fixpoint_analysis.py

# “””
DLE Smooth-Core Fixed-Point & Local Stability Analysis

Final version for paper inclusion.

Result summary:

- Z* and S* have clean closed-form fixed points
- R grows to ecological ceiling (boundary equilibrium)
- B is stabilized only by discrete guards (debt sink, jubilee)
- The smooth-core alone is NOT locally stable in B
- This is the paper’s central thesis: stability is architectural
  “””

import numpy as np
from scipy.optimize import fsolve, brentq

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# PARAMETERS

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P = dict(
alpha=0.02, gamma_base=0.5, rho_raw=0.15, psi=0.9, theta=1.2,
iota=0.002, kappa=0.5, zeta=0.03, Z_cap=0.6,
r_S=0.015, eta=0.00004, K=450.0, R_max_coeff=1100.0,
tau_sf=0.06, tau_vat=0.08, tau_pi=0.15, tau_dem=0.002,
maint_rate=0.03, pub_frac=0.15, ubi_share=0.45, G_frac=0.10,
r_interest=0.01,
lam=0.10, Omega=5000.0, chi=0.8,
)

def gamma_eff(Z, p): return p[“gamma_base”] * (1 + p[“psi”] * Z)
def rho_eff(Z, p):   return p[“rho_raw”] / (1 + p[“theta”] * Z)
def A_star(p):        return p[“Omega”] * (1 - p[“chi”]*p[“tau_sf”]/p[“lam”])

print(”=” * 72)
print(”  DLE SMOOTH-CORE: FIXED-POINT & LOCAL STABILITY ANALYSIS”)
print(”  Reduced system: x = (R, Z, S, B), A quasi-stationary”)
print(”=” * 72)

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# PROPOSITION 1: Z* (closed form)

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A_ss = A_star(P)
T_sf = P[“tau_sf”] * A_ss
I_pub = P[“pub_frac”] * T_sf
Z_uncapped = P[“iota”] * P[“kappa”] * I_pub / P[“zeta”]
Z_star = min(Z_uncapped, P[“Z_cap”])

print(f”””
PROPOSITION 1 (Z fixed point).
In the post-rebuild regime (I_rebuild = 0), the resilience capital
dynamics dZ/dt = ι·κ·I_pub − ζ·Z yield a unique fixed point:

```
Z* = min(ι·κ·I_pub* / ζ,  Z_cap)
   = min({P['iota']}·{P['kappa']}·{I_pub:.1f} / {P['zeta']},  {P['Z_cap']})
   = min({Z_uncapped:.4f}, {P['Z_cap']})
   = {Z_star:.4f}
```

Z converges monotonically with rate ζ = {P[‘zeta’]}.
Stability: ∂(dZ/dt)/∂Z = −ζ = {-P[‘zeta’]:.3f} < 0.  ✓
“””)

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# PROPOSITION 2: S* (piecewise closed form)

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print(f”””  PROPOSITION 2 (S fixed point, conditional on R*).
dS/dt = r_S·(1 − S) − η·max(0, R − K) = 0 yields:

```
Case 1: R* ≤ K = {P['K']:.0f}  ⇒  S* = 1.0
Case 2: R* > K  ⇒  S* = 1 − (η/r_S)·(R* − K)
                      = 1 − {P['eta']/P['r_S']:.6f}·(R* − {P['K']:.0f})
```

Stability: ∂(dS/dt)/∂S = −r_S = {-P[‘r_S’]:.3f} < 0.  ✓
Coupling: ∂(dS/dt)/∂R = −η = {-P[‘eta’]:.6f} < 0 (R pressures ecology).
“””)

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# PROPOSITION 3: R* (boundary equilibrium)

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g_star = gamma_eff(Z_star, P)
r_star = rho_eff(Z_star, P)
Pi_coeff = g_star - r_star  # profit per unit R
I_priv_coeff = (1 - P[“tau_pi”]) * Pi_coeff
drift_coeff = (P[“alpha”] - r_star) + I_priv_coeff + P[“maint_rate”]

print(f”””  PROPOSITION 3 (R boundary equilibrium).
The capital accumulation equation in the smooth core:
dR/dt = (α − ρ_eff)·R + I_priv(R) + I_pub + I_maint(R)

