The weak-base twisted-cocycle set of an ARBITRARY continuous hom
into the target, relative to an elementary kernel — the weak-base
generalization of TorsorProgram.Z1 (no surjectivity onto the quotient
imposed; the basepoint is the hom itself). Carrier chosen to match the
EdgeTwistLocalCore twist layer verbatim.
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The g-INTRINSIC invariant duals: functionals on the kernel
module invariant under the source-action THROUGH the hom — only the
operators in the image are quantified. At weak (non-surjective) homs
this can be STRICTLY larger than the target-level conjInvDuals E
(audit/CITABILITY_AUDIT_20260708.md §G1: hard-coding conjInvDuals
at weak bases would be a smuggled invariance transfer).
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prop:localduality consumer keep, general (obstructed) form,
with the scalar-pushout pinning (NSW (7.2.6) local Tate duality +
NSW (7.3.1) local Euler–Poincaré characteristic, 2nd ed. — both quantify
over ARBITRARY finite G_k-modules, verbatim-verified in
audit/CITABILITY_AUDIT_20260708.md §D4/§G; manuscript l.1974–1994): at
an elementary kernel, the local lifting problem carries an obstruction
valued in the dual of the invariant-dual space X = (R^∨)^C (consumer
spelling: conjInvDuals); it vanishes iff framed lifts exist; the framed
twisted-cocycle count is |N|²·|X| on every lift fibre
(dim Z¹_{F,ρ} = 2 dim A + dim (A^∨)^C, l.1979). THIRD CLAUSE
(two-part provenance): the per-λ cover pinning is the local half of
lem:obstructionseparation (l.4219–4238) — functoriality of the
embedding-problem obstruction under the pushout R ↠ R/ker λ plus the
Hoechsmann vanishing criterion for the resulting 𝔽₂-kernel problem
(NSW (3.5.9) + (3.5.11); K. Hoechsmann, Zum Einbettungsproblem,
J. reine angew. Math. 229 (1968) 81–106), derived by the manuscript
from prop:localduality + lem:collapse in five lines (l.4224–4237).
The cover data (ker λ normal, B_λ framed, R/ker λ central of order
2) is IN-TREE PROVEN (Lifting/ScalarPushout), so the clause's
arithmetic content is exactly the duality pairing. See BLOCKR_P1_DESIGN
§5.2.
FOURTH CLAUSE (d) — the weak-base count (audit/BLOCKR_P8_DESIGN.md
§14-S8.1, graded audit/CITABILITY_AUDIT_20260708.md §G1): the
twisted-cocycle count |Z¹| = |N|²·|X_g| at the base induced by an
ARBITRARY continuous hom h : G_{ℚ₂} → Y (weak lifts included; no
surjectivity), with X_g := mapInvDuals E h the g-INTRINSIC
invariant-dual space of the induced module — the SAME
NSW (7.2.6) + (7.3.1) content: both theorems quantify over arbitrary
finite G_{ℚ₂}-modules, and the module here is N with the action
induced through h; dim Z¹ = dim B¹ + dim H¹ = (dim N − h⁰) + (h⁰ + h² + 2·dim N − dim N) = 2·dim N + dim (N^∨)^{G_{ℚ₂}} by (7.3.1)
with h² = dim (N^∨)^{G_{ℚ₂}} by (7.2.6). conjInvDuals is
deliberately NOT used here (§G1 refutation guard).
The pinned local obstruction (stable choice on the keep).
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Clause (d) projection — the weak-base twisted-cocycle count
(o-free; the frame argument only selects the keep instance).
prop:localduality at vanishing invariants (the historical keep
statement, now DERIVED from the general form): with no nonzero invariant
duals the obstruction space is trivial, so framed lifts always exist and
the count degenerates to |N|².
The full mstage cochain statement from sharp inputs: the exact
four-clause form of MStage.sec7_mstage_cochainData, derived from the
radical hit (GammaA half) and the invariant-dual vanishing (GQ2 half,
through the canonical local-duality keep).