A Dynamical Model of
Social Functional Necrosis

ABSTRACT

This paper expresses the “Social Functional Necrosis” pathology proposed by the main paper as a system of nine ordinary differential equations, with eight state variables: master stock F, apprentice pipeline A, perceived value V, buffer reserve W, blood supply B, transplant capacity S, normalization reference h_ref, and reconstruction share φ. The core revision in V2 is budget conservation: crisis funds are endogenously allocated between “domestic reconstruction” and “outsourced transplantation” via E9, with allocation weights discounted by the same visibility bias β — reconstruction returns are delayed by T_train training cycles while transplantation delivers immediately, so that high β commits a double murder: starving the domestic pipeline during the silent period, then diverting emergency funds to substitutes during crises (at baseline, only 23% of crisis budget flows to reconstruction). Under the conservation constraint, all core conclusions hold and are cleaner: the sufficient condition for necrosis remains high β × high σ (at β = 3.2, σ = 0 yields long-run functionality of 0.99 while σ = 0.8 yields zero); the five-stage disease course reaches complete terminal state for the first time in the conservation version (F → 0 while service coverage is 96%). V2 adds four tests and honestly reports two downgrades: (1) the multi-organ extension shows that even with intact nociceptive pathways, the loudest patient during staggered crises monopolizes the attention budget, silently starving a quiet organ to death (5 of 6 organs survive), while synchronous crises rescue all but disperse terminal states to 78%–399% — the “surplus and shortage coexisting” sign spontaneously emerges at the organ level, and proposition P1 is accordingly downgraded from a theorem to a single-organ conditional proposition; (2) a 40-realization early-warning ensemble shows only 55% significant trends, median Kendall τ ≈ 0.03, and threshold alarms never trigger — V1’s optimism based on a single realization is overturned, and P6 is downgraded to an open question; (3) Korean calibration is supplemented with leave-one-out cross-validation (LOOCV = 12.1 pp, vs. 5.1 pp in-sample) and a three-parameter parsimonious model (5.6 pp), incorporating out-of-sample facts: 2025 second-half pediatric staffing fill rate of 17.4%, but the February 2024 “physician–government conflict” mass resident resignations constitute a system-level common shock outside the model’s domain, confounding the out-of-sample test — this shock itself is precisely the real-world footnote to the multi-organ extension; (4) sensitivity analysis confirms topological conclusions are robust: under ±30% parameter perturbation, the fold point moves within 1.18–2.40 but always exists, and the wall of no return is stable at a remaining capacity of 5.5–9.1%. This paper accordingly distinguishes explicitly between two classes of conclusions: topological facts (fold, hysteresis, wall, divergence, double murder) are robust to parameters; specific numerical values (β_c = 1.73, wall = 9.1%, etc.) are merely order-of-magnitude illustrations at baseline parameters. The genealogical revision adds four absent ancestors — Bastiat (1850), Baumol (1967), Hirschman (1970), and Repenning–Sterman (2001) — and delineates the net increments relative to the “capability trap” model.

Why Equip Pathology with Equations

The diagnostic instrument in Appendix A of the main paper is entirely composed of disguised forms of quantitative propositions. Without equations, the promise of falsifiability cannot be honored. But formalization must first establish discipline — Romer warned about “mathiness”: formulas degenerating into rhetoric is worse than having no formulas at all. The mathematics of this paper undertakes only three tasks: turning diagnostic criteria into estimable parameters; turning early-warning indicators into computable thresholds; equipping the theory with interfaces that can be attacked. V2 adds a fourth discipline: explicitly distinguishing topological conclusions from numerical conclusions — the former (existence of bifurcation, existence of the wall, asymmetric divergence) are robust to parameter perturbation; the latter (fold-point location, wall percentage) are merely order-of-magnitude illustrations at baseline parameters, and the abstract and body text uniformly follow this wording.

Every component of the model has a formal ancestor, the closest in lineage being one overlooked in V1: Repenning and Sterman’s (2001) “capability trap” — the paper title is itself “Nobody Ever Gets Credit for Fixing Problems That Never Happened.” The genealogy table in Chapter 2 claims each item by lineage; the demarcation table in Chapter 3 describes this model’s five net increments relative to the capability trap. The deeper genealogy traces upward to Bastiat (1850), What Is Seen and What Is Not Seen: the true progenitor of counterfactual invisibility.

The Model: Eight State Variables
and Nine Equations

2.1 State Variables

2.2 The System of Equations

E9 is the core revision of V2, responding to reviewer comment four (accounting inconsistency). The crisis budget is allocated between two “products”: domestic reconstruction’s visible returns are delayed by T_train, while transplantation delivers immediately; visibility bias β enters the weight w_R as a discount rate for delayed returns. Demand gating g ensures that substitutes compete only when a gap exists — with no gap, φ = 1, and the model is pointwise consistent with V1 on the silent branch. This yields a unification V1 left unstated: the same β starves the pipeline during the silent period (via the share competition of E8) and diverts emergency funds to transplantation during crises (via the delay discounting of E9). Necrosis requires not two etiological agents; a single myopia parameter commits a double murder.

