Supplement: the T−1 gap amplitude in closed form (the F8 recursion’s seed) - Power-law decay of the buffer-stock consumption gap

Status: RECORD, SUPPLEMENTAL (2026-07-12; owner placement ruling: “maybe as supplemental material or an appendix; not in the main body”). This file records one validated data point for open item 2 of grid_design_final_spec.md (derive the exact (λ_t, F_t) of the F8 amplitude recursion). It changes no main-body document and anchors no shipped machinery: per the F11 uniformity ruling, B_t stays DIAGNOSTIC-ONLY (level continuity removed amplitude-anchoring), and the deep-tail region has negligible agent occupancy — the practical weight of this constant is validation/derivation bookkeeping, nothing more.

The result¶

Proposition 1 (The T−1 gap amplitude in closed form)

For the transitory-only case (ψ ≡ 1, LivPrb = 1, Γ = 1; discretized θ with E[θ] = 1), one period before the c(m) = m terminal — writing h_{T−1} = 1 + 1/R for human wealth at T−1 (BST’s convention: the current period’s income is included), b := m − 1 for BST’s bank balances, w̄ := b + h_{T−1} for perfect-foresight total wealth (the bar marks the PF agent’s upper-bound object, as in BST’s c̄), and x_{T−1}(w̄) := c̄_{T−1}(m) − c_{T−1}(m) for the gap (precautionary saving — the corpus-wide \psav = x per the 2026-07-16 registry ruling, defined with the gap in final_proof_myst.md §0/§1, label eq-gap-def) below the perfect-foresight rule c̄_{T−1}(m) = κ(b + h_{T−1}), κ = R/(R+Þ):

\wbar \cdot \psav_{T-1}(\wbar) \to B_{T-1} = (\rho+1) \cdot \mathrm{Var}(\theta) / (2 \cdot \Thorn \cdot R), \qquad \Thorn = (\beta R)^{1/\rho}.

(1)

Since B_T = 0, the F8 recursion B_t = λ_t·B_{t+1} + F_t gives B_{T−1} = F_{T−1}: this is the exact forcing term at the recursion’s seed, and a falsifiable target for any candidate (λ_t, F_t) derivation.

Derivation sketch (verified numerically; see Validation)¶

T−1 Euler with the terminal policy: c^{−ρ} = βR·E[(Ra + θ)^{−ρ}], a = m − c.
Expand at large a using E[θ] = 1: E[(Ra+θ)^{−ρ}] = (Ra)^{−ρ}· [1 − ρ/(Ra) + ρ(ρ+1)E[θ²]/(2R²a²) + O(a^{−3})]. Inverting through (·)^{−1/ρ}, the E[θ]-terms assemble the perfect-foresight line c̄(a) = (Ra+1)/Þ exactly, and the surviving second-order coefficient is −(ρ+1)(E[θ²]−1)/(2R²) = −(ρ+1)Var(θ)/(2R²): c(a) = (Ra+1)/Þ − (ρ+1)Var(θ)/(2ÞRa) + O(a^{−2}).
The gap is defined at MATCHED m, not matched a: inverting m = a + c(a) and comparing with c̄(m) = κ(b + h_{T−1}) cancels the (1−κ) bookkeeping exactly and leaves x(m) = D/m + O(m^{−2}) with D = (ρ+1)Var(θ)/(2ÞR); w̄ = b + h_{T−1} ⟹ w̄·x → D. (A matched-a reading gives a spurious (Þ+R)/Þ-type factor — the inversion step is load-bearing.)

Validation (2026-07-12, adversarial-refuter pass of the downstream¶

mapping workflow; all three routes independent)

At the public MoM notebook calibration (ρ = 2, β = 0.96, R = 1.02, θ ~ 7-atom lognormal(σ = 1) discretization, Var(θ_disc) = 1.035381): B_{T−1} = 1.538706, and

measured w̄·x at the deepest Euler-certified knot of a dense T−1 solve: ratio to B_{T−1} = 0.999946;
a 1/X-form extrapolation of the measured w̄·x: W_∞ = 1.538706 (rel. dev. −1.4e−07);
a solver-independent direct bisection of the T−1 Euler equation at m = 10⁶: w̄·x / B_{T−1} = 1.000010 (and |c_direct/c_solved − 1| = 0).

Provenance pin: logistic-vs-powerlaw-unit-interval-mapping @ b4225a1 :: checks/refuter_chi_slope_checks.py :: R6 (+ committed refuter_chi_slope_checks_out.txt).

Scope and open remainder¶

Un-derived here: the general-ψ / LivPrb < 1 / Γ ≠ 1 form of F_{T−1} (the expansion generalizes mechanically — ψ enters the return factor R/(Γψ) and the adjustment becomes a joint-moment expression — but no general form has been written down or validated), and everything about λ_t and F_t for t < T−1. Open item 2 of grid_design_final_spec.md remains open; this supplement just gives its future derivation a validated seed and a registered target.