Alternative proof γ: the compactified domain and the boundary fixed point

AUTHOR PASS (2026-07-07) + INDEPENDENT REFUTER PANEL RF1/RF2 (2026-07-08): ZERO BROKEN, ZERO MODERATE. RF1 (proof-logic lens): every PROVEN-HERE item re-derived line-by-line from the declared imports (γ0/γ1/γ2/γ3, γ-A, γ-R1/γ-R, γ-T, γ-C1/γ-C2, γ-B1/γ-B2/γ-B3, γ-B — all VERIFIED; no petitio principii at the boundary, no uniqueness-class gap, no hidden uniformity in γ3, no illegitimate ψ-limit exchange); the GAP-γ-equicont quarantine HOLDS (consumer audit clean). Two MINOR bookkeeping findings (RF1-F1/F2) and two display notes (RF1-F3/F4) are repaired in place below, each marked “(repaired per review RF1-…)”; no constant moves. RF2 (numerics lens): all theorems VERIFIED on fresh/hostile designs the author never ran — Pareto-θ at the L4′ moment boundary × Þ_Γ = 0.95, ρ = 6, zero-atom θ, skewed deep-forward ψ_min = 0.4 ≪ Þ_Γ, a wrong-weight plateau sweep pinning the (γ7.1) ψ²-weight (root ŵ = 1.9999, margin 6647×), and exact detection rates for γ-T; findings RF2-N1–N3 (evidence-route and remark-level) applied below. The two registered RT1 FAILs stand, adjudicated by BOTH packs as honest crossover phenomenology (crossover-corrected intercepts land on B to 0.006–0.10%), not masked defects. 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.md (theorem statements and the shared assumption block), stage_A_proof.md §§1–5 (the imported foundations) and stage_B_proof.md §§B0–B3 (their ψ-general forms). Numerical verification with pre-registered falsifiers: verify_altproof_compactified_checks.py (+ saved output _out.txt beside it); results summarized in §10, including two registered near-resonance checks that FAILED as registered and are reported, not tuned away.

§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 := min(1, q_∞). The theorem then says: W extends to a continuous function on the compact interval [0, z_a], and at the added point the functional equation degenerates to one scalar equation,

W(0) = λ·W(0) + F(0),        λ = 1/(ℛÞ_Γ) < 1,   F(0) = c_J/ℛ,

whose unique solution is the owner’s “one number”,

B = F(0)/(1−λ) = κ̲(ρ+1)σ²/(2(ℛÞ_Γ−1)).

That is the q_∞ > 1 case (§3). The document is equally explicit about where the one-number picture degenerates: at q_∞ = 1 the boundary value is +∞ and the finite number is a slope (§4); at q_∞ < 1 the natural boundary is not a point but a circle — 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 — and the finite object is a boundary function on that circle, proven here only at the envelope level, with the equicontinuity step honestly tagged GAP-γ-equicont (§6).

Standing assumptions. A0–A6 of statement.md (ψ ≡ 1, bounded θ, σ² > 0) for §§1–6 and §8; the Stage-B block B-A0/A3–A6 for §7. 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);   Stage B adds  σ_B² = E[ψ²]σ_θ² + h²σ_ψ²  and  q_∞ = the (E)-root.

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 stage_A_proof.md: 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/§7). From stage_B_proof.md: (B3.1)/(B3.2), Lemma B-5.1 ((B5.2)–(B5.5)), Corollary B-5.2, the x₀^B threshold block (B5.0). As benchmarks only (targets to match, never used in a proof): Theorem A3’s B, Theorem B3’s B_ψ, Theorem B-res’s constant, Theorem A2’s P.

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. One honest near-miss is flagged where it occurs: the Stage-B boundedness lemma γ-B2 uses a stopped expectation unroll — a path-sum-flavored argument (no tilt, no renewal, no change of measure), the single place this route borrows that flavor. (Disclosure sharpened per review RF1-F4: the device is structurally the same as stage_B §B5’s own B1(i) boundedness iteration — chain + stopping time + per-step weight + E[ψ]=1 kill; no text or statement is imported, so the firewall stands, but the credit is owed to stage_B §B5’s boundedness proof specifically. The γ-route’s genuinely new Stage-B content is the boundary-stability assembly (γ-B3), not the boundedness technique.)

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 this version was committed, an adversarial self-pass ran the brief’s attack list (shell-index bookkeeping; the phase-displacement control of §6; the Cesàro step; the boundary cases q_∞ ∈ {1±δ}; the Stage-B forward window; fat-tailed θ). It caught and fixed two drafting defects — the Stage-B backward-band lemma’s shell width (γ-B1: the guaranteed backward step is ln(ψ_min/Þ_Γ), not −þ_g, so (−þ_g)-shells self-refer whenever ψ_min < 1; recorded in place) and a missing cross-term factor in (γ6.1) — and it kept two registered numerical falsifiers FAILED (RT1, §10) rather than repairing them post hoc. The independent (non-author) refuter pass was completed 2026-07-08 (RF1/RF2 — see the header block above); its repairs 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 := min(1, q_∞). 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 Stage-B analogue genuinely loses this property — see §7.)

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_∞,     and at s = 1:  λ(1) = 1/(ℛÞ_Γ).

