Statement: power-law decay of the buffer-stock consumption gap (Stage A: ψ≡1)

Status: candidate theorem set with proof draft (stage_A_proof.md). Every claim carries its ledger status from 00_THEOREM_PLAN.md. Notation follows Carroll’s buffer-stock papers (BST). Lemmas are cited by number (L0, L1, …, from the proof documents); primes (e.g. L9′) mark strengthened variants. Reading this for the first time? Start with final_proof_myst — the synthesis presentation of record (backward-induction spine, §2; the compactified boundary-fixed-point core re-cast as one of its four rigor engines, §5; full economics literature fabric, BST terminology, illustrative figures on HAFiscal’s estimated calibrations); it imports every proof body by reference from the four proof documents.

1. Model and assumptions¶

A consumer solves the infinite-horizon income-fluctuation problem

max E₀ Σ_{t≥0} β^t L^t · Γ_t-normalized CRRA utility,   u(c) = c^{1−ρ}/(1−ρ)  (ρ>0; ρ=1 is log),

in permanent-income-normalized form: given market resources m > 0 (beginning of period, including current income), choose consumption c ∈ (0, m] (equivalently end-of-period assets a = m − c ≥ 0), with next-period normalized resources

m_{t+1} = (R/\Gamma) \cdot a + \theta_{t+1}.

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Assumptions.

(A1) u′(c) = c^{−ρ}, ρ > 0.
(A2) Constant gross return R > 0, constant permanent-income growth factor Γ > 0, discount factor β ∈ (0,1). (Survival probability L ∈ (0,1] may be folded into β; see Remark 9.)
(A3) {θ_t} i.i.d., E[θ] = 1, σ² := Var(θ) ∈ (0, ∞), with bounded support supp θ ⊆ [θ_min, θ_max], 0 ≤ θ_min < 1 < θ_max < ∞ — the bounded-support, mean-one transitory clause of BST’s Friedman–Muth income process (Assumption 1). (In BST the zero-income event θ = 0 w.p. ℘ > 0 is part of the maintained income-process assumption; here θ_min = 0 is permitted throughout, the results of §§2–4 hold with or without the atom, and §5’s constraint-end results require the worst-atom mass ℘ > 0, which enters the maximal MPC κ̄ = 1 − ℘^{1/ρ}Þ_R directly. For unbounded θ see Remark 8. Primitive form, per BST: when the zero-income event is active the transitory draw is TWO-PART — first the unemployment state (probability ℘, income 0), then, if employed, the employed shock on its strictly positive support, scaled so E[θ] = 1 unconditionally; the one-part mixture stated above is the derived marginal law, interchangeable in every expectation. The simulation machinery draws two-stage, and newborns’ first period uses the employed branch with the UNSCALED mean-one draw — see the ergodic-coverage companion.) No permanent shocks: ψ ≡ 1 (Stage A).
(A4) FHWC (finite human wealth): ℛ := R/Γ > 1.
(A5) RIC (return impatience): Þ_R := (βR)^{1/ρ}/R < 1; write κ̲ := 1 − Þ_R ∈ (0,1).
(A6) GIC (growth impatience): Þ_Γ := (βR)^{1/ρ}/Γ < 1. (Note Þ_Γ = ℛ·Þ_R; under FHWC, GIC neither implies nor is implied by RIC.) — Not needed for the forcing floor (stage-A Lemma 6.1).
(A0) The problem admits a unique solution c: (0,∞) → (0,∞) with the standard properties (continuous, strictly increasing, 0 < c(m) ≤ m; Euler equation with equality wherever a(m) > 0; c is the globally-stable fixed point of the Coleman–Reffett operator). [PROVEN-CITED: under A1–A6 the conditions FVAC and WRIC of Carroll (Theoretical Foundations of Buffer Stock Saving, Quantitative Economics; henceforth BST) hold — stage_A_proof.md §1 shows GIC+FHWC ⟹ FVAC and RIC ⟹ WRIC — and existence/uniqueness/stability also follow from Li–Stachurski 2014 JEDC / Ma–Stachurski–Toda 2020 JET 187 (CITE-CHECK on the exact hypothesis mapping is flagged in the ledger).]

Derived objects.

h  := 1/(1−ℛ⁻¹) = R/(R−Γ)    (normalized human wealth, current income included — BST eq-HDef)
w̄  := b + h              (perfect-foresight total wealth, human and market — the PF agent's wealth)
c̄(m) := κ̲·(m−1+h) = κ̲·w̄     (BST's perfect-foresight consumption function eq-cFuncPFUnc; the asymptote)
x(m) := c̄(m) − c(m) = κ̲·(m−1+h) − c(m)   (THE GAP — precautionary saving; x ≥ 0 by the forcing floor (stage-A Lemma 6.1)/L2; the gap g(w̄) := κ̲·w̄ − c(m(w̄)) = x(m(w̄)) is the same quantity in the PF-wealth coordinate)
þ_g := ln Þ_Γ                (the log growth-patience rate, NEGATIVE under GIC, so −þ_g = ln(1/Þ_Γ) > 0 is the per-period descent rate — the log-ladder step)
q_∞ := ln(ℛ)/(−þ_g) > 0       (the transitory-only Kesten root, eq. (E0) of the derivation)

Here h is BST’s human wealth (eq-HDef): its PDV series starts at the current period’s income, so the future-income-only quantity, where genuinely needed, is h − 1. w̄(m) := b + h = m − 1 + h is perfect-foresight total wealth (human and market) — a function of the state, not a constant — viewed at the decision moment after this period’s returns have been realized on the kapital saved last period (BST’s k_t = a_{t−1}; b = ℛ·k at ψ ≡ 1 — BST’s bank balances). The bar marks an upper-bound object, matching BST’s own perfect-foresight bound c̄(m) = (b + h)κ̲ (eq-cFuncPFUnc) and the overbar-signifies-an-upper-bound convention (as in κ̲, the lower bound); w is unused in BST, so there is no clash. The symbol x denotes the consumer’s precautionary saving — the eXtra saving induced by precaution: the shortfall of consumption below the perfect-foresight rule, x(m) := c̄(m) − c(m); the gap g(w̄) := κ̲·w̄ − c(m(w̄)) carries the same quantity in the PF-wealth coordinate, w̄(m) := b + h (with b = m − 1, BST’s bank balances).

q_∞ is equivalently the unique positive root of Þ_Γ^{−q} = ℛ — the ψ≡1 case of the eigenvalue equation (E): E[ψ^{1+q}] = ℛ·Þ_Γ^q.