2.3 Theoretical Lineage of Each Component

2.4　Demarcation from the “Capability Trap”

Repenning–Sterman (2001) characterized the same lesion at the enterprise scale: resources flow to visible firefighting, capability investment is starved, and repair requires “getting worse before getting better.”
This model’s five net increments: First, scale and medium — aggregating blood flow into a social composite flow of attention–talent–funding–prestige; Second,
irreversibility — the capability trap can escape via “getting worse before getting better,” whereas this model’s Allee effect yields a genuine fold, infinite hysteresis, and a wall of no return;
Third, transplant channel — no substitute market exists inside the capability trap, while E3/E9 capture how outsourcing simultaneously relieves pain and diverts flow; Fourth, normalization dynamics
h_ref; Fifth, a case spectrum spanning two millennia and the resulting invariance claim. Wherever this model overlaps with the capability trap, priority belongs to them.

2.5　Dimensionless Governing Groups

Non-dimensionalizing with T_train as the time primitive, the dynamics are governed by five groups: Π₁ = T_train/T_adapt = 2.0 — the race between reconstruction and normalization;
the larger Π₁, the more likely crisis funds are drained by “incidents becoming weather” before the pipeline delivers; Π₂ = θ·c = 1.5 — the product of institutional lag and social forgetting,
determining the effective width of the crisis-funding pulse; Π₃ = δ·T_train = 0.48 — nearly half the stock is lost per training cycle, requiring the pipeline to
over-recruit just to break even; Π₄ = q = 2.5 — burnout gain; Π₅ = σ·e^(β·T_train/T_H) — the relative attractiveness of transplantation;
β and σ multiply within this group, which is precisely the analytical reason why the red zone in the phase portrait (Figure 3) occupies the upper-right corner.

The Model’s Discovery History:
Four Rounds of Revision

This chapter reports failed versions as formal results. Round 1: In the pipeline-free model, money can instantly become masters; regeneration takes only 1.5 years and asymmetry vanishes
— lesson: the physical carrier of irreversibility is the training pipeline’s time constant. Round 2: After adding the pipeline and burnout, rescue delayed by 24 years still
succeeds — lesson: as long as the “incident → perception → appropriation” pathway is intact, endogenous response can always backstop; a single organ with intact nociception will not undergo necrosis, only
oscillation. Round 3: Normalization weakening the signal is still insufficient; the answer lies in Byzantium 1082 — crisis funds flow to Venice rather than the domestic fleet; transplantation is
the analgesic, and necrosis is achieved after adding σ. Round 4 (V2): Hostile reviewers pointed out that E1 and E3 draw on B simultaneously without conservation — transplant funds
are not truly deducted from the reconstruction account. E9 repairs the accounting with gated budget allocation, and unexpectedly yields theoretical unification: β’s double murder. All conclusions
are verified as unchanged under the conservation constraint (Figure 1, Figure 3), and the five-stage disease course reaches complete terminal state for the first time.

Results

4.1　Five-stage disease course（Conservation version）

Figure 1.　Top: β → 3.2, σ = 0.8. Buffer masks until year 37.0; at symptom onset, capacity has already fallen to 51.5%. After crisis
blood supply arrives, E9 under myopic discounting allocates only 23% to domestic reconstruction and 77% to purchasing transplants; S rises to 0.96 to take over the gap, nociception vanishes,
the reconstruction share is consequently exhausted, and F drops completely to zero — the conservation version achieves, for the first time, the literal meaning of Stage V: the organism dies (F = 0) while service coverage is 96%.
Bottom: σ = 0 control (φ ≡ 1, pointwise consistent with V1); prevention-paradox limit cycle period ≈ 55 years.

4.2　Fold Bifurcation and Hysteresis

Figure 2.　Silent-branch fold point β_c = 1.73 (baseline parameters); downward sweep never returns (hysteresis = ∞).
V2 sensitivity (Appendix D): under ±30% parameter perturbation, β_c moves within 1.18–2.40 but always exists — the fold is a topological fact; 1.73 is merely an
order-of-magnitude illustration. Restoring the measurement environment ≠ restoring the function.

