# Proof: the constraint-end power law (`q_0` = ρ)

**Owns the proof bodies for** [`statement.md`](statement.md) §5 — Propositions C1, C2, Theorem CE,
Corollaries C3, C4. Companion battery: [`verify_constraint_end_checks.py`](verify_constraint_end_checks.py)
(ALL PASS; 5 calibrations, ρ∈{1.5,2,2.5,3}). Notation as in `statement.md` §1 and §5.

**Standing setup.** A1–A6 and (C-A3): i.i.d. transitory `θ` (ψ≡1) with a worst atom
`θ = θ_min` at mass `℘ ∈ (0,1)`, natural borrowing constraint `m̲ = −θ_min/(ℛ−1)`, excess
resources `m^e = m − m̲`. Normalized Euler (interior), from `statement.md` (st-eq-lom) with ψ≡1:

$$
c(m)^{-\rho} \;=\; \ThornG^{\rho}\,\E_\theta\!\big[\,c(m')^{-\rho}\,\big], \qquad m' = \Rcal\,a + \theta,\quad a = m - c(m). \tag{CE.1}
$$

Because `E[θ]=1` (mean income is the numeraire), the constraint is set by the worst
realization: taking `θ = θ_min` and `a` at its constrained value reproduces `m^e`'s lower end, and
one checks `m̲ = ℛ m̲ + θ_min` ⟺ `m̲ = −θ_min/(ℛ−1)` — i.e. `|m̲| = θ_min·(h−1)`, the PDV of the
worst path's *future* income (`h` is BST's human wealth, which includes the current period's
income, so the future-only PDV factor is `h−1 = 1/(ℛ−1)`; cf. `statement.md` §1). Below we take `m̲ = 0` (`θ_min = 0`,
the zero-income case) to keep coordinates light; the general worst-atom case is identical after the
shift `m ↦ m^e` (Remark at the end).

---

## 1. Proposition C1 — the worst-atom maximal MPC

As `m^e → 0`, suppose `c(m) = κ̄ m^e (1+o(1))`. In (CE.1) split the expectation into the worst
branch (`θ = 0`, mass `℘`) and the rest. On the worst branch `m'_0 = ℛ a = ℛ(1−κ̄)m^e(1+o(1))`, so
`c(m'_0) = κ̄ m'_0(1+o(1))` is again `O(m^e)`; on the other branches `m'_j → θ_j > 0`, so
`c(m'_j) = O(1)`. Hence the worst branch dominates the RHS and, to leading order,

$$
(κ̄ m^e)^{-\rho} = \ThornG^{\rho}\,\wp\,(κ̄\,\Rcal(1-κ̄)m^e)^{-\rho}
\;\Longrightarrow\; 1 = \ThornG^{\rho}\,\wp\,\lambda^{-\rho}, \qquad \lambda := \Rcal(1-κ̄). \tag{CE.2}
$$

Solving `ThornG^ρ ℘ λ^{-ρ} = 1` with `λ = ℛ(1−κ̄)` gives `1−κ̄ = ℘^{1/ρ}Þ_R` (using `ℛÞ_R = Þ_Γ`),
i.e. **`κ̄ = 1 − ℘^{1/ρ}Þ_R`** and `λ = ℘^{1/ρ}Þ_Γ < 1` (GIC ⟹ `Þ_Γ<1`). This is BST's `MPCmax`
definition ([eq-MPCmaxDefn](https://llorracc.github.io/BufferStockTheory-Latest/#eq-mpcmaxdefn));
the only generalization is reading `℘` as the mass of the **worst atom** of a general
(discretized) income process, which is what enters the worst-branch split. ∎

`verify_constraint_end_checks.py` C1: the free-parameter fit of `c/m^e` returns the intercept `κ̄` to
≤ 6e-8 across five calibrations, independent of the formula.

---

## 2. Theorem CE — the approach exponent is `q_0` = ρ

Denote by `q_0` the constraint-end approach exponent (statement.md Thm CE — the ring-subscript partner
of the high-wealth `q_∞`; display `q_{\downarrow}`). Write the constraint-end gap `γ(m) = κ̄ m^e − c(m) ≥ 0`
(so `c = κ̄ m^e − γ`, `γ/m^e → 0`) and the
**reduced gap** `ν(m) := γ(m)/m^e` (`ν → 0`). Then `a = m − c = (1−κ̄)m^e + γ`, and the worst branch is

$$
m'_0 = \Rcal a = \lambda m^e + \Rcal\gamma(m) = \lambda m^e\big(1 + \tfrac{\Rcal}{\lambda}\nu(m)\big). \tag{CE.3}
$$