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+K̂/x)F_s(x). For x ≥ 2C₀/Þ_Γwe haveu ≤ ½, and for s ≤ 1, (1−u)^{−s} ≤ (1−u)^{−1} ≤ 1 + 2u = 1 + 2C₀/(Þ_Γx). Then (1+K̂/x)(1+2C₀/(Þ_Γx)) ≤ 1 + D̂/xforx ≥ max(K̂, 2C₀/Þ_Γ)(the cross term2K̂C₀/(Þ_Γx²)is ≤min(K̂, 2C₀/Þ_Γ)/xthere, and the sum of the three terms is ≤D̂/xby the choiceD̂ = 2(K̂ + 2C₀/Þ_Γ)). Since the bracket multiplying it is nonnegative, the enlarged factor can be pulled out front. (γ-dn): (1+u)^{−s} ≥ 1 − su ≥ 1 − u(Bernoulli,s ≤ 1), and (1−K̂/x)(1−u) ≥ 1 − D̂/x; the forcing keeps a factor ≥ (1−D̂/x). (i) is the display; (ii) is (5.3)/(5.4) multiplied by ℛ^{−1}x^

  , plus, on shell

  X_n^ζ, x ≥ ½x_aÞ_Γ^

  , giving the geometric rate; (iii) is Lemma γ0; (iv) is one division:

  1/(Þ_Γx ∓ C₀) = (z/Þ_Γ)(1 ∓ C₀z/Þ_Γ)^{−1}`. ∎

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. Three facts carried forward:

Σ_n δ_n = 2D̂Þ_Γ/(x_a(1−Þ_Γ)) < ∞           (geometric — the summable perturbation budget);
at s = 1:  F_n⁺, F_n⁻ → c_J/ℛ               (the shells recede, so (5.4) pinches both);
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.) Once a boundary limit of W₁ is known to exist (that is γ3’s job), (γ1.6) passes to the limit by dominated convergence and becomes the scalar boundary equation of §3. The monotone bracket (γ-up)/(γ-dn) is the workhorse; (γ1.6) is the punchline.

§2. The two core lemmas: boundedness and boundary stability¶

Both lemmas are self-contained: they consume only (γ1.5) and elementary analysis. In particular Theorem A1 is NOT imported — boundedness is proven internally, which keeps this route free-standing.

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; the resonant combination λ = 1 with F_n → F_∞ > 0 is precisely the excluded case, and indeed there W₁ is unbounded (§4). Nothing beyond g ≤ ḡ (for the anchor) and (γ1.5) was used.

Proof. Let S := limsup M_n < ∞. Fix ε > 0; choose N with: M_k ≤ S + ε for all k ≥ N, δ_n ≤ ε, F_n⁺ ≤ F_∞ + ε for all n ≥ N. For n ≥ N + b: M_{n+1} ≤ (1+ε)[λ(S+ε) + F_∞ + ε]. Taking limsup and then ε ↓ 0: S ≤ λS + F_∞, so S ≤ F_∞/(1−λ). Symmetrically, with I := liminf m_n > −∞: for n large, m_{n+1} ≥ (1−ε)[λ(I−ε) + F_∞ − ε], so I ≥ λI + F_∞, so I ≥ F_∞/(1−λ). Then

F_∞/(1−λ) ≤ I = liminf m_n ≤ liminf M_n ≤ limsup M_n = S ≤ F_∞/(1−λ),

and the same chain with limsup m_n in the middle. All four limits coincide. ∎

Remark (why this half-page is the engine). The lemma is one-directional stability at a boundary: an inward-marching recursion with asymptotically constant coefficients forgets its data at the anchor and is trapped at the fixed point F_∞/(1−λ) of the limiting scalar map w ↦ λw + F_∞. The window b is free (finite windows do not move a limsup); in Stage A we only need b = 0 (Lemma γ0), while Stage B uses a genuine window (§7). No rate for F_n → F_∞ and no monotonicity are needed — this is exactly why the compactified route is short: (5.4) is consumed as a limit, never as an expansion.

Function-level form (used in §7). The same four lines run directly on S̄ := limsup_{x→∞} W(x) and s̲ := liminf_{x→∞} W(x) when the recursion refers to points → ∞: one obtains S̄ ≤ λS̄ + F(0) and s̲ ≥ λs̲ + F(0), hence both equal F(0)/(1−λ). Shells are needed only where the count of shells matters (§4) or the phase — the position within the shell (§6) — matters.

§3. Theorem γ-A: the boundary value at q_∞ > 1 — the owner’s one number¶

Proof. Boundedness: Lemma γ2(i) at s = 1 (hypotheses: λ(1) = 1/(ℛÞ_Γ) < 1; F̄ ≤ J̄/ℛ < ∞ by Lemma γ1(ii)). Stability: the sequences M_n(1), m_n(1) satisfy Lemma γ3’s hypotheses with b = 0 (by (γS-up)/(γS-dn), absorbing the outer (1±δ_n) into the lemma’s δ_n), λ = 1/(ℛÞ_Γ), and F_n^± → c_J/ℛ (Lemma γ1(ii): the shells recede, so (5.4) pinches sup and inf). Hence M_n, m_n → F(0)/(1−λ) = B. Every x ≥ x_a lies in some plain shell X_{n(x)} ⊂ X_{n(x)}^ζ with n(x) → ∞ as x → ∞, and m_{n(x)} ≤ W(x) ≤ M_{n(x)}, so lim_{x→∞} W(x) = B, i.e. lim_{z→0} W(z) = B. Define W(0) := B; continuity on (0, z_a] holds since c is continuous (L0/L1); continuity at 0 is the limit just proven. For (γ3.1): in (γ1.6), η_L, η_R → 0 by (5.5); x_{t+1}/x → Þ_Γ uniformly a.s. by (3.2); W₁(x_{t+1}) → B a.s. (since x_{t+1} ≥ Þ_Γx − C₀ → ∞ a.s.) and W₁ is bounded, so dominated convergence gives E_t[(x/x_{t+1})W₁(x_{t+1})] → Þ_Γ^{−1}B; and xJ(x) → c_J (5.4). The limit equation reads ℛB = Þ_Γ^{−1}B + c_J, which is (γ3.1) after dividing by ℛ, and solves to B = c_JÞ_Γ/(ℛÞ_Γ−1). Substituting c_J = κ̲(ρ+1)σ²/(2Þ_Γ) (5.4) gives the closed form; the Þ_Γ cancels — why the formula is clean: the forcing constant carries a Þ_Γ^{−1} from the rescale Jacobian and the resolvent carries a Þ_Γ from the one-shell delay, and they are inverse to each other. ∎

Remark 3.2 (the geometric resolvent). Unrolling (γ3.1): B = Σ_{k≥0} λ^k F(0) — the boundary value is the geometric sum of the forcing received one shell at a time, discounted by the boundary multiplier. This sum is the shared skeleton of all four proof routes (§9).

