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Alternative proof γ: the compactified domain and the boundary circle

SIMPLIFIED EDITION (2026-07-19), under the standing assumption GIC-policy: ℛÞ_Γ < 1, equivalently (via stage_A’s identity (I4) Þ_Γ^{−q_∞} = ℛ) q_∞ < 1. The full edition of this engine — which also proves the supercritical boundary value at q_∞ > 1 (Theorem γ-A’s one number B; Stage-B Theorem γ-B’s B_ψ) and the resonance q_∞ = 1 (Theorem γ-R) — is served at the archive: https://llorracc.github.io/BufferStockTheory-Latest/powerlaw-decay-theory/alt-proof-compactified/. Dropped material is cited there, never re-proved here. Provenance: the full edition passed an author pass (2026-07-07) and an independent refuter panel RF1/RF2 (2026-07-08) with ZERO BROKEN, ZERO MODERATE; every item surviving into this edition (γ0/γ1/γ2, γ-C1/γ-C2, γ-T restated at q < 1) was re-derived line-by-line by RF1 and exercised on hostile designs by RF2, and the in-place repairs touching surviving text (RF1-F3, RF2-N2) are retained below, each marked “(repaired per review …)”. The panel refereed the FULL document; this simplified restatement has not been separately refereed — its claim is consistency with the archived edition under the standing assumption. Packs: review/RF1_compactified_logic.md, review/RF1_altproof_compactified_verdicts.md, review/RF2_altproof_compactified_numerics.md (+ their RF*_check_* scripts).

Companion to statement_simpler.md (theorem statements and the shared assumption block, including the standing GIC-policy assumption) and stage_A’s foundations (BufferStockTheory-Latest:theory/powerlaw-decay/stage_A_proof.md §§1–5, the imported one-step apparatus; an archived engine file, cited as a repo path). Numerical verification with pre-registered falsifiers: verify_altproof_compactified_checks.py (+ saved output _out.txt beside it); results summarized in §10 — the suite re-registered for the q < 1 corpus, including a near-boundary crossover check (RT1 at q_∞ = 0.95) kept precisely because it measures the pre-asymptotic transient rather than hiding it.

§0. Scope, status, and the firewall

What this document is. The repo owner asked for a proof route in these words:

“compress the space from 0 to infinity by assuming the power law holds in the limit, so that there is a finite number (probably 1) that represents the exact limit as assets approach infinity — it might be easier to prove, and easier to understand, than the other proofs.”

This document makes that intuition rigorous. The coordinate z := 1/x compresses the wealth ray [x_a, ∞) to the half-open interval (0, z_a]; the point at infinity becomes the honest boundary point z = 0, which we adjoin. “Assuming the power law holds in the limit” is implemented as a choice of units: we study the compensated gap W(z) := x^q·g(x) with q := q_∞ (under GIC-policy q_∞ < 1, so the archived edition’s clip min(1, q_∞) is vacuous and never appears here). The headline theorem is then a geometry, not a single number: at the correct units the boundary multiplier is exactly neutral — λ(q_∞) = 1, the definition of q_∞ — and the forcing fades geometrically, so W settles into a bounded envelope

0 < m_∞ ≤ liminf_{z→0} W ≤ limsup_{z→0} W ≤ M_∞ < ∞

with a strictly positive floor. But the natural boundary is not a point: in log wealth the dynamics repeat with period −þ_g (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), so what survives at infinity is the position of ln x modulo −þ_g: a circle of phases, and the finite object is a boundary function on that circle, proven here at the envelope level, with the equicontinuity step honestly tagged GAP-γ-equicont (§6). The owner’s “one number (probably 1)” is thus, under GIC-policy, honestly TWO numbers — the envelope pair (m_∞, M_∞) — plus a phase. (In the archived supercritical case q_∞ > 1 the circle collapses to a point and one number B is exact; that case is excluded by the standing assumption and cited at the archive — see the §3 stub below.)

Standing assumptions. A0–A6 of statement_simpler.md (ψ ≡ 1, bounded θ, σ² > 0), plus the standing GIC-policy assumption ℛÞ_Γ < 1 ⟺ q_∞ < 1, throughout. Notation as in the companions:

ℛ = R/Γ > 1,  Þ_Γ = (βR)^{1/ρ}/Γ < 1,  κ̲ = 1 − (βR)^{1/ρ}/R,  h = ℛ/(ℛ−1),
x = m − 1 + h,  g(x) = κ̲x − c(m) ∈ [0, ḡ],  ḡ = κ̲(h−1),  þ_g = ln Þ_Γ,  q_∞ = ln ℛ/(−þ_g),
σ² = Var(θ);   GIC-policy:  ℛÞ_Γ < 1  ⟺  q_∞ < 1
(for general ψ, q_∞ = the (E)-root — cited only in Remark 5.2's aside).

Here h is BST’s human wealth — the Γ-normalized PDV of labor income including the current period’s unit, with limiting value h = 1/(1 − ℛ^{−1}) = ℛ/(ℛ−1) — so the PDV of future labor income alone is h − 1 = 1/(ℛ−1). Two conversions connect this page’s letters to the reader-facing statement and proof pages. First, the wealth variable x := m − 1 + h is exactly their w̄: perfect-foresight total wealth (human and market), the consumer’s total perfect-foresight wealth, the bar marking the PF agent’s upper-bound object as in BST’s own c̄(m) = κ̲·(m − 1 + h); this page keeps the letter x because its coordinate apparatus (z := 1/x) and the chart convention it cites (v = 1/x) are built on it. Second — the notation bridge for the working dialect of this engine document: here x denotes the wealth coordinate and g(x) the gap g(x) = κ̲x − c(m) = c̄(m) − c(m), exactly their precautionary saving, which the reader-facing pages write as \psav = x(w̄) per the 2026-07-16 registry ruling — declared bridge, same object (their x(w̄) is this page’s g(x), NOT this page’s x). On this page the gap keeps the letter g because x carries the coordinate apparatus and the letter s is reserved for the free compensation exponent of §1.2.

