# BSD Phase 3 — P5 Rank-2 Scalar Collapse and Archimedean Comparison Boundary **Version:** v0.3 **Date:** 2026-08-14 **Curve:** $E=389.a1$ **Prime:** $p=11$ **Status:** `P5_REDUCED_TO_ARCHIMEDEAN_RANK2_PERIOD_COMPARISON` ## 0. Claim discipline This note does **not** claim the Birch--Swinnerton-Dyer conjecture for $389.a1$ and does not promote numerical agreement to proof. It imports one project-level theorem-chain state from Phase 2: $$ \Sha(E/\mathbb Q)[11^\infty]=0. $$ The imported certificate is stored in `dependencies/389a1_p11_kurihara_certificate.json`. The goal here is narrower: identify the minimal remaining obstruction to the $11$-primary **leading-term** formula in rank $2$ and compile that obstruction into finite theorem gates. --- ## 1. Fixed arithmetic data For $E=389.a1$, LMFDB records $$ \operatorname{rank}E(\mathbb Q)=2, \qquad E(\mathbb Q)_{\mathrm{tors}}=0, $$ and $$ \prod_{\ell}c_\ell=1. $$ It also records $$ \operatorname{Reg}^{\mathrm{NT}}(E) \approx 0.15246017794314375162432475705, $$ $$ \Omega_E \approx 4.9804251217101101506427155839, $$ and $$ \frac{L^{(2)}(E,1)}{2!} \approx 0.75931650028842677023019260790. $$ The last three quantities are used below only as numerical evidence unless an independent rigorous interval certificate is attached. --- ## 2. Scalar collapse Define the archimedean BSD quotient $$ \mathcal B_\infty(E) := \frac{L^{(2)}(E,1)/2!} {\Omega_E\operatorname{Reg}^{\mathrm{NT}}(E)}. $$ For a rank-$2$ elliptic curve the strong BSD leading-term formula predicts $$ \mathcal B_\infty(E) = \frac{\#\Sha(E/\mathbb Q)\prod_\ell c_\ell} {\#E(\mathbb Q)_{\mathrm{tors}}^2}. $$ For $389.a1$, the exact torsion and Tamagawa factors are both $11$-adic units, and the imported Phase-2 result gives $$ \Sha(E/\mathbb Q)[11^\infty]=0. $$ Hence the arithmetic side of the $11$-primary leading-term formula has valuation $$ 0. $$ This yields the following reduction. ### Proposition 2.1 — $11$-primary scalar reduction Assume that $\mathcal B_\infty(E)$ is given a canonical rational realization $$ \mathcal B_\infty(E)\in\mathbb Q^\times. $$ Then, for $E=389.a1$, the $11$-primary valuation statement in the strong BSD leading-term formula is equivalent to $$ \boxed{ v_{11}(\mathcal B_\infty(E))=0. } $$ ### Proof The arithmetic factor is $$ \frac{\#\Sha(E/\mathbb Q)\prod_\ell c_\ell} {\#E(\mathbb Q)_{\mathrm{tors}}^2}. $$ Its $11$-primary contribution is trivial because $$ \Sha(E/\mathbb Q)[11^\infty]=0, \qquad \prod_\ell c_\ell=1, \qquad \#E(\mathbb Q)_{\mathrm{tors}}=1. $$ Therefore its $11$-adic valuation is zero. Once $\mathcal B_\infty(E)$ is canonically rational, equality of the $11$-primary valuations is exactly the displayed condition. $\square$ --- ## 3. The rationality gate comes before the valuation gate The expression $$ \mathcal B_\infty(E) $$ is initially a real number obtained from a complex $L$-value derivative, a real period and a Néron--Tate regulator. For a generic real number there is no canonical operation $$ v_{11}:\mathbb R^\times\to\mathbb Z. $$ Thus the statement $$ v_{11}(\mathcal B_\infty(E))=0 $$ is not even well-typed until a comparison theorem places the normalized leading term in an algebraic field with a specified embedding at $11$; in the classical BSD normalization the desired target is $\mathbb Q^\times$. Accordingly P5 separates into two gates: $$ \boxed{ \mathrm{P5\!\!\!-RAT}: \mathcal B_\infty(E)\in\mathbb Q^\times } $$ and, only after that, $$ \boxed{ \mathrm{P5\!\!\!