2. Results¶

Theorem 1 (Theorem A1 (the power law: two-sided order). [PROVEN-HERE, L5–L7, L11; review-verified R2 (full re-derivation + 52/52 pointwise constant audit on 4 models incl. θ=0-atom and exact-resonance)])

Under A0–A6 and σ² > 0 there are constants 0 < c ≤ C < ∞ (explicitly constructible from the primitives — the q_∞<1 lower constant via chaining the forcing floor (stage-A Lemma 6.1) at the ladder base, repair per review R2 F-3; the explicit constants are deliberately crude, see Remark 7) and w̄_A1 < ∞ such that for all w̄ ≥ w̄_A1:

(i) if q_∞ > 1: c·σ²·w̄^{−1} ≤ g(w̄) ≤ C·w̄^{−1}; (ii) if q_∞ < 1: c·w̄^{−q_∞} ≤ g(w̄) ≤ C·w̄^{−q_∞}; (iii) if q_∞ = 1: c·σ²·w̄^{−1}·ln w̄ ≤ g(w̄) ≤ C·w̄^{−1}·ln w̄.

In particular the leading exponent is min(1, q_∞) (with a log correction exactly at the resonance q_∞ = 1), and x is O-regularly varying with both Matuszewska indices equal to −min(1,q_∞) in cases (i)–(ii). The (iii) resonance case now has a SHARP constant (2026-07-06): w̄·g(w̄)/ln w̄ → κ̲(ρ+1)σ²/(−2þ_g) — the ψ≡1 corollary of Stage-B Theorem B-res (L11-const closed; see §3).

Theorem 2 (Theorem A2 (sharp form for q_∞ < 1: the Kesten channel). [PROVEN-HERE, L9; review-verified R3 incl. a constants-level measurement of the telescope increment. P’s fine structure (constancy; oscillation size) is NOT required here — owned by periodic_factor_fine_structure.md])

Under A0–A6, σ² > 0 and q_∞ < 1 there is a positive, Lipschitz, (−þ_g)-periodic function P: ℝ → [c, C] such that

g(\wbar) = \wbar^{-\qup} \cdot ( P(\ln \wbar) + O(\wbar^{-(1-\qup)}) ) \qquad \text{as } \wbar \to \infty.

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Equivalently g(w̄) = w̄^{−q_∞}P(ln w̄) + O(w̄^{−1}) — the second (income/forcing) channel appears exactly at the predicted order w̄^{−1}. The amplitude object P is determined by the boundary data of x at the edge of the linearization region (moderate w̄) and is not available in closed form. P is genuinely (generically) NON-constant — so x is O-regularly varying but NOT regularly varying (cf. the lattice caveat in renewal theory: Beare–Toda Remark 2.3 vs their eq. (1.5); Kevei 2017).

The FINE STRUCTURE of P (whether it is constant; how large its oscillation) is NOT required for this or any theorem here, and is factored out to the standalone appendix periodic_factor_fine_structure.md. In brief: non-constancy is measured (R3, osc(P)/P̄ ≈ e^{−α/(−þ_g)}, where osc(P) := sup P − inf P is the oscillation — the total amplitude — of the log-periodic prefactor P, and P̄ its mean) and reduced to an explicit Fourier criterion (generic non-constancy is a theorem; universal is probably false — tiling obstruction); the oscillation size has a PROVEN unconditional linear bound osc(P) ≤ (−þ_g)·TV(P′)/6 (⟹ effective constancy on the grids of the estimated calibrations — all the application needs) and an open sharp exponential bound. See the appendix.

Theorem 3 (Theorem A3 (sharp form and closed-form amplitude for q_∞ > 1: the income channel). [PROVEN-HERE mod F1 (F2 since TRANSCRIBED), L10; review-verified R4: independent re-derivation with NO (I)-identity imposed and ASYMMETRIC atoms; 6 new calibrations to ≤0.2%; μ₃ proven to enter only one order deeper])

Under A0–A6, σ² > 0 and q_∞ > 1:

\lim_{\wbar \to \infty} \wbar \cdot g(\wbar) = B, \qquad B = \kap \cdot (\rho+1) \cdot \sigma^2 / ( 2 \cdot (\Rcal \cdot \ThornG - 1) ) > 0.

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So x is genuinely regularly varying of index −1 — no lattice caveat — and the amplitude is a closed-form prudence (Jensen) coefficient: ρ(ρ+1)/2 is the second-order Taylor coefficient of u′ (relative prudence ρ+1, times ρ/2), and the 1/(ℛÞ_Γ−1) factor is the resolvent of the renewal sum, diverging as q_∞ ↓ 1 (crossover/resonance). Two flags in stage_A_proof.md §8: F1 = the compounded constant bookkeeping of the L4→L10 chain (now machine-verified: the exact rearrangement identity and the B-formula are independently reproduced by a CAS from the raw Euler equation — appendix §2); F2 = citation transcription for L0/L1 — since TRANSCRIBED AND MAPPED (stage_A §8 flag table + stage_B §B11; residuals bibliographic only). Neither is structural. [Numerical status: CONFIRMED to 0.2% at the guard-clean depth — independent-EGM deep-tail measurement at case A, m = 3e4 (x/c ≈ 8e-10, comfortably above the 1e-10 fp64 cancellation guard; the previously-quoted m = 1e5 point sits marginally below the guard and is kept only as corroboration, per review R2 F-9): w̄·x = 0.02811/0.02975/0.03117 vs B(Var(θ_disc)) = 0.02805/0.02970/0.03111 at N_θ = 11/21/81 (amplitude tracks each discretization’s own Var(θ), ∝ σ² as claimed). Review R4 independently confirms B at six new calibrations (ρ ∈ {0.7, 1.5, 2, 5}, q_∞ ∈ {1.3, 2, 5}) to ≤ 0.2%, under strong skew of both signs and a θ=0 atom to ≤ 0.03%, with the (8.1)-residual ≤ 1e-6 certifying a and c_J jointly on the real solution. HARK 0.17 agrees to ~5%, and that residual is grid/interpolation bias, not solve tolerance (correction per R4 A3-5b: tightening tol 1e-6 → 1e-12 changes nothing; aXtraCount 800 → 3000 collapses the excess to +1% — a piecewise-linear cFunc under a concave c over-states the gap between knots; this strengthens A3: HARK also converges to B as its grid refines). The previously reported fitted amplitudes (≈0.6 HARK / ≈0.2 EGM) are crossover-window fit artifacts: replicated on ONE solution by the old windows (0.607 / 0.217), while the tail-only window returns 0.02806 at R² = 1.0000 — numerics_appendix.md §3. The w̄^{−1} exponent was always solid; the amplitude is now identified too, and the approach rate to B matches Remark 8.3’s w̄^{−(q_∞−1)} (R4: measured slopes match −(q_∞−1) to 4 digits across q_∞ ∈ {1.05, …, 1.5}).]