Remark 3.3 (approach rate — observed, not proven here). The recursion heuristic e_{n+1} ≈ λ·e_n + (F_n − F_∞) predicts |W − B| ≍ x^{−min(1, q_∞−1)} provided F_n − F_∞ = O(Þ_Γ^n); since (5.4) is imported rate-free, this document does not prove a rate, and none of §§2–7 needs one. The prediction matches the falsifier F1a (fitted x-power 0.597 vs q_∞−1 = 0.6 at q_∞ = 1.6, §10) and statement.md Remark 8.3 / review R4. Curiosity: at q_∞ = 2 the two rates collide (λ = Þ_Γ) — a second-order resonance in the error term, invisible in the value.

Remark 3.4 (honesty: uniformity near the resonance). B ∝ 1/(ℛÞ_Γ−1) → ∞ as q_∞ ↓ 1, and the shell contraction λ = e^{−(q_∞−1)(−þ_g)} ↑ 1: the number of shells to enter the plateau scales like ((q_∞−1)(−þ_g))^{−1}, i.e. the theorem’s content becomes visible only for ln x ≫ 1/(q_∞−1). (Quantitatively — added per review RF2-N3 — the onset is ln x ≳ ln x_c + O(1)/(q_∞−1) with the crossover scale x_c = |b₁/B|^{1/(q_∞−1)} calibration-dependent and GROWING as Þ_Γ ↑ 1: RF2 measured x_c ≈ 30 at Þ_Γ = 0.95 vs ≈ 730 at Þ_Γ = 0.96 at fixed q_∞ = 1.6, leaving 3.5% raw residuals at x ≈ 2e5 that the crossover-corrected intercept resolves to 0.05%; cf. Cor. A4.2’s additive window.) γ-A is a fixed-q_∞ statement, NOT uniform in q_∞ near 1. The registered near-resonance falsifier RT1 FAILED as registered precisely because of this (the author’s own pre-registered window criterion mis-modeled the crossover; §10, kept as a FAIL): at q_∞ = 1 ± 0.05 and ln x ≤ 12 the quantity (q_∞−1)·ln x is order one, and the measured behavior interpolates between §4’s line and §3’s plateau. The correct uniform-in-q_∞ objects near the resonance are the crossover statements of Cor. A4.2 and Theorem B3’s divergence law B·(q_∞−1) → κ̲(ρ+1)σ²/(−2þ_g·E[ψ²])|_{ψ≡1} (statement.md §3), consistent with §4 below.

§4. Theorem γ-R: the resonance q_∞ = 1 — a slope, not a value¶

At q_∞ = 1 (⟺ ℛÞ_Γ = 1), λ(1) = 1: the boundary multiplier is neutral — equal to one, so the recursion neither contracts nor expands — and the forcing does not fade (F_n^± → c_J/ℛ > 0). A neutral recursion with persistent forcing grows linearly in the shell index — the boundary value is +∞, and the finite invariant is the slope per shell. The Cesàro lemma:

Proof. (i) For K large and n > K, unroll: a_n ≤ Π_{j=K}^{n−1}(1+δ_j)·a_K + Σ_{j=K}^{n−1} Π_{i=j}^{n−1}(1+δ_i)·F_j ≤ P_K·[a_K + Σ_{j=K}^{n−1}F_j], with P_K := Π_{j≥K}(1+δ_j). Divide by n; the Cesàro mean of a convergent sequence converges: limsup a_n/n ≤ P_K·F_∞. Let K → ∞: P_K ↓ 1 (summability), giving (i). (ii) Same unroll downward: a_n ≥ Π(1−δ_j)a_K + Σ_j Π_{i≥j}(1−δ_i)F_j ≥ Q_K·Σ_{j=K}^{n−1}F_j with Q_K := Π_{j≥K}(1−δ_j) ↑ 1; divide by n, let K → ∞. ∎

This is the Stolz–Cesàro mechanism (classical; e.g. Muresan 2009) run through a summable multiplicative perturbation — the perturbation products P_K, Q_K → 1 are exactly the summable perturbation budget Σδ_n < ∞ of §1.5.

Proof. At s = 1, λ(1) = 1 and (γS-up)/(γS-dn) are exactly the hypotheses of Lemma γ-R1 for a_n = M_n (upper) and a_n = m_n (lower; note (γS-dn) has the referred point in X_n^ζ by Lemma γ0 — band width zero, so no window enters and the constant is sharp), with F_n^± → c_J/ℛ (Lemma γ1(ii)). Hence M_n/n, m_n/n → c_J/ℛ. For x ∈ X_n, m_n ≤ W(x) ≤ M_n and |n − ln(x/x_a)/(−þ_g)| ≤ 1, so W(x)/ln x → c_J/(ℛ(−þ_g)). Finally c_J/(ℛ(−þ_g)) = c_JÞ_Γ/(−þ_g) = κ̲(ρ+1)σ²/(−2þ_g) by (5.4)'s constant and ℛÞ_Γ = 1. ∎

Benchmark. The constant equals the ψ≡1 corollary of Stage-B Theorem B-res (statement.md §3), proven there by the tilted-walk engine; γ-R is an independent, elementary proof of that corollary at ψ ≡ 1. Falsifier F1b/F3 (§10): per-shell increment measured 0.036476 vs c_J/ℛ = 0.036449 (0.07%); log-slope 0.224075 vs 0.224276 (0.09%).

Remark 4.1 (reconciliation with γ-A). As q_∞ ↓ 1, γ-A’s boundary value blows up at the law B·(ℛÞ_Γ−1) = c_JÞ_Γ, while its plateau recedes (Remark 3.4); expanding x^{−(q_∞−1)} = e^{−(q_∞−1)ln x} in the crossover window (q_∞−1)ln x = O(1) gives W ≈ B(1 − x^{−(q_∞−1)}) ≈ ln x · c_JÞ_Γ/(−þ_g) — the resonance slope is the q_∞ → 1 limit of “boundary value × approach rate”. The failed RT1 registration and its post-hoc diagnostics (§10) measure exactly this interpolation.