Import list (everything consumed, nothing else). From the archived stage_A_proof.md (repo path in the companion block): L0/L1 (existence; c continuous, increasing, concave), Lemma 2/2′ (sandwich 0 ≤ g ≤ ḡ, strict g > 0 on m > m̄, g non-increasing convex, Euler equality on m > m̄), Lemma 3.1/Cor 3.2 (exact recursion (3.1) and excursion bound (3.2) with C₀), Lemma 5.1 (the one-step identity (5.2) with (5.3)–(5.5)), Corollary 5.2 (the upper/lower one-step comparison bounds, with K̂), Lemma 5.3 (= L4′, cited only in a robustness remark), Lemma 10.1 (= A5, cited only for the convex geometry of 𝔏 in §5). As a benchmark only (a target to match, never used in a proof): Theorem A2’s P. (The archived full edition additionally imports the Stage-B one-step block for its §7 and benchmarks against Theorem A3’s B, Theorem B3’s B_ψ, and Theorem B-res’s constant — all supercritical/resonance apparatus sitting outside GIC-policy.)

Firewall (engines deliberately NOT used). None of: stage_A §6–§8 (ladder/renewal machinery, Lemmas 6.2/6.3/7.2/8.1), Feller/AMN/Blackwell key-renewal theorems, the stage_B §B4 tilt toolkit (T1)–(T6) or its deployments, Goldie/Kesten implicit renewal (§B7), the B2-arith lattice Fourier apparatus. Every lemma below is proven from the import list plus elementary real analysis. (The full edition’s one honest near-miss — the Stage-B boundedness lemma γ-B2’s stopped expectation-unroll, a path-sum-flavored argument — is archived with its §7 and does not occur in this edition.)

Kinship note. Two sibling alternative-proof documents (a probabilistic path-sum route and a recursion-tree/Akra–Bazzi route) are being written independently; by construction this document shares with them only the one-step identity and the geometric resolvent — see §9 for why that sharing is forced. Their files were not read.

Status vocabulary. PROVEN-HERE / PROVEN-CITED / SKETCHED / GAP / OPEN, as in the ledger.

Internal red-team disclosure. Before the full edition was committed, an adversarial self-pass ran the brief’s attack list (shell-index bookkeeping; the phase-displacement control of §6; fat-tailed θ; plus items pertaining to the archived sections — the Cesàro step, the boundary cases q_∞ ∈ {1±δ}, the Stage-B forward window). Of its catches, the one touching surviving text is a missing cross-term factor in (γ6.1), fixed in place; and it kept a registered numerical falsifier FAILED (RT1 at q_∞ = 0.95, §10) rather than repairing it post hoc — that measurement survives here, restated as the registered crossover/transient expectation of Remark 6.2. The independent (non-author) refuter pass was completed 2026-07-08 (RF1/RF2 — see the header block above); its repairs to surviving text are applied in place.

Reading order if you only read one page: §9.

§1. The compactified coordinate, the compensation, and the shell equation

1.1 The coordinate and the boundary point

Fix an anchor x_a ≥ x₀ (pinned in (γ1.4) below; x₀ is stage_A’s (5.0) threshold) and set

z := 1/x,     z_a := 1/x_a,

mapping [x_a, ∞) homeomorphically onto (0, z_a]. Adjoin the boundary point z = 0 (“infinite wealth”). This is the one-dimensional instance of the classical Poincaré compactification of dynamical systems — the chart at infinity there is literally v = 1/x, with the line at infinity {v = 0} (Dumortier–Llibre–Artés 2006, ch. 5; coordinates verified against the source, see References). Everything below happens on [0, z_a]; we freely write W(z) and W(x) for the same object.

1.2 The compensation: “the power law holds in the limit” as a choice of units

For a free exponent s > 0 define the s-compensated gap

W_s(x) := x^s·g(x)   ≥ 0,                                                  (γ1.1)

and reserve W := W_q for the theorem’s compensation q := q_∞ (GIC-policy makes the archived edition’s clip min(1, q_∞) vacuous). Two elementary facts, used silently: W_s ≤ x^s·ḡ < ∞ on every bounded x-set (Lemma 2), and W_s is continuous on (0, z_a] (c is continuous, L0/L1). The whole content of the theorems is the behavior of W_s at z = 0.

1.3 Shells marching to the boundary, and the ζ-enlargement

The dynamics rescale wealth by Þ_Γ per period ((3.1): x_{t+1} = Þ_Γx + (θ−1) + ℛg(x)), so the natural decomposition of (0, z_a] is into geometric shells. In x-coordinates:

X_n := [x_a·Þ_Γ^{−n}, x_a·Þ_Γ^{−(n+1)}),    n = 0, 1, 2, …                 (γ1.2)

In z these are S_n = (z_aÞ_Γ^{n+1}, z_aÞ_Γ^n]: n → ∞ is exactly z → 0. One period of ln x per shell: −þ_g.

The one-step comparison (Cor 5.2) evaluates g at the displaced points Þ_Γx ∓ C₀, and the displacement C₀ is constant in x ((3.2)); relative to the shell width (which grows like Þ_Γ^{−n}) it vanishes, but near a shell edge it can cross the boundary. The clean fix is to enlarge every shell by the lifetime-displacement constant

ζ := C₀/(1−Þ_Γ)      (already named ζ in stage_A (5.0a)),                   (γ1.3)

whose defining property is that it is the fixed point of the displacement recursion s ↦ Þ_Γ·s + C₀: Þ_Γζ + C₀ = ζ. Why this is the right constant: ζ = C₀ + Þ_ΓC₀ + Þ_Γ²C₀ + … is the largest total displacement the affine recursion can accumulate over an entire trajectory, so a ζ-margin absorbs the worst case exactly, with no slack to iterate. Define the enlarged shells

X_n^ζ := [x_a·Þ_Γ^{−n} − ζ,  x_a·Þ_Γ^{−(n+1)} + ζ].