Corollary 1 (Corollary A4 (two channels, crossover, MPC rate). [PROVEN-HERE given A1–A3; expansions SKETCHED])

Under A0–A6, σ² > 0:

(Leading exponent.) −ln g(w̄)/ln w̄ → min(1, q_∞), with q_∞ from (E0); with permanent shocks the same statement is conjectured with q_∞ from (E) (Stage B).
(Crossover.) For q_∞ > 1, on any window [w̄_a, w̄_b] with w̄_b ≪ w̄_cross := (A/B)^{1/(q_∞−1)} (A the effective Kesten-channel amplitude from the boundary data) the local log-log slope reads ≈ −q_∞, migrating to −1 beyond w̄_cross — this is the grid-depth migration of Q_emp observed in the extrapolation harness.
(MPC rate.) c′(m) − κ̲ = −x′(w̄) = min(1,q_∞)·g(w̄)/w̄·(1+o(1)) — i.e. ≍ w̄^{−(1+min(1,q_∞))} [PROVEN-HERE via the convex-differentiation squeeze given the sharp forms; at q_∞<1 the constant genuinely oscillates with P — non-constancy is now a measured fact (R3), but the ORDER statement is unaffected: the sufficient margin holds ≥20× at every probed −þ_g, and osc(P)/P̄ ≈ e^{−α/(−þ_g)} collapses it at the estimated calibrations’ −þ_g].
(Knife-edge.) As Þ_Γ ↑ 1 (GIC knife-edge), −þ_g ↓ 0 so q_∞ → ∞: the Kesten channel steepens away and x ≍ w̄^{−1} purely; moreover the one-step rescaling w̄ ↦ Þ_Γw̄ − O(1) is near-additive on any bounded window w̄ ≲ 1/(1−Þ_Γ), which is exactly why exponential fits look locally adequate on bounded grids near the knife-edge (e.g. COL-TOP (code id CAL-CTOP) — the top, most patient atom of HAFiscal’s estimated college discount-factor distribution, described in final_proof_myst: growth impatience holds for it, but only barely, Þ_Γ sitting just below 1, so the near-additive window covers every grid depth used in estimation). Asymptotically the exponential is still wrong: the decay is a power law, bounded below at order w̄^{−1} by the forcing floor (stage-A Lemma 6.1).

Lemma 1 (Lemma A5 (eigenvalue uniqueness and selection; stated for general ψ, used in Stage B). [PROVEN-HERE, L13–L14; review-verified R2, and STRENGTHENED per R2 F-5])

Let ψ ≥ 0, E[ψ] = 1, E[ψ^{1+q}] < ∞ for q in a right-neighborhood of the relevant root. Define 𝔏(q) := ln E[ψ^{1+q}] − ln ℛ − q·ln Þ_Γ. Then 𝔏 is convex, 𝔏(0) = −ln ℛ < 0, and under plain GIC — no further hypothesis — 𝔏(q) → +∞ automatically (Jensen: E[ψ^{1+q}] ≥ 1, so 𝔏(q) ≥ −lnℛ + q(−þ_g); if the moment explodes at finite q_c, monotone convergence gives 𝔏 → +∞ there instead — either way a root exists; the earlier draft’s “GIC-Mod-type conditions” hedge is deleted). Hence there is a unique positive root q_∞, and automatically 𝔏′(q_∞) ≥ lnℛ/q_∞ > 0 — and the exact q_∞-tilted step mean equals 𝔏′(q_∞) itself (R2’s identity, verified to 1.9e-16), which is precisely the positive-drift condition the tilted renewal argument needs. Selection: solutions of the homogeneous gap equation behaving like w̄^{+|q|} are excluded by the a-priori bound x ≤ κ̲(h−1) — and this exclusion has bite: R2 exhibited a second, negative root q₋ = −1.327 of (E) for a ψ with mass near 0, whose branch grows like w̄^{+1.33}. The decaying branch with the positive root is the only admissible one, and its amplitude is pinned by matching at moderate w̄ (constructively: L9’s telescope).

3. Stage B: permanent shocks — the Kesten channel proper. [ATTEMPTED PROOFS in `stage_B_proof.md`; supersedes the former Conjecture B. Statuses per result; algebra machine-verified in `verify_algebra_B.py`]¶

Model: m_{t+1} = (R/(Γψ_{t+1}))a + θ_{t+1}; normalized Euler c^{−ρ} = Þ_Γ^ρE_t[ψ_{t+1}^{−ρ}c(m_{t+1})^{−ρ}]. Assumptions: A1–A2; (B-A3) (ψ,θ) i.i.d., ψ ⊥ θ, E ψ = E θ = 1, supp ψ ⊆ [ψ_min, ψ_max] ⊂ (0,∞) (ψ_min > 0), supp θ ⊆ [0, θ_max] (zero-income atom permitted, as in A3), nondegeneracy σ_B² := Var(ψ(θ+h−1)) = E[ψ²]σ_θ² + h²σ_ψ² > 0; FHWC+RIC+GIC; (B-A0) existence for the ψ-model (BST route; FVAC^ψ βΓ^{1−ρ}E[ψ^{1−ρ}] < 1 — genuinely stronger than FVAC, assumed, finite since ψ_min > 0); (B-NA) (only for Thm B2) the step S := ln(ψ/Þ_Γ) is non-arithmetic: supp S ⊆ λℤ for no λ > 0. [Restated per review RB3: the first draft’s “ln ψ non-arithmetic” was the wrong object — RB3’s solved counterexample ψ = Þ_Γe^{±0.2} satisfies it while the walk is arithmetic (span 0.2) and the KRT fails. Any ψ with a density satisfies (B-NA); sufficient Þ_Γ-free version: ln ψ in no shifted lattice a + λℤ.] κ̲ and h are UNCHANGED by ψ (the ψ-weights cancel in the limiting-MPC computation; E ψ = 1 in human wealth); q_∞ = the unique positive (E)-root E[ψ^{1+q}] = ℛÞ_Γ^q (Lemma A5, already ψ-general, with the corrected tilted mean 𝔏′(q_∞)). Master identity (machine-verified): ψ_{t+1}w̄_{t+1} = Þ_Γw̄ + W_{t+1} + ℛg(w̄) with W := ψ(θ+h−1) − h, E[W] = 0, Var(W) = σ_B² — permanent shocks force the gap through human-wealth revaluation (the h²σ_ψ² term), and the ψ^{−ρ} Euler weight cancels the rescale Jacobian exactly, so Stage-A’s linearization bookkeeping carries over verbatim.