Remark 4.2 (why the resonance is where the one-number picture first degenerates). The scalar boundary equation (γ3.1) at λ = 1 reads W(0) = W(0) + F(0) with F(0) > 0 — no finite solution. The compactified picture survives, but the continuous extension is to W/ln(1/z) rather than W; the finite number the owner asked for is the coefficient of the divergence.

§5. Theorem γ-T: the trichotomy — the compactified problem detects the exponent¶

Compensation is a dial, not a hypothesis. Turning s away from q = min(1, 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: W_q is bounded above (γ2) and eventually bounded below by a positive constant — at q_∞ > 1 by W₁ → B > 0 (γ-A), at q_∞ = 1 by m_n ≥ (c_J/ℛ − ε)n → ∞ (γ-R), at q_∞ < 1 by 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, and the two dials). 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: λ(s) crosses 1 exactly at s = q_∞ — below it the recursion contracts, above it expands;

• the forcing dial: F_s stays bounded exactly for s ≤ 1 — for s > 1 the forcing itself diverges like x^{s−1} and no contraction can absorb it (at q_∞ > 1, for s ∈ (1, q_∞], λ(s) ≤ 1 yet W_s → ∞: the income/prudence channel caps the usable compensation at 1 — this is the q_∞ > 1 subtlety the min(1, q_∞) formula encodes).

Whichever dial binds first sets q = min(1, q_∞). (Bookkeeping note: Lemma γ1 derives (γ-up)/(γ-dn) only for s ≤ 1 — its D̂ would become D̂(s) = 2(K̂ + 2sC₀/Þ_Γ) for s > 1 — but no s > 1 recursion is ever used: every s > 1 claim in γ-T is read off the s ≤ 1 results through the identity W_s = x^{s−q}·W_q.)

Remark 5.2 (relation to Lemma A5 / branch selection). For general ψ (Stage B) the boundary multiplier at compensation s is λ_B(s) = E[ψ^{1+s}]/(ℛÞ_Γ^s) = e^{𝔏(s)} (§7), 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): all four wrong-compensation per-shell log-rates, in units of −þ_g, measured within ±0.012 of (s−q), correct signs (the q_∞ = 1.6 cases carry the predicted small deficit from the still-decaying W₁ − B factor).

§6. γ-C: q_∞ < 1 — the honest geometry: a boundary circle, not a boundary point¶

At q = q_∞ < 1: 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 (how the owner’s one number degenerates, exactly). At q_∞ > 1 the boundary circle is still there, but γ-A forces the boundary function to be the constant B — the circle collapses to a point and “one number” is exact. At q_∞ < 1 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). 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 (Stage-A resonance q_∞ = 1 sits between). There the circle also exists but the divergence is common to all fibers (γ-R’s slope is phase-free); the phase question only affects the O(1) intercept — visible in F3’s fitted intercept, not pursued.

§7. γ-B: Stage B (permanent shocks), q_∞ > 1 — the boundary value B_ψ¶

Setting: stage_B_proof.md §§B0–B3. The recursion is now a random rescale (B3.1): ψ_{t+1}x_{t+1} = Þ_Γx + W_{t+1} + ℛg(x), |W_{t+1}| ≤ C_W, and the one-step identity carries the Euler weight ψ inside the expectation ((B5.2), Cor B-5.2). Two structural changes against Stage A, both honest:

1. the referred point y_ψ := (Þ_Γx − C₀^B)/ψ is random: in logs it lands in a bounded window ln x − (−þ_g) + [−ln ψ_max, −ln ψ_min] + O(1/x) — the band is back (its correct shell-binning is by the guaranteed step μ = ln(ψ_min/Þ_Γ), NOT by −þ_g — see γ-B1);

2. if ψ_min < Þ_Γ — the generic case at real calibrations (stage_B (B5.0a) note) — the window reaches forward (y_ψ > x with positive probability): the shell-indexed sup-induction has no well-founded order to induct on, and the band-sup route of §2 does NOT close verbatim. This corrects the design brief’s expectation (“γ2/γ3 close verbatim in band-sup form”): they do only when ψ_min > Þ_Γ (γ-B1); the generic case needs one genuinely new (still elementary) argument (γ-B2).

7.1 The compensated Stage-B one-step¶

Multiply Corollary B-5.2’s upper and lower one-step bounds by x and compensate at s = 1. Using ψ·x/y_ψ = ψ²·(1 − u_B)^{−1}/Þ_Γ, u_B := C₀^B/(Þ_Γx), and folding (1 ± K̂_B/x) and (1 ∓ u_B)^{−1} into one factor exactly as in Lemma γ1 (constant D̂_B := 2(K̂_B + 2C₀^B/Þ_Γ)):

(γB-up)  W₁(x) ≤ (1 + D̂_B/x)·[ (ℛÞ_Γ)^{−1}·E[ψ²·W₁(y_ψ)] + F_B(x) ],
(γB-dn)  W₁(x) ≥ (1 − D̂_B/x)·[ (ℛÞ_Γ)^{−1}·E[ψ²·W₁(y′_ψ)] + F_B(x) ],       (γ7.1)

with y′_ψ := (Þ_Γx + C₀^B)/ψ, F_B(x) := ℛ^{−1}xJ(x) → c_J^B/ℛ ((B5.4); bounded by (B5.3)). Validity threshold (stated per review RF1-F1): (γ7.1) holds for x ≥ x₀^B — its folding needs u_B ≤ ½ (i.e. x ≥ 2C₀^B/Þ_Γ) and x ≥ K̂_B for the D̂_B-collapse, and both are dominated by x₀^B because stage_B’s (B5.0a)/(B5.0) contain the entries 8(ρ+1)C₀^B/Þ_Γ ≥ 2C₀^B/Þ_Γ and 2K̂_B (domination re-verified by RF1 against stage_B_proof.md); Lemma γ-B2’s X₁ secures this through its entry (a) x₀^B — the dependence was previously silent, and a future edit of stage_B’s threshold display should re-check it here. The boundary multiplier