Proof. Upper end: Þ_Γ·(x_aÞ_Γ^{−(n+2)} + ζ) + C₀ = x_aÞ_Γ^{−(n+1)} + (Þ_Γζ + C₀) = x_aÞ_Γ^{−(n+1)} + ζ, the top of X_n^ζ, using the fixed-point property of ζ. Lower end: Þ_Γ·(x_aÞ_Γ^{−(n+1)} − ζ) − C₀ = x_aÞ_Γ^{−n} − (Þ_Γζ + C₀) = x_aÞ_Γ^{−n} − ζ, the bottom of X_n^ζ. Displacements between −C₀ and +C₀ land between these. ∎

The payoff: the shell recursion below refers only to the previous enlarged shell — band width zero, no self-reference, no sliver bookkeeping. (This is where a naive version of the argument invites attack; the ζ-fixed-point makes the bookkeeping exact rather than approximate. The archived Stage-B analogue genuinely loses this property — see the §7 stub.)

Domain admissibility: fix

x_a := ζ + max{ x₀,  2C₀/Þ_Γ,  K̂,  D̂ }        (D̂ from Lemma γ1 below),      (γ1.4)

so every point of every X_n^ζ (n ≥ 0) lies in [x₀, ∞) where the one-step identity (5.2) and Corollary 5.2’s comparison bounds hold, and additionally x_a ≥ 3ζ (stage_A’s (5.0a) already contains the entry 2ζ, so x₀ ≥ 2ζ), whence inf X_{n+1}^ζ ≥ ½·x_aÞ_Γ^{−(n+1)}. Why each entry: x₀ = the imported validity threshold of §5; 2C₀/Þ_Γ makes the compensation ratio expandable (u ≤ ½ below); K̂, D̂ make the products of (1 + small/x) factors collapsible to one (1 + D̂/x).

1.4 The compensated one-step inequalities

Multiply Corollary 5.2’s upper and lower one-step bounds by x^s and write the displaced compensation ratio as

x^s·g(Þ_Γx ∓ C₀) = (x/(Þ_Γx ∓ C₀))^s · W_s(Þ_Γx ∓ C₀)
                 = Þ_Γ^{−s}(1 ∓ u)^{−s} · W_s(Þ_Γx ∓ C₀),      u := C₀/(Þ_Γx).

Define the boundary multiplier and the compensated forcing

λ(s) := Þ_Γ^{−s}/ℛ  =  Þ_Γ^{q_∞−s}  =  e^{−(q_∞−s)(−þ_g)},        F_s(x) := ℛ^{−1}·x^s·J(x).

Why λ(s) has this form: one step of the dynamics divides x by Þ_Γ^{−1} (one shell), so the compensation x^s gains a factor Þ_Γ^{−s}, while the Euler identity (5.2) divides the right side by ℛ; the second equality is stage_A’s identity (I4) Þ_Γ^{−q_∞} = ℛ. Note

λ(s) < 1 ⟺ s < q_∞,     λ(s) = 1 ⟺ s = q_∞,     λ(s) > 1 ⟺ s > q_∞.

In particular, under GIC-policy the s = 1 evaluation gives λ(1) = 1/(ℛÞ_Γ) > 1: over-compensating past q_∞ puts the recursion in its EXPANDING regime — that divergence is γ-T’s wrong-exponent signature (§5), not a contraction. (In the archived supercritical case the same number 1/(ℛÞ_Γ) sits below 1 and serves as the contraction device of the one-number theorem; under GIC-policy it certifies that over-compensation diverges.)

Proof. (γ-up): from Corollary 5.2’s upper bound, `W_s(x) ≤ (1+K̂/x)·λ(s)ℛ·ℛ^{−1}(1−u)^{−s}W_s(Þ_Γx−C₀)

1.5 The shell sequences

For n ≥ 0 define

M_n(s) := sup_{X_n^ζ} W_s,     m_n(s) := inf_{X_n^ζ} W_s,     δ_n := 2D̂·Þ_Γ^{n+1}/x_a.

Each M_n(s) is finite (W_s ≤ x^sḡ on the compact shell) and each m_n(s) ≥ 0. By Lemma γ1 (take sup/inf over x ∈ X_{n+1}^ζ; the referred points range in X_n^ζ by γ0, and on X_{n+1}^ζ, D̂/x ≤ δ_n by x_a ≥ 2ζ — a consequence of (γ1.4), which derives x_a ≥ 3ζ; citation repaired per review RF1-F3: (γ1.4) has no literal x_a ≥ 2ζ entry):

(γS-up)   M_{n+1}(s) ≤ (1 + δ_n)·[ λ(s)·M_n(s) + F_n⁺(s) ],
(γS-dn)   m_{n+1}(s) ≥ (1 − δ_n)·[ λ(s)·m_n(s) + F_n⁻(s) ],                 (γ1.5)

with F_n⁺(s) := sup_{X_{n+1}^ζ} F_s, F_n⁻(s) := inf_{X_{n+1}^ζ} F_s. Two facts carried forward:

Σ_n δ_n = 2D̂Þ_Γ/(x_a(1−Þ_Γ)) < ∞           (geometric — the summable perturbation budget);
at s < 1:  F_n⁺ ≤ (J̄/ℛ)(½x_aÞ_Γ^{−(n+1)})^{s−1} → 0 geometrically, Σ_n F_n⁺ < ∞.

This — a scalar recursion per shell, marching n → ∞ toward the boundary point — is the entire apparatus. Every theorem below is a statement about what such a recursion can do.

Remark 1.6 (the exact equation, for the narrative). Multiplying the exact identity (5.2) by x gives the equality version at s = 1:

(1 + η_L(x))·ℛ·W₁(x) = (1 + η_R(x))·E_t[ (x/x_{t+1})·W₁(x_{t+1}) ] + x·J(x),     (γ1.6)

with η_L, η_R = O(1/x) (5.5), x/x_{t+1} → Þ_Γ^{−1} a.s. (3.2), x·J(x) → c_J (5.4). (The remainders η_L(x), η_R(x) are Lemma 5.1’s: they collect everything the leading-order reading drops — the evaluation-point shifts and linearization curvature of the exact gap identity — and obey 0 ≤ η_L ≤ K_L/x, |η_R| ≤ K_R/x (5.5), so they vanish as x → ∞: every neglected effect is one order higher in 1/x than the terms kept.) In the archived supercritical edition, (γ1.6) passes to the limit by dominated convergence and becomes the scalar boundary equation of the one-number theorem (archive §3). Under GIC-policy W₁ = x·g diverges (≍ x^{1−q_∞}, §5), so the s = 1 limit equation carries no boundary information here; the informative object is the recursion at the correct compensation s = q_∞, read per fiber (§6). The monotone bracket (γ-up)/(γ-dn) is the workhorse throughout.