Theorem 4 (Theorem B1 (two-sided order; leading exponent min(1, q_∞), q_∞ from (E)). [per-direction statuses, §B5])

(i) q_∞ > 1: cσ_B²w̄^{−1} ≤ g(w̄) ≤ Cw̄^{−1} — both PROVEN (F-B1 CLOSED: the upper bound now runs on a pathwise per-step multiplicative bound + one-step moments with geometric slack e^{𝔏(1)/2}; no restart, no Hölder transfer — §B5); (ii) q_∞ < 1: c·w̄^{−q_∞} ≤ g(w̄) ≤ C_δ·w̄^{−(q_∞−δ)} for every δ > 0 — both PROVEN (lower: q_∞-tilt first-passage with pathwise overshoot ≤ S_max and explicit constants, base anchored by the forcing floor (stage-B Lemma B-6.1); upper: defective-tilt supersolution). Exact-exponent upper now in BOTH step regimes: ⟸ Thm B2 under (B-NA), and ⟸ Thm B2-arith in the arithmetic case (GAP-B-arith CLOSED 2026-07-06); (iii) q_∞ = 1: cσ_B²w̄^{−1}ln w̄ ≤ g(w̄) ≤ w̄^{−1}(a ln w̄ + b) — both directions PROVEN, LOSS-FREE (GAP-B-res-upper CLOSED 2026-07-06): the missing lemma — exponential moments of Σ_{j<ν}1/w̄_j under the tilt, uniform in the start — is proven as (T6) via new band-occupation exponential tails on the ladder ((T5): one-sided domination + λ = q_∞ Chernoff + layer-cake), with the admissible λ scaling LINEARLY in the stopping threshold; the assembly runs on an exact resonance cancellation (e^{−U_i}/w̄_i = w̄^{−1}∏(1−ε_j)^{−1} pathwise), the mirror bound at the stopped step, and Cauchy–Schwarz against the geometric P̌(ν > i). Every constant explicit; machine-verified with pre-registered falsifiers (verify_resupper_checks.py, ALL PASS). The η-loss first pass (x ≤ C_ηw̄^{−1+η}ln w̄ ∀η > 0, an honest self-correction of the first draft’s overclaim) is retained in §B5 as history. [RB2’s exact-resonance numerics — w̄·x = C ln w̄ + D at R² = 0.99998 over three guard-clean decades, effective η ≤ 0.0064 — are now confirmations of a proven form.]

All probabilistic inputs reduce to 𝔏’s convex geometry: the q-tilted step cgf is Ě[e^{λS}] = e^{𝔏(q_∞+λ)} (S = ln(ψ/Þ_Γ)), giving drift, supermartingale, occupation, and exponential-functional bounds (§B4, all PROVEN).

Theorem 5 (Theorem B2 (sharp constant, q_∞ < 1, non-arithmetic STEP). [PROVEN-HERE — F-B2 CLOSED and SIMPLIFIED per review RB3 (direct pointwise route; smoothing demoted to a robustness alternative); residual = the section-level KRT cite, transcribed with hypothesis checklist in §B11])

w̄^{q_∞}·g(w̄) → A ∈ (0, ∞),    A = 𝔏′(q_∞)^{−1}·∫_ℝ F̂(s) ds
(boundary-data-dependent; the display is LITERAL — F̂ ∈ L¹(ℝ)).

A TRUE constant — Stage B is CLEANER than Stage A here: the random log-step S is non-arithmetic under (B-NA), killing the lattice pathology, so Theorem A2’s periodic prefactor degenerates to a constant. Proof: reduce (B5.1) to a whole-line renewal equation under the exact q_∞-tilt; pointwise envelopes for ALL forcing pieces — the jitter piece Δ₂ has a DIRECT envelope from the one-sided-slope convexity inequality fed the proven B1(ii)-δ bound (review RB3’s simplification: the first draft’s premise “Δ₂ has no pointwise envelope without x-regularity”, the entire rationale of former FLAG F-B2, was FALSE) — then the key renewal theorem on the UNSMOOTHED equation. No doubling lemma anywhere (no circularity with the order bounds). The Goldie-smoothing + monotone-de-smoothing route (the original F-B2 closure) is retained in §B7 as a robustness alternative. Numerics (RB3): the A-representation reproduces the measured constant end-to-end to 0.04% on a solved model; kernel-independence to 4.5e-5.

Theorem 6 (Theorem B3 (sharp closed-form amplitude, q_∞ > 1). [PROVEN-HERE (F-B1 closed ⟹ no inherited flag; residual = Stage-A-style constant-bookkeeping only, machine-verified at both pillars), §B8; closed form CAS-verified from the raw ψ-Euler])

\wbar \cdot g(\wbar) \to B_\psi = \kap(\rho+1) \cdot \sigma_B^2 / ( 2(\Rcal\ThornG - \E[\psi^2]) ), \qquad \sigma_B^2 = \E[\psi^2]\sigma_\theta^2 + h^2\sigma_\psi^2.