With I_priv = (1−τ_π)·max(0, γ_eff·R − ρ_eff·R) and I_maint = m·R,
the net drift coefficient is:

```
c_R = (α − ρ_eff) + (1−τ_π)·(γ_eff − ρ_eff) + m
    = ({P['alpha']:.4f} − {r_star:.4f}) + (1−{P['tau_pi']})·({g_star:.4f} − {r_star:.4f}) + {P['maint_rate']}
    = {P['alpha']-r_star:.4f} + {I_priv_coeff:.4f} + {P['maint_rate']}
    = {drift_coeff:.4f}
```

Since c_R > 0, capital drift is positive at ALL interior levels:
dR/dt > 0 for all R in (0, R_max(S)).

⇒ The system has NO interior fixed point for R.
⇒ R grows monotonically to the ecological ceiling R_max(S) = {P[‘R_max_coeff’]}·S^1.5.
⇒ The steady state is a BOUNDARY EQUILIBRIUM at carrying capacity.

Interpretation: The DLE economy is not demand-constrained in the long
run. It is ecologically constrained. Growth stops because nature sets
a ceiling, not because investment falters.
“””)

# Solve R*-S* jointly at boundary

def RS_system(x):
R, S = x
return [R - P[“R_max_coeff”] * S**1.5,
P[“r_S”] * (1 - S) - P[“eta”] * max(0, R - P[“K”])]

R_star, S_star = fsolve(RS_system, [580, 0.65])
Y_star = g_star * R_star

print(f”””  Joint boundary equilibrium (R* = R_max(S*)):
R* = {R_star:.2f}
S* = {S_star:.6f}
Y* = γ_eff·R* = {Y_star:.2f}
R_max(S*) = {P[‘R_max_coeff’]*S_star**1.5:.2f}  (verification)
“””)

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# PROPOSITION 4: B* (requires discrete stabilizers)

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outlays = P[“G_frac”]*Y_star + P[“ubi_share”]*T_sf + I_pub + P[“maint_rate”]*R_star
Pi = Y_star - r_star * R_star
revenues = T_sf + P[“tau_vat”]*Y_star + P[“tau_pi”]*max(0,Pi) + P[“tau_dem”]*R_star
primary = revenues - outlays

print(f”””  PROPOSITION 4 (B dynamics — instability of smooth core).
The government budget in the smooth core:
dB/dt = Outlays + r·B − Revenues

At the boundary equilibrium:
Outlays (ex-interest) = {outlays:.2f}
Revenues (ex-sink)    = {revenues:.2f}
Primary balance       = {primary:+.2f} (surplus)

However, the smooth-core B equation has:
∂(dB/dt)/∂B = r = {P[‘r_interest’]:+.4f} > 0

This means: in the smooth core, government debt is locally UNSTABLE.
Any perturbation in B grows exponentially at rate r.

The primary surplus ({primary:.2f}) can offset interest costs up to a
critical debt level B_crit = primary / r = {primary/P[‘r_interest’]:.0f}.
Above this level, debt diverges.

⇒ The smooth-core system is NOT locally stable in the B dimension.
⇒ Stability in B requires the discrete stabilizers:
- Debt sink (activated when B/Y exceeds threshold)
- Jubilee mechanism (emergency deleveraging)
- Peak guard (hard ceiling on debt)

This is the paper’s central architectural claim formalized:
smooth dynamics alone cannot ensure fiscal stability. The guard
architecture is structurally necessary.
“””)

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# JACOBIAN OF THE 3D STABLE SUBSYSTEM (R, Z, S)

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print(”=” * 72)
print(”  JACOBIAN OF THE STABLE SUBSYSTEM (R, Z, S)”)
print(”=” * 72)

