Grid design: final spec + the measured findings that forced it - Power-law decay of the buffer-stock consumption gap (simplified capstone under GIC-policy)

Status: DESIGN CONCLUSIONS OF RECORD (2026-07-10/11, owner dialogue following the G1 KILL). This document owns the post-KILL grid-design conversation: eight measured findings, the agreed final scheme (owner-proposed, measurement-sharpened), and the open items. Companion records, archived in the full corpus at BufferStockTheory-Latest:theory/powerlaw-decay/: grid_placement_design.md (the killed candidate’s pre-registration), grid_placement_p1_frontier_of_failure.md (the KILL), the P1 harness proto_grid_placement.py (all measurements below ran against it and hark_egm_adapter.py on the committed truth apparatus; consolidation into a single verify script is open item 1).

The findings ledger (each measured this arc, HS-mean/COL-TOP/GIC-CAP anchors)¶

Anchors and notation, once. HS-mean = a calibration using the estimated parameters of a representative high-school-educated household from HAFiscal (the mean discount factor of the High-School group); COL-TOP = the most patient college-educated type from the same estimation (the top atom of the College group’s discount-factor distribution); GIC-CAP = the patience-ceiling college calibration (the “GIC-cap” defined precisely in the archived full corpus’s final_proof, https://llorracc.github.io/BufferStockTheory-Latest/powerlaw-decay-theory/: an agent so patient that no target level of wealth exists). In the frozen verification code these appear under their original identifiers CAL-HS, CAL-CTOP, CAL-CCAP, which are pinned instruments and keep their names. Wealth is indexed by w̄ := b + h = m − 1 + h, perfect-foresight total wealth in BST’s notation (b := m − 1, BST’s bank balances; h = human wealth, the current period’s income included; the bar marks the PF agent’s upper-bound object, as in BST’s c̄). Scoping note for the simplified corpus: GIC-CAP is retired as a document calibration — the simplified exhibits use COL-MID (β = 0.99160, q_∞ = 0.5513) in its place; CAL-CCAP stays a pinned instrument in the frozen archive code, and the GIC-CAP numbers below stand as measured record.

Remark 2 (F2 — The owner’s mortality-adjusted stable points solve the pure-GIC anchor problem.)

With perpetual-youth replacement (newborns at a=0 drawing the same transitory income), the cross-sectional mean dynamics factor mortality EXACTLY as a return shave R → L·R in the stable-point loci. Existence conditions become GIC-Mod-Liv / GIC-Raw-Liv (the latter = BST’s mortality-adjusted growth impatience, the population-GPF<1 condition already in find_ergodic_distribution). Measured at the patience-ceiling college atom GIC-CAP (GIC-Mod fails, unadjusted loci: Trg nonexistent, StE 340.9): the L·R-adjusted balanced-growth root m̌_mort = 40.18 vs measured neutral-measure ergodic mean 41.23 (2.5%, Jensen sign correct since a(m) is convex); the Δm=0 version (59.4) anchors the raw-measure mean. The measure decides which locus gets the shave: Harmenberg-neutral aggregation ⇒ E_N[ψ⁻¹]=1 ⇒ the StE locus.

Remark 5 (F5 — Form convergence is unreachable; fitted exponents must not be trusted.)

Q₂ agreement with q_∞ to 10% needs m ≈ 3,200–3,800; to 5%, m ≈ 2×10⁶ (HS-mean); never in-window at COL-TOP/GIC-CAP. Measured drift rates w̄^(−0.14/−0.18/−0.25) — the slow pre-asymptotic drift near the q_∞→1 boundary (the amplitude-climb panel of the full corpus’s compactified-boundary figure, archived at https://llorracc.github.io/BufferStockTheory-Latest/powerlaw-decay-theory/, replayed in exponent space; approach from BELOW at depth, the signed-subleading prediction). The stabilization variant (|dQ/d ln w̄| small) certifies calm, not correctness: it triggers at a pseudo-exponent 20–40% from q (GIC-CAP: a placid Q≈0.63 vs true 1). Consequence: the tail exponent is the ONE object that must be imported from the theorem rather than measured — the forcing-floor clamp (stage-A Lemma 6.1 / stage-B Lemma B-6.1, in the archived engine files BufferStockTheory-Latest:theory/powerlaw-decay/stage_A_proof.md and BufferStockTheory-Latest:theory/powerlaw-decay/stage_B_proof.md) and the guarded_fit mode exist for exactly this.

Remark 6 (F6 — Sparse tail knots self-contaminate; zero tail knots dominates.)

