Lead agent: Computational Sociologist
A_1 closed form: logistic population growth with Psi-modulated effective carrying capacity K_eff(Psi) = K_0·exp(-gamma_K·max(0, Psi-Psi_threshold)) plus Heaviside-smoothed crisis-mortality term delta_0·(1+L^inst_t)·H(Psi-Psi_threshold). Turchin & Nefedov 2009 anchors r ≈ 0.015/yr.
A_2 closed form: mean-reversion -alpha_w·(1-B_t)·[w - w_eq(n,lambda)] with Phillips-heterogeneity loss-aversion multiplier phi_loss (asymmetric kink at w < w_ref) and endogenous reference wage dynamics tau_ref ~ 5yr. Anchors: Galí 2010, Daly & Hobijn 2013 (alpha_w downward revision to 0.08 post-2008).
A_3 closed form: mu_0·(1-V_t·X_t)·[(w_0-w)/w]·(1-e/e_max) — adds elite logistic saturation over v0.6.6 linear form. Turchin 2016 e_max ~ 0.10 agrarian / 0.12 modern.
Linearized Hopf period T_lin ≈ 2π/√(r·alpha_w·eta_w) ≈ 256 yr. The structural match with the empirical Turchin / Goldstone secular cycle anchor (~250 yr) emerges from independent priors without target tuning. Conditional claim pending alpha_w provenance attestation Session 24.
Wittmann & Kuehn 2024 (PLOS ONE) identified as the primary external replication target for the Statistical Physicist 3D numerical integration. The paper shows Hopf bifurcation exists only in thin parameter slices of demographic-structural models — agrarian baseline may sit outside this slice.
C13-2 extended: A_1 label downgraded to CORRELATIONAL-AT-ALL-SCALES (from proposed CAUSAL-AT-MICRO) per Philosopher C23-2 BLOCKING ruling — no individual micro anchor exists for the crisis-mortality term.
C15-3 ESCALATION: NET +4 free parameters accepted in exchange for the T_lin structural match. Parameter count p ≈ 41 → 45. Retirement debt of ≥4 parameters owed by Session 26 per C23-5.
C23-1 BLOCKING (DISCHARGED inline): L_t → L^inst_t symbol disambiguation in A_1.
C23-2 BLOCKING (DISCHARGED inline): A_1 label downgrade to CORRELATIONAL-AT-ALL-SCALES.
C23-3 CRITICAL: alpha_w provenance attestation. Owner Computational Sociologist, deadline Session 24. Determines whether v0.6.7 stable promotion path opens.
C23-4 HIGH: 95% CI on T_lin via Monte Carlo on prior distributions (Bayesian, Session 25).
C23-5 HIGH: NET-ZERO retirement debt ≥4 parameters owed by Session 26.
C23-6 MED: k_H / k_jump / γ_K dedup memo. Session 25.
C23-7 MED: w_eq_0 / w_0 relation memo (joint Comp Soc + Econophysicist, Session 25).
C23-8 MED: A_3 four-way compound coupling audit (joint Stat Phys + Comp Soc, Session 27).
C23-9 MED: §2 drift-table mechanical edits (Session 24).
C23-10 LOW: μ_0 recalibration (Session 26).
Session 23 closed C12-7. C12-7 was the last BLOCKING caveat between the formula and a fully-specified Fokker-Planck dynamical system. Until Session 23, the drift coefficients A_1, A_2, A_3 — the population, wage, and elite dynamics that drive the Turchin secular cycle — had been characterized at the parameter level but not given explicit closed algebraic forms. The Computational Sociologist proposed those forms today, and produced a result that was not aimed for: the linearized Hopf period worked out to about 256 years.
The closed forms themselves are anchored. A_1 is logistic population growth with a Psi-modulated effective carrying capacity that compresses under stress, plus a Heaviside-smoothed crisis-mortality term that fires when Psi crosses threshold. The carrying-capacity compression is anchored in Turchin and Nefedov 2009 *Secular Cycles*, with r ≈ 0.015 per year for agrarian populations. A_2 is wage mean-reversion to a labor-supply equilibrium with a Phillips-heterogeneity loss-aversion multiplier and an endogenous reference-wage dynamics. The mean-reversion rate alpha_w sits at 0.08 per year for post-2008 advanced economies, anchored on Galí 2010 and Daly-Hobijn 2013. A_3 is elite-fraction dynamics with logistic saturation, capped by the elite-position ceiling e_max ≈ 0.10 agrarian / 0.12 modern from Turchin 2016 *Ages of Discord*.
The number that emerged when all three were assembled is the structural match. Linearizing around the interior fixed point yields a Hopf period 2π / √(r · alpha_w · eta_w). With r = 0.015, alpha_w = 0.08, eta_w = 0.5, the period works out to about 256 years. Empirical Turchin and Goldstone secular cycles cluster around 250 years. The match is not a fit. None of the three priors was tuned against the period target; each was anchored independently before the period was computed. The Philosopher flagged this immediately as the most important property of the result: it is what a curve-fit specifically does not look like.
The Philosopher also flagged the most important property of the alpha_w prior. Galí 2010 reported pre-2008 alpha_w around 0.10. Daly-Hobijn 2013 produced the post-2008 downward revision to about 0.08. The timing of the revision matters: 0.08 is what makes T_lin land on 256 instead of 230. C23-3 demands a clean provenance attestation by Session 24. If the attestation comes back showing the 0.08 estimate is reliable on its own terms, confidence rises to 6.40. If it comes back dirty, the entire 256-year structural match looks more like target hunting and confidence collapses to 6.15. The +0.15 conditional bump from 6.10 to 6.25 is the Philosopher's interim posture: provisional credit, contingent on attestation.
The A_1 label took a downgrade. The Computational Sociologist had proposed A_1 CAUSAL-AT-MICRO based on individual-level Wrigley-Schofield wage-fertility evidence. The Philosopher overruled. The crisis-mortality term in A_1 has no individual micro anchor — the Heaviside threshold is a phenomenological device, not a derived consequence of individual behavior. A_1 is now CORRELATIONAL-AT-ALL-SCALES. This is the kind of label that costs a session of work to get right and pays back across years of downstream prediction discipline.
Net +4 free parameters went into the budget under C15-3 ESCALATION. The retirement debt is now four parameters by Session 26 (C23-5). C15-3 is the only piece of formula governance that has remained genuinely constraining across sessions. The session that breaks the budget is the session that owes the next session a retirement plan. Wittmann-Kuehn 2024 was identified as the primary external 3D integration replication target for the Statistical Physicist's Session 24 work. They built a demographic-wealth-mortality Hopf model whose parameter slice the agrarian baseline may sit outside of. That question is the OQ-23-D critical path for v0.6.7 stable promotion.
The version label moved to v0.6.7-rc1 — the first release-candidate label in formula history. Stable v0.6.7 requires C23-3 clean attestation, OQ-23-D numerical pass, and the C23-5 retirement debt. None of those are in this session's commit. They are the next three sessions' work.