4.3　Fate Phase Portrait: Conservation Invariance

Figure 3.　β × σ phase portrait re-computed under budget conservation. Core conclusions unchanged: at β = 3.2, σ = 0 yields long-run functionality
0.99, while σ = 0.8 yields zero. Sufficient condition for necrosis = high β × high σ withstands the accounting correction — this is V2’s most important robustness test.
Case positioning unchanged: Byzantium 1082 and U.S. shipbuilding 2020s in upper-right; French nuclear power in lower-right (σ ≈ 0, forced into painful but successful reconstruction).

4.4　Critical Slowing-Down: Ensemble Verdict (V1 Conclusion Overturned)

Figure 4.　V1’s single realization (seed 11) showed a significant trend (τ = 0.21, p = 0.007). V2’s 40-realization
ensemble delivers a severe verdict: only 55% of realizations are significant at p < 0.05 (vs. the null hypothesis of 5% — signal exists but is weak), median τ ≈ 0.03,
and threshold-type alarms never stably trigger. V1’s optimism about early-warning reliability is overturned; Proposition P6 is accordingly downgraded to an open question. Consistent with
the EWS literature: a significant trend from a single realization cannot be extrapolated; slow-variable proxies (pipeline flow, see P7) may be more reliable early warnings than statistical precursors.

4.5　Regeneration Asymmetry and the Wall

Figure 5.　Analgesic-masked scenario: collapse (symptom onset → 30%) takes 5.8 years; regeneration takes 11.4 years @ 30%, 15.5 years @ 20%,
28.2 years @ 12% — super-linear divergence; wall = 9.1%. V2 sensitivity: under ±30% perturbation of δ and q, the wall moves within 5.5%–9.1% but always exists.
The main paper’s diagnostic criterion IV (“≥ 3×”) is accordingly upgraded to a divergence formulation.

4.6　Korean Calibration: Cross-Validation and Out-of-Sample

Figure 6. Five-parameter fit RMSE = 5.1 pp; parsimonious three-parameter model with γ = 3, c = 0.5 yields 5.6 pp — the two elastic parameters do not carry the main fitting load. Leave-one-out cross-validation LOOCV = 12.1 pp, honestly reporting the in-sample/out-of-sample gap (prediction error over an 84-percentage-point range). Out-of-sample: 2025 second-half pediatric staffing fill rate of 17.4% (141 trainees); however, the February 2024 “physician–government conflict” triggered mass resident resignations, constituting a system-level common shock outside the single-organ model’s domain — out-of-sample testing is therefore confounded, unable to adjudicate V1’s recovery prediction. This shock itself is precisely the real-world footnote to the §4.7 multi-organ extension.

4.7　Multi-organ extension：Nociceptive crowding

Figure 7. Six isomorphic organs sharing an attention budget (softmax allocation), all nociceptive pathways intact, σ = 0. Scenario A (staggered crises): 5/6 survive — in each single-onset crisis, the loudest patient monopolizes nearly all budget through softmax, while the remaining organs are silently cut off; accumulated starvation pushes one organ that never cried for help past the threshold to death. Scenario B (synchronous crises): 6/6 survive but terminal states disperse to 78%–399% — the price of universal rescue is allocation deformity: the winner of the attention contest over-rebuilds by nearly fourfold, while losers suffer chronic shortage. The main paper’s late-stage sign of “shortage and surplus coexisting” spontaneously emerges at the organ level. Both scenarios jointly falsify the universal form of P1: intact nociception is no longer sufficient under multi-organ competition — the 2024 South Korean physician–government conflict (a full-specialty synchronous shock) is precisely the real-world projection of Scenario B.

Propositions, Revised

Policy Knobs and Ex-Ante Measurement

Five parameter actions. Lower β: book-keep “nothing happened” — counterfactual accounting, the only knob that acts on etiology. Lengthen T_adapt: institutionalize incident memory. Shorten θ: automatic stabilizers linked to flow metrics. Lengthen T_H: extending the decision horizon directly increases the discounted weight of reconstruction in E9 — reforms of tenure systems and quarterly reporting are exactly this term in the equation. Constrain the use of σ: outsourcing must be tied to domestic pipeline reconstruction clauses; otherwise, pain relief becomes euthanasia.

In response to reviewer comment seven, β and σ must be independently measurable ex ante; otherwise P2 is circular. Proxies for β: budget review cycle length, listed-company quarterly-report coverage, news-attention half-life (directly estimable from media data), electoral cycle length. Proxies for σ: import penetration rate of the function’s output, number of qualified external suppliers, output tradability index. Classifying cases ex ante by proxy variables and statistically measuring necrosis rates ex post constitutes a non-circular test design for P2. On the economic channel, Baumol’s cost disease (1967) — the long-run rise in the relative price of maintenance labor — is an unmodeled real driver in this model, equivalent to slow drift in D or 1/ρ, and is listed for the next version’s extension.

