{
  "round": "RUN-100",
  "gate": "src102",
  "C7": "PASS / canonical",
  "kappa_det": "1",
  "actual_independent_routes": 1,
  "determinant_category": "rank-1 projective R_S-modules in cohomological degree 0, both ends included",
  "why": "Det's functoriality on rank-1 projective R_S-modules: Det(Psi_4 Psi_3 Psi_2 Psi_1^{-1}) = Det(Psi_4)Det(Psi_3)Det(Psi_2)Det(Psi_1)^{-1}, one route, the same functor, the same grading, dual, localization, orientation and induced H^1 maps (Eq. 11); (18.1) applies, and (18.2) on h = id gives 1",
  "category": {
    "R": "H_Gamma",
    "localization": "R_S = H_Gamma[1/X^n, 1/L_E, 1/L_p(Ad g)] (RUN-098's S; nothing more inverted)",
    "inverted_only_temporarily": [
      "X^n",
      "L_p(g x psi)",
      "L_p(Ad g)"
    ],
    "why": {
      "X^n and L_p(g x psi) = L_E": "Psi_1^{-1} (RUN-098: injective, not surjective, inverse over H_Gamma[1/L_E][1/X^n])",
      "L_p(Ad g)": "Psi_3's fraction as written (RUN-098)"
    },
    "cancel_in_composite": [
      "L_p(g x psi)",
      "L_p(g x tau)",
      "L_p(Ad g)",
      "C_d"
    ],
    "named_extractions_remaining": {
      "log^{[r+1]}": "C1.8 divides kappa-hat_d by it",
      "X^n": "the leading term at X = 0 (C1.7 area; C1.10's n)"
    },
    "residual_as_written": "log^{[r+1]} / X^n",
    "final_category": "rank-1 projective R_S-modules in cohomological degree 0, both ends included",
    "K1_lives_in": "Aut(L_aux (x) R_S') = R_S'^x for the S' at which both routes invert - not needed: one route, h = id",
    "non_unit_divisors_are_not_K1_defects": [
      "L_E's zero (jet / L-slot)",
      "X^n (jet order (14.1))",
      "log^{[r+1]} (C1.8 extraction)",
      "lattice indices (L)"
    ],
    "same_localization_both_ends": true,
    "nothing_more_inverted": true,
    "agrees": true
  },
  "objects": {
    "K_aux": {
      "is": "C_aux = D_cris(F+ D(f)^*(-1-k)) framed by the unique omega_f (RUN-097 L0)",
      "kind": "rank-1 projective module",
      "degree": 0,
      "det_exponent": 1,
      "det": "det_R(L) = (L, 1): the identity on lines (Knudsen-Mumford)"
    },
    "K_mid": {
      "is": "the Beilinson-Flach family line in H^1(Q, V(f,g)^* (x) H_Gamma) (L1)",
      "kind": "rank-1 projective module",
      "degree": 0,
      "det_exponent": 1,
      "det": "det_R(L) = (L, 1): the identity on lines (Knudsen-Mumford)"
    },
    "K_lead": {
      "is": "its leading line at X = 0, image in H^1(Q, V(g)^*(psi^{-1}) (x) H_Gamma(-j)) (L2)",
      "kind": "rank-1 projective module",
      "degree": 0,
      "det_exponent": 1,
      "det": "det_R(L) = (L, 1): the identity on lines (Knudsen-Mumford)"
    },
    "K_kato": {
      "is": "the Kato line: H^1(Q, V(g)^*(psi^{-1}) (x) H_Gamma(-j)), free of rank 1 (Prop C1.1), generated by kappa(g x psi) (L3)",
      "kind": "rank-1 projective module",
      "degree": 0,
      "det_exponent": 1,
      "det": "det_R(L) = (L, 1): the identity on lines (Knudsen-Mumford)"
    },
    "K_arith": {
      "is": "C_arith = the L_p(g x psi) line inside H_Gamma (L4)",
      "kind": "rank-1 projective module",
      "degree": 0,
      "det_exponent": 1,
      "det": "det_R(L) = (L, 1): the identity on lines (Knudsen-Mumford)"
    }
  },
  "routes": {
    "A": {
      "path": "C_aux <-Psi_1- BF family line -Psi_2-> leading line -Psi_3 (C1.13)-> Kato line -Psi_4 (Kato)-> C_arith",
      "theorems": [
        "C1.9 / LZ16 regulator",
        "C1.6-C1.8",
        "C1.13 (Prop C1.1 lift)",
        "Kato reciprocity (through LR)"
      ],
      "status": "the route"
    },
    "A'": {
      "path": "leading line -regulator (C1.9 with C1.7/C1.8)-> C_arith, skipping C1.13 and Kato",
      "status": "the same map as Psi_4 Psi_3: C1.13 is PROVED by equating the two regulator images (Eq. 11 + Kato) and lifting by Prop C1.1",
