Preprint · Not peer reviewed

Multiply by a Million: Scale Invariance in LLM Negotiators

1Introduction

People do not treat a bargain a million times larger as the same bargain. They gamble on small stakes and turn cautious on large ones (Markowitz, 1952; Weber and Chapman, 2005), grow more risk averse as payoffs are scaled up (Holt and Laury, 2002), and discount large delayed amounts less than small ones (Thaler, 1981); Prelec and Loewenstein trace these effects to a common source (Prelec and Loewenstein, 1991). Large language models learn from human text and may inherit such stake effects.

LLM agents are increasingly evaluated as negotiators (Bianchi et al., 2024; Abdelnabi et al., 2023; Zhu et al., 2026), usually within one range of prices. A basic robustness question is whether the same bargain, written with every monetary amount a million times larger, produces the same relative offers and the same division of surplus. For negotiators whose utility is linear in money, rescaling leaves the Nash and Rubinstein solutions unchanged (Nash, 1950; Rubinstein, 1982); a person with concave utility need not be indifferent. Our task removes that reason. It gives the negotiator no wealth, budget or time cost, only a private limit and the goal of the best price, so a deal at the same share of the surplus is equally good at every scale. Invariance is the benchmark the task itself sets, and a departure means the model brings in a stake effect, or a number-handling effect, that the task does not ask for.

One observed bargain shows what can go wrong (Figure 1). Gemma 4 31B, selling with a private floor of $83 to a buyer with a ceiling of $118, opened at $120 and kept 69% of the surplus. With limits of $83M and $118M, the same words and the same sampling seed, it opened at $95M and kept 11%.

Earlier exploratory and registered studies of this game (the discovery phase, Section 3 and Appendix D) suggested a specific account: at large amounts the opener asks for less; deals settle near the midpoint of the two first offers; and a public reference number such as an appraisal may remove the effect. This paper reports Study 3, a single account, under one protocol, on five models and on bargains no earlier run used. Its hypotheses, analysis code and scripted opening values were frozen and hashed before the first game, and every registered test is reported.

We find:

• Opening modesty at scale is shared (Section 5.1). Without the appraisal, all five models open closer to their own limit at $100M than at $100. The appraisal reduces this on four models; on Haiku it does not.
• Closing shifts follow the openings (Section 5.3). Scripting both first offers at their $100M positions shifts deals by the change in the scripted midpoint, within ±0.05 of the surplus, on all five models (informatively on Gemma, Luna and Gemini); with the openings fixed, concessions add a small shift on Gemma and Luna and a larger one on Haiku.
• A reference number is not a general fix (Section 5.2). The appraisal removes Gemma's shift and shrinks Luna's, but reverses Gemini's and enlarges Haiku's.
• Inside Gemma, the reference takes over late (Section 6, exploratory). The appraisal becomes decisive at layer 41 of 60, on the reply's price-writing tokens, through attention heads that largely also carry the scale and private-range effects, while the scale signal persists with less effect on the price.
• The scale signal looks like numerical magnitude, not stakes (Section 6.1, exploratory). Gemma reads amounts on one number line in five notations and loses its preference for better gambles at $100M, but the direction that shifts its openings moves its risky choices by 5% of that gap.

The scope is specific: five models, one prompt family, one single-issue game, self-play, thinking off or minimised. We separate registered primary tests, registered secondaries and exploratory analyses throughout.

2Related work

Inside bargaining theory, the more risk-averse party receives less under the Nash solution (Kihlstrom et al., 1981), so a stake-dependent risk attitude is one route by which scale could move a split. Human bargaining studies find mixed stake-size effects (Andersen et al., 2011; Larney et al., 2019; Karagözoğlu and Urhan, 2017). For LLMs, multiplying every payoff in a repeated Prisoner's Dilemma while preserving its structure produces model-specific responses (Huynh et al., 2026); dictator and ultimatum studies likewise find heterogeneous stake effects (Polachek et al., 2026; Araujo and Uhlig, 2026). We extend the fixed-structure manipulation to a continuous price negotiated over multiple turns.

LLM negotiators have been studied in self-play and in arenas (Fu et al., 2023; Bianchi et al., 2024; Abdelnabi et al., 2023; Zhu et al., 2026). Two bargaining studies vary the value at stake, and in both the magnitude moves together with the geometry: doubling a retail price with cost fixed triples the margin (Liang and Xu, 2026), and limits drawn from items' historical prices make the dollar level and the width of the zone of agreement vary together (Bergemann et al., 2026). A self-play study that varies the zone's width finds model-specific crossovers between buyer and seller advantage (Ríos et al., 2025); a house-sale study varies whether a description of the house and comparable sales are shown, close to our appraisal contrast, but not the magnitude (Shah et al., 2025).

First offers predict negotiated outcomes in people (Galinsky and Mussweiler, 2001; Schweinsberg et al., 2023; Lipp et al., 2023) and can anchor LLM negotiations (Takenami et al., 2025). This motivates the midpoint account but does not establish it for our game, which the opening clamp tests directly. Inside models, number values are linearly decodable (Zhu et al., 2024), alongside digit-wise and helical codes (Levy and Geva, 2024; Kantamneni and Tegmark, 2025); irrelevant numeric anchors have been localised to sparse circuits whose depth differs by model family (Owusu et al., 2026). Our mechanistic methods build on activation patching and difference-vector steering (Heimersheim and Nanda, 2024; Turner et al., 2023; Zou et al., 2023).