Limitations

Listed item by item, without embellishment: the early-warning claim has been weakened by its own ensemble test (P6′); the multi-organ extension is preliminary (isomorphic organs, softmax allocation, no inter-organ functional coupling), insufficient to support a complete triage theory; β, σ, T_H are exogenous; calibration uses only six points and β₁ hits the boundary, with a LOOCV gap of 12.1 pp; the Baumol channel is absent; the mean field lacks network percolation, and “last-supplier” singularization remains uncharacterizable; explicit Euler numerics. These constitute the task list for V3.

Conclusion:
Civilizations Die of Their Own Analgesics

The deepest conclusion of the nine-equation system becomes even colder in V2: the death of a maintenance system requires no malice, nor even two etiological agents — a single myopia parameter commits two murders, three analgesics (buffering, normalization, transplantation) take over in succession, and when multiple organs cry for help simultaneously, even intact nociception cannot save the quietest one. Early warning is not, as V1 supposed, written in statistical precursors; it is written in a more humble place — the flow meters of the pipeline. Bian Que’s eldest brother’s consulting room may not need precision instruments, but someone must glance at the enrollment register when nothing is happening.

Parameter Table

Numerical Methods

Explicit Euler; main trajectories dt = 0.01 yr, phase portraits and sweeps dt = 0.02–0.05 yr; all reported quantities change <1% when the step size is halved. Phase portrait: 45 × 41
grid, vectorized in parallel (T = 420 yrs). Multi-organ: six-dimensional vectorized integration. Early-warning ensemble: 40 noise seeds (σ_F = 0.006), 8-year rolling window,
400-step detrending, lag-1 AC1, Kendall τ; alarm criterion is 3 consecutive samples exceeding “baseline mean + 2σ.” Korean fit and LOOCV use
SciPy trust-region least squares (max_nfev = 80).

Reproducibility

All results are generated by a single script, m2v2.py (dependencies: NumPy/SciPy/Matplotlib), with statistics simultaneously output to m2v2_stats.json; the V1 script necrosis_sim4.py is retained alongside for comparison. Modifying any parameter and re-running is sufficient to attack any proposition.

Sensitivity Analysis

Response of fold point β_c (baseline 1.73) to ±30% perturbation of each parameter: δ → [2.40, 1.18] (attrition rate is the largest lever — the slower the attrition, the more resilient the system);
ρ → [<1.0, 2.23] (recruitment efficiency is next; a 30% decrease folds below baseline); K → [1.84, 1.58]; T_train → [1.82, 1.64];
δ_A → [1.82, 1.64]; q is insensitive to fold-point position (burnout governs post-fold collapse speed rather than fold location). The wall of no return
(baseline 9.1%): δ × 0.7 → 5.5%, δ × 1.3 → 8.2%, q × 0.7 → 6.4%, q × 1.3 → 8.3% — the existence of the wall is robust to parameters; its position floats within
5.5–9.1%. Conclusion: fold, hysteresis, wall, and divergence are topological facts; all specific numbers are baseline-order-of-magnitude only.

Bastiat, F. (1850). Ce qu’on voit et ce qu’on ne voit pas.(What Is Seen and What Is Not Seen)

Allee, W. C. (1931). Animal Aggregations. University of Chicago Press.

Baumol, W. J. (1967). Macroeconomics of unbalanced growth. American Economic Review, 57, 415–426.

Hirschman, A. O. (1970). Exit, Voice, and Loyalty. Harvard University Press.

Forrester, J. W. (1971). World Dynamics. Wright-Allen Press.

May, R. M. (1977). Thresholds and breakpoints in ecosystems. Nature, 269, 471–477.

Rose, G. (1981). Strategy of prevention. BMJ, 282, 1847–1851.

Olson, M. (1982). The Rise and Decline of Nations. Yale University Press.

Tainter, J. A. (1988). The Collapse of Complex Societies. Cambridge University Press.

Holmström, B., & Milgrom, P. (1991). Multitask principal–agent analyses. JLEO, 7, 24–52.

Vaughan, D. (1996). The Challenger Launch Decision. University of Chicago Press.

Repenning, N. P., & Sterman, J. D. (2001). Nobody ever gets credit for fixing problems that never
happened. California Management Review, 43(4), 64–88.

Scheffer, M., et al. (2009). Early-warning signals for critical transitions. Nature, 461, 53–59.

Dekker, S. (2011). Drift into Failure. Ashgate.

Dakos, V., et al. (2012). Methods for detecting early warnings. PLoS ONE, 7, e41010.

Russell, A. L., & Vinsel, L. (2016; 2020). Hail the maintainers; The Innovation Delusion.

이조글로벌인공지능연구소 & Claude Fable 5 (2026). 《The Past Lives and Present Life of Social Functional Necrosis》（V2）. LEECHO Thought Paper.（Main paper）