With a dense body and n_tail ∈ {1..8} log-spaced knots to the corridor top, the solved VALUES at the sparse knots are wrong by 3–9% (their own EGM expectations interpolate across e-fold-wide cells); body metrics are unaffected at HS-mean and actively HURT at GIC-CAP (body-only M1 4.6e-3 beat body+8-tail 1.4e-2). A post-solve representation swap (log-gap-in-log-w̄ between the same knots) cannot help — both representations are capped by knot fidelity, not interpolation. The one clean knot is the body top (1.2e-4), and the analytic theory tail from it delivers ~10% over the whole segment (5× inside its own corridor). Consequence: NO tail knots, ever; the region between the body and any deep consumer is served by the law. (In-solve transformed interpolation could reclaim ~3× on tail knots only ≈ 10–20% total budget — not pursued.)

Remark 8 (F8 — Life cycle unifies through a backward amplitude recursion (OPEN: derive + validate).)

At each age the gap decays with exponent EXACTLY 1 (a finite sum of discounted adjustments), amplitude B_t = λ_t·B_{t+1} + F_t with B_T = 0 — a regime-independent finite-horizon fact. Under GIC-policy the per-step multiplier exceeds one (λ_B = E[ψ²]/(ℛÞ_Γ) > 1: the trio’s values are 1.0096/1.0047/1.0026), so as the horizon recedes the recursion DIVERGES: no finite infinite-horizon amplitude exists, the exponent-1 reading fails in the limit, and the infinite-horizon object is instead the w̄^{−q_∞} power law with its value accumulated at the journey’s end. The convergent case (λ < 1, a genuine fixed point B — the closed-form boundary value of the supercritical branch) is archived with the full corpus (https://llorracc.github.io/BufferStockTheory-Latest/powerlaw-decay-theory/). T−1 anchor = the one-period Kimball precautionary premium. Falsifiable check: w̄·g_t(w̄) → B_t age by age on dense finite-horizon solves.

THE SPEC (owner-proposed scheme, sharpened by F1–F8)¶

Placement: stock exp_mult (never tuned — F1) from the (age-varying, HARK- automatic via BoroCnstNat_t) constraint bottom to a_max.
a_max (infinite horizon, once-and-for-all): the wealth-mass-negligibility rule — the smallest top with negligible WEALTH-weighted mass above (agent-mass is the wrong measure: 0.66% of agents held 17.6% of wealth). Patience sensitivity enters automatically via ζ* (tail exponent: 9.19 HS-mean / 3.17 COL-TOP / none at cap), the mortality-adjusted anchors m̌_mort (F2) at the pure-GIC edge, and the most-patient- type-binds rule for heterogeneous populations (= the rule the existing production_aMax helper implements for the estimated calibrations, now with its theory). Constant grid (required by back-in-time-fast / Newton-FTI, TM methods, and estimation smoothness).
Life cycle: same spec; adapt only on the backward-recursive theory objects (κ̲_t, h_t, B_t, BoroCnstNat_t — computable BEFORE each age’s solve: no chicken-and- egg; h_t here follows HARK’s hNrm convention, the PDV of income beyond the current period, i.e. one less than BST’s age-t human wealth — the full corpus’s archived T−1 amplitude supplement writes its h_{T-1} in BST’s convention, = hNrm_{T-1} + 1); one union-span constant grid unless profiling demands per-age.
Above a_max (a GIVEN, per the owner): the power law — with the THEORY exponent q_∞ (< 1 under GIC-policy; infinite horizon) or the recursive amplitude B_t/w̄ (life cycle); never the fitted exponent (F5).
Tests (always run, the owner’s two regions): (i) near-kink Euler-residual panel (direct, cheap — where error originates); (ii) self-convergence at the visited set (one warm-started 2N re-solve + Richardson — F7; model-revision-robust).
Reports (never gates): corridor rel_gap(a_max), form distance |Q_fit/q − 1| (the honesty dial: 11% at HS-mean’s tol-1e-1 top, 37% at GIC-CAP’s), and the tail-law mode in use.
Parameter count: a_min, the a_max rule’s mass tolerance, N, two test tolerances — five interpretable numbers plus frozen shape and theorem-supplied law.

Definition 1 (F9 — The operator eigen-probe: a model-agnostic numerical estimator of q_∞ (owner request: exponent computable when no closed form exists).)