-VAL}_{11}: v_{11}(\mathcal B_\infty(E))=0. } $$ For the present curve, `P5-VAL11` is arithmetically predetermined by the Phase-2 $\Sha[11^\infty]$ certificate, but logically blocked by `P5-RAT`. --- ## 4. Literature-route audit ### 4.1 Burns--Kurihara--Sano / Generalized Perrin--Riou The Generalized Perrin--Riou framework is extremely close to the desired bridge: it connects higher Darmon derivatives of Kato zeta elements with higher derivatives of the complex $L$-function. However, the arbitrary-rank **leading-term** comparison is not currently an unconditional theorem that can be inserted here as a non-circular proof of BSD. The later Mazur--Tate paper explicitly records that in general rank the relevant Generalized Perrin--Riou statement follows, up to a $\mathbb Z_p^\times$ factor, from the relevant BSD statement together with a generalized Iwasawa main conjecture. Its main Mazur--Tate theorem also assumes BSD over $\mathbb Q$ when proving the leading-term component. Classification for P5: ```text order-of-vanishing machinery EXTERNAL_THEOREM_INPUT higher-rank leading-term bridge CIRCULAR_WITH_BSD for present goal archimedean rank-2 comparison NOT_CLOSED ``` ### 4.2 Castella--Hsieh rank-$2$ generalized Kato classes Castella--Hsieh prove rank-$2$ nonvanishing results for generalized Kato classes and obtain a leading-term formula for an **anticyclotomic $p$-adic $L$-function** in terms of a derived $p$-adic height and an enhanced $p$-adic regulator. This closes important $p$-adic height/Selmer geometry, but it is not the required identity $$ \frac{L^{(2)}(E,1)}{2!} \leftrightarrow \Omega_E\operatorname{Reg}^{\mathrm{NT}}(E). $$ Classification: ```text generalized Kato class technology EXTERNAL_THEOREM_INPUT rank-2 p-adic height bridge EXTERNAL_THEOREM_INPUT complex-to-Neron-Tate bridge TYPE_MISMATCH / NOT_CLOSED ``` ### 4.3 Chan-Ho Kim: higher Gross--Zagier / Selmer structure Kim's higher Gross--Zagier framework removes low-rank restrictions from structural comparisons between Kolyvagin systems, modular symbols, Heegner systems and Selmer structure. This is precisely why the Phase-2 Kurihara route can control the $11$-primary Selmer quotient. But the structural formula does not provide the missing rank-$2$ complex leading coefficient against the classical Néron--Tate regulator over $\mathbb Q$. Classification: ```text Selmer structure CLOSED_ENOUGH_FOR_P4 Kurihara/Kolyvagin comparison CLOSED_ENOUGH_FOR_P4 archimedean leading coefficient NOT_PROVIDED ``` ### 4.4 Kim--Pollack refined Tamagawa number conjectures Kim--Pollack give a refined BSD-type description of Bloch--Kato Selmer groups using Kolyvagin derivatives of $L$-values, under the Iwasawa main conjecture localized at the augmentation ideal. This is a powerful **discrete** analogue of Beilinson--Bloch--Kato and determines exact Selmer rank/module structure. Its strength is also the reason it does not by itself solve P5: the construction deliberately accesses the Selmer side through discrete Kolyvagin derivatives rather than supplying the missing archimedean comparison $$ L^{(2)}(E,1)/2! \longleftrightarrow \Omega_E\operatorname{Reg}^{\mathrm{NT}}(E). $$ Classification: ```text discrete Selmer determinant EXTERNAL_THEOREM_INPUT rank/module structure EXTERNAL_THEOREM_INPUT archimedean period comparison NOT_PROVIDED ``` --- ## 5. P5 verdict After the Phase-2 $11$-primary $\Sha$ closure, the strongest audited routes no longer leave a diffuse rank-$2$ Selmer obstruction. They leave one sharply typed missing map: $$ \boxed{ \text{complex higher derivative} \longrightarrow \text{classical Neron--Tate determinant with algebraic normalization}. } $$ Equivalently: $$ \boxed{ \mathrm{P5} \rightsquigarrow \mathrm{P5\!\!\!-RAT} + \mathrm{P5\!\!\!-VAL}_{11}. } $$ Current status: ```text P5-RAT OPEN / ARCHIMEDEAN RANK-2 PERIOD COMPARISON P5-VAL11 ARITHMETIC TARGET = 0, BUT BLOCKED BY P5-RAT P5 OVERALL REDUCED, NOT PROVED ``` This is a genuine reduction: the remaining wall is not ``prove more Selmer finiteness'' and not ``compute another numerical BSD quotient.'' --- ## 6. Numerical scalar witness Using the displayed LMFDB decimals, $$ \Omega_E\operatorname{Reg}^{\mathrm{NT}}(E) \approx 0.759316500288426770230192607899779387254353517170616391495, $$ so the displayed-data quotient is $$ \mathcal B_\infty^{\mathrm{disp}}(E) \approx 1.000000000000000000000000000000290541224328304419\ldots. $$ Thus the natural candidate is $$ \boxed{\mathcal B_\infty(E)=1.