§2. The core lemma: boundedness on the compactified domain

The lemma is self-contained: it consumes only (γ1.5) and elementary analysis. In particular Theorem A1 is NOT imported — boundedness is proven internally, which keeps this route free-standing. (The full edition proves a second core lemma here — γ3, the λ < 1 trapping engine, “an inward-marching recursion with asymptotically constant coefficients forgets its anchor and is trapped at F_∞/(1−λ)” — whose only consumers are the archived supercritical theorems γ-A and γ-B; under GIC-policy no λ < 1 recursion at the theorem’s compensation ever arises, so γ3 is cited at the archive, not reproduced: https://llorracc.github.io/BufferStockTheory-Latest/powerlaw-decay-theory/alt-proof-compactified/, §2.)

Proof. (i) Let λ′ := (1+λ(s))/2 < 1 and pick n₂ with (1+δ_n)λ(s) ≤ λ′ for n ≥ n₂ (possible: δ_n → 0; explicitly n₂ = ⌈(−þ_g)^{−1}·ln( 4D̂λ/(x_a(1−λ)) )⌉₊, from solving δ_n = 2D̂Þ_Γ^{n+1}/x_a ≤ (1−λ)/(2λ) — why: it is the first shell where the coefficient perturbation δ_n can no longer bridge the gap between λ and the midpoint λ′). Set

K := max( M_{n₂},  F̄′/(1−λ′) ),     F̄′ := (1 + sup_n δ_n)·F̄.

If M_n ≤ K for some n ≥ n₂ then M_{n+1} ≤ λ′K + F̄′ ≤ λ′K + (1−λ′)K = K (the second inequality is the definition of K). Induction from n₂; the finitely many shells before n₂ are individually finite. (ii) Unroll (γS-up) with λ(s) ≤ 1:

M_n ≤ Π_{j<n}(1+δ_j) · [ M₀ + Σ_{j<n} F_j⁺ ] ≤ P_∞·[ M₀ + Σ_j F_j⁺ ] < ∞,

with P_∞ := Π_j(1+δ_j) ≤ e^{Σδ_j} < ∞. ∎

Remark. The two hypotheses are exactly the two ways a recursion M_{n+1} ≈ λM_n + F_n can stay bounded — and under GIC-policy the operative combination is case (ii) at the neutral setting: λ(q_∞) = 1 with geometrically FADING forcing, hence bounded. The combination the lemma excludes — λ = 1 with forcing that does not fade (F_n → F_∞ > 0) — grows linearly in the shell index instead; under GIC-policy the forcing at the theorem’s compensation always fades, so that combination never arises here (it is the archived resonance case q_∞ = 1, §4 stub). Nothing beyond g ≤ ḡ (for the anchor) and (γ1.5) was used.

§3. Archived: the supercritical boundary value (Theorem γ-A)

Outside GIC-policy — at q_∞ > 1, i.e. ℛÞ_Γ > 1 — the boundary multiplier at s = 1 satisfies λ(1) = 1/(ℛÞ_Γ) < 1, and the full edition proves there that W₁ = x·g(x) extends continuously to the boundary point with the scalar value B = κ̲(ρ+1)σ²/(2(ℛÞ_Γ−1)) — the “one number” of the owner’s original request — together with its geometric-resolvent reading B = Σ_k λ^k F(0) and the observed approach rate. That case is excluded by the standing assumption and is cited, not re-proved: https://llorracc.github.io/BufferStockTheory-Latest/powerlaw-decay-theory/alt-proof-compactified/, §3. (Its Remark 3.4 — the constants are not uniform near q_∞ = 1 — survives in restated, q < 1-native form as Remark 6.2 below.)

§4. Archived: the resonance q_∞ = 1 (Theorem γ-R)

At the knife-edge ℛÞ_Γ = 1 the boundary multiplier is neutral AND the forcing persists (F_n^± → c_J/ℛ > 0): the full edition proves there that the compensated gap grows linearly in the shell index — (x/ln x)·g(x) → κ̲(ρ+1)σ²/(−2þ_g) — via a perturbed Stolz–Cesàro lemma (γ-R1). GIC-policy is a strict inequality, so the resonance sits outside this document; but its proximity is what governs the transient constants of Remark 6.2 — which is why COL-TOP’s 1.2×10⁻⁴ margin to the boundary matters there. Cited, not re-proved: https://llorracc.github.io/BufferStockTheory-Latest/powerlaw-decay-theory/alt-proof-compactified/, §4.

§5. Theorem γ-T: wrong-exponent detection — the compensation dial

Compensation is a dial, not a hypothesis. Turning s away from q = q_∞ makes the boundary behavior degenerate in a diagnostic direction:

Proof. Everything follows from W_s = x^{s−q_∞}·W_{q_∞} and the two-sided control of W_{q_∞}: bounded above (γ2(ii)) and eventually bounded below by a positive constant, m_n → m_∞ > 0 (Lemma γ-C1 below). The per-shell factors are Þ_Γ^{−(s−q_∞)} = e^{(s−q_∞)(−þ_g)} by construction of the shells. ∎

Remark 5.1 (the dynamical reading — one dial is live, the other is slack). Intrinsically, the recursion (γS-up/dn) at compensation s has boundary multiplier λ(s) = e^{−(q_∞−s)(−þ_g)} and forcing scale F_s ≍ x^{s−1}. The homogeneous dial is the live one: λ(s) crosses 1 exactly at s = q_∞ — below it the recursion contracts, above it expands, and the per-shell rates of γ-T are Þ_Γ^{q_∞−s̃} for a probed exponent s̃. The forcing dial (F_s bounded exactly for s ≤ 1) never binds under GIC-policy: the crossing sits at q_∞ < 1, so every relevant compensation satisfies s ≤ 1, and the income/prudence cap at 1 — the origin of the archived edition’s min(1, q_∞) formula — is slack. (Accordingly, Lemma γ1’s restriction to s ≤ 1 costs nothing anywhere in this document.)

