Goal-Content Integrity Decay in Recursively Self-Improving Systems

Question. In a recursively self-improving system whose Storage is its weights — written for generation g+1 by generation g, with the current generation's weights influencing the writing — which decay curve describes goal-content integrity at a fixed error rate E?

Answer. None of the usual three (linear / multiplicative / exponential). With weights in the loop, the decay is super-exponential: double-exponential in the weakest coupling, a (k+1)-tower with the verifier in storage, factorial if the verification stack deepens. Linear is falsified; multiplicative ≡ exponential is only the degenerate no-influence case; a floor exists only with an exogenous anchor. The full statement, with attributions, is in CORE.md; the lay version is SIMPLE.md.

Documents

• SIMPLE.md — the result as a 10-line proof for a lay audience. Start here.
• CORE.md — the one-page technical statement: model, master theorem, survival threshold, exit taxonomy, genomic mapping, falsifier. Single source of truth for the technical claims.
• REVIEW.md — independent fleet audit (2026-10-02): no major issues; includes the resolution log for all findings.
• This README — orientation only; intentionally thin so it cannot drift from the sources it points to.

Verdict summary

Regime	Curve	Key theorems
No weights-influence (degenerate)	exponential I₀(1-E)^g	noFeedback_eq, noFeedback_exp
Weakest coupling (fidelity ∝ integrity)	double-exponential (I₀(1-E))^(2^g)/(1-E)	feedback_mul, feedback_eq
Verifier-in-storage, depth k ≥ 1	tower (1-E)^(((k+1)^g−1)/k)	feedbackPow_mul, feedbackPow_catastrophe
Deepening stack, m i ≥ 2	factorial-or-worse exponent P g = ∏ m i ((g+1)! ≥ (g/e)^g for m i = i+2)	feedbackGrow_eq, growP_factorial, feedbackGrow_catastrophe
Any law in the generous class	band: tower ≤ I g ≤ exponential	sandwich, goal_content_integrity
Exogenous restoration B	exponential relaxation to floor B/E	affineStep_eq
Linear, ever	falsified	geometric_not_affine, linear_dies, linear_le_geometric

At E = 9.1%: fatal (≤ 0.25% integrity) by generation 6 under the weakest coupling; below 1/2 at generation 2 at depth k = 6 (gen_two_k6_fails); below 1/5 at generation 3 under the factorial stack (grow_fact_gen3). Survival threshold: E* ≈ k·ln(1/τ)/(k+1)^G (survival_threshold) — no E > 0 tolerates an unbounded horizon. Details and attribution in CORE.md.

Layout

• GoalDecay/Models.lean — base recurrences, closed forms, bounds.
• GoalDecay/Theorems.lean — linear falsified; exponential form; sandwich; catastrophe.
• GoalDecay/Stronger.lean — verifier-in-storage k-family.
• GoalDecay/Growing.lean — deepening stack, factorial regime, 9.1% numerals.
• GoalDecay/Repair.lean — von Neumann multiplexing; common-mode lock-in; no-floor-without-anchor; behavioral-eval ceiling.
• GoalDecay/Core.lean — self-contained distillation (imports only Mathlib).
• GoalDecay.lean — library root.
• Verify.lean — axiom receipt printer (44 theorems).
• sim/ — numerical harness: simulate{,2,3,4,5}.py + results{,2,3,4,5}.csv. All closed forms verified to float dust; fully deterministic and byte-reproducible (see REVIEW.md).

Reproduce

lake exe cache get
lake build
lake env lean Verify.lean

Toolchain: leanprover/lean4:v4.34.1, mathlib4 @ v4.34.1 (pinned in lean-toolchain, lakefile.lean, lake-manifest.json). All 44 receipted theorems depend only on [propext, Classical.choice, Quot.sound]; two depend on no axioms at all (see REVIEW.md).