{ "attack": "C4 — the actual identification of Θ_aux from the Kummer unit ε = 1 + √2", "verdict": "UNMEASURED", "existing_source": "κ(ε), the Kummer class of ε = 1 + √2, spanning the line H¹_f(Q, Q₁₁(χ₈)(1)); q-blind and non-circular; log₁₁(ε)/(11√2) ≡ 5 (mod 11)", "existing_target": "Θ_aux ∈ Sym^m(T_X*) ⊗ 𝒞_aux — the X-leading tensor of the meromorphic Eichler–Shimura comparison at the critical-slope Eisenstein point, Attack 09 (16). Neither m nor its value is known", "missing_comparison_object": "the map that places the ES leading tensor INTO H¹_f(Q, Q₁₁(χ₈)(1)) — concretely the non-split extension class carried by the generalised eigenspace at the θ-critical Eisenstein point, together with its Bloch–Kato exponential identification. RUN-091's weight jet is NOT that class: it is the deformation along the reducible locus (field-blind), so the transverse direction is what must be built", "missing_theorem": "a critical-slope Eisenstein comparison of Bellaïche–Dasgupta / Loeffler–Rivero A4.5 type, identifying that extension class with the Kummer class up to an explicit L-invariant — CITED as the shape of what is needed, not as something established here", "smallest_constructible_precursor": "the generalised eigenspace at the θ-critical Eisenstein point of E₂(1,χ₈) at level 88 — its dimension decides whether a non-split extension exists at all. Needs overconvergent forms of slope 1 with nebentypus χ₈; the tree's Manin engines are trivial-character only", "next_executable_attack": "either that generalised eigenspace, or — if a peer line can supply the ES leading tensor's value in ANY stated frame — the single scalar κ(ε) ↦ Θ_aux, which is now all that separates them, since the target is a line", "compression_this_round": "from 'is there an identification between two unrelated objects?' to 'does Θ_aux land in this ONE line, and what is the ONE scalar?' — plus the refutation of the naive arrow through RUN-091's trace jet", "finding_1_target_is_a_line": { "statement": "H¹_f(Q, Q₁₁(χ₈)(1)) is a ONE-DIMENSIONAL Q₁₁-line spanned by the Kummer class κ(ε); any nonzero class in it is a scalar multiple of κ(ε)", "chi8_is_the_character_of_Q_sqrt2": true, "unit_rank": 1, "sigma_is_minus_inverse": true, "chi8_eigenspace_dimension": 1, "eleven_inert": true, "log_over_11_sqrt2_mod_11": 5, "label": "COMPUTED_HERE for the unit arithmetic; CITED_THEOREM (Soulé / Bloch–Kato) for H¹_f" }, "finding_2_jet_is_not_the_class": { "statement": "the trace jet is NOT the Kummer class: different groups, and the jet is blind to the quadratic field while κ(ε) is not. Refuted on type grounds, not by a numerical mismatch.", "what_the_jet_is": "char₂ ⊗ (log₁₁ ∘ χ_cyc) — the X-derivative of the character χ₈·χ_cyc^{1+X}, a deformation WITHIN the reducible locus; it lies in H¹(Q, Q₁₁) ≅ Hom(G_Q, Q₁₁)", "what_kappa_epsilon_is": "a class in H¹_f(Q, Q₁₁(χ₈)(1)), specific to Q(√2)", "field_blindness_rows": { "3": { "char2_chi8": -3, "char2_comparison": -3, "X_over_char2_chi8": 847, "X_over_char2_comparison": 847, "log_11": 847, "same": true }, "7": { "char2_chi8": 7, "char2_comparison": -7, "X_over_char2_chi8": 220, "X_over_char2_comparison": 220, "log_11": 220, "same": true }, "13": { "char2_chi8": -13, "char2_comparison": -13, "X_over_char2_chi8": 374, "X_over_char2_comparison": 374, "log_11": 374, "same": true }, "17": { "char2_chi8": 17, "char2_comparison": -17, "X_over_char2_chi8": 935, "X_over_char2_comparison": 935, "log_11": 935, "same": true }, "19": { "char2_chi8": -19, "char2_comparison": 19, "X_over_char2_chi8": 517, "X_over_char2_comparison": 517, "log_11": 517, "same": true }, "23": { "char2_chi8": 23, "char2_comparison": -23, "X_over_char2_chi8": 1111, "X_over_char2_comparison": 1111, "log_11": 1111, "same": true }, "29": { "char2_chi8": -29, "char2_comparison": 29, "X_over_char2_chi8": 517, "X_over_char2_comparison": 517, "log_11": 517, "same": true } }, "label": "COMPUTED_HERE" }, "q_blindness": { "fixed_under_q_to_uq": true, "non_circular": true, "construction_inputs": [ "ε = 1 + √2", "the prime 11", "√2 / the field Q(√2)", "χ₈" ], "label": "COMPUTED_HERE" }, "frames": { "Kummer frame": { "what": "ε ∈ O_K^×, and κ(ε) ∈ H¹_f(Q, Q₁₁(χ₈)(1)) via the Kummer map", "canonical": true, "comparison_scalar_needed": false, "why": "ε is canonical up to ±1 and up to ε ↦ ε^{-1}; the line it spans is canonical" }, "Eisenstein family frame": { "what": "E_k(1,χ₈) with the weight coordinate X, k = 2 + X", "canonical": true, "comparison_scalar_needed": false, "why": "Eisenstein series are explicit at every weight (RUN-091)" }, "Eichler–Shimura frame": { "what": "the meromorphic ES comparison of Attack 09 (16) and its X-leading term", "canonical": false, "comparison_scalar_needed": true, "why": "Attack 09 §5 fixes it only after the period frame is chosen; its leading constant C is the unknown" }, "quotient frame": { "what": "the Eisenstein quotient transport, carrying q_f", "canonical": true, "comparison_scalar_needed": false, "why": "RUN-091: the two Jordan–Hölder characters are distinct, so the one-dimensional Eisenstein quotient is canonical — but q_f itself is still unknown, which is why Θ_aux must not be built through it" } }, "claim_labels_legend": [ "COMPUTED_HERE", "CITED_THEOREM", "DERIVED_FROM_VERIFIED_INPUTS", "UNMEASURED", "FAILED" ], "status": { "C1": "UNMEASURED — auxiliary side built, partner maps missing", "C2": "UNMEASURED — 11-adic valuation/nonvanishing improved, exact unit unresolved", "C3": "PASS", "C4": "UNMEASURED — target is a line, naive arrow refuted", "C5-C7": "BLOCKED BY C4", "C8": "PARTIAL", "C9-C10": "NOT YET READY", "BSD": "OPEN" } }