Erratum, 2026-11-06: the scale-free-cascade discriminator is withdrawn
This is the erratum as issued with the three CNBF papers. It is shown here unabridged. File names in it refer to the author's working folders.
Paper: The Dropzone: Convergent Nodes, Bifurcating Failure, and the Governance of the Committed Gate DOI: 10.5281/zenodo.21534956 Issued: 2026-11-06 Status: erratum, not a silent revision. The changes below are applied in CNBF-Dropzone.md and are listed here as the changeset going into the paper.
One sentence: the series typed nodes by the shape of their precursor fluctuation distribution — Gaussian-with-rising-variance at a fold, a scale-free cascade at a hinge — and that pairing does not discriminate; typing falls to the structural channel, and the cascade turns out to belong to a coupled population at loop gain approaching one rather than to a node of any particular type.
Changeset
| Site | Withdrawn or amended | Replacement |
|---|---|---|
| §4, discriminating-precursor paragraph | "The discriminating precursor is therefore the fluctuation statistics: approximately Gaussian variation with diverging variance ahead of a fold, a scale-free cascade ahead of a hinge." | Withdrawn and replaced. Records that §9 kill test 1 was written against this mapping and has fired. Discriminating work falls to the structural channel; the two geometries remain distinguishable before either resolves, but through a structural rather than a statistical channel. |
| §9, kill test 1 | "A structurally typed sensor node whose precursor events are distributed without a characteristic scale … falsifies the geometry mapping of §2 and §4." | Marked FIRED. The first clause is satisfied by the mean-field bundle. The geometry mapping of §2 and §4 is falsified as stated, and §4 is amended rather than defended. Three riders stated: the evidence is a simulated canonical family and not this paper's tissue, so the pairing is withdrawn rather than inverted for the sphincter; the falsifier carried no tissue restriction and declining to count a model in the same class would be the retreat the series forbids (C8); and the test's second clause did not fire, which is why the load-bearing case is left with an absence rather than the other half of a swap. |
| A.2, closing fence | "The load-bearing discriminator is therefore the fluctuation statistics — Gaussian with diverging variance at a fold, scale-free cascade at a hinge" | Amended to the structural channel, with presence-or-absence of divergence as the robust reading and the cascade assigned to the coupled population. |
What survives, unchanged
§2's flattening-and-lengthening argument is untouched — slowing needs no loop, and the local-load bundle's absence of slowing is the clause these data confirm. The condition-(iii) collar treatment at A.2, the strongest in the set, is unaffected. Kill test 6 (the fold/queue exponent test, added at v4) is untouched and becomes more load-bearing, since the structural channel is now carrying the discrimination. Kill tests 2–5 are unaffected.
The finding
CNBF-LoopMap/CNBF-DiscriminatorStress-v1.md §3 and §3A ran the two canonical fibre-bundle models of fracture through the series' own typing criteria. The result:
| Mean-field (equal-load-sharing) bundle, N = 10⁶ | Local-load-sharing bundle, N = 4096 | |
|---|---|---|
| Transfer curve | saturating (slope → 0 at peak) | stiff to rupture (0.97 early, 0.69 at peak) |
| Relaxation vs distance to failure | diverges, 9 → ~130 rounds; exponent −0.42 measured, 1/2 analytic | flat, 4–5 rounds down to 10⁻⁵ of σ_c |
| Structural type | sensor / fold | load-bearing / hinge |
| Intact at failure | 50% | 86%, single snap |
| Precursor burst sizes | scale-free: slope −2.62 (5–500), −1.70 within 1% of failure | steep and bounded: slope −4.5; largest of ~11 000 bursts across 20 bundles = 10 fibres; P(Δ ≥ 10) = 0.009% |
The two structural criteria agree with each other in both models. The fluctuation-shape criterion disagrees with both, in both.