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Denominator = ℛÞ_Γ(1 − e^{𝔏(1)}) > 0 ⟺ q_∞ > 1; diverges at the q_∞ = 1 resonance — with the divergence law now pinned: B_ψ·(q_∞−1) → κ̲(ρ+1)σ_B²/(2·E[ψ²]·𝔏′(1)) as q_∞ ↓ 1, identifying the resonance constant’s normalization as E[ψ²]𝔏′(1) = E[ψ²S] — the ψ-general analogue of Stage-A’s descent rate −þ_g (DERIVED two ways in the RB fix pass — the (B8.1) resonance balance and this boundary law — and measured to 0.02%/0.3%; reconciles RB2-vs-RB4, §B8 remark + fixpass_checks.py; the q_∞ = 1 LIMIT itself is now PROVEN — Theorem B-res below). Reduces to Theorem A3 at ψ ≡ 1; depends on ψ only through E[ψ²] (ψ-skew-blind — machine-verified with skewed rational atoms). Notably, (B-NA) is NOT needed: the w̄^{−1} channel is forcing-anchored for random rescales too (the associated homogeneous equation would need E[ψ²] = ℛÞ_Γ, i.e. q_∞ = 1) — genuinely tested by RB4’s exact-lattice ψ designs (supp S ⊆ λℤ by construction, spans 0.2–1.0: amplitude matches to ≤ 0.1%, lattice tones ≤ 2.3e-6 and depth-decaying; the appendix’s earlier “arithmetic 2-atom” evidence was mislabeled — that ψ is non-arithmetic under the corrected criterion below). (Signed subleading at q_∞ < 1: B_ψ < 0 — SKETCHED as an asymptotic statement; the u²-order ALGEBRA producing the signed closed form is now machine-exact (RB4). It drives the Q_emp non-monotonicity reading of the HARK-default calibration, appendix §8. Bonus evidence (RB1): a GIC-VIOLATING calibration’s measured tail amplitude matches B_ψ to 0.02% — B3’s GIC hypothesis looks weakenable; recorded as conjecture-with-evidence in §B8, NOT claimed.)

Theorem 7 (Theorem B2-arith (q_∞ < 1, ARITHMETIC step: the lattice companion). [PROVEN-HERE 2026-07-06, §B7 — GAP-B-arith CLOSED, mod the classical arithmetic whole-line renewal cite (Gut/Asmussen, per RC3); REFUTER-REVIEWED (RC3, 0 broken; verified on a harder 3-atom aperiodic lattice); battery verify_arith_checks.py ALL PASS])

Under B-A0–A6, σ_B² > 0, q_∞ < 1, and supp S ⊆ λℤ with span λ (the arithmetic complement of (B-NA)): uniformly over the phase (the position of ln w̄ modulo the lattice span λ),

\wbar^{\qup} \cdot g(\wbar) = P_B(\ln \wbar) + o(1), \qquad P_B(\phi) = (\lambda/\Lfun'(\qup)) \cdot \sum_{m \in \mathbb{Z}} \hat{F}(\phi + m\lambda),

(5)

P_B λ-periodic, continuous, positive; mean over one period = Theorem B2’s constant formula 𝔏′(q_∞)^{−1}∫F̂; Fourier coefficients EXACT: c_k(P_B) = 𝔏′(q_∞)^{−1}F̂^(2πk/λ). The §B7 reduction is exact with lattice steps (all drift corrections live inside F̂), so the phase is exactly preserved and the classical lattice renewal theorem applies verbatim — stronger than the “subsequence version” previously anticipated in the bridge note below. At ψ ≡ 1 this DEGENERATES to Stage-A Theorem A2’s telescope (P_B = P; span λ = −þ_g), giving Stage-A’s periodic factor the exact Fourier representation c_k(P) = (−þ_g)^{−1}F̂^(2πk/(−þ_g)) — the rigorous form of L9′(a)'s “k = ±1 Mellin residue” object. Oscillation SIZE deliberately not quantified (the L9′(b)-analogue; still open). End-to-end phase-resolved verification on an exact-lattice design: 0.116% at m ≈ 3e5, decreasing in depth.

Theorem 8 (Theorem B-res (q_∞ = 1 sharp log-law). [PROVEN-HERE 2026-07-06, §B8 — GAP-B-res CLOSED; consumes the loss-free B1(iii); REFUTER-REVIEWED (RC2, 0 broken — first adversarial pass on the B-res apparatus; C_B confirmed by two independent symbolic derivations at skewed resonance); battery verify_resonance_const_checks.py, 7 pre-registered falsifiers, ALL PASS])

Under B-A0–A6, σ_B² > 0 and q_∞ = 1 (⟺ E[ψ²] = ℛÞ_Γ):

\wbar \cdot g(\wbar)/\ln \wbar \to C_B := \kap(\rho+1) \cdot \sigma_B^2/(2 \cdot \E[\psi^2] \cdot \Lfun'(1)), \qquad |\wbar \cdot x/\ln \wbar - C_B| \leq C_{\text{fin}}/\ln \wbar,

(6)

𝔏′(1) = E[ψ²S]/E[ψ²] the tilted drift. Proof: the exact tilted one-step identity W₁ = Ě_t[W₁(w̄_{t+1})] + c_J^B/ℛ + δ, |δ| ≤ C(1+ln w̄)/w̄, telescoped along the TRUE tilted chain to a stopping band; Wald pins the expected rung count (u + O(1))/𝔏′(1); the error sum is w̄-free by (T5′)-banded first moments. Measured: C_fit = 33.923 vs C_B = 33.929 (0.02%) at the exact-resonance calibration. The ψ ≡ 1 corollary closes Stage-A L11-const: w̄·x/ln w̄ → κ̲(ρ+1)σ²/(−2þ_g) at q_∞ = 1 (R2’s Richardson-corroborated 0.0272294 target is now a theorem’s value; stage_A_proof.md Lemma 6.4 updated by reference).

Side conditions and the discretized-ψ bridge (§B9)¶

E[ψ^{1+q_∞}|ln ψ|] < ∞, E[ψ^{−ρ}] < ∞: trivial under (B-A3); stated and moved past. (B-NA) for computational (discrete) ψ [criterion CORRECTED per reviews RB3/RB4 — the first draft’s version was wrong in both directions]: the condition is on the step values s_i := ln ψ_i − þ_g — an N-atom ψ is arithmetic iff all s_i lie in one lattice λℤ (equivalently: all pairwise ratios s_i/s_j rational); it involves Þ_Γ, not the ψ-atoms alone. Consequences: a 2-atom ψ is arithmetic iff s₁/s₂ ∈ ℚ — generically FALSE (the old “always arithmetic for 2 atoms” is retracted); incommensurable log-atom DIFFERENCES are sufficient for non-arithmeticity, never necessary; where genuinely arithmetic, the span is the gcd-type generator of the {s_i} (S = ±λ has span λ, not |ln(ψ₁/ψ₂)| = 2λ) and w̄^{q_∞}x carries a span-periodic prefactor — now a THEOREM (B2-arith above), with the prefactor explicit: P_B = (λ/𝔏′)Σ_mF̂(·+mλ). Its SIZE is expected ≈ e^{−α/span}-small by the R3 law, and RB3’s solved true-lattice case measured < 1e-7 at span 0.2 (≥ 3 orders below the naive transfer) — the size bound remains the open L9′(b)-analogue. Numerics may treat A as constant; a theorem for a specific discretization checks the s_i ∈ λℤ condition and, where arithmetic, applies B2-arith. (L9′ itself is a separate queue item.)

