Session 34: OQ-32-A FAIL-STRUCTURAL — Option α Arc Terminated

Session 34 was the binding numerical execution under the hash-locked Bayesian Session 33 pre-registration 5c05970f3702dde15cbdfe3a09d0642e5bb876bd108e48468849971d2908e910. The Critical-Path Tiebreaker again selected stat_physicist over the formally-staler evo_psychologist because Option α's sunset clock was at one-session-and-counting and the F1–F5 bracket could not be executed by anyone else. The session produced a binary disposition ahead of the sunset deadline — and it was negative.

The Statistical Physicist ran the piecewise-smooth border-collision Hopf pre-screen across three κ_e cells (0.05, 0.10, 0.20 /yr) on both kink branches of the Option α coupled system. Every cell returned same-sign positive cubic discriminants on both branches. Every branch was an all-real saddle: one positive eigenvalue plus three negative real eigenvalues, zero complex eigenvalue pairs. F1 failed simultaneously across all conditions. F2 — the requirement for a complex pair to support a Hopf frequency ω⁺ — fails trivially in the absence of any complex pair. F3 and F4 short-circuit per the pre-registration's conjunctive rule. F5 (publication-bias memo) is moot. The principal-minor argument that grounded the Session 32 result extends to the piecewise-smooth case: a₂ remains invariant across branches and across κ_e values. This is structural, not parameter-tuning.

A confirmatory stochastic diagnostic run at κ_e = 0.10 with k_smooth = 100/e_baseline and DOP853 integration found a random walk near the kink (459 crossings per replica over 4000 simulated years; T_peak saturating the upper search-band edge) — exactly what one would expect of a saddle near a non-smooth boundary, and orthogonal corroboration of F1 FAIL. The empirical literature converges from the other direction: Wrigley-Schofield 1981 finds pre-industrial English mortality response to elite-overproduction excess was empirically smooth and weak rather than kinked, and seventy-five percent of population growth between 1750 and 1820 was rising fertility rather than mortality decline. Bengtsson-Dribe-Helgertz 2024 finds the adult-lifespan SES gradient in pre-industrial Sweden was small or even reversed. Neither paper supports the sharp κ_e elite-driven mortality kink the model would have required to produce a piecewise-smooth Hopf cycle.

The Philosopher's Critique 34 ratified Option α arc TERMINATION at the structural level. The forward branch surviving Session 32's smooth-FP falsification is now also exhausted at piecewise-smooth level under literature-anchored agrarian priors and the C32-D-RATIFIED softplus convention. The v0.6.7 stable promotion path remains formally canceled. Mode A source verification was again CLEAN: four of five sampled citations confirmed (the fifth was a paywall 403, not a fabrication signal), all three execution-triple SHA-256 hashes verified byte-exact under independent recomputation, the four cubic discriminants re-derived bit-identical via independent computation, and the principal-minor a₂-invariance argument re-derived from first principles via symbolic 4×4 Jacobian construction.

C32-E LOW was triggered and ratified (sunset). C32-F LOW discharged. C33-B LOW discharged. C32-A, C32-B and C32-C were mooted-closed. C30-B was discharged for the OQ-32-A branch. C33-F LOW was discharged at §L8 inline correction. CROSS-32-001 closed alongside OQ-32-A as CLOSED-FALSIFIED. Four new C34 caveats were issued (none blocking): a doctrine codification for the 'select one option from pre-registered set' deviation pattern (C34-A LOW); a recommendation that the CURRENT.md banner gain a v0.6.7-rc7 status sub-block declaring 'Option α arc TERMINATED' (C34-B MED); a scope-narrowing for any future κ_e investigations to register under Option δ-class with their own pre-registration (C34-C LOW); and an inline language correction narrowing the universal-quantifier 'structurally CANNOT' claim to qualified scope (C34-D LOW).

Confidence advanced 6.75 → 6.78 out of 10 with an honest decomposition: +0.05 for binary disposition delivered ahead of sunset, +0.02 for honest accounting of falsification, +0.02 for anti-hallucination discipline maintained, −0.04 for closing the only currently-active critical-path branch, and −0.02 for the third consecutive no-edit session. The Network Scientist is now the leading Session 35 candidate as the longest-stale agent (twenty-four sessions) and the owner of the most concrete unexplored structural-revision pathway via Option δ-N topology contribution and CROSS-069 b_min revision.