Remark 5.2 (relation to Lemma A5 / branch selection). For general ψ the boundary multiplier at compensation s is λ_B(s) = E[ψ^{1+s}]/(ℛÞ_Γ^s) = e^{𝔏(s)} (the Stage-B one-step; BufferStockTheory-Latest:theory/powerlaw-decay/stage_B_proof.md), so “λ_B(s) = 1” IS the eigenvalue equation (E), and A5’s convex geometry (𝔏 convex, 𝔏(0) < 0, unique positive root) is the statement that the boundary map has exactly one neutral compensation. γ-T is the dynamical restatement of A5’s root-selection: the growing branch x^{+|q|} excluded there by g ≤ ḡ is here excluded by γ2’s boundedness — the same a-priori sandwich, used once. (Kesten-theory kinship, one line: q_∞ is the Kesten exponent of the rescale x_{t+1} ≈ (Þ_Γ/ψ_{t+1})x weighted by the Euler ψ-factor; the detection statement is the primal, compactified face of that root.)

Falsifier F4 (§10): detection probes are registered at s = q_∞ ± 0.15 (the corpus’s 2026-07-19 detection registration — only s̃ = q_∞ stabilizes); the frozen-stack run at the prior ±0.10 probes measured the two per-shell log-rates, in units of −þ_g, at −0.0999/+0.1001 vs the predicted ±0.1 — exact to ±0.0001, correct signs.

§6. γ-C: the main theorem — a boundary circle, not a boundary point

Under GIC-policy this is the standing case: the document’s headline result. At the theorem’s compensation q = q_∞: λ(q_∞) = 1 exactly (definition of q_∞), and the forcing fades geometrically (F_n^± ≤ CÞ_Γ^{(1−q_∞)n}, Lemma γ1(ii)). A neutral recursion with summable forcing converges — but to what depends on where you stand within the shell: the map z ↦ z/Þ_Γ (one shell inward) preserves the log-phase

φ(x) := (ln x mod (−þ_g)) ∈ ℝ/(−þ_g)ℤ,

the position of ln x modulo the period −þ_g (the quotient ℝ/(−þ_g)ℤ is a circle of circumference −þ_g). In log wealth the dynamics repeat with period −þ_g, so what matters asymptotically is this position alone: the compactification’s natural boundary here is not the point z = 0 but the circle of phases ℝ/(−þ_g)ℤ. Call the set of wealth levels sharing one phase — the geometric sequence {x = x̄·Þ_Γ^{−j}, j ∈ ℕ}, equal ln x modulo −þ_g — a fiber: distinct fibers march to the boundary without ever mixing (up to the small per-step displacement of phase, the “phase blur”, bounded next). What is proven here, and what is imported:

Proof. Boundedness: γ2(ii). Upper convergence: by (γS-up) with λ = 1, M_{n+1} − M_n ≤ δ_n·M̄ + (1+δ₀)F_n⁺ =: s_n with Σ s_n < ∞ (M̄ := sup M_n). The sequence a_n := M_n − Σ_{k<n}s_k is non-increasing and bounded below, hence convergent; therefore M_n converges. Lower convergence: by (γS-dn), m_{n+1} ≥ (1−δ_n)m_n, so m_{n+1} − m_n ≥ −δ_n·M̄, and b_n := m_n + Σ_{k<n}δ_kM̄ is non-decreasing and bounded, hence m_n converges. Positive floor: m_{n+1} ≥ (1−δ_n)·m_n iterates to m_n ≥ m_{n₃}·Π_{k≥n₃}(1−δ_k) > 0, provided m_{n₃} > 0 for some n₃ with δ_k < 1 beyond it — and m_{n₃} = inf_{X_{n₃}^ζ} W_{q_∞} > 0 because g is continuous and strictly positive on the compact shell (Lemma 2 strict; take n₃ ≥ 1 so the shell’s bottom x_aÞ_Γ^{−n₃} − ζ > x_a − ζ ≥ x₀ ≥ (h−1) + m̄ sits strictly above the strictness threshold, i.e. m > m̄ everywhere on the shell). Identification of limsup/liminf: every x lies in X_{n(x)}, so all accumulation points lie in [m_∞, M_∞]; both endpoints are attained along maximizing/minimizing sequences; and since W is continuous on (0, z_a], the intermediate value theorem fills the interval. ∎

Lemma γ-C2 (summable phase blur). PROVEN-HERE.

Along a fiber x_j := x̄·Þ_Γ^{−j} (x̄ ∈ X₀^ζ), the one-step comparison at x_{j+1} refers to the points Þ_Γx_{j+1} ∓ C₀ = x_j ∓ C₀, whose log-phase differs from φ(x_j) by

|Δφ_j| ≤ 2C₀/(Þ_Γ·x_j)      (for x_j ≥ 2C₀/Þ_Γ);

this per-step displacement Δφ_j of the consulted phase is the phase blur, and the total blur accumulated from shell N inward is geometrically small: Σ_{j≥N}|Δφ_j| ≤ (2C₀/(Þ_Γx̄))·Þ_Γ^N/(1−Þ_Γ) → 0. Likewise the coefficient perturbations sum to Σ_{j≥N}δ_j → 0 and the forcing to Σ_{j≥N}F_j → 0. Proof: |ln(1∓u)| ≤ 2u for u ≤ ½ applied to u = C₀/(Þ_Γx_j); geometric sums. ∎

So the recursion asymptotically decouples across phases: each fiber talks only to o(1)-neighborhoods of itself. What follows from γ-C1/γ-C2 by elementary means: every fiber sequence W(x_j) has convergent subsequences, all with limits in [m_∞, M_∞]; and, writing osc(f; A) := sup_A f − inf_A f for the oscillation of a function over a set, the per-step fiber increment obeys

|W(x_{j+1}) − W(x_j)| ≤ δ_j·M̄ + (1+δ_j)·[ F_j + osc( W ; x_j·e^{[−u_j, u_j]} ) ],   (γ6.1)

with everything except the last term summable. The last term is the local oscillation of W over the multiplicative window x_j·e^{[−u_j, u_j]} — an interval of phases of width 2u_j, u_j = 2C₀/(Þ_Γx_j).

