The two can run entirely in parallel. --- # 48. Relationship with AMRAL Round38+ AMRAL finite closure still handles the global necessity atlas: $$ 66/77 \rightarrow 77/77. $$ 09D does not automatically close the AMRAL residual. Thus: $$ \boxed{ a_{\rm Leb}\ge0.835 \text{ NOT CERTIFIED globally}. } $$ --- # 49. When Should the Final Stress Test Begin? When: 1. 09B first-order charts certified； 2. 09D scalar box instantiated； 3. local radius remainder certified； 4. the AMRAL global residual has been compressed further; 5. root-131 local alignment is complete; the current geometric atlas can be frozen as: $$ \mathcal K. $$ Then use the Projection Possibility Space / AGC series actively to search for: $$ \texttt{UNKNOWN}. $$ --- # 50. Current status ```text PAPER09C SECONDARY LP: ANALYTICALLY ELIMINATED SIDE PAIR NORMAL FORM: STRUCTURALLY DERIVED / INTERVAL REPLAY OPEN NOMINAL SECOND-ORDER LOWER: 0.225586328... COARSE ACCEPTANCE BOX LOWER: 0.202 TARGET: 0.1 COEFFICIENT-LEVEL ROBUST MARGIN: 0.102 RHO MAX: 0.003 ACCEPTED REMAINDER CONSTANT: C3 <= 30 FINITE-RADIUS CONDITIONAL LOWER: 0.112 FINITE-RADIUS TARGET MARGIN: 0.012 INTERVAL COEFFICIENT INSTANTIATION: OPEN TOPOLOGY EXHAUSTION: OPEN FIRST-ORDER MULTICHART CERTIFICATE: OPEN GLOBAL LEBESGUE THEOREM: OPEN ``` --- # 51. Next node The next genuinely valuable node is no longer new calculus in 09E. It should instead be: $$ \boxed{ \textbf{LESR Paper 09E — Canonical Interval Instantiation and First-Order Chart Exhaustion}. } $$ Two goals: ## Lane A — scalar interval instantiation Actually use interval arithmetic to enclose: $$ [c_0], [\gamma_w], [A], [D], [\lambda_S] $$ within the 09D acceptance box. ## Lane B — topology / first-order closure Complete: $$ \text{CERT-09B.4} $$ together with secondary latent-wall exhaustion. If both lanes PASS, only the finite-radius remainder remains for the local barrier proof. --- # 52. Conclusion Paper 09C compressed multiscale second-order geometry into a convex LP. Paper 09D goes one step further: $$ \boxed{ \text{convex LP} \rightarrow \text{one-dimensional kink inequality}. } $$ The core is: $$ \boxed{ |X+H| + |X-H| \ge 2|H|. } $$ Mirror-paired side slivers therefore directly control horizontal secondary drift: $$ w_2. $$ Provided: $$ \frac{\gamma_w}{D} < 2\lambda_S, $$ we obtain: $$ \boxed{ \Phi_2 \ge c_0 - \frac{\gamma_w A}{D}. } $$ nominal： $$ \boxed{ 0.225586328\ldots } $$ Even with the deliberately loose publication acceptance box, we still have: $$ \boxed{ \Phi_2 \ge 0.202. } $$ Allowing a very coarse: $$ |R_3| \le 30\rho^3 $$ up to: $$ \rho=0.003, $$ still gives: $$ \boxed{ A(\rho) \ge A_0 + 0.112\rho^2 > A_0 + 0.1\rho^2. } $$ The arithmetic difficulty of root-6217's local second-order gate has therefore been substantially reduced. The actual remaining responsibilities are now concrete: $$ \boxed{ \text{interval instantiate the acceptance box} + \text{certify topology completeness} + \text{prove a very coarse remainder bound}. } $$ This is proof engineering, no longer unknown local calculus. --- # Canonical Source Declaration This file is the UTF-8 Markdown canonical source for `LESR-LUC-09D-v0.1`. The only mathematical delimiters used are `$...$` and `$$...$$`. This paper supersedes Paper 09C's descriptions of secondary-minimizer uniqueness and interval-LP necessity, while retaining Paper 09C's second-scale topology reduction, nominal coefficient and numerical replay. This paper's acceptance box is a conditional proof target, not a purported completed interval measurement. This paper does not claim that the global Lebesgue universal covering theorem has been proved. ===END===