The claim
Take three communities A, B, C and transports between their response spaces, given as stochastic kernels. Compare the direct route K_{C←A} with the relay K_{C←B}K_{B←A} by their path discrepancy, the expected distance between the two routes’ outputs on the source’s response distributions (A4’s Δ_{A,B,C}, with d total variation). Set beside it the directed task loss of CLAIM-259: the positive part of the rise in a receiver’s optimal risk on a declared decision problem.
Agreement with total loss. Let the source utter a promise or a prediction with probability ½ each, rendered so that a source reader names the act with zero error. Let every kernel on both routes send every input to one fixed output, the same ambiguous expression. The two routes then give the same output for every input, so Δ = 0. A target reader sees one expression whatever was said, so the best guess has risk ½ on the act-identification task with 0–1 loss, and the directed loss is ½, its maximum for two equiprobable acts.
Disagreement with no loss. Let the target’s outputs be pairs (act, register), with register formal or informal. Let the direct route keep the act and render it formal, and the relay keep the act and render it informal. For every input the two routes’ outputs have disjoint supports, so Δ = 1, its maximum. A target reader on either route names the act with zero error, so the directed loss on act identification is 0.
So neither measure bounds the other. A low path discrepancy is evidence that the routes are coherent with each other, and says nothing about whether either keeps what a receiver needs. A high one can mean that the routes keep different things a task does not need. A study of translation paths needs both measures, and needs the task declared before the loss is read (CASE-185).
What it does not say
- That consistency is worthless. In settings where the truth is known to be consistent, as in LIT-1820’s synchronization benchmark, disagreement between routes does locate error (THEORY-1158).
- That the collapse is only a toy. It is the extreme of a real pressure. A model trained on consistency between routes alone is rewarded by moving toward it, as THEORY-1088’s discourse sheaves reach consensus by changing their expression maps.
- That a disagreement which loses nothing on one task is harmless. Register may be what a different task needs. The loss is relative to the declared family of tasks, as CLAIM-348 warns of coarse-grainings generally.
- Anything about holonomy in the gauge sense. Stochastic kernels are generally not invertible, so path discrepancy is not a Wilson loop and carries none of THEORY-1140’s invariance.
Note of 2026-10-31: coexisting meanings, and the criterion they lack
Bencherki, Brummans and Vézy (LIT-1902, THEORY-1278) hold that organizational becoming “is not constituted by the erasure of differences in meanings (or the reduction of equivocality)” (p. 1358). They take Massumi’s aim for micropolitics, “a maximally inclusive co-patterning of contrasting tendencies” (Massumi 2017, p. 64; LIT-1910, THEORY-1283), whose beauty “requires an affirmation of discord”. Both say that a family of readings can lack coherence, in TERM-097’s sense, without that being a failure.
This claim’s second construction has the same shape in the line’s terms. Two routes that keep the act and differ in register disagree completely and lose nothing the act-identification task needs. The parallel is of structure, not of mechanism (ADR-032). The record’s family is routes of translation judged on a declared task. Theirs is the significations of one action in an organization.
The construction supplies what the readings lack. Neither says when coexisting meanings are harmless and when one of them is a loss. Massumi calls the outcome “always as-yet unknowable” (THEORY-1283). On this claim, whether a disagreement costs anything can be answered only relative to a declared task, by CLAIM-259’s directed loss, and coherence alone answers nothing either way. The first construction is the converse warning, also by analogy: a reduction of equivocality pushed to its limit is the constant rendering, perfectly coherent and with no fidelity.
No status, title or warrant changes.