      "induces_same_H1_map_as_A": true
    },
    "B": {
      "path": "a direct Eichler-Shimura / de Rham comparison C_aux -> C_arith not through the BF class",
      "status": "not a route: no theorem in the source connects an f-side crystalline line to a g-side L_p line except through the one object that sees both, the Rankin-Selberg class"
    },
    "C": {
      "path": "Kato's class -> the tree's modular-symbol series (src83) as a second arrow into C_arith",
      "status": "not a route: a change of GENERATOR of the target line by q = [gamma_Kato : gamma_tree] (RUN-099: tag L, bucket q_D); stays out of C7"
    }
  },
  "coherence": {
    "same determinant convention": {
      "yes": true,
      "evidence": "Knudsen-Mumford det on rank-1 projective modules, det(L) = (L, 1), on both sides"
    },
    "same cohomological grading": {
      "yes": true,
      "evidence": "every object in degree 0; no det R-Gamma on either side"
    },
    "same dual convention": {
      "yes": true,
      "evidence": "the * lives inside the objects (V(g)^*, V(f,g)^*, LR Part C throughout); no dualisation step occurs along the route; Psi_4 is covariant; the target is scalar (H_Gamma)"
    },
    "same localization": {
      "yes": true,
      "evidence": "every arrow's scalar taken in the one R_S of RUN-098, both ends included"
    },
    "same orientation": {
      "yes": true,
      "evidence": "the generator choices (b_f+, omega_f, kappa_Kato, the tree's series), sign included, are G/L slots of the C5 ledger (RUN-096/097); Round 030 (19): deferred to final scalarisation, even exponent"
    },
    "same induced H^i maps": {
      "yes": true,
      "evidence": "the one route's two descriptions agree on H^1 iff Eq. 11 - a theorem; checked on exact instances"
    }
  },
  "k1": {
    "formula": "kappa_det(h) = prod_i det(H^i(h))^{(-1)^i}  (18.2)",
    "positive_control_A": {
      "h": "identity, same functor, same route",
      "kappa": "1",
      "accepted": true
    },
    "positive_control_B": {
      "h": "explicit automorphism, det H^0 = 2, det H^1 = 5",
      "kappa": "2/5",
      "by_hand": "2/5",
      "accepted_at_its_value": true
    },
    "chain_homotopic_to_identity": {
      "declared": true,
      "kappa": "1",
      "consistent": true
    },
    "raw_ring_injection": {
      "Psi_1 over H_Gamma": "c_f C_d L_p^g: an injection between two different lines, not an automorphism of one; not in K_1(H_Gamma)",
      "taken_as_K1": false
    },
    "q_kept_out": true
  },
  "not_claimed": [
    "C9 (the residual neutral line's monodromy) and C10 - untouched as instructed",
    "the characteristic ideal of any H^2 (main-conjecture content; not a K_1 matter)",
    "v_11(q) (RUN-099: L, q_D, outside-certificate obligation)"
  ],
  "claim_labels": {
    "COMPUTED_HERE": [
      "A vs A' induce the same H^1 map (Eq. 11) on exact instances",
      "the K_1 formula on the controls and on h = id",
      "the grading/parity bookkeeping"
    ],
    "CITED_THEOREM": [
      "LR C1.13 and its proof (Eq. 11 + Kato's reciprocity + Prop C1.1)",
      "Round 030 (18.1), (18.2), (19)",
      "Knudsen-Mumford determinant"
    ],
    "DERIVED_FROM_VERIFIED_INPUTS": [
      "RUN-097's chain",
      "RUN-098's localization and composite",
      "RUN-099's q (kept out)"
    ],
    "UNMEASURED": [
      "C9 (the residual neutral line's monodromy) and C10 - untouched as instructed",
      "the characteristic ideal of any H^2 (main-conjecture content; not a K_1 matter)",
      "v_11(q) (RUN-099: L, q_D, outside-certificate obligation)"
    ],
    "FAILED": []
  },
  "status": {
    "C1-C6": "PASS",
    "C7": "PASS / canonical",
    "C8": "PASS",
    "C9-C10": "UNMEASURED",
    "R0": "PASS (category C)",
    "BSD": "OPEN"
  }
}