} $$ But this is **not** a proof. The inputs are rounded real approximations, and no theorem in the present package proves that $\mathcal B_\infty(E)$ lies in a discrete rational set. This demonstrates the exact verification obstruction: $$ \text{arbitrarily strong real approximation} \not\Rightarrow \text{exact equality} $$ without a discreteness/rationality theorem. --- ## 7. Rational-reconstruction escape The obstruction immediately suggests a finite escape route. ### Lemma 7.1 — denominator-bound reconstruction Suppose a theorem establishes $$ \mathcal B_\infty(E)=\frac{a}{b}\in\mathbb Q, \qquad \gcd(a,b)=1, \qquad 1\le b\le B. $$ Any two distinct reduced rationals whose denominators are at most $B$ differ by at least $$ \frac{1}{B^2}. $$ Therefore, if a rigorous interval computation proves $$ \left|\mathcal B_\infty(E)-1\right|<\frac{1}{2B^2}, $$ then necessarily $$ \boxed{\mathcal B_\infty(E)=1.} $$ ### Consequence A future theorem need not necessarily hand us the full equality directly. It would already be enough to provide both: 1. rationality of the normalized rank-$2$ leading term; 2. an explicit denominator bound $B$. Then a rigorous ball/interval computation of the complex $L$-derivative, real period and Néron--Tate regulator can close the equality by finite rational reconstruction. The included script `scripts/p5_rational_reconstruction_gate.py` implements this final finite gate once a **rigorous** interval and a proved denominator bound are supplied. --- ## 8. Proof-enclosure interpretation The present rank-$2$ example exhibits a strong form of proof-space collapse: $$ \text{large BSD statement} \to \text{prime-local Selmer certificate} \to \text{exact }\Sha[11^\infty]\text{ control} \to \text{one normalized real scalar}. $$ The verification space is therefore small, but it is not yet discrete. The missing mathematical operation is a **discretizer**: $$ \mathbb R\text{-valued leading term} \longrightarrow \mathbb Q\text{- or algebraic-valued normalized invariant}. $$ Once such a map is proved with effective denominators, numerical verification becomes finite and exact. This is the precise sense in which the current obstruction is a **space-domain proof-enclosure boundary**, rather than a lack of additional raw computation. --- ## 9. Next research route Do **not** proceed to all-prime gluing as though P5 were closed. The next high-value route is a representation escape focused on the missing discretizer. Priority order: ```text P5-E1 Audit ETNC/determinant-line formulations for an unconditional rationality statement weaker than full BSD. P5-E2 Search congruence-ideal / modular-degree / integral determinant techniques for an explicit denominator bound on the normalized rank-2 archimedean leading term. P5-E3 Build a rigorous Arb/Dokchitser interval certificate for L^(2)(E,1), Omega_E and Reg(E), but keep it as evidence until P5-E1/P5-E2 supplies discreteness. P5-E4 Independently compute the cyclotomic p-adic height/regulator and p-adic L-leading term at p=11 as a consistency certificate. This does not substitute for P5-RAT. ``` Stop rule: > Any route that requires the classical rank-$2$ BSD leading-term formula, or a conjecture equivalent to it, must be labeled `CIRCULAR_WITH_BSD` and may not be promoted to proof. --- ## 10. External primary references audited - D. Burns, M. Kurihara, T. Sano, *On derivatives of Kato's Euler system for elliptic curves*, arXiv:1910.07404. - D. Burns, M. Kurihara, T. Sano, *On derivatives of Kato's Euler system and the Mazur--Tate Conjecture*, arXiv:2103.11535. - F. Castella, M.-L. Hsieh, *On the non-vanishing of generalized Kato classes for elliptic curves of rank 2*, arXiv:1809.09066, current arXiv version dated 2026-03-22. - C.-H. Kim, *A higher Gross--Zagier formula and the structure of Selmer groups*, arXiv:2203.12161, current arXiv version dated 2026-03-22. - C.-H. Kim, R. Pollack, *The refined Tamagawa number conjectures for GL2*, arXiv:2505.09121. - LMFDB, elliptic curve $389.a1$ data page.