4. Remarks¶

(What is new.) The literature proves c(m)/m → κ̲ and c′(m) → κ̲ (BST lemma-MPCBoundsConvg; Ma–Toda 2022 JMathE, vol. 98 art. 102562, under regularly-varying u′) and the dual wealth-tail Pareto exponents (Beare–Toda 2022 ECMA; MST 2020 JET Thm 3.3 (CITE-CHECK number); Stachurski–Toda 2019 JET), but no rate or form for the primal gap. Even the LEVEL convergence x → 0 (our L3) appears not to be stated in the literature (the ratio and derivative limits do not imply it); it falls out of the gap equation in two lines.
(Primal ≠ shadow of dual.) At ψ≡1 with bounded θ under GIC, the ergodic wealth distribution has compact support (the wealth recursion is a contraction with bounded innovations), so there is NO wealth Pareto tail — yet the consumption gap is a power law. The two power laws are distinct objects sharing a root family: dual root E[(Þ_Γ/ψ)^ζ] = 1 (no positive root at ψ≡1) vs primal root E[ψ^{1+q}] = ℛ·Þ_Γ^q (root exists at ψ≡1 thanks to the level ℛ > 1).
(Where the power law comes from.) The gap equation’s rescaling w̄ ↦ Þ_Γw̄ + O(1) is multiplicative; in ln w̄ it is a random-walk/renewal structure — so the decay is a power law, not an exponential, which would require additive-in-m dynamics, i.e. Þ_Γ = 1, the GIC knife-edge (Cor. A4.4). This is the rigorous version of the derivation’s §2.
(Sharpness of A1’s constants.) The σ² in the lower bounds is sharp in order: as σ² → 0 the model degenerates to PF and x ≡ 0. After the R2 F-3 repair, the w̄^{−q_∞} lower bound in (ii) carries its σ² explicitly (the ladder base is priced by the forcing floor’s c₋σ²/y_bdry — stage-A Lemma 6.1 — rather than by qualitative strict-Jensen positivity), so the σ²-degeneration of the (ii)-constant is now a formula, not an allusion to boundary data.
(Both constraints allowed.) The artificial constraint a ≥ 0 is assumed for concreteness; with the natural borrowing constraint (θ_min > 0) all statements hold verbatim above the respective m̄ — the asymptotics never see the constraint directly (excursion bound), only through the boundary data (hence through P and the q_∞<1 constants, NOT through q_∞ or B).
(Fine structure of P — factored out; NOT required for any theorem here.) Whether P is constant and how large its oscillation is are refinements owned by the standalone appendix periodic_factor_fine_structure.md. In brief: “P constant” is numerically REFUTED (R3, osc(P)/P̄ ≈ e^{−α/(−þ_g)}, α ≈ 0.8–1.2) and reduced (via Theorem B2-arith at ψ≡1) to the explicit criterion c_k(P) = (−þ_g)^{−1}F̂^(2πk/(−þ_g)); generic (a.e.) non-constancy is a theorem, universal is probably false (tiling obstruction). The oscillation SIZE has a PROVEN unconditional linear bound osc(P) ≤ (−þ_g)·TV(P′)/6 (⟹ effective constancy at the estimated calibrations’ −þ_g, extrapolated ~10^{−10}–10^{−41}, which is all the application needs) and an open sharp exponential bound osc(P) ≤ Ce^{−c/(−þ_g)} (reduced to strip-analyticity of F̂, c = 2πw with w the analyticity-strip half-width — a local symbol, not the perfect-foresight-wealth coordinate w̄ — measured þ_g-independent w ≈ 0.15). None of this is consumed by A1/A2/A3, B1–B3, B-res, the forcing floors, or the HAFISCAL/HARK application.
(Effective, not practically tight — rephrased per review R2 F-8.) All upper/lower bounds are effective: every constant is an explicit function of (ρ, β, R, Γ, θ_min, θ_max, σ²) via the proof’s threshold (written x₀ in the proof documents; our w̄₀ — the proof documents write the wealth coordinate as x = m + h and the gap as g(x); their g(x) is our g(w̄)), C₀, K̂ (no compactness or soft arguments anywhere in L2–L11). But the compounded constants are astronomically conservative (R2 measured e.g. Π̄ = e^{K̂S̄} ≈ 1e14 at case A, giving upper constants ~19 orders above truth; the F-3-explicit q_∞<1 lower constant costs ~9 orders), so the finite-w̄ BOUNDS are practically vacuous on estimation grids. What is practically meaningful is the ORDER content plus the tight local up/down ladder inequalities (the displays tagged (five-up)/(five-dn) in final_proof.md; MyST labels eq-five-up/eq-five-dn in final_proof_myst.md), which hold at their design tightness 1 + K̂/w̄ (R2’s audit). The effectiveness claim is about the absence of soft steps, not about usable finite-w̄ error bars.
(Unbounded θ.) L4′ (Lemma 5.3): if supp θ is unbounded (θ ≥ 0) but E[θ^k] < ∞ for some k > max(1, q_∞) + 1, Theorems A1–A3 hold verbatim (PROVEN-HERE, REFUTER-REVIEWED RC4 — RC4-F1 fix incorporated: the anchor carries an explicit y_var entry for the fat-tailed-θ variance floor): (A) the workhorse holds on the good event A(w̄) = {θ ≲ Þ_Γw̄} with truncated moments → the full moments (A^c-tails at relative order w̄^{−(k−1)}, below every ladder slot iff k > max(1,q_∞)+1); (B) the UPPER chain survives verbatim (it consumes only the pathwise lower bracket w̄_{t+1} ≥ Þ_Γw̄ − 1, θ ≥ 0); (C) the LOWER chain runs on a depth-adapted truncation ladder y_{j+1} = Þ_Γy_j + y_j^{1/2} + C anchored at a FIXED level y_anc ≥ 2y★ (it stalls at the fixed point y★, not at w̄₀) — exponent-preserving and with a positive-constant probability product, both numerically confirmed (verify_L4prime_checks.py). The lognormal benchmark has all moments; the discretized-lognormal numerics are covered by the bounded-support theorems as stated.
(Mortality.) With survival probability L (perpetual-youth), replace β by βL throughout: Þ_Γ = (βLR)^{1/ρ}/Γ, q_∞ = ln ℛ/ln(1/Þ_Γ) — matching qstar_discrete in the harness. The Blanchard annuity variant additionally rescales R; either way mortality enters ONLY through Þ_Γ (and Þ_R).
(Practical implication, unchanged.) The principled tail extrapolator is x ≈ C·w̄^{−q} with q = min(1, q_∞) (Theorems A1–A3), as implemented in the HARK PR (decay_extrap_form='powerlaw'); the slope-matched Q_emp = B·(m_top−1+h) sits near q_∞ on short grids and migrates to min(1,q_∞) on deep ones (Cor. A4.2) — the harness’s observed migration is now a theorem-backed diagnostic, not a stylized fact.

