The Geometry of the Critical Line: A Reader's Map (Sixth Edition)

Sixth Edition of the Reader's Map for the "Geometry of the Critical Line" programme (May 2026). This is the navigation guide to a 51-paper, 52-research-note open-access programme that investigates the spectral geometry of the Riemann critical line through the SCT 5-manifold, the chiral spectral obstruction, the connection matrix, the asymptotic Evans zero law, the assembled prime-side Weil kernel, and — in the Phase VII reset that defines this edition — the spatial sterility of the σ-bubble carrier, the closed scalar Weil evaluator, and the restricted Weil positivity sign problem. The programme proves five independent characterisations of the critical line σ = 1/2, isolates a chiral spectral obstruction in the Friedrichs domain, derives an asymptotic Evans zero law for the connection-matrix entry M₂₁, assembles all five factors of the Weil prime-side kernel from the SCT spectral geometry (Paper 47), reduces the trace-formula problem to two analytic inputs (Paper 48), proves the σ-bubble carrier is spatially sterile (Paper 49), establishes rigidity and no-go pruning around the resulting Arithmetic Lift Conjecture (Paper 50, RN50), closes the scalar Weil evaluator analytically (RN52), proves restricted Weil positivity at the prime-free boundary a₂ = ½ log 2 with a local extension into the first prime-active window (RN51), and develops the finite-window continuation framework for the restricted Weil minimum m (a) (Paper 51). No claim is made that the Riemann Hypothesis is proved or disproved. The published frontier is the global sign of m (a) — bidirectional sign infrastructure that, if resolved, would yield either resolution. This edition supersedes the Fifth Edition's Paper 48 endpoint/KMS frontier framing. Paper 49 proves the raw σ-bubble carrier is exactly flat after gauge elimination and Liouville change of variables, with spectrum λ₌, ₍ = m² + (nπ/4) ² — so the nontrivial arithmetic information cannot live in raw carrier eigenvalues and must enter through an evaluator, arithmetic lift, or Weil-positivity mechanism. The earlier two-input endpoint/KMS frame is replaced, within the published corpus, by the single Arithmetic Lift Conjecture; the live published frontier moves to the sign of m (a). New in this edition. Phase VII added: Spatial Sterility, Scalar Evaluation, and Restricted Weil Positivity (Papers 49–51, RN50–RN52). The Phase VII reset reframes the published boundary at the scalar layer and at the finite-window sign problem. Paper 49. Proves the Gauge Elimination Theorem and the Spatial Sterility Theorem: in every winding sector, the σ-bubble operator is unitarily equivalent to −d²/dx² + m² on the flat interval −2, 2 with Dirichlet boundary conditions. The reduced spectral zeta is Z_σ (u) = (4/π) ^2u ζ (2u). This replaces Paper 48's endpoint/KMS two-input frame by the Arithmetic Lift Conjecture. Paper 50 and RN50. Establish arithmetic-lift governance: rigidity (uniqueness of any admissible extension, if it exists) and no-go pruning (angular-only/Hecke-only factorisations, spectral-diagonal lifts, and post-expectation derivations are ruled out). Any successful prime-part evaluator must be non-diagonal, displacement-sensitive, and logarithmic-derivative-like. RN52. Closes the scalar Weil evaluator analytically. The reduced scalar observable produced by sterility is fixed by Jacobi reduction; the log-derivative decomposition of the completed reduced zeta produces endpoint, archimedean, and prime terms as a closed scalar theorem — without invoking a crossed-product algebra. RN51 and Paper 51. RN51 proves restricted archimedean positivity at the prime-free boundary a₂ = ½ log 2 (via Slepian concentration), with a nonquantitative local extension into the first prime-active window. Paper 51 then develops the general finite-window continuation framework for m (a): threshold turn-on, compactness, exact minimisers, continuity, the self-adjoint family Kₐ with compact resolvent, the first-touch/scalar-obstruction theorem, and renormalised central-mode root stiffness. Epistemic symmetry framing. Global positivity m (a) > 0 for all a > 0 implies RH via the Weil-positivity route; a constructive certified parameter a* with m (a*) < 0 implies its negation in the same framework. The published corpus establishes neither and supplies the bidirectional machinery for studying both. Dictionary-entry table updated. 9 entries proved (RN43, RN45, Papers 47–48), 1 closed at the reduced scalar layer (endpoint values h (±i/2) via Paper 49 + RN52), 1 reframed via the Arithmetic Lift (KMS/sector normalisation, Papers 49–50), and 1 open at the operator-valued internal-realisation layer (Paper 50, RN50). New "Positive Claim" section. Makes the m (a) -sign frontier explicit, alongside what global positivity or a constructive negative witness would mean. Falsification record extended. Raw carrier zero-identification is retired (Paper 49), naive arithmetic packaging routes are pruned (Paper 50, RN50), and Paper 48's endpoint/KMS framing is reframed at the scalar level rather than serving as the live frontier. Complete paper inventory extended to all 51 papers and 52 research notes with DOIs. The programme lies at the intersection of spectral geometry, analytic number theory, non-self-adjoint ODE theory, complex dynamics, and operator-algebraic models inspired by the Connes/Bost–Connes framework. Part of a 51-paper open-access programme on the geometry of the Riemann zeta function's critical line, anchored by the SCT 5-manifold and the cover equation Φ + e^iπ − 1/Φ = 0.