GAP-γ-equicont (the honest missing step). GAP.

To convert (γ6.1) into convergence along each fiber one needs an asymptotic modulus of continuity in the phase direction — a uniform bound on how much W can move under a small multiplicative shift of wealth: some ω with ω(0+) = 0,

|W(x·e^u) − W(x)| ≤ ω(|u|)   for all large x, |u| ≤ −þ_g,

and summability of Σ_j ω(u_j) along the geometric sequence u_j ≍ Þ_Γ^j (a Dini-type condition ∫₀ ω(t)/t·dt < ∞; any Hölder modulus qualifies). Given such an ω, (γ6.1) is summable, each fiber sequence is Cauchy, and the fiber limits assemble into a function P_γ: ℝ/(−þ_g)ℤ → [m_∞, M_∞] with W(x) − P_γ(φ(x)) → 0; the compensated gap then extends continuously to the compactified space whose boundary is the circle, with W|_{boundary} = P_γ, max P_γ = M_∞, min P_γ = m_∞. This equicontinuity control is exactly the L8/L9 (doubling / log-Lipschitz) territory of the Stage-A engine, which is off-limits for this self-contained route — so it is tagged here as a GAP and NOT claimed. The sharp form is imported as Theorem A2 (g = x^{−q_∞}(P(ln x) + O(x^{−(1−q_∞)})), P positive, Lipschitz, (−þ_g)-periodic): under that import, P_γ = P and the circle picture above is a theorem. Without the import, this document proves the envelope (γ-C1), the blur budget (γ-C2), and subsequential fiber limits — no more.

Remark 6.1 (the boundary object, and its honest scalar summaries). The boundary object is a function on a circle; the honest scalar summaries are the pair (m_∞, M_∞) (proven here) or the mean of P (owned by the B2-arith Fourier representation, off-limits here). (One-line historical aside: in the archived supercritical case the circle is still there but collapses to a point — γ-A forces the boundary function to be the constant B — which is why the full edition could answer the owner’s request with literally one number.) Whether M_∞ > m_∞ — i.e. whether the circle is visible — is the fine structure of P: measured non-constant with relative amplitude osc(P)/P̄ ≈ e^{−α/(−þ_g)} (osc(P) := sup P − inf P, the total amplitude of the periodic factor; P̄ its mean) where α is calibration-dependent (softened per review RF2-N2: R3 measured α ≈ 0.8–1.2 on its lognormal designs, but RF2 resolved a grid-independent oscillation, locked to the phase φ, of amplitude 7.8e-4 at −þ_g = 0.693 on an atomic-θ design — implying α ≈ 4.9, so the prefactor-1 α≈1 extrapolation over-predicts ~250–400× there; positive side: that oscillation is the first direct resolution of a non-constant P in this route’s numerics, confirming the imported A2 circle picture); analyzed in periodic_factor_fine_structure.md (owner document; not needed by anything here — the RF2 data point is recorded there). At the −þ_g of the estimated calibrations the oscillation is far below numerical visibility — falsifier F1c′ measured a phase profile flat to 3.4e-5 at −þ_g = 0.288, an upper bound contaminated by residual envelope drift, consistent with both “effectively constant” and the (calibration-dependent) R3 law (§10).

Remark 6.2 (transients honesty: the constants are NOT uniform as q_∞ → 1). γ-C1 is a fixed-q_∞ statement. As q_∞ ↑ 1 — i.e. as ℛÞ_Γ ↑ 1, the GIC-policy boundary — the forcing’s per-shell fade factor Þ_Γ^{(1−q_∞)} tends to 1, so the number of shells the envelope needs scales like ((1−q_∞)(−þ_g))^{−1}: the theorem’s content becomes visible only for ln x ≫ 1/(1−q_∞). Below that depth the measurement is dominated by the still-live transient — the x^{1−q_∞}-amplitude crossover channel. The registered near-boundary check RT1 (q_∞ = 0.95, §10) measures exactly this and is kept as standing evidence: the split-window local slope of W₁ on ln x GROWS with depth (0.519 shallow → 0.746 deep) — the expected pre-asymptotic signature, reported rather than hidden. Nor is this an academic caveat: among the corpus’s estimated calibrations, COL-TOP sits 1.2×10⁻⁴ INSIDE the GIC-policy boundary (ℛÞ_Γ ≈ 1 − 1.2×10⁻⁴) — squarely in the regime where the constants are enormous and every accessible wealth level is pre-asymptotic. In plain terms: the closer a calibration sits to the boundary, the longer the power law takes to show its true slope; at patience that extreme, any slope you can measure at observable wealth is still in transit toward −q_∞, and honest reporting says so.

§7. Archived: Stage B (permanent shocks) at q_∞ > 1 — the boundary value B_ψ

For general bounded ψ the referred point becomes random and the compensated one-step carries the weight ψ², with boundary multiplier λ_B = E[ψ²]/(ℛÞ_Γ). The full edition proves, under the hypothesis q_∞ > 1 (⟺ λ_B < 1), that W₁ → B_ψ = κ̲(ρ+1)σ_B²/(2(ℛÞ_Γ − E[ψ²])) — boundedness via a stopped expectation-unroll (γ-B2), stability via a four-line function-level argument (γ-B3). Under GIC-policy that hypothesis never holds; on the contrary, λ_B = E[ψ²]/(ℛÞ_Γ) > E[ψ²] ≥ 1 at every calibration — the s = 1 Stage-B recursion is expanding, the same over-compensation signature as λ(1) > 1 in §1.4. General-ψ tail theory at q_∞ < 1 belongs to Theorems B2/B2-arith (BufferStockTheory-Latest:theory/powerlaw-decay/stage_B_proof.md), not to this route. Cited, not re-proved: https://llorracc.github.io/BufferStockTheory-Latest/powerlaw-decay-theory/alt-proof-compactified/, §7.