# At the boundary, R is clamped to R_max(S). The effective dynamics

# become 2D in (Z, S) with R enslaved to R_max(S).

# But for the full picture, we compute the 3D Jacobian with the

# ceiling as a restoring force.

def rhs_3d(x):
R, Z, S = x
Zc = min(Z, P[“Z_cap”])
g = gamma_eff(Zc, P)
rho = rho_eff(Zc, P)
Rm = P[“R_max_coeff”] * max(S, 0.01)**1.5
# Ceiling as restoring force: strong drag when R > R_max
ceil = 0.5 * max(0, R - Rm)
Y = g * R
Pi = Y - rho * R
Ip = max(0, (1 - P[“tau_pi”]) * max(0, Pi))
Im = P[“maint_rate”] * R

```
dR = (P["alpha"] - rho) * R + Ip + I_pub + Im - ceil
dZ = P["iota"] * P["kappa"] * I_pub - P["zeta"] * Zc
dS = P["r_S"] * (1 - S) - P["eta"] * max(0, R - P["K"])
return np.array([dR, dZ, dS])
```

x3_star = np.array([R_star, Z_star, S_star])
res3 = rhs_3d(x3_star)
print(f”\n  Fixed point: R*={R_star:.2f}, Z*={Z_star:.4f}, S*={S_star:.6f}”)
print(f”  Residual: |f(x*)| = [{res3[0]:.4f}, {res3[1]:.4f}, {res3[2]:.6f}]”)

# Jacobian via finite differences

J3 = np.zeros((3, 3))
for i in range(3):
h = 1e-5 * max(1, abs(x3_star[i]))
dx = np.zeros(3); dx[i] = h
J3[:, i] = (rhs_3d(x3_star + dx) - rhs_3d(x3_star - dx)) / (2*h)

labs = [“R”, “Z”, “S”]
print(f”\n  Jacobian J₃(x*):”)
print(f”  {’’:>10}”, end=””); [print(f”{‘∂’+l:>12}”, end=””) for l in labs]; print()
for i in range(3):
print(f”  d{labs[i]}/dt  “, end=””)
for j in range(3):
print(f”{J3[i,j]:>12.6f}”, end=””)
print()

evs3 = np.linalg.eigvals(J3)
print(f”\n  Eigenvalues:”)
for i, ev in enumerate(sorted(evs3, key=lambda x: x.real)):
re, im = ev.real, ev.imag
tag = “stable” if re < 0 else “UNSTABLE”
if abs(im) < 1e-10:
print(f”    λ_{i+1} = {re:>+12.6f}  [{tag}]”)
else:
print(f”    λ_{i+1} = {re:>+12.6f} ± {abs(im):.6f}i  [{tag}]”)

all_stable_3d = all(ev.real < 0 for ev in evs3)
print(f”\n  ▶ 3D subsystem (R,Z,S): {‘LOCALLY STABLE ✓’ if all_stable_3d else ‘UNSTABLE ✗’}”)

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# PARAMETER SWEEP ON 3D SUBSYSTEM

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print(f”\n{’=’ * 72}”)
print(f”  STABILITY OF (R,Z,S) SUBSYSTEM ACROSS PARAMETER SPACE”)
print(f”{’=’ * 72}”)

print(f”\n  {‘ψ’:>5} {‘θ’:>5} {‘ρ’:>6} | {‘R*’:>7} {‘S*’:>6} {‘ρ_eff’:>7} | {‘λ₁’:>10} {‘λ₂’:>10} {‘λ₃’:>10} | Stable”)
print(f”  {‘─’*80}”)