λ_B := E[ψ²]/(ℛÞ_Γ) = e^{𝔏(1)} < 1  ⟺  q_∞ > 1

(A5’s convex geometry: 𝔏 convex, 𝔏(0) = −ln ℛ < 0, unique root q_∞, so 𝔏(1) < 0 iff 1 < q_∞). Why ψ² and not ψ: one ψ is the Euler marginal-utility weight of (B5.2), the second arises because the compensation x/y_ψ reintroduces the rescale — the same reason Theorem B3’s denominator carries E[ψ²]. The review harness’s delta_B81 diagnostic (wrong -weight controls) certifies on solved models that the ψ²-weight is the one carrying signal (F5, §10: main residual 3.5–4 orders below the ψ¹ control and a factor ≈8/≈47 below the Stage-A-shaped control; the registered margin |main| ≤ 0.2·min(controls) holds in both designs). Evidence-route caveat and robust replacement (per review RF2-N1): the Stage-A-shaped control ctrlB is NOT an O(1)-plateau control — it decays like x^{−min(1,q_∞−1)} and vanishes with the ψ-spread (law RF2-D: ctrlB − main = −C(Þ_Γx)^{−p}(E[ψ^{2+p}]−E[ψ²])/(ℛÞ_Γ), verified power 0.594 vs 0.6, amplitude ratio 1.03), so the 0.2·min(controls) margin is probe-depth- and grid-fragile in mild ψ-designs (the author’s 1.66× headroom at Na = 5000 inverts to 0.46× at Na = 3000 at the aMax/50 probe). The ROBUST certification of the ψ²-weight is RF2’s wrong-weight plateau sweep: ctrl_w plateaus at B_ψ(E[ψ²]−E[ψ^w])/(ℛÞ_Γ) (law RF2-P, measured to 0.5–0.9% across w ∈ {0, 1, 1.5, 2.5, 3}), the root of w ↦ ctrl_w is ŵ = 1.9999, margin 6647× (review/RF2_check_5_bpsi_discrimination.py). The identification itself is robust; this caveat is about the evidence route only.

Lemma γ-B1 (boundedness, backward-band case ψ_min > Þ_Γ). PROVEN-HERE (shell width corrected by the internal red-team pass).¶

Let μ := ln(ψ_min/Þ_Γ) > 0, the guaranteed backward log-step: pathwise ln y_ψ = ln x − ln(ψ/Þ_Γ) + ln(1 − u_B) ≤ ln x − μ (the jitter term is ≤ 0 on this side). Red-team catch, recorded: the first draft binned by the (−þ_g)-shells of §1 and claimed a backward window [n − b_ψ, n]; that is FALSE whenever ψ_min < 1 (the generic sub-case), because then μ < −þ_g and the referred point can land in shell n+1 itself — the guaranteed step is μ, not −þ_g. The correct bins are μ-shells: Y_k := [X₀e^{kμ}, X₀e^{(k+1)μ}), X₀ := max(x₀^B, 4C₀^B/(Þ_Γμ)) (the second entry keeps the downward log-jitter 2u_B ≤ μ/2 for the window’s bottom edge). For x ∈ Y_{k+1}: ln y_ψ ≤ ln x − μ < top(Y_k), so the referred point lies in Y_j with j ≤ k (no self-reference), and ln y_ψ ≥ ln x − ln(ψ_max/Þ_Γ) − μ/2 bounds the window depth by b_w := ⌈ln(ψ_max/Þ_Γ)/μ⌉ + 1. With M_k^Y := sup_{Y_k} W₁ and P_k := max_{j∈[k−b_w, k]} M_j^Y, (γB-up) gives M_{k+1}^Y ≤ (1+δ_k)[λ_B·P_k + F̄_B] (δ_k := D̂_B e^{−kμ}/X₀, geometric), and the window-max induction of Lemma γ2(i) closes: P_{k+1} ≤ max(P_k, λ′P_k + F̄′) ≤ max(P_{k₂}, F̄′/(1−λ′)) for k ≥ k₂, where k₂ := max(⌈the γ2(i)-type shell index⌉, b_w) — base and sub-X₀ bookkeeping made explicit per review RF1-F2: (i) the displayed recursion needs the whole window [k−b_w, k] to consist of defined bins, so the induction starts no earlier than k = b_w (the finitely many bins below k₂ are individually finite, W₁ ≤ ḡ·x on each compact Y_j, so P_{k₂} < ∞); (ii) for the few early steps whose referred points fall below X₀, the crude bound W₁(y) ≤ ḡ·y ≤ ḡ·X₀ (from L2^B) covers them. Hence sup W₁ < ∞ on [X₀, ∞). As ψ_min ↓ Þ_Γ, μ ↓ 0 and the shells degenerate — the quantitative signal that the forward case is a genuine phase change, not a bookkeeping inconvenience. (γ-B1 is in any case redundant for Theorem γ-B: γ-B2 covers all bounded ψ including this one.) ∎

Proof (stopped expectation-unroll; the one path-sum-flavored argument in this document — no tilt, no renewal, no change of measure: only (γB-up) iterated plus i.i.d. factorization and one stopping time).

Let (ψ_j)_{j≥1} be i.i.d. copies of ψ and define the comparison chain

y₀ := x,     y_{j+1} := (Þ_Γ·y_j − C₀^B)/ψ_{j+1},

(the bracket points of (γB-up) composed), and the stopping index τ := inf{ j : y_j < X₁ }. Choose

X₁ := max{ x₀^B,  (C₀^B + hψ_max)/Þ_Γ,  D̂_B·2λ_B/(1−λ_B),  D̂_B·2/(ℛ−1) },

whose entries buy, in order: (a) admissibility of (γB-up) at every alive point; (b) domain safety one overshoot step below X₁ (if y_{τ−1} ≥ X₁ then y_τ ≥ (Þ_ΓX₁ − C₀^B)/ψ_max > h > 0, so g(y_τ) is defined and W₁(y_τ) ≤ ḡ·X₁ since y_τ < X₁); (c) (1 + D̂_B/X₁)·λ_B ≤ λ̄ := (1+λ_B)/2 < 1; (d) (1 + D̂_B/X₁)/ℛ ≤ r̄ := (1 + ℛ^{−1})/2 < 1.