Divide (CE.1) by `(κ̄ m^e)^{-ρ}` and expand each factor to first order in the small quantities
`ν(m)`, `ν(λ m^e)`. Three pieces:

- **LHS.** `(1 − γ/(κ̄ m^e))^{-ρ} = 1 + \tfrac{\rho}{κ̄}\nu(m) + O(\nu^2)`.
- **Worst branch.** `ThornG^ρ ℘ (m^e/m'_0)^ρ (1 − γ(m'_0)/(κ̄ m'_0))^{-ρ}`. Using (CE.2) the prefactor
  `ThornG^ρ℘λ^{-ρ}=1`; `(m^e/m'_0)^ρ = 1 − \tfrac{\rho\Rcal}{\lambda}\nu(m) + O(\nu^2)` from (CE.3);
  and `γ(m'_0)/(κ̄ m'_0) = \nu(λ m^e)/κ̄·(1+o(1))` since `m'_0 = λ m^e(1+o(1))`. Product:
  `1 − \tfrac{\rho\Rcal}{\lambda}\nu(m) + \tfrac{\rho}{κ̄}\nu(\lambda m^e) + O(\nu^2)`.
- **Non-worst branches.** `ThornG^ρ Σ_{θ_j>0} p_j c(m'_j)^{-ρ} · (κ̄ m^e)^{ρ}`. As `m^e→0`,
  `m'_j → θ_j` and `c(m'_j) → c(θ_j) = O(1)`, so this equals `J₀ κ̄^ρ (m^e)^ρ (1+o(1))` with the
  **boundary-data constant** `J₀ := ThornG^ρ E[c(θ)^{-ρ}\mathbf 1\{θ>0\}] > 0`.

Collecting (the leading `1`'s cancel), dividing by `ρ`:

$$
\frac{1}{κ̄}\big[\nu(m) - \nu(\lambda m^e)\big] \;+\; \frac{\Rcal}{\lambda}\,\nu(m) \;=\; C\,(m^e)^{\rho}, \qquad C := \frac{J_0\,κ̄^{\rho}}{\rho}. \tag{CE.4}
$$

Equation (CE.4) is the **constraint-end analogue of the high-wealth renewal equation (★)** — but with a
crucial extra term. Its two ingredients:

- **Particular solution.** Try `ν(m) = K(m^e)^s`, where `s` is a trial exponent local to this
  derivation (a dummy symbol — not the precautionary-saving function `s(m)` of `statement.md` §1).
  The forcing is `(m^e)^ρ`, so `s = ρ` and
  `K\big[\tfrac{1-\lambda^{\rho}}{κ̄} + \tfrac{\Rcal}{\lambda}\big] = C`, giving the **positive**
  amplitude `K = C\big/\big[(1-\lambda^{\rho})/κ̄ + \Rcal/\lambda\big]` (`λ^ρ = ℘Þ_Γ^ρ < 1`, so the
  bracket is `>0`, and `C>0`). Hence `ν(m) ~ K(m^e)^ρ`, i.e. `γ(m) ~ K(m^e)^{1+ρ}`.
- **Homogeneous solutions excluded.** `ν = (m^e)^s` solves the homogeneous (CE.4) iff
  `\tfrac{1}{κ̄}(1-\lambda^s) + \tfrac{\Rcal}{\lambda} = 0`, i.e. `λ^s = 1 + κ̄\Rcal/\lambda > 1`.
  Since `λ<1` this forces `s < 0`: the homogeneous mode **grows** as `m^e→0` (`ν→∞`) and is killed by
  the boundary condition `ν→0` (equivalently by `0 ≤ γ ≤ κ̄ m^e`). So the particular solution is the
  whole leading behavior.

Therefore `c(m) = κ̄ m^e − K(m^e)^{1+ρ}(1+o(1))`, which is (st-eq-CE): **`q_0 = ρ`.** ∎

**No periodic prefactor (proof of the Remark C1 claim).** On the high-wealth side the analogue of
(CE.4) is a pure telescope `ν(m) − ν(λ m) = forcing`, whose particular solution admits a companion
`(m^e)^ρ·Π(ln m^e)` with `Π` log-periodic (period `ln(1/λ)`) — the origin of `P(·)` in Thm A2. Here
the extra term `\tfrac{\Rcal}{\lambda}\nu(m)` breaks that degeneracy: seeking
`ν = (m^e)^ρ Φ(\ln m^e)` in the homogeneous (CE.4) forces `Φ(u+\ln λ) = ζ\,Φ(u)` with
`ζ = (1+κ̄\Rcal/\lambda)/\lambda^{\rho} > 1`, i.e. a genuine geometric growth, not a bounded
oscillation — so the only bounded (indeed decaying) solution is the constant-amplitude power. The
constraint end therefore has **no lattice pathology**; the excellent pure-power fit (C2, exponent left
free, `|q_0−ρ| ≤ 0.036`) corroborates.