§8. The forcing floor from the boundary picture — a remark, not a new claim

One application of (γ-dn) at s = 1, dropping the (nonnegative) W-term, gives for x ≥ x_a:

W₁(x) ≥ (1 − D̂/x)·F₁(x) ≥ (1 − D̂/x)·j₋σ²/ℛ,     i.e.   g(x) ≳ σ²/x,

the forcing floor with a one-step constant j₋σ²/ℛ — the floor is the first Neumann term of the boundary forcing sum. (In the archived supercritical case, iterating instead of dropping resums the geometric series and recovers γ-A’s B; under GIC-policy the iterated s = 1 sum diverges along with W₁, and only the one-step floor itself is used.) The sharp, GIC-free version of the floor (weaker hypotheses, better constant, the no-o(1/x) corollary, Stage-B form with σ_B²) is the forcing-floor lemma pair, proved inline in the stage proofs (stage-A Lemma 6.1 / stage-B Lemma B-6.1); nothing new is claimed here. PROVEN-CITED (the one-step display above is PROVEN-HERE but adds nothing to the owning lemmas).

§9. Why this is the easy proof

(Plain language; one page; the section the owner’s request was really asking for.)

Compress. Wealth runs over [x_a, ∞). The change of variable z = 1/x squeezes this ray into the bounded interval (0, z_a], and infinite wealth becomes an ordinary boundary point, z = 0, which we glue on. This move is a one-dimensional Poincaré compactification — the standard trick from dynamical systems for studying behavior “at infinity” with finite tools (the classical planar chart at infinity is literally v = 1/x).

Choose units. The gap g itself vanishes at the boundary — too little information. So we measure it in units that are expected to make the limit finite: W = x^q·g(x) with q = q_∞. That unit choice is the rigorous meaning of “assume the power law holds in the limit”: no assumption is made — if the units are right, the compensated gap has a finite nonzero boundary behavior, and if they are wrong, the boundary behavior degenerates to 0 or ∞ and tells you the true exponent by the rate at which it does so (§5).

Neutral units, fading forcing. The Euler equation, rewritten in these coordinates, links each shell of the interval (a geometric band z ≈ z₀Þ_Γ^n) to the next-outer shell:

W(shell n+1) ≈ λ·W(shell n) + F_n,       λ = λ(q_∞) = 1,   F_n ≤ C·Þ_Γ^{(1−q_∞)n} → 0.

Under GIC-policy, at exactly the units that could make the limit finite the multiplier is NEUTRAL — λ(q_∞) = 1 is the definition of q_∞ — while the forcing fades geometrically. March inward (n → ∞ = z → 0): a neutral recursion with summable forcing converges — the compensated gap is trapped in a bounded envelope [m_∞, M_∞] with a strictly positive floor. What a neutral recursion no longer does is mix phases: one step moves you exactly one shell, preserving your position ln x mod (−þ_g), so what survives at the boundary is a function on the circle of phases, not a single number. The proof has exactly three moving parts, each elementary: coefficients converge at the boundary (§1, from the imported one-step identity), the sequence is bounded (§2, a three-line unroll), and neutral multiplier + summable forcing ⟹ convergent envelope (§6, a telescoping-series argument).

The one regime under GIC-policy. A linear recursion at a boundary can contract (λ < 1), drift (λ = 1 with persistent forcing), or converge neutrally (λ = 1 with fading forcing). The standing assumption puts this document always in the third: the “number” is a function on the circle of phases, with elementary envelope [m_∞, M_∞] and pointwise form the periodic factor P (§6). (The contracting and drifting regimes arise only outside GIC-policy — the archived edition’s §§3–4.)

The exponent is detected, not guessed. Compensating with the wrong s flips the boundary multiplier λ(s) = e^{−(q_∞−s)(−þ_g)} across 1: W_s → 0 below the true exponent, → ∞ above it, at predicted geometric rates (§10). q_∞ is the unique dial setting with a finite nonzero reading; and the equation “λ(s) = 1” is, for general ψ, literally the eigenvalue equation (E) of Lemma A5 — with λ_B(s) = e^{𝔏(s)}, convexity of 𝔏 says the dial crosses 1 once.

A master’s-class one-liner. “In the units where the limit could be finite, the boundary multiplier is exactly neutral and the forcing at infinite wealth has already faded; so the compensated gap settles — and the only thing infinity remembers is where you stand on the circle of log-wealth phases.” Everything else in this document is the bookkeeping certifying that the two words “at infinity” are legitimate.

Kinship (why four proofs share one skeleton). Unrolled at neutral multiplier, the boundary object is the summed forcing Σ_k F_k collected one shell at a time — and under GIC-policy (ℛÞ_Γ < 1) that sum is end-dominated: the gap at huge wealth is the pile-up of the small prudence corrections (J ≈ c_J/x) collected while wealth decays geometrically toward its target, dominated by the terms gathered near the destination rather than near the starting wealth (the 0.051 head/tail signature of §10’s prior-art check). Every correct proof computes this same end-dominated sum; the routes differ ONLY in the engine that justifies the exchange of limits: renewal/ladder theorems (stage_A/B), a probabilistic path-sum, a recursion-tree/Akra–Bazzi induction, or — here — continuity at the boundary of a compactified domain, where the exchange of limits is a telescoping envelope argument. The compactified route is the shortest because it asks the weakest question: not “how fast?” (rates), not “along which paths?” (fluctuations), only “what survives at the boundary?”.

§10. Numerical verification (pre-registered falsifiers; simplified-edition registration)

Harness: verify_altproof_compactified_checks.py (bare python3 + numpy, longdouble EGM solver imported from review/R4_egm_lib.py; per-point g/c > 1e-10 cancellation guard AND per-point Euler-residual gap-certificate ≤ 1e-3 on every shell sample; Þ_Γ ≤ 0.9 for all amplitude checks). The full edition registered 15 checks, 13 PASS / 2 FAIL — both FAILs the author’s own near-resonance window predictions (RT1), reported rather than tuned away; that harness, its supercritical/resonance rows, and their bookkeeping are archived with the full edition. This edition’s registered suite is the q < 1 subset, restated where the 2026-07-19 rulings moved a registration: F1c and F1c′ (registrations unchanged — results below carried from the frozen-stack run), F4 (detection probes widened to s = q_∞ ± 0.15), F6′ (fat-tail robustness re-targeted from the archived supercritical B to the envelope/floor), and RT1 (q_∞ = 0.95) re-registered as a crossover/transient expectation rather than a test of a resonance law this edition no longer states; the saved output verify_altproof_compactified_checks_out.txt is regenerated when the re-registered rows are re-run. The corpus’s battery of record for the GIC-policy trio is verify_gicpolicy_trio_checks.py (30/30 PASS), asserting among other things COL-MID’s q_∞ = 0.5513, ℛÞ_Γ = 0.997809, and the guarded-window tail slope −0.5290 on [2e4, 2e6] vs −0.5513 (rel. err. 4.0% — the pre-asymptotic transient of Remark 6.2, reported honestly).