5. The constraint end (bottom knot): the `q_0` power law¶

Sections 2–4 characterize how c(m) approaches its high-wealth asymptote (the perfect-foresight line κ̲(m−1+h), from below, as a power law with the eigenvalue exponent min(1,q_∞) — Theorem A1, Corollary A4). This section is the mirror at the other end: how c(m) approaches its constraint-end asymptote (the maximal-MPC line, from below, as m falls to the borrowing constraint). The two ends turn out to be structurally different (Remark C1): the high-wealth end is governed by a nontrivial eigenvalue with a possible log-periodic prefactor (Theorem A2); the constraint end is governed by the utility curvature ρ alone, with no periodic prefactor (Theorem CE). Full derivation and proofs: constraint_end_proof.md; pre-registered battery verify_constraint_end_checks.py (ALL PASS).

Setup and objects (transitory worst atom; ψ≡1). Add to A1–A6:

(C-A3) {θ_t} i.i.d. with a worst atom: θ = θ_min ≥ 0 with probability ℘ ∈ (0,1), and supp θ ⊆ [θ_min, θ_max]. The natural borrowing constraint is m̲ := −θ_min/(ℛ−1) ≤ 0 (the most one can owe and still repay under the worst income path; m̲ = 0 in the zero-income case θ_min = 0). Write excess resources m^e := m − m̲ > 0.

Derived (the constraint-end analogues of κ̲, h):

κ̄  := 1 − ℘^{1/ρ}·Þ_R      (the MAXIMAL MPC — BST eq-MPCmaxDefn; the m^e→0 limit of c/m^e)
λ   := ℛ·(1 − κ̄) = ℘^{1/ρ}·Þ_Γ < 1   (the worst-branch contraction rate toward the constraint)
γ(m) := κ̄·m^e − c(m) ≥ 0    (THE CONSTRAINT-END GAP; γ/m^e → 0)

κ̄ is the m→m̲ mirror of κ̲ (BST eq-MPCmaxDefn): c(m)/m^e → κ̄ as m^e → 0 (BST lemma-MPCBoundsConvg; eq-cBounds gives κ̲_t m ≤ c_t(m) ≤ κ̄_t m per period). Note λ = ℛ(1−κ̄) < 1 is the exact factor by which the worst-income branch maps m^e toward the constraint.

Proposition 2 (Proposition C2 (finite-horizon recursion; the T−1 anchor). [PROVEN-CITED (BST eq-MPCmaxInvApndxIter), MoM restatement; verified C3, |meas−pred| ≤ 4e-14])

The period-t maximal MPC obeys BST’s backward recursion, written here in constraint-end form:

\bar\kappa_{T-n}^{-1} = 1 + \mathsf{M}\,\bar\kappa_{T-n+1}^{-1}, \quad \mathsf{M} := \wp^{1/\rho}\ThornR = 1-\bar\kappa, \quad \bar\kappa_T = 1,

(7)

with closed form κ̄_{T−n} = κ̄/(1 − 𝖬^{n+1}). In particular the one-from-terminal MPC is κ̄_{T−1} = 1/(1+𝖬) = 1/(2−κ̄), materially above the stationary κ̄ — this is why a solver read at the terminal-adjacent horizon measures a larger constraint-end slope than the ∞-horizon formula (the recursion, not the stationary value, is the correct anchor at finite horizon).

Theorem 9 (Theorem CE (the constraint-end power law: the approach exponent q_0 = ρ). [PROVEN-HERE (local Euler analysis, constraint_end_proof.md); verified C2 (free-exponent fit), 5 calibrations ρ∈{1.5,2,2.5,3}, |q_0−ρ| ≤ 0.036])

Under A1–A6 and (C-A3), there is a constant K > 0 (explicit modulo boundary data — Remark C2) such that, as m^e → 0,

\cFunc(m) = \bar\kappa\,m^e - K\,(m^e)^{1+\rho}\,(1+o(1)), \qquad\text{equivalently}\qquad \bar\kappa - \frac{\cFunc(m)}{m^e} = K\,(m^e)^{\rho}\,(1+o(1)).

(8)

Write q_0 for the constraint-end approach exponent — the 0 subscript names the constraint (lower) end of the state space, the partner of the high-wealth q_∞ (subscript ruling 2026-07-17, superseding the 2026-07-14 arrows and the earlier q° ring form; code and the wider literature write q* for the high-wealth root — HARK’s API is unchanged). Then γ(m) = κ̄ m^e − c(m) ≍ (m^e)^{1+q_0}, equivalently the MPC deficit κ̄ − c/m^e ≍ (m^e)^{q_0}, and in the Method-of-Moderation coordinates μ = ln m^e, χ = ln((1−ω)/ω), χ(μ) = μ + b₀ + O(e^{q_0·μ}) as μ → −∞. Then q_0 = ρ, the CRRA (the smooth c ↦ χ change of variables preserves the exponent; b₀ = ln((κ̄−κ̲)/C), C = κ̄(h − 1 + m̲)). Unlike the high-wealth Theorem A2, there is no log-periodic prefactor: the worst-branch map is the deterministic contraction λ = ℘^{1/ρ}Þ_Γ, and the current-gap feedback term lifts the telescope degeneracy that produces P(·) on the high-wealth side (proof §3).

Corollary 3 (Corollary C4 (the bottom grid-design rule — mirror of q·hEx). [PROVEN-HERE given Thm CE; verified C4, predicted vs measured inside-asymptote m^e within 16%])

The high-wealth grid rule “the top knot must reach w̄ ≳ (A/B)^{1/(q_∞−1)}” has a constraint-end mirror. Since the relative deviation of c from the κ̄-line is (κ̄ − c/m^e)/κ̄ = (K/κ̄)(m^e)^ρ (Thm CE), the bottom knot m^e_0 sits inside the constraint asymptote to relative tolerance tol iff

m^e_0 \;\lesssim\; \bigl(\text{tol}\cdot\bar\kappa/K\bigr)^{1/\rho}.

