# When agreement outlives the frame

Foray 245 · Essai 3 · 23 October 2026 · Original synthetic construction by Codex

## The question advanced

Essai 2 showed that two detailed states can have the same exact summary but demand different actions. This attempt adds a way to search for a replacement grouping when relations change. The search objective is structural; the subsequent action test is separate. It does not prove a universal rule for finding abstractions.

The pursued end remains finding useful abstraction layers, keeping useful approximations within their range, and changing layers when needed. The way here is a finite exhaustive search plus a counterexample and negative controls. The means are six invented queues, a known transition matrix, an averaging operator, six fixed dispatch actions and an explicit benefit function. No client evidence or operational data appears in the construction.

## Source contributions and boundaries

- **Herbert A. Simon (1962), *The Architecture of Complexity*, Proceedings of the American Philosophical Society 106(6), 467–482.** [CMU archive](https://iiif.library.cmu.edu/file/Simon_box00064_fld04924_bdl0001_doc0001/Simon_box00064_fld04924_bdl0001_doc0001.pdf). The full paper links hierarchic structure, stable intermediate forms, selective search, near-decomposable dynamics and economical description. The particular bridge here is pp. 473–478: stronger within-system relations, weak external relations, and time scales that can make aggregate descriptions useful. Simon also explicitly considers the possibility that what we can describe shapes what we notice. The pairing search and dispatch assessment below are this essai's construction, not Simon's algorithm or result.
- **Robert Gibbons, Marco LiCalzi and Massimo Warglien (2021), *What Situation Is This? Shared Frames and Collective Performance*, Strategy Science 6(2), 124–140.** [DOI](https://doi.org/10.1287/stsc.2021.0120). Their agents categorise a space of mixed-motive games and act rationally within the perceived situations. Coarse frames may improve or worsen payoffs relative to detailed perception. Repeated interaction, frame changes, exemplar memories and sincere dialogue add further conditions. This motivates testing performance separately from shared perception. The present model does not reproduce their games, equilibria or learning mechanism; unlike their unaware agents, its reviewer explicitly compares frames.
- **Whitman Richards, Brendan D. McKay and Diana Richards (2002), *The Probability of Collective Choice with Shared Knowledge Structures*, Journal of Mathematical Psychology 46, 338–351.** [DOI](https://doi.org/10.1006/jmps.2001.1391). The journal year is 2002; copyright and online publication are 2001. Their graph represents relationships between alternatives, and distance from an ideal point constrains each agent's weak preference order. Analytical results for trees and complete multipartite graphs and simulations for random graphs concern the existence of collective winners/cycles. They do not establish the winner's external quality. Our interaction graph represents queue dynamics, not a preference-domain graph, so neither their probabilities nor their graph theorems transfer. We calculate the actual pairwise votes directly.

All three source texts were read in full afresh for this attempt. The 2021 article's separately hosted online supplement was not included in the source record and was not read. No omitted proof is claimed as independently verified.

## Fixed model and comparison contract

There are six queues A–F. A nonnegative vector x gives their starting load. Total load is always 12 in the eight audit cases. A column vector evolves as y(t+1) = P y(t). P is symmetric and stochastic, so total load is conserved. Its diagonal is 0.50; each designated partner receives 0.50 − 4ε; each of the other four queues receives ε. The interactive domain is 0 ≤ ε ≤ 0.10, horizon t = 0,…,20.

Original strong pairs are AB, CD, EF. Changed strong pairs are AC, BE, DF. Switching the relation scenario changes the underlying system for **all** comparisons. Within any displayed comparison, the initial readings, P, horizon, options, travel costs, decision rule and quality objective are identical. Changing frame does not remove a dispatch option or alter the evidence provided to a reviewer.

The eight audit inputs are six separate single-queue surges of 12, a B+D surge [0,8,0,4,0,0], and broad load [1,2,3,1,2,3]. They are declared synthetic cases, not sampled evidence. Their results never enter the grouping search. For prediction, the six basis surges also provide a linear-system check: mixtures are linear, but the chosen-action operation is not.

A dispatch reaches the whole true load in one chosen queue, less travel cost c = [0,0.1,0.2,0.3,0.4,0.5]. Thus true benefit for action i is y_i(t) − c_i. There is no uncertainty, capacity constraint, redistribution after dispatch or second action. The costs avoid many indifferent choices; they are invented assumptions visible in the interface.

## Derive a grouping, do not rename one

Enumerate all 15 ways to partition six items into three unordered pairs. For each partition g, compute the cut weight: sum P_ij over i < j whose endpoints lie in different groups. Minimise it. This criterion rewards containing strong interactions within a group; it never sees x, dispatch actions, costs or their outcomes. Fixing three pairs is a declared representational budget, not a discovered optimal number of categories.

