The claim
Take a chain u_0 → u_1 → … → u_n produced by maps f_1, …, f_n on a metric space, u_i = f_i(u_(i−1)), and a reference chain v_i = g_i(v_(i−1)). Let e_i = d(u_i, v_i). Let ε_i = d(f_i(v_(i−1)), g_i(v_(i−1))) be the discrepancy step i introduces at the reference point, and L_i a Lipschitz constant of f_i. By the triangle inequality,
e_i ≤ d(f_i(u_(i−1)), f_i(v_(i−1))) + d(f_i(v_(i−1)), g_i(v_(i−1))) ≤ L_i e_(i−1) + ε_i,
so
e_n ≤ (∏(j=1..n) L_j) e_0 + Σ(i=1..n) ε_i ∏_(j>i) L_j.
The impact of a local error depends on the dynamics after it. There are three regimes: contractive (L < 1, errors are damped), neutral (L = 1, they accumulate, giving the familiar d(u_0, u_n) ≤ nε), and expansive (L > 1, they amplify). Closeness at every step is compatible with large divergence end to end, because similarity is not transitive.
One restriction is exact: for stochastic kernels in total variation, contraction coefficients do not exceed one. Amplification needs another metric, an enlarged state, or nonlinear, state-dependent dynamics.
Where the record applies it
CLAIM-056 applies the recursion to translation and reconstruction chains, with the discrepancy read as a transport error against a reference transport, which rests on CLAIM-106’s graded fidelity. CLAIM-191 applies it to restatements of an argument across sessions. Both now rest on this claim for the recursion and keep their own premises for the application.
THEORY-289, filed from LIT-971 (read in NOTE-815), is an exact closed-form case of the bounded regime. When each round of a linear regression is refit to labels from the previous fit and earlier rounds’ data are kept, the per-round discrepancy is scaled by 1/i, the weighted sum converges, and test error is bounded at π²/6 times the first fit’s. When the data are replaced, the discrepancies add and error grows linearly. Error is accumulated but bounded: every round still adds some.
What it does not say
- It does not say any departure from the reference chain is an error in a normative sense. The bound measures departure relative to a reference.
- It does not say drift is random. Attraction toward conventions is a further claim (CLAIM-068).
- It does not cover directed, non-metric distortion (QUESTION-017).
Note of 2026-10-19: witnesses for the three regimes
The bound is an upper bound. Each regime is also attained, on ℝ with the reference g_i(x) = c·x and the actual chain f_i(x) = c·x + ε, from u_0 = v_0. Then e_i = c·e_(i−1) + ε exactly, and every step introduces the same discrepancy ε.
- c = 1/2: e_n = 2ε(1 − 2⁻ⁿ) < 2ε. Each error is damped, and the total stays bounded.
- c = 1: e_n = nε. The errors accumulate, so each step is within ε of the reference while the endpoints are nε apart.
- c = 2: e_n = (2ⁿ − 1)ε. The errors are amplified.