(9)

So aXtraMin must be pushed below this scale for a solver to read the true constraint-end slope κ̄; a bottom knot placed above it reports a slope biased by O((m^e_0)^ρ), and a fitted-tangent extrapolation below such a knot mis-specifies the tail exponent (the constraint-end analogue of the Q_emp grid-depth migration, Cor. A4.2). This is the theory pin for the Method-of-Moderation bottom tail: its decay constant is ρ, not a fitted quantity.

Remarks (constraint end)¶

C1. (The two ends are structurally different.) High-wealth end: the one-step map m ↦ (Þ_Γ/ψ)m is random-multiplicative, so the gap solves a renewal/eigenvalue problem (root q_∞ of E[ψ^{1+q}] = ℛÞ_Γ^q) with a lattice/periodic prefactor. Constraint end: the binding one-step map is the single deterministic contraction m^e ↦ λ m^e (only the worst atom returns you to the constraint), so there is no eigenvalue to solve and no periodicity — the exponent is fixed by the curvature of u′ at the level of the Euler expansion, giving exactly ρ.
C2. (The amplitude K.) K > 0 is explicit modulo one boundary-data constant — the non-worst branches’ marginal-utility mass J₀ := Þ_Γ^ρ E_t[c(m_{t+1})^{-ρ}\mathbf 1\{θ_{t+1}>θ_min\}] at m^e = 0 — exactly as the high-wealth amplitude A (Thm A2) depends on boundary data; the closed skeleton is K = J₀κ̄^ρ/(ρ[(1−λ^ρ)/κ̄ + ℛ/λ]) (proof §2). The exponent q_0 (= ρ) needs no boundary data.
C3. (Permanent shocks — CLOSED 2026-07-14, see §5b below.) The ψ-general theory (Theorem CE-ψ, st-thm-CE-psi) proves q_0 = ρ under the uniform-contraction criterion ℘_eff^{1/ρ}Þ_Γ < ψ_min and characterizes the complementary regime as a renewal/eigenvalue problem — confirming both halves of the original conjecture. The remaining rigor item is GAP-CE-ψ-II (regime-II amplitude; st-rem-CE-regime).
C4. (Coordinate invariance.) Thm CE is stated invariantly in c-space (κ̄ − c/m^e ≍ (m^e)^ρ, verified directly). The MoM χ-coordinate rendering χ = μ + b₀ + O(e^{ρμ}) follows because ω, χ are smooth non-degenerate functions of the position of c between its bounding lines; the exact b₀/χ constants use the MoM ω-definition (consumed by the MoM implementation workstream, out of scope here).

5b. The permanent-shock extension (GAP-CE-ψ closure, 2026-07-14)¶

Proof bodies: constraint_end_proof_psi.md (F1/F2, the (CE-ψ) equation, the (E-ψ↓) eigen-equation, the two regimes). Pre-registered battery: verify_ce_psi_checks.py. Status tags as in §2.

Objects. Worst transitory atom ξ_min with mass ℘_w (zero-income case: ξ_min = 0, ℘_w = ℘); ψ ∈ [ψ_min, ψ_max], E[ψ] = 1; ℛ̃(ψ) = R/(Γψ).

m_min = −ξ_min/(ℛ̃_max − 1),  ℛ̃_max = R/(Γψ_min);   m^e = m − m_min
λ(ψ)  = ℘_w^{1/ρ}·Þ_Γ/ψ      (branch-wise contraction on the worst-transitory event)
℘_eff = the worst-JOINT-atom mass: ℘_w when ξ_min = 0; ℘_w·P[ψ = ψ_min] for discrete ψ
        with ξ_min > 0 (the fiber-selection effect, proof §4) — exactly HARK's
        `WorstIncPrb`.

Proposition C1-ψ (κ̄ is ψ-invariant). [PROVEN-HERE, proof §0(F2); battery gate B2] The growth-normalization weight (Γψ)^{−ρ} cancels exactly against the (Γψ)^{+ρ} carried by next-period resources on the reachable worst fiber, so for EVERY ψ-distribution

κ̄ = 1 − ℘_eff^{1/ρ}·Þ_R,

the ψ≡1 formula with the joint worst mass (matches BST eq-MPCmaxDef and HARK’s calc_mpc_max accounting).

Theorem CE-ψ (regime I: uniform contraction). [PROVEN-HERE at the ψ≡1 proof’s rigor level, proof §§1–2; battery gates B1/B4] If the worst-branch map contracts through the smallest permanent shock,

℘_eff^{1/ρ}·Þ_Γ < ψ_min        (equivalently  λ(ψ_min) < 1),

and the generic non-resonance w·E[λ(ψ)^ρ] ≠ 1 holds, the constraint-end approach exponent is unchanged by permanent shocks:

q_0 = ρ:   γ(m^e) = κ̄m^e − c(m) ≍ (m^e)^{1+ρ},   MPC → κ̄,   no log-periodic prefactor

(the ψ-mixture’s homogeneous root is strictly negative and excluded by 0 ≤ γ ≤ κ̄m^e, exactly as at ψ≡1; the mixture additionally smooths).

Characterization (regime II) + the remaining gap. [DERIVED; amplitude rigor OPEN = GAP-CE-ψ-II; battery gate B5 adjudicates empirically] If λ(ψ_min) > 1 (an expanding worst fiber — e.g. any fixed ℘_w with ψ-support reaching low enough ψ_min; the precise sense in which a continuous permanent component reinstates renewal), the homogeneous equation

(E-ψ↓)   w·(℘_eff^{1/ρ}Þ_Γ)^s·E[ψ^{−s}] = 1,   w ∈ (0,1) the feedback weight,

acquires a positive root s*₊, and the constraint end becomes a random-multiplicative (renewal) problem with

q_0 = min(ρ, s*₊).

Whether the (m^e)^{s*₊} mode carries generic nonzero amplitude is the open rigor item (GAP-CE-ψ-II); the battery’s regime-II specs test it empirically: fitted exponents at s*₊ confirm amplitude-genericity, fits pinned at ρ would refute it. The regime boundary ℘_eff^{1/ρ}Þ_Γ = ψ_min is a primitive, checkable criterion (HS-mean (code id CAL-HS): the worst JOINT atom is the lowest employed-income atom — θ_min ≈ 0.581 undercuts the unemployment income 0.7 — giving ℘_eff ≈ 0.020 and λ(ψ_min) ≈ 0.15: deep in regime I, so the estimated calibration keeps q_0 = ρ).