Iterate (γB-up) k times along the chain, applying it only at alive indices j < τ∧k (where y_j ≥ X₁, hence (1 + D̂_B/y_j) ≤ 1 + D̂_B/X₁); at the stopped index bound W₁(y_τ) ≤ ḡX₁; at the un-stopped horizon bound W₁(y_k) ≤ ḡ·y_k. Writing w_j := (1 + D̂_B/X₁)·ψ_{j+1}²/(ℛÞ_Γ) for the alive weight factors:

W₁(x) ≤ E[ Π_{j<τ∧k} w_j · W₁(y_{τ∧k}) ] + Σ_{i<k} E[ 1{i<τ}·Π_{j<i} w_j ]·(1+D̂_B/X₁)·F̄_B,

with F̄_B := ℛ^{−1}·sup_x xJ(x) < ∞ (B5.3). Three estimates:

1. Forcing. E[1{i<τ}Π_{j<i}w_j] ≤ E[Π_{j<i}w_j] = ((1+D̂_B/X₁)λ_B)^i ≤ λ̄^i (drop the indicator — all factors positive; i.i.d. factorization E[Πψ_j²] = E[ψ²]^i; entry (c)). Sum: ≤ F̄_B(1+D̂_B/X₁)/(1−λ̄).

2. Stopped mass. E[Π_{j<τ∧k}w_j·1{τ ≤ k}]·ḡX₁ ≤ ḡX₁·Σ_{i≤k} λ̄^i ≤ ḡX₁/(1−λ̄) (decompose over {τ = i}, bound each layer as in 1).

3. Alive linear term. On {τ > k} every y_j ≥ X₁, so the (1+·) factors are controlled, and pathwise y_k ≤ x·Π_{j≤k}(Þ_Γ/ψ_j) (the −C₀^B only helps). Hence

   E[Π_{j<k}w_j·ḡy_k·1{τ>k}] ≤ ḡx·(1+D̂_B/X₁)^k·E[ψ²·(Þ_Γ/ψ)]^k/(ℛÞ_Γ)^k
                              = ḡx·((1+D̂_B/X₁)/ℛ)^k ≤ ḡx·r̄^k → 0   (k → ∞),

   using E[ψ²·(Þ_Γ/ψ)] = Þ_Γ·E[ψ] = Þ_Γ and entry (d). This is the step the band-sup route cannot reproduce: the weight ψ² and the position (Þ_Γ/ψ)x are correlated, and any sup-over-window bound throws the correlation away exactly when ψ_min < Þ_Γ.

Let k → ∞: W₁(x) ≤ [ḡX₁ + (1+D̂_B/X₁)F̄_B]/(1−λ̄) =: K_B for every x ≥ X₁. ∎

Proof (function-level γ3 — direction-agnostic, so the forward window is harmless). S̄ := limsup_{x→∞}W₁ < ∞ (γ-B2; γ-B1 in the backward case). Fix ε > 0 and X with W₁(y) ≤ S̄ + ε for y ≥ X. For x ≥ (X·ψ_max + C₀^B)/Þ_Γ, every window point satisfies y_ψ ≥ (Þ_Γx − C₀^B)/ψ_max ≥ X a.s., so E[ψ²W₁(y_ψ)] ≤ E[ψ²](S̄+ε); with (γB-up) and x → ∞ then ε ↓ 0: S̄ ≤ λ_B·S̄ + F_B(0) (also using F_B(x) → F_B(0), (B5.4)). Symmetrically s̲ ≥ λ_B·s̲ + F_B(0) via (γB-dn). Hence S̄ ≤ F_B(0)/(1−λ_B) ≤ s̲ ≤ S̄. ∎

Proof. γ-B2 (or γ-B1) + γ-B3 + c_J^B = κ̲(ρ+1)σ_B²/(2Þ_Γ) (B5.4). ∎

Remark 7.1 (the compactification’s gift). The band-sup crudeness that costs sharpness at finite depth evaporates at the boundary: as x → ∞ every window position converges to the same boundary point, so E[ψ²W₁(y_ψ)] → E[ψ²]·W₁(0) exactly — no correlation loss in the limit. This is why γ-B3 is four lines while the finite-depth Stage-B machinery (off- limits §B4–§B8) is heavy: the compactified route only ever asks questions AT the boundary.

Falsifiers (§10): F5a (backward design, ψ_min = 0.94 > Þ_Γ = 0.85): deepest W₁/B_ψ − 1 = +0.22%, M_n decreasing; F5b (generic/forward design, ψ_min = 0.75 < Þ_Γ): +0.35%, approach from below (M_n increasing to the plateau — the registered “M_n non-increasing” wording anticipated only the from-above case and passed via its ≤1e-3 wiggle clause; reported honestly, see §10); delta_B81 weight-controls discriminate in both designs (ψ¹ control 3.5–4 orders above the main residual; Stage-A-shaped control ≈8×/≈47×).

Stage-B resonance and q_∞ < 1: OPEN-here (proven elsewhere).¶

At q_∞ = 1 (E[ψ²] = ℛÞ_Γ) the shell displacement ln(ψ/Þ_Γ) makes the shell index a random walk, not a march: the per-fiber/per-shell bookkeeping above needs occupation and drift control for the walk under the ψ²-weight — exactly the (T)-toolkit’s business (off-limits here), and Theorem B-res proves the sharp log-law with it. Likewise q_∞ < 1 Stage B (Theorems B2/B2-arith: KRT/lattice-renewal engines). This document does not attempt either; the γ-route’s honest scope at Stage B is q_∞ > 1. Status: OPEN-here, PROVEN in stage_B_proof.md by the other engine.

§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; iterating instead of dropping resums the geometric series and recovers γ-A’s B — the floor is the first Neumann term of the boundary fixed point. 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 = min(1, 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).

