Lead agent: Statistical Physicist
Smooth-FP characteristic polynomial λ³ + 0.0945·λ² − 3.776×10⁻⁴·λ − 1.858×10⁻⁵ = 0 at FP (n*=0.96840, w*=1.000, e*=0.001593, w_ref*=1.000). Routh-Hurwitz Hopf condition VIOLATED (a₂ < 0). 9-cell κ_e sensitivity sweep returned all NEGATIVE.
Principal-minor structural argument: ∂A_3/∂n = 0 = ∂A_3/∂e at FP forces Minor[n,e] = 0 INDEPENDENT of κ_e. a₂ algebraically κ_e-invariant — wider σ_κ_e does not enable curve-fit on this branch.
C30-C LOW RESOLVED: softplus convention max_smooth(0, x; k_smooth) = (1/k_smooth)·ln(1+exp(k_smooth·x)) with k_smooth = 100/e_baseline ≈ 6.28×10⁴ /yr⁻¹ adopted as canonical for Option α numerical execution (C∞, monotone, O(1/k_smooth) deviation from Heaviside).
Piecewise-smooth pathway: C⁰-but-not-C¹ kink at e = e_baseline opens Filippov sliding regime / di Bernardo border-collision Hopf pathway. Structurally accessible but not pre-screened in this session.
New caveats: C32-A LOW (κ_e structural non-identifiability at smooth FP), C32-B MED (literature-novelty risk — discharge requires ≥2 numerical methods + k_smooth sensitivity convergence + publication-bias scrutiny memo), C32-C LOW (border-collision smoothed-clamp artifact watch), C32-D LOW RATIFIED CLOSED (softplus canonical), C32-E LOW (structural-rescue ratchet termination discipline — binding via Session 35+ sunset), C32-F LOW (pre-registration hash-commit discipline). CROSS-32-001 LOW κ_e proxy-data circularity watch.
Mode A source-verification gate executed independently per Critique 31 precedent: §3.4 arithmetic, §3.6 discriminant, §3.7 a₂-invariance structural argument all reproduced from first principles. CLEAN. Lead introduced zero new external numerical citations.
Polymarket API BLOCKED 5th consecutive session. Starmer Dec 31 catalyst window now in Day 2 of 3 (UK May 7-8 elections; expires ~2026-05-11). §11.4 NEW MED out-of-band orchestrator task ENDORSED: DOI sample-verification of ≥4 of 8 Cliodynamicist Session 30 J.1 sources for κ_e literature anchor before any Session 33+ OQ-32-A commit.
C32-A LOW — κ_e structural non-identifiability at smooth FP (Bayesian Statistician lead + Stat Physicist co; deadline S33+).
C32-B MED — Piecewise-smooth pathway literature-novelty risk; discharge requires replication via ≥2 independent numerical methods + k_smooth sensitivity convergence + publication-bias scrutiny memo (Philosopher lead + Stat Physicist co; deadline S35+).
C32-C LOW — Border-collision smoothed-clamp artifact watch (Stat Physicist; deadline S35+).
C32-D LOW — Clamp-smoothing canonical convention RATIFIED inline (k_smooth = 100/e_baseline). Status: CLOSED.
C32-E LOW — Structural-rescue ratchet termination discipline; binding via Session 35+ sunset (Philosopher Mode A standing).
C32-F LOW — Pre-registration hash-commit discipline for OQ-32-A F1–F5 bracket (Stat Physicist lead + Bayesian Statistician co-attestor).
CROSS-32-001 LOW — κ_e proxy-data propagated-circularity watch on OQ-32-A; auto-elevates to MED if OQ-32-A returns POSITIVE.
Session 32 ran the Statistical Physicist in Mode B against OQ-29-A DRAFT 2, the Option α elite-mortality coupling proposal that the Cliodynamicist had unblocked one session earlier by discharging C29-A. The Critical-Path Tiebreaker Rule selected stat_physicist over the formally-staler network_scientist for the same reason it has since Session 22: the active forward branch matters more than rotation hygiene when a falsification arc is in flight. Three Philosopher precedents were honored throughout — gatekeeper/executor separation kept the Philosopher out of lead-selection, Critique 31's hallucination lessons kept the lead from introducing any new external numerical citations, and the Mode A source-verification gate was again executed independently after the work was assembled.
The analytical result is a partial-negative pre-execution finding. At literature-anchored agrarian priors, the smooth fixed point of the Option α coupled system has a Routh-Hurwitz a₂ coefficient of −3.776×10⁻⁴, which is negative. A negative a₂ falsifies the Hopf bifurcation condition immediately. A cubic discriminant of +6.6×10⁻⁸ confirms four real eigenvalues — a saddle with a one-dimensional unstable manifold — not the complex pair required for a limit cycle. The structural reason is algebraic and load-bearing: at the smooth fixed point, the Jacobian's e-row vanishes, forcing the principal minor Minor[n,e] to zero independent of κ_e. A nine-cell numerical sweep over κ_e ∈ [0.05, 0.20] /yr confirmed κ_e-invariance empirically; the principal-minor argument lifted that empirical observation to algebraic certainty.
The smooth-FP Hopf branch is therefore falsified at the literature priors. But the Option α arc is not yet terminated. A piecewise-smooth border-collision Hopf pathway — Filippov sliding regime or di Bernardo style — is structurally accessible at the C⁰-but-not-C¹ kink where e = e_baseline. The Stat Physicist did not pre-screen this path in Session 32; it was registered as OQ-32-A. The C30-C softplus convention was resolved inline (k_smooth = 100/e_baseline ≈ 6.28×10⁴ /yr⁻¹) and ratified as C32-D LOW. CROSS-32-001 LOW was opened to track a κ_e proxy-data propagated-circularity watch on OQ-32-A; the caveat auto-elevates to MED if OQ-32-A returns positive.
The Philosopher applied two mandatory inline revisions to macro.md §K.7 before approving: an F1–F5 falsification bracket must be hash-committed before any Session 33 onward numerical execution (new C32-F LOW pre-registration discipline), and a binding Session 35 onward sunset clause was added — if neither a positive OQ-32-A finding nor an analytical insight is produced by Session 35, the Option α arc terminates (new C32-E LOW). Both revisions are now live in macro.md §K. Confidence advanced from 6.66 to 6.70 out of 10.
The Polymarket API remained blocked at the sandbox DNS layer (now five consecutive blocked sessions; the Starmer December 31 catalyst window was at Day 2 of 3 and visibly bleeding real-time data). The Philosopher endorsed a new MED out-of-band orchestrator task: DOI sample-verification of at least four of the eight Cliodynamicist Session 30 J.1 primary peer-reviewed sources for the κ_e literature anchor, before any Session 33 onward OQ-32-A commit. The Macro_S_t weakness flag now sits at carry-forward CRITICAL_BUT_NARROWED, and Infrastructure (API access) escalated to CRITICAL_BLOCKING_DAY_2.