n_stable = 0
n_total = 0

for psi in [0.6, 0.9, 1.2]:
for theta in [0.8, 1.2, 1.6]:
for rho in [0.10, 0.15, 0.20, 0.25, 0.30]:
pv = dict(P); pv[“psi”]=psi; pv[“theta”]=theta; pv[“rho_raw”]=rho

```
        Zs = min(pv["iota"]*pv["kappa"]*I_pub/pv["zeta"], pv["Z_cap"])
        g = gamma_eff(Zs, pv)
        rho_e = rho_eff(Zs, pv)
        
        try:
            def RS2(x):
                R, S = x
                return [R - pv["R_max_coeff"]*max(S,0.01)**1.5,
                        pv["r_S"]*(1-S) - pv["eta"]*max(0,R-pv["K"])]
            Rs, Ss = fsolve(RS2, [580, 0.65])
            if Rs < 0 or Ss < 0 or Ss > 1.01: raise ValueError
            
            def rhs3v(x):
                R, Z, S = x
                Zc = min(Z, pv["Z_cap"])
                gv = gamma_eff(Zc, pv); rv = rho_eff(Zc, pv)
                Rm = pv["R_max_coeff"]*max(S,0.01)**1.5
                ceil = 0.5*max(0, R-Rm)
                Y = gv*R; Pi = Y - rv*R
                Ip = max(0,(1-pv["tau_pi"])*max(0,Pi))
                Im = pv["maint_rate"]*R
                return np.array([
                    (pv["alpha"]-rv)*R + Ip + I_pub + Im - ceil,
                    pv["iota"]*pv["kappa"]*I_pub - pv["zeta"]*Zc,
                    pv["r_S"]*(1-S) - pv["eta"]*max(0,R-pv["K"])
                ])
            
            x3 = np.array([Rs, Zs, Ss])
            Jv = np.zeros((3,3))
            for i in range(3):
                h = 1e-5*max(1,abs(x3[i]))
                dx = np.zeros(3); dx[i]=h
                Jv[:,i] = (rhs3v(x3+dx) - rhs3v(x3-dx))/(2*h)
            
            evs = np.linalg.eigvals(Jv)
            evs_s = sorted(evs, key=lambda x: x.real)
            st = all(ev.real < 0 for ev in evs)
            n_total += 1
            if st: n_stable += 1
            
            ev_strs = []
            for ev in evs_s:
                ev_strs.append(f"{ev.real:>+10.4f}")
            
            print(f"  {psi:>5.1f} {theta:>5.1f} {rho:>6.2f} | {Rs:>7.1f} {Ss:>6.3f} {rho_e:>7.4f} | {' '.join(ev_strs)} | {'✓' if st else '✗'}")
        except:
            n_total += 1
            print(f"  {psi:>5.1f} {theta:>5.1f} {rho:>6.2f} | {'— no convergence —':>60}")
```

print(f”\n  Summary: {n_stable}/{n_total} parameter combinations locally stable in (R,Z,S)”)

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# SUMMARY

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print(f”””
{’=’ * 72}
SUMMARY OF RESULTS
{’=’ * 72}

1. CLOSED-FORM FIXED POINTS
   Z* = {Z_star:.4f} (capped at Z_cap)   — stable (rate −ζ = {-P[‘zeta’]})
   S* = {S_star:.6f} (ecological health)  — stable (rate −r_S = {-P[‘r_S’]})
   R* = {R_star:.2f} (at ecological ceiling) — boundary equilibrium
1. 3D SUBSYSTEM (R, Z, S): {‘LOCALLY STABLE ✓’ if all_stable_3d else ‘UNSTABLE ✗’}
   The physical economy converges to a well-defined boundary
   equilibrium determined by ecological carrying capacity.
1. B DYNAMICS: UNSTABLE WITHOUT GUARDS
   ∂(dB/dt)/∂B = r = {P[‘r_interest’]:+.4f} > 0
   The smooth-core fiscal dynamics are inherently unstable.
   Stability requires the discrete stabilizer architecture
   (debt sink, jubilee, peak guard).
1. ARCHITECTURAL IMPLICATION
   The DLE’s resilience is NOT a property of the smooth dynamics.
   It emerges from the combination of:
   (a) a stable physical core (R, Z, S),
   (b) discrete fiscal guards that bound B, and
   (c) regime switching that activates stabilizers under stress.
   
   This separation — stable physics + engineered fiscal guards —
   is the paper’s central architectural claim, now formally verified.
   “””)