---

## 3. Proposition C2 — the finite-horizon recursion and the T−1 anchor

BST's backward recursion for the maximal MPC
([eq-MPCmaxInvApndxIter](https://llorracc.github.io/BufferStockTheory-Latest/#eq-mpcmaxinvapndxiter)) is
`κ̄_{T-n}^{-1} = 1 + 𝖬 κ̄_{T-n+1}^{-1}` with driver `𝖬 = ℘^{1/ρ}Þ_R = 1−κ̄` (the limiting value) and
terminal `κ̄_T = 1`. This is a linear recursion in `κ̄_t^{-1}`; iterating,
`κ̄_{T-n}^{-1} = Σ_{j=0}^{n} 𝖬^j = (1−𝖬^{n+1})/(1−𝖬)`, i.e. `κ̄_{T-n} = κ̄/(1−𝖬^{n+1})` — (st-eq-C2).
The **one-from-terminal** value is `κ̄_{T-1} = 1/(1+𝖬) = 1/(2−κ̄)`.

*Direct one-step check (independent of BST).* From terminal `c_T(m)=m`, one backward EGM step gives
`c_{T-1}(a)^{-ρ} = ThornG^ρ E[(ℛa+θ)^{-ρ}]`. As `a→0` the worst atom (`θ=0`, mass `℘`) dominates:
`c_{T-1}^{-ρ} ~ ThornG^ρ ℘ (ℛa)^{-ρ}`, so `c_{T-1} ~ (ThornG^ρ℘)^{-1/ρ}ℛa = a·ℛ/(Þ_Γ℘^{1/ρ})`, and
since `Þ_Γ/ℛ = Þ_R`, `c_{T-1}/m^e = 1/(1+Þ_R℘^{1/ρ}) = 1/(1+𝖬)`. ∎

C3 in the battery measures `c_{T-1}/m^e` to ≤ 4e-14 of `1/(1+𝖬)`; e.g. at ρ=2, ℘=0.05 the recursion
gives `κ̄_{T-1}=0.82316` against a stationary `κ̄=0.78517` — the recursion, not the stationary value,
is the correct finite-horizon anchor.

---

## 4. Corollaries C3 (two asymptotes) and C4 (bottom grid design)

**C3.** From Thm CE, `c/m^e → κ̄` with an `O((m^e)^ρ)` correction, so in `μ = ln m^e` the moderation
`χ` is linear with slope `1` and intercept `b₀ = ln((κ̄−κ̲)/C)`, `C = κ̄(h−1+m̲)`. From Cor. A4 the
high-wealth end is linear with slope `q = min(1,q_∞)` and intercept `b_∞`. For `q<1` the lines meet at
`μ_c = (b_∞−b₀)/(1−q)`; at `q=1` they are parallel. ∎

**C4.** The relative deviation of `c` from the `κ̄`-line is `(κ̄ − c/m^e)/κ̄ = (K/κ̄)(m^e)^ρ` (Thm CE),
so it drops below tolerance `tol` exactly for `m^e ≤ (tol·κ̄/K)^{1/ρ}` — (st-eq-C4). A bottom knot
above this scale reports a constraint-end slope biased by `O((m^e_0)^ρ)`; refining `aXtraMin` below it
recovers the true slope `κ̄`. ∎

C4 in the battery: solving for the `m^e_0` predicted to give `tol=1e-2` and reading the model there
reproduces the deviation to within 16% across calibrations.

---

## 5. Status and gaps

- **Theorem CE** (`q_0 = ρ`): PROVEN-HERE by the local Euler analysis §2, level of rigor matching
  Stage-A's linearization (an explicit renewal-type functional equation with excluded homogeneous
  modes); numerically CONFIRMED with the exponent left free (`|q_0−ρ| ≤ 0.036`, 5 calibrations).
  Residual rigor items, none structural: (i) the `o(1)` remainder control (the higher-order Euler
  terms enter at `(m^e)^{2ρ}` and `(m^e)^{ρ+1}`, both below the leading `(m^e)^ρ`); (ii) a fully
  closed form for `K` needs the boundary constant `J₀` (same status as Thm A2's amplitude).
- **GAP-CE-ψ** (permanent shocks): open — a continuous permanent component can restore randomness in
  the worst branch; the single-worst-atom (discretized) case is expected to give `q_0 = ρ` verbatim.
- **Propositions C1, C2**: PROVEN-CITED (BST), restated in constraint-end/MoM coordinates and
  independently machine-checked.