ERRATUM to AXN:02E3 — Monotonicity Per Realization Is Not Established: Re-scoping Appendix A (ii)–(iii) and the Bistability Claim of Deposit #783
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Deposit #783 (Fear and Trembling: Diversity Contraction Across Substrates, v9.1, Nobel Glas) asserts in Appendix A that under axioms A1–A4 Shannon entropy is non-increasing and modal mass non-decreasing at each step of the chain ΞΌ_{t+1} = S(RΞΌ_t). Both properties hold for the selection kernel S alone and for the composed step in expectation and in the long run; they do not hold per realization, because finite resampling R can flatten the distribution before S concentrates it. Counterexample: (0.90, 0.10) → resample (0.80, 0.20) → concentrate (0.85, 0.15); entropy rises 0.325 → 0.423 nats, modal mass falls 0.90 → 0.85. Property (i), support non-increasing with probability 1, stands. The §6 bistability sentence is likewise re-scoped: super-linear vanishing of regeneration makes the zero state absorbing (local); a second stable equilibrium requires g(D) − pD to change sign twice on (0, 1], which the paper's saturating example supplies. No numerical result depends on per-step monotonicity; Figure 4 is the deterministic ODE. Canonical bytes unchanged; registry-level cross-reference per #1554. Counterexample supplied by an external machine reader on 2026-09-05 and recomputed.
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ERRATUM to AXN:02E3 — Monotonicity Per Realization Is Not Established: Re-scoping Appendix A (ii)–(iii) and the Bistability Claim of Deposit #783
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ERRATUM to AXN:02E3 — Monotonicity Per Realization Is Not Established
status: DEPOSITED v1.0
type: ERRATUM
corrects: Fear and Trembling: Diversity Contraction Across Substrates, v9.1 — deposit #783, AXN:02E3.EMPIRICAL, 2026-06-03; Nobel Glas
subject: Appendix A, properties (ii) and (iii); §6 case 3 (bistability)
severity: Mathematical overstatement — the properties hold for the selection kernel alone and for the composed chain in expectation / in the long run; they are asserted for the realized chain at each step, and that is false. The paper's qualitative argument and its numerical results are unaffected
verification: Appendix A fetched from /data/texts/AXN-02E3-text.md 2026-09-05; counterexample recomputed
1. The claims
Appendix A, under axioms A1–A4 (finite sampling; non-expansion of support; strict selection; in-support stochasticity):
(ii) Shannon entropy H(ΞΌ_{t+1}) ≤ H(ΞΌ_t) when S is monotonically type-concentrating …
(iii) The mode concentrates: the probability mass on the highest-weight type increases monotonically.
Both are stated for the step ΞΌ_t → ΞΌ_{t+1} of the realized chain, where ΞΌ_{t+1} = S(RΞΌ_t) and R is a finite resample (A1).
2. The counterexample
Two types. ΞΌ_t = (0.90, 0.10).
1. R, finite sample: a realization with empirical frequencies (0.80, 0.20). Permitted by A1 and A4; support unchanged.
2. S, monotonically type-concentrating on the sample: (0.85, 0.15). Mass moves to the heavier type, as (ii) requires of S.
H(0.90, 0.10) = 0.325 nats. H(0.85, 0.15) = 0.423 nats. Entropy rose. Modal mass fell from 0.90 to 0.85. Both (ii) and (iii) fail for this step.
The mechanism is plain once seen: selection concentrated the sampled distribution, but finite sampling had first flattened it. (ii) and (iii) are properties of S; the appendix asserts them of S∘R.
3. The correction
Appendix A (ii) and (iii) are to be read as holding:
- for the selection step S applied to any distribution — exactly as stated;
- for the composed step S∘R in expectation over the sampling in R, and for the chain in the long run (almost surely, as t → ∞, support contracts by (i) and the chain concentrates), not at every realized step.
(i) — support non-increasing with probability 1 — stands as written; it depends on A2 alone.
The paper's "standing qualifier" (§7) already reads contraction as a statement about support and entropy under the axioms; this erratum adds that, for entropy and modal mass, "under the axioms" means in expectation and asymptotically, not per step. No numerical result in the paper depends on per-step monotonicity; the Figure 4 dynamics are computed from the deterministic ODE in D, to which the counterexample does not apply.
4. The bistability claim
§6, case 3: super-linear vanishing of regeneration (g(0) = 0, g′(0) = 0) is said to give "a bistable trap: a stable high-diversity equilibrium and a stable low-diversity equilibrium (or zero), separated by an unstable threshold."
What super-linear vanishing establishes on its own is local: D = 0 is an attractor, since near zero αΈ ≈ −pD. A second, high-diversity stable equilibrium and the unstable threshold between them require a global property of g — that g(D) exceeds pD on some interval and falls below it again above it. The paper's saturating example supplies that property, so the bistable picture in Figure 4 is correct for that g. The general sentence should read: super-linear vanishing makes the zero state absorbing; bistability follows when, additionally, g(D) − pD changes sign twice on (0, 1]. This is the strong-Allee case as the paper says, and the strong-Allee taxonomy carries the same global condition.
5. Scope
Canonical bytes of #783 unchanged; AXN and hash untouched. Registry-level cross-reference at the record, per the mechanism of 2026-08-27 (#1554), so the June appendix meets the September re-scoping at the point of use.
6. Record of discovery
The counterexample was supplied by an external machine reader (ChatGPT) in an unprimed traversal on 2026-09-05 and recomputed here. The reader's own gloss is the right one and is adopted: the theorem must distinguish individual realizations, expectations, and long-run behavior.