One equation at the boundary. 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 = 1/(ℛÞ_Γ),   F_n → F(0) = c_J/ℛ.

March inward (n → ∞ = z → 0): the recursion forgets the anchor data and is trapped at the fixed point of the limiting map. AT the boundary, the whole functional equation collapses to one line of high-school algebra:

W(0) = λ·W(0) + F(0)        ⟹        W(0) = F(0)/(1−λ) = B.

That is the owner’s finite number: B = κ̲(ρ+1)σ²/(2(ℛÞ_Γ−1)). 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 induction), and bounded + asymptotically-autonomous ⟹ trapped at the fixed point (§2, a four-line limsup argument). The theorem is the continuity of one function on one compact interval at one point.

The three regimes are the three things a linear recursion can do at a boundary.

• λ < 1 (q_∞ > 1): contract — the boundary value is F(0)/(1−λ); one number (§3).

• λ = 1 with persistent forcing (q_∞ = 1): drift — linear growth in the shell index, ≍ ln x; the finite number is the slope, κ̲(ρ+1)σ²/(−2þ_g) (§4).

• λ = 1 with fading forcing (q_∞ < 1): neutral — the recursion converges but no longer mixes phases, so the “number” is a function on the circle of phases; its envelope [m_∞, M_∞] is elementary, its pointwise form is the periodic factor P (§6).

The exponent is detected, not guessed. Compensating with the wrong s flips the boundary multiplier λ(s) = e^{−(q_∞−s)(−þ_g)} across 1, or blows up the forcing x^{s−1} (for s > 1): W_s → 0 below the true exponent, → ∞ above it, at predicted geometric rates — verified to ±0.012 (in units of −þ_g) per shell (§10). min(1, q_∞) is the unique dial setting with a finite nonzero reading; and the equation “λ(s) = 1” is, at Stage B, 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 units where the limit is finite, the Euler equation at infinite wealth reads W = W/(ℛÞ_Γ) + c_J/ℛ; solve for W.” Everything else in this document is the bookkeeping certifying that the two words “at infinity” are legitimate.

Kinship (why four proofs share one constant). B = Σ_k λ^k F(0) is a geometric sum — and it must be: the gap at huge wealth is the discounted pile-up of the small prudence corrections (J ≈ c_J/x) collected once per period while wealth decays geometrically toward its target. Every correct proof computes this same 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 four-line limsup argument. The compactified route is the shortest because it asks the weakest question: not “how fast?” (rates), not “along which paths?” (fluctuations), only “what number at the boundary?”.

§10. Numerical verification (pre-registered falsifiers; author-pass)¶

Harness: verify_altproof_compactified_checks.py (bare python3 + numpy 1.26.4, longdouble EGM solvers imported from review/R4_egm_lib.py and review/RB4_egm_lib_B.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). Falsifiers F1–F6, CC, RT1 were registered in the script docstring BEFORE any measurement; saved output: verify_altproof_compactified_checks_out.txt. 13 PASS / 2 FAIL of 15 registered checks (+5 post-hoc INFO controls PH1–PH3, added after the first registered run and marked as such in the script) — both FAILs are the author’s own registered near-resonance window predictions (RT1), reported as failures and diagnosed below (not tuned away).

Main calibrations: 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_∞ ∈ {1.6, 1.0, 0.6} via ℛ = e^{q_∞(−þ_g)}. Shell geometry as in §1 (ζ reported per run; measurement bins are the plain geometric shells).

Reading the passes.

• F1a/F2/CC (γ-A): the shell means contract to B at the predicted per-shell rate e^{−(q_∞−1)(−þ_g)} (x-power 0.597 ≈ 0.6), land within 0.9% at the guard ceiling, and the one-step residual W₁(x) − λW₁(Þ_Γx) − c_J/ℛ — the finite-z version of the scalar boundary equation (γ3.1), and the same object as the review libs’ delta81 diagnostics — vanishes with depth. The mechanism, the constant, and the coefficient-limit lemma are all exercised independently.

• F1b/F3 (γ-R): at exact resonance (ℛÞ_Γ = 1 to 1e-12), the per-shell increment of the shell means matches c_J/ℛ to 0.07% and is flat across the last 40 shells — the cleanest possible signature of Lemma γ-R1’s mechanism (neutral multiplier + convergent forcing); the log-slope matches the resonance constant κ̲(ρ+1)σ²/(−2þ_g) to 0.09%.

• F1c/F1c′ (γ-C): envelope converges (increments 2.4e-5 and shrinking), floor positive. The deepest-shell phase profile at −þ_g = 0.288 is flat to 3.4e-5 — far below the crude e^{−α/(−þ_g)} extrapolation of the R3 oscillation law (~2e-2..6e-2 at α ∈ [0.8, 1.2]): the measured number is an UPPER bound on osc(P)/P̄ contaminated by residual envelope drift (the profile is a monotone micro-ramp, not a resolved oscillation), so the run is consistent with “P effectively constant here” and does NOT resolve the circle-vs-point question — which belongs to periodic_factor_fine_structure.md, not to this route.

• F4 (γ-T): the wrong-exponent dial responds at the predicted geometric rates with the predicted signs in all four registered cases; the q_∞ = 1.6 pair carries the expected small deficit/excess from the still-decaying W₁ − B factor (measured −0.1118/+0.0882 vs ±0.1; the q_∞ = 0.6 pair is exact to ±0.0001 because there the envelope has already converged).

• F5 (γ-B): both ψ-designs land on B_ψ (+0.22% / +0.35%) — including the GENERIC forward-window design ψ_min = 0.75 < Þ_Γ = 0.85 where the band-sup induction provably does not close and boundedness rests on Lemma γ-B2’s stopped unroll. Honest note: design (b) approaches the boundary value from BELOW (M_n increasing), while the registered wording (“M_n eventually non-increasing”) anticipated only the from-above case; the check passed through its ≤1e-3 late-wiggle clause, and the from-below approach is the correct reading (the subleading coefficient changes sign between the designs). The wrong-weight controls exceed the main residual by 3.5–4 orders (ψ¹-weight) and by factors ≈8 / ≈47 (Stage-A-shaped argument, designs a/b) — the registered |main| ≤ 0.2·min(controls) margin holds in both designs: the E[ψ²] multiplier of (γ7.1) is what the data selects. (Caveat per review RF2-N1: the Stage-A-shaped control decays like x^{−min(1,q_∞−1)} rather than plateauing, so this margin is probe-depth- and grid-fragile in the mild design (a) — it inverts at Na = 3000; the robust ψ²-weight certification is RF2’s wrong-weight plateau sweep, root ŵ = 1.9999 with 6647× margin — see §7.1.)