It is not an inversion, and the distinction matters. The original pairing made two positive claims. One half relocates: the scale-free cascade belongs to the fold. The other half has no replacement: the hinge shows neither a cascade nor Gaussian-with-rising-variance, but many small events steeply distributed. So a hinge is diagnosed by an absence, which is weaker evidence than a presence and is indistinguishable from an instrument below its event-count floor unless that count is checked. Reading the result as a label swap would wrongly imply the discriminator still works with the arms exchanged.
The mechanism, which is why the relocation is not a coincidence. A precursor burst is a branching cascade whose branching ratio is the round-trip gain G of the loop containing the node. Mean burst size is 1/(1 − G) and diverges as G → 1; at G = 1 a mean-field branching cascade gives sizes distributed as s^(−3/2), which is the measured near-failure exponent. For the mean-field bundle G = σ/U*², reaching exactly 1 at the failure load σ_c = 1/4, and the gain read directly off the event stream as the ratio of breaks in successive redistribution rounds matches that formula to within 2.2% across seven loads. Scale-free bursts are the event-stream view of a fold in a coupled population.
A cascade is not exclusive to a population of load-bearing elements. What it requires is discrete threshold-like transitions in the constituents, coupling between them, and gain near one. Rupture supplies discreteness most naturally, but coupled reversible units flip discretely and also cascade. What the event stream can type is the constituents and their coupling, read off reversibility: irreversible events alone imply non-returning constituents; a stream carrying reverse events implies reversible ones (a healing bundle under the same ramp gives ≈20× more break events, DiscriminatorStress §7); constituents that merely accumulate should give no discrete events at all.
Why precision does not repair it. The burst distribution is set jointly by the node's geometry and by the coupling of the population the node sits in, so distinct combinations of the two are indistinguishable in it. That is an identifiability failure, not the estimability failure the series already carried from Carpenter & Brock (2011), and no amount of sampling or precision addresses it. Series style sheet §1.5 carries the distinction as of v15.
What replaces it
Typing falls to the structural channel, which is where the two criteria that agreed already live:
| Reading | Type |
|---|---|
| Saturating transfer curve; relaxation diverges, exponent in [1/2, 1) | fold (sensor) |
| Saturating step; relaxation diverges, exponent in (1, ∞), pool behind the step rising | queue (sensor, queueing) |
| Stiff transfer curve to rupture; no divergence at all | hinge (load-bearing) |
The sign of the recovery-time trend remains corroborating and not deciding (C2, first half, unchanged), and the more robust reading is whether divergence occurs at all rather than which way τ moves — the local-load bundle was flat, not falling. Avalanche statistics remain admissible for typing only where the coupling topology is held fixed or recorded in the structural channel.
Scope of the evidence, stated as a limit
One model family, two bundles, simulated, canonical in the fracture literature and not the tissue of any paper in this series. Accordingly: the distribution-shape pairing is withdrawn, not reversed, in all three papers. The relocation of the cascade to the fold is asserted for coupled populations at gain approaching one, where it has a derivation and a measurement; nothing is asserted about the direction of the pairing in metabolic or institutional nodes.
What would defeat this correction
- A structurally hinge-typed model — stiff transfer curve, no slowing — with a scale-free precursor distribution measured above the event-count floor. That would restore the pairing for that class.
- A mean-field cascade in which the successive-round event ratio departs from the linearised round-trip gain by more than finite-N scatter while the bursts stay scale-free. That would break the gain reading.
- A coupled population at gain approaching one that does not produce scale-free bursts, which would remove the mechanism.
Provenance
Raised by the 2026-10-20/21 pressure test recorded in CNBF-LoopMap/CNBF-DiscriminatorStress-v1.md (§3, §3A, §7) and its pre-registration DiscriminatorStress/PREREG.md, written before the simulations ran. The misattribution of cascade to the node rather than the network was opened as series style sheet §5.18 at v12 and deliberately left unactioned at the freeze; this erratum actions it. The convention that prescribed the withdrawn discriminator, C2, is repaired in the same pass, and prescription P2 in the sheet's §8.1 register is superseded by it.