The exponent is the eigenvalue condition of the model’s OWN one-period backward operator on power-law perturbations of the PF line; measure it numerically: apply one Euler-inversion step to the trial c(m') = kappa*(m'+h-1) - eps*c(x0)*((m'+h-1)/x0)^(-s) (written with BST’s h; the committed script’s own h variable equals h−1), difference two eps (isolates the linear response, cancels the adjustment term), root-find the unit multiplier. Grid-free (the trial is analytic — probe at x0 = 1e6..1e8), hence immune to both diseases of estimation-from-solved-values (sparse-knot contamination; pre-asymptopia). Validated 6/6 by verify_eigen_probe_checks.py — a HARK-coupled battery, archive-only now (frozen record at BufferStockTheory-Latest:theory/powerlaw-decay/): q^ matches the analytic q_∞ to 5.6e-6 / 8.6e-6 / 5.0e-5 at HS-mean/COL-TOP/GIC-CAP with depth-consistency ~1e-6. Trial normalization at the probe point is load-bearing (an unnormalized epsx^-s trial at depth is ~1e-40, beneath float64 — the first implementation failed exactly there). Estimation FROM SOLVED VALUES is confirmed unreliable even in its best form (two-term A·w̄^−q + B·w̄^−1 fit on m-windows to 1e4: 15–40% off — with h ≈ 200 the identifying w̄-variation is not in any reachable window). Reported open question: outside GIC the probe (true finite-w̄ operator) and the analytic (E)-root (idealized limit) disagree (4.68 vs 7.33 at beta1.02); within hypotheses they agree to 1e-5. Portability requirements: the PF asymptote (kappa, h) + one backward step applied to a supplied next-period function — i.e., anything a time-iteration solver already has. THE SPEC amendment: for model variants without the analytic eigen-equation, the tail exponent comes from this probe (the eigen-equation is the probe evaluated on paper).

Definition 2 (F10 — The aXtraMax mass rule, made algorithmic (closes the “inconclusive aXtraMax conversations”): measure the wealth quantile; the closed form only covers; earlier estimation practice is reproduced as special cases of the explicit dial.)

The keystone object is the MORTALITY-AUGMENTED DUAL ROOT ζ_L, the unique ζ > 0 solving LivPrb · E[(Þ_Γ/ψ)^ζ] = 1: at LivPrb = 1 it is the Kesten root ζ*; for LivPrb < 1 it EXISTS whenever an expanding branch exists (P(Þ_Γ/ψ > 1) > 0) — including at the patience ceiling GIC-CAP where ζ* does not (mortality is what truncates the patient tail; measured: ζ_L = 9.74 / 4.34 / 1.92 at HS-mean / COL-TOP / GIC-CAP, ordering with patience as required). ζ_L > 1 ⟺ aggregate wealth finite = the existence condition for a wealth-measure top; ζ_L → 1 means wealth concentration makes any grid top economically arbitrary — refuse, don’t grid. The CLOSED-FORM Pareto inversion of ζ_L is a poor quantile estimator in exactly the regimes that matter (measured seed/truth: 0.38× at HS-mean — UNSAFE undershoot, the quantile sits only ~3× the anchor so the tail regime has not begun; 128× over at GIC-CAP’s ζ_L→1 hypersensitivity) — so it serves ONLY as the generous covering-grid top for the measurement. THE RULE IS MEASURED: coarse solve → deterministic neutral-measure stationary distribution with mortality reset (the same kernel the F2 loci are exact for) → the (1−ε_w) WEALTH-weighted a-quantile × small safety; certificate = re-measure on the final solve. Measured quantiles (truth solves, cover 1e5):

anchor	measure	ε=1e-2	ε=1e-3	ε=1e-4
HS-mean	agent	2.4	3.1	3.8
HS-mean	wealth	2.8	3.6	4.3
COL-TOP	agent	4.6	6.3	8.1
COL-TOP	wealth	5.6	7.6	9.8
GIC-CAP	agent	217	491	1,078
GIC-CAP	wealth	679	2,206	7,108

The dial is now explicit where earlier practice chose it implicitly: the agent-measure variant at conventional tolerances reproduces the quantile definitions previously used to size the estimation grids, and the wealth measure is the one the moment-consumers (Lorenz-type statistics) actually weight — impatient atoms need only body-scale tops (HS-mean ~4, COL-TOP ~10 even wealth-weighted at 1e-4) while the binding patient atom’s top is set by (measure, ε_w) alone. With the theory tail law attached, wealth above the grid is PRICED (corridor accuracy), not lost, so moderate ε_w (1e-2, cap-atom top ≈ 700) induces wealth-moment bias of order 1e-5 of aggregate wealth. Populations: max over atoms (most-patient binds). Implementation: pf_decay.aXtraMax_from_wealth_mass (+ dual_root(..., LivPrb=...)). The validation battery verify_wealth_mass_rule_checks.py was HARK-coupled and is archive-only now (frozen record at BufferStockTheory-Latest:theory/powerlaw-decay/); the ζ_L anchors and the wealth-quantile table above stand as archived record. The simpler corpus adds the COL-MID classical dual (Kesten) root ζ* = 4.8481, the unique ζ > 0 solving E[(Þ_Γ/ψ)^ζ] = 1 (battery verify_dualroot_checks.py).

Definition 3 (F11 — The C1 two-term attachment (owner ruling 2026-07-11: DEFAULT for explicit-exponent tails; motivation = SSJ-type Jacobian robustness).)