Main q<1 calibration of this harness: Þ_Γ = 0.85 (−þ_g = 0.16252), ρ = 2, G = 1, θ = lognormal(0.2, N=7) + 5% unemployment atom (σ²_disc = 0.087565, θ_min = 0), q_∞ = 0.6 via ℛ = e^{q_∞(−þ_g)}; the fiber check runs at Þ_Γ = 0.75. Shell geometry as in §1 (ζ reported per run; measurement bins are the plain geometric shells).

falsifiersectionregistered criterionresult
F1c§6envelope Cauchy (≤2e-2), positive floor, at q_∞ = 0.6PASS: increments ≤ 2.4e-5; envelope [4.28234, 4.28240]
F1c′§6fiber flattening at −þ_g = 0.288 (signature; profile reported)PASS: drift 4.6e-4 → 2.6e-4; profile flat to 3.4e-5
F4§5wrong-s per-shell rates at s = q_∞ ± 0.15 within tolerance of (s−q_∞), correct signsre-registered 2026-07-19 (probes widened from ±0.10, where the run measured −0.0999/+0.1001 vs ±0.1)
F6′§2/§6fat-tail θ (Pareto α=3 tail), θ_max ∈ {10,40,160} at q_∞ = 0.6: envelope Cauchy + positive floor at every escalation; constants degradation reportedre-registered 2026-07-19 (the archived F6 targeted the supercritical B)
RT1 (q_∞=0.95)§6crossover/transient expectation: local slope of W₁ on ln x GROWS with depth (the x^{1−q_∞} amplitude regime)CONSISTENT: split-window slope 0.519 shallow → 0.746 deep

Reading the results.

Controls (INFO, not falsifiers). PH1 (solver quality) is re-targeted at the q<1 headline: grid-doubling (Na 6000 → 12000) and a half-shell anchor shift (x_b → x_b·e^{(−þ_g)/2}) must leave the F1c envelope inside its Cauchy tolerance (the archived run’s PH1 established grid- and binning-invariance of the full edition’s headline at far tighter margins). PH2 is the RT1 split-window diagnostic quoted above.

Prior art. The orchestrator’s pre-brief mechanism check (mech_check_unroll.py, 2026-07-07 — committed alongside this document; quoted for provenance, nothing below depends on it; near-knife-edge calibrations Þ_Γ ≈ 0.99) already measured this skeleton via the forward unroll of Cor 5.2: at q_∞ = 0.6 the unroll/solved ratio was 1.047 with per-term head/tail ratio 0.051 — the END-DOMINATED (fading-forcing) signature of §6, the sum carried by the terms collected near the destination. This suite reproduces that signature on a large-(−þ_g) calibration with registered tolerances and adds the envelope, fiber, and detection checks.

Solver-quality note (honesty). The Stage-A solves stop on the library’s plateau guard (it ≈ 700, sup-rel policy-churn ≈ 2e-2 concentrated at the constraint-boundary nodes); deep- tail solution quality is certified NOT by that global norm but per point, by the Euler-residual gap-certificate (er/ρ)(c/g) ≤ 1e-3 enforced on every sample (the library’s own design: “tail quality is certified separately per point”), and independently by the envelope/fiber/detection agreements above and the grid-doubling control (PH1).

References

Internal: statement_simpler.md; BufferStockTheory-Latest:theory/powerlaw-decay/stage_A_proof.md §§1–5 (imports; incl. the stage-A forcing-floor Lemma 6.1); BufferStockTheory-Latest:theory/powerlaw-decay/stage_B_proof.md (cited only for the stage-B forcing-floor Lemma B-6.1, Remark 5.2’s general-ψ multiplier, and the §7 scope note); periodic_factor_fine_structure.md (owner of P’s fine structure); review harness review/R4_egm_lib.py. Archived full edition of this document (served): https://llorracc.github.io/BufferStockTheory-Latest/powerlaw-decay-theory/alt-proof-compactified/.

External (kept deliberately minimal — simplicity is this document’s product; BufferStockTheory-Latest:theory/powerlaw-decay/alt_proof_econlit.md owns the literature fabric):

Document status recap (simplified edition, GIC-policy standing): γ0/γ1/γ2, γ-T (restated at q < 1), γ-C1/γ-C2: PROVEN-HERE. §6 circle-limit: IMPORTED (Theorem A2) modulo GAP-γ-equicont (tagged). §8: PROVEN-CITED (owners: the stage-proof forcing-floor lemmas, stage-A 6.1 / stage-B B-6.1). §§3, 4, 7 (supercritical one number, resonance, Stage-B B_ψ) and Lemma γ3: ARCHIVED — cited at the served full edition, never re-proved here. Numerical suite: F1c/F1c′ carried PASS from the frozen-stack run; F4/F6′ re-registered 2026-07-19 (re-run pending); RT1 (q_∞ = 0.95) kept as the registered crossover/transient evidence of Remark 6.2; corpus battery of record verify_gicpolicy_trio_checks.py 30/30 PASS. Provenance: the FULL edition was refereed by the independent panel RF1/RF2 (2026-07-08; every PROVEN-HERE status CONFIRMED, quarantine HOLDS, RT1 FAILs adjudicated as honest crossover phenomenology); this simplified restatement has not been separately refereed — the panel’s repairs touching surviving text (RF1-F3, RF2-N2) are retained in place (marked).

Power-law decay of the buffer-stock consumption gap (simplified capstone under GIC-policy)
Statement: power-law decay of the buffer-stock consumption gap (Stage A: ψ≡1)
Power-law decay of the buffer-stock consumption gap (simplified capstone under GIC-policy)
Grid design: final spec + the measured findings that forced it