At ε = 0.01, the changed relation matrix has a unique minimiser AC · BE · DF with cut weight 0.12. The inherited AB · CD · EF cuts all three strong edges and has cut weight 1.47. At ε = 0.10 every off-diagonal relation equals 0.10; all 15 candidates tie at 1.20. The interface reports no preferred grouping, rather than selecting an arbitrary winner and calling it discovered.

Let R_g replace every pair's two readings by their mean. The coarse forecast is (R_g P)^t R_g x. In each step it applies the same true P and then averages again. This is the lifted quotient dynamics; it is not the full model with labels merely drawn around it. Group sums are preserved by averaging, while within-pair contrasts are lost. Full detail is the singleton partition.

Three reviewers use the same deterministic ranking rule: predicted load minus travel cost. The interface compares all-old, all-derived, all-detail and one reviewer of each type. Equal frames and equal rules force unanimous agreement in this model. Agreement is therefore not an estimated advantage of social sharing. The mixed-frame team is a diversity illustration; its success here is not a theorem that diversity is always helpful.

## Distinct quantities

- **Agreement:** fraction of the three reviewer pairs with identical first-choice sets. It can be 0, 1/3 or 1.
- **Collective stability:** one option strictly wins every pairwise majority comparison. Indifferent reviewers contribute zero to that pairwise margin. A tied or cyclic result gives no chosen action and is not assigned a quality score. This deliberately uses a strict winner, narrower than the weak non-defeat condition in Richards et al.
- **Action quality:** lost benefit = max_i[y_i(t) − c_i] − [y_chosen(t) − c_chosen]. Zero is optimal for the stated objective. It is independent of agreement and does not measure fairness or adoption.
- **Approximation error:** maximum absolute queue forecast error among the selected team's reviewers. A small error need not preserve a decision close to a tie.
- **Recovery:** change in the eight-case action loss when old boundaries are replaced by the relation-derived boundaries. This is a comparison after known counterevidence, not a model of time to learn, persuade or implement.

## Constructed result

At ε = 0.01 and t = 3, original relations give the original pairing. It has zero lost benefit in all eight cases. After the relations change, keeping those boundaries yields 100% agreement and a strict stable winner in all eight cases, but only 3/8 optimal decisions. Mean lost benefit is 2.794576 and worst lost benefit is 4.883888. Recomputing boundaries derives AC · BE · DF; all 8/8 decisions are optimal with zero lost benefit. The mixed-frame team also reaches 8/8, with only 58.33% average pair agreement.

The default B surge makes the failure legible. The old frame recommends A unanimously and loses 4.883888 benefit units. The derived frame recommends B unanimously and loses zero. Its maximum forecast error is 0.000384; the old frame's is 3.35917125. A and B surges have exactly the same old-frame input summary, so no computation that retains only that summary can distinguish their different true future loads. The replacement separates A from B using their changed partners, without fitting to this dispatch result.

This does **not** mean a structural optimum suffices. At t = 0 the same derived grouping is optimal in only 4/8 cases, with mean loss 4.6375: its within-pair averaging discards distinctions before they have decayed. At ε = 0.10 the relation search cannot identify a privileged pairing at all. At long horizons all queues approach the same mean; fine distinctions become less relevant for this particular objective.

## Independent arithmetic check

For a true partner pair, let μ be the overall mean and m the pair mean. The full exact solution is:

    y_i(t) = μ + (1 − 6ε)^t (m − μ) + (4ε)^t (x_i − m).

The correct paired forecast omits only the final contrast term. With changed relations and old boundaries, every old group has the same aggregate transition matrix: diagonal 0.50 + ε and off-diagonal (0.50 − ε)/2. Its forecast is:

    old_i(t) = μ + (0.25 + 1.5ε)^t (old_pair_mean − μ).

The independent Python check uses these closed-form expressions and exact fractions, not the JavaScript transition loop. It compares dynamics, projections and action regret against every recorded run. Tests separately check conservation, all 15 candidates, the A/B witness, immediate-decision failure, a constructed majority cycle, ties and invalid controls. No claim of empirical validation follows from these software checks.

## What to try next, and what remains unknown

The bounded proposed heuristic is to name the decision and horizon, audit the interactions cut by the current grouping, search candidate boundaries without rewarding the favored answer, then check lost action-relevant distinctions using independent outcomes. A structural tie should remain a tie. An immediate or boundary-sensitive decision may need detail even when the aggregate dynamics are accurate.

This finite search has a built-in pair budget, homogeneous known weights and no noisy observations. Real relationships may be directed, nonlinear, unobserved or contested; structural cut weight might be the wrong objective. Learning a replacement from finite evidence would need held-out observations, uncertainty and switching costs. A different objective could judge the same unanimous outcome differently. None of these extensions is established by this attempt.

Replay from a repository checkout with `node abstraction-layers/essai-03/replay.cjs --verify`, `python3 abstraction-layers/essai-03/verify_analytic.py`, and `node --test tests/abstraction-frames*.test.cjs`. Public downloads include the model and recorded results; verification utilities remain in repository source.