The owner accepted the Jacobian case for slope-matching and ruled: keep the one-term tail available, implement a two-term tail, make two-term the default, and name the reason — guarding against Jacobian pathologies in SSJ-type (sequence-space Jacobian) pipelines, where policy derivatives are primitive inputs and a C1 kink at the attachment point makes them discontinuous for queries crossing the cut (measured one-term kink at tol=1e-1 cuts: ~0.4% of κ at HS-mean, ~2.9% UPWARD at GIC-CAP — the upward case also locally violates MPC-monotonicity).

Form and closed-form amplitudes: with z = (x+h−1)/(x_cut+h−1) (x an m-like grid coordinate, h BST’s human wealth), level gap G and fitted rate Q_fit = S·(x_cut+h−1)/G (S = body slope minus κ) read at the cut, gap(z) = A·z^(−Q) + B·z^(−(Q+1)) with the two matching conditions (level A+B=G; slope Q·A+(Q+1)·B = Q_fit·G) giving

B = G\cdot(Q_{\mathrm{fit}} - Q), \qquad A = G\cdot(1 + Q - Q_{\mathrm{fit}}).

(1)

Collapse: Q_fit = Q ⟹ B = 0 — the one-term tail is the exact special case. Second exponent Q+1, NOT the theory-subleading pair {q_∞, 1} (the general treatment’s {min,max}(1,q_∞), collapsed under GIC-policy): that pair’s spacing |q_∞−1| can be small near the q_∞→1 boundary (every HAFiscal anchor is within 1% of it, F4) ⟹ amplitude blow-up and catastrophic cancellation; Q+1 is uniformly conditioned, and the second term is an ATTACHMENT (boundary-layer) object absorbing exactly the one-term kink (Q−Q_fit)·G/p — not an asymptotic claim (near the q_∞→1 boundary the true subleading acquires a log factor — main-chain Lemma L11 of the full corpus, archived at https://llorracc.github.io/BufferStockTheory-Latest/powerlaw-decay-theory/).

Properties (healthy knot G>0, Q_fit>0, A>0 ⟺ Q_fit<Q+1): gap > 0 everywhere (c below the PF line); c′ > κ everywhere (the MPC floor is preserved — the bracket Q·A + (Q+1)·B/z runs monotonically between Q_fit·G > 0 and Q·A > 0); concavity c″<0 holds at the cut iff Q_fit ≥ Q/2 (below that, a local convexity blip — document, don’t guard); leading exponent Q at depth. GUARD: Q_fit ≥ Q+1 (A ≤ 0: the body locally decays steeper than theory+1 — a coarse/broken top) ⟹ warn + one-term fallback. Rescue case (Q_fit ≤ 0): attaches smoothly with B<0, A>G; inherits the body’s sub-κ MPC on an initial segment (the C0 one-term rescue instead “corrected” it with an upward kink; smooth inheritance is the C1-default price, documented). Amplitude note: the deep amplitude is A = (1+Q−Q_fit)·G (≈12% under the one-term G at the HS-mean tol=1e-1 handoff, where Q_fit≈0.5 vs q=0.38) — the reason the two-term battery accuracy band is pre-registered at 1e-3 rather than inheriting the one-term 6e-4.

Life-cycle/infinite-horizon uniformity (owner ruling, same day): both settings use this same form with one number changed — Q = q_∞ infinite horizon, Q = 1 at every finite age; B_t stays diagnostic-only (the continuity invariant removed amplitude-anchoring symmetrically).

Open items¶

Consolidate the dialogue measurements (F1–F7) into a committed verify_grid_design_checks.py battery (they currently exist as inline session runs against the committed harness; each finding above states its construction).
Derive the exact (λ_t, F_t) of the F8 amplitude recursion; validate w̄·g_t(w̄) → B_t on dense life-cycle solves (T−1 Kimball-adjustment anchor; its closed form is recorded in the full corpus’s T−1 amplitude supplement, archived at https://llorracc.github.io/BufferStockTheory-Latest/powerlaw-decay-theory/t1-amplitude-supplement/); then it joins pf_decay as a per-age quantity next to the MPCmin/hNrm recursions.
HARK wiring of THE SPEC as a constructor + the two tests + the reports (a future small commit on the decay branch; the mNrm_stable_points helper with the L·R loci belongs in the same commit).
Union-vs-per-age grid profiling for life-cycle models (only if the union count is ever measured to be binding).

Footnotes¶

Document prose states the theory tail exponent as q_∞ itself throughout, per the simplified corpus’s standing GIC-policy assumption (ℛÞ_Γ < 1, hence q_∞ < 1). Library code keeps the general clip min(1, q_∞); under GIC-policy the clip is inactive.
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