• F6 (γ2 robustness, refuter lesson honored): a fat-tailed θ (discretized Pareto α = 3.0 tail, L4′-admissible) with θ_max escalated 10 → 160 degrades the constants exactly as the proof says it must (C₀: 9 → 140; ζ: 60 → 934; usable shells: 40 → 23) while the CONCLUSION is untouched (W₁ → B(σ²_disc) to 0.16–0.37% at every escalation). Bounded-θ theorems applied to the discretization; the unbounded case is L4′'s import, not re-proven here.

Reading the failures (RT1) — kept as failures. The author’s registered criterion said the resonance slope law should govern the whole guard-clean window at q_∞ = 1 ± 0.05 to 10%. It does not: at q_∞ = 0.95 the measured ln-x slope is 2.9× the resonance constant (the window is already feeling the growing x^{1−q_∞} Kesten-channel amplitude); at q_∞ = 1.05 it is 8× SMALLER (the window is already 90% of the way to the plateau at B — mostly past crossover). Diagnosis: the registered criterion implicitly required |q_∞−1|·ln x ≪ 1, but on this window |q_∞−1|·ln x ≈ 0.2–0.7 — order one, mid-crossover. The post-hoc diagnostics quantify it (PH2): the split-window local slope at q_∞ = 0.95 GROWS with depth (0.519 shallow → 0.746 deep, against the resonance constant 0.220 — the x^{1−q_∞} amplitude regime), while at q_∞ = 1.05 it SHRINKS toward zero (0.031 → 0.024 — flattening onto B). The honest uniformity statement (now Remark 3.4/4.1): γ-A’s plateau needs ln x ≫ 1/(q_∞−1); γ-R holds exactly at q_∞ = 1; in between, behavior interpolates along the crossover of Cor. A4.2, and no fixed-window law with a 10% tolerance exists for |q_∞−1| ≈ 0.05 at accessible depths (|q_∞−1| ≲ 0.01 would be needed). The failed registration is retained in the script and the output verbatim.

Post-hoc controls (PH1–PH3; INFO, not falsifiers). PH1 (solver quality): doubling the EGM grid (Na 6000 → 12000) moves the q_∞ = 1.6 deepest-shell mean by −4.6e-5 relative; shifting the shell anchor by half a shell (x_b → x_b·e^{(−þ_g)/2}) moves the measured |mean−B|/B from 0.897% to 0.942% — both far inside the 2% falsifier tolerance (grid- and binning-invariance of the headline). PH2: the RT1 diagnosis above. PH3: the Stage-B approach to B_ψ is from ABOVE in design (a) (mean/B_ψ: 1.231 → 1.0022) and from BELOW in design (b) (0.712 → 0.9965) — documenting the sign flip behind the F5b wording note.

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 the same skeleton via the forward unroll of Cor 5.2: unroll/solved ratios {1.006, 1.032, 1.047} at q_∞ ∈ {1.6, 1.0, 0.6} and per-term head/tail ratios {19.7, 0.87, 0.051} — respectively the λ < 1 contraction, the Cesàro flatness, and the end-dominated (fading-forcing) signatures of §§3, 4, 6. This suite reproduces all three signatures on large-(−þ_g) calibrations with registered tolerances and adds the boundary-value, detection, Stage-B, and fat-tail 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 six theory-target agreements at 0.07–0.9% and by PH1’s grid-doubling control (−4.6e-5 relative shift of the headline under Na 6000 → 12000).

References¶

Internal (this repo): statement.md; stage_A_proof.md §§1–5 (imports; incl. the stage-A forcing-floor Lemma 6.1); stage_B_proof.md §§B0–B3 (imports; incl. the stage-B forcing-floor Lemma B-6.1); periodic_factor_fine_structure.md (owner of P’s fine structure); review harnesses review/R4_egm_lib.py, review/RB4_egm_lib_B.py.

External (kept deliberately minimal — simplicity is this document’s product; alt_proof_econlit.md owns the literature fabric):

• F. Dumortier, J. Llibre, J.C. Artés, Qualitative Theory of Planar Differential Systems, Springer Universitext, 2006, ch. 5 (Poincaré compactification; the chart at infinity is (u, v) = (y/x, 1/x) with the line at infinity {v = 0} — the coordinate z = 1/x used here is the one-dimensional instance). Chart convention verified against the source before citing (2026-07-07).

• M. Muresan, A Concrete Approach to Classical Analysis, Springer, 2009 (the Stolz–Cesàro theorem; any standard real-analysis text serves).

Document status recap: γ0/γ1/γ2/γ3/γ-A/γ-R1/γ-R/γ-T/γ-C1/γ-C2/γ-B1/γ-B2/γ-B3/γ-B: PROVEN-HERE. §6 circle-limit: IMPORTED (Theorem A2) modulo GAP-γ-equicont (tagged). §7 Stage-B resonance and q_∞ < 1: OPEN-here (PROVEN in stage_B by the off-limits engine). §8: PROVEN-CITED (owners: the stage-proof forcing-floor lemmas, stage-A 6.1 / stage-B B-6.1). Two registered numerical falsifiers (RT1) FAILED as registered; diagnosis in §10; no proof statement depends on them. Refuter panel 2026-07-08 (RF1/RF2): every PROVEN-HERE status CONFIRMED, the quarantine HOLDS, the RT1 FAILs adjudicated as honest crossover phenomenology; repairs RF1-F1/F2/F3/F4 and RF2-N1/N2/N3 applied in place (marked).