3Hypotheses and where they came from

Study 3's hypotheses were generated in a discovery phase of two registered studies, three registered addenda and exploratory probes, all reported in Appendix D. Table 1 lists what each suggested and which Study 3 test checks it. No discovery data enter a primary test; they fix only the hypotheses, the margins and the scripted opening values, and the descriptive secondary S8 compares Study 3 with them.

Two gaps in the discovery phase motivated a new study rather than another addendum. In the registered studies, Gemma and the API models never saw an appraisal, so the remedy had been tested only on Qwen, together with the limit ranges. And the protocol changed between runs (one game per cell or four, blocked or interleaved order, 512 or 1,024 output tokens), so cross-model comparisons were confounded with procedure.

4Study 3 design

Game. Two copies of a model negotiate one price. The seller knows a private floor F, the buyer a private ceiling C, and both know a deal is possible. They alternate JSON moves (OFFER a whole-dollar price, ACCEPT the standing offer, or WALKAWAY) with a message of at most 500 characters, for at most 16 messages. Each call is rebuilt from the rules, the side's own limit and the public history. Invalid replies forfeit the game without repair. For a deal at P, seller share is s = (P − F)/(C − F).

Fixtures and scales. Forty fresh fixtures (G01–G40) from a public seeded generator place floors at 75–95% and ceilings at 105–125% of a reference value V. Each is played at V = $100, $100K and $100M. Money is always written as digits followed by "dollars"; with the digits masked, the prompts at the three scales are identical.

Conditions (Table 2). The core grid crosses the appraisal (A0: no appraisal, Study 2's prompt; A1: one added line, "An independent appraisal values the lot at V. Both sides know this.") with scale and opener. The opening clamp, in A0 only, scripts both first offers into the public history and lets the model begin playing at message 3, at game scale G across the scripted opening scale O. The scripted positions are each model's mean first-offer positions in its own earlier no-appraisal games, in zone units. At $100M they are the $100 prices with six zeros appended.

Models. Qwen3.8 27B (FP8) and Gemma 4 31B (bf16) run on pinned weights in vLLM 0.30.0 with each family's non-thinking sampling settings and no repetition penalties. gpt-6-luna, claude-haiku-5.5 and gemini-3.5-flash-lite (hereafter Luna, Haiku and Gemini) run through OpenRouter on each vendor's own endpoint with vendor sampling defaults and thinking off; Gemini's lowest setting, "minimal", does not guarantee zero reasoning, so its reasoning tokens were counted (none were reported). Sampling stays model-native; "one protocol" means every procedural rule is identical. Seeds are shared across scale, appraisal and clamp cell, so every within-fixture contrast shares them (Anthropic receives none; Appendix C).

Sessions. Each model ran in two independent sessions (samples 1–2 and 3–4): separate GPU pods for the open-weight models, separate days for the API models. Each session interleaved all cells in one fixed shuffle.

Tests. The fixture is the unit of analysis; a cell's value is the mean over its four games (agreements only, for seller share).

• P1 (openings). With g the opener's first-offer gap from its own limit, h_f(A) = mean over openers of g($100M) − g($100), and I_f = h_f(A1) − h_f(A0). One-sided Wilcoxon, I > 0.
• P2 (closing shift). d_i = s($100M) − s($100), averaged over openers, per condition. A two-sided Wilcoxon (H1) and equivalence if the 90% bootstrap CI lies within ±0.10 (H2) (Lakens, 2017). A non-significant H1 is never read as equivalence.
• P3 (decomposition, A0). The concession channel κ is the scale effect with both openings held at the same scripted positions, averaged over the opening scale; "openings suffice" predicts κ equivalent to zero within ±0.05. The opening channel ω is the effect of moving the scripted openings from their $100 to their $100M positions, averaged over the game scale; "deals follow the scripted midpoint" predicts ω − Δm equivalent to zero within ±0.05, where Δm is the shift in the scripted offers' midpoint.

Wilcoxon p-values use 200,000 random sign flips; bootstrap intervals resample fixtures 10,000 times; Holm correction runs across the five models within each test. Nine registered secondaries (S1–S9) are reported in Appendix B.

Registration and data access. The registration, the files the study executes or reads (37 in all, including the analysis code and the scripted opening values) and their SHA-256 hashes were committed before any registered game; every session recorded the commit and checked the hashes. The registration was not lodged with an external registry (Appendix A). Format pilots on separate fixtures were judged on format only. One rule was set after the first API pilot, which was then rerun under it, and before the freeze: text after the first complete JSON object is ignored and flagged, rather than forfeiting the game. After the open-weight sessions, and before the second API session, we looked at Gemma and Qwen through the frozen analysis; this access was committed to a log first, the API outcome data stayed unexamined, and nothing in the protocol changed (Appendix A).

5Results

All 16,000 games were scored and every game record reproduces exactly from the raw calls. Luna forfeited 35 games (34 truncated replies); no other model forfeited. Table 3 scores every registered primary test against its prediction: 21 of 25 are met.

5.1Openings shrink at scale; the appraisal attenuates this on four models

Without the appraisal, every model's opener moves toward its own limit at $100M (S1: h from −0.101 on Luna to −0.247 on Gemini, all Holm-adjusted p < 0.001; Figure 2). The registered appraisal × scale interaction I is positive on Gemma (+0.114), Qwen (+0.101), Luna (+0.048) and Gemini (+0.195). On Haiku it is −0.020 [90% CI −0.041, +0.003]: with the appraisal shown, Haiku's sellers open at 0.55 of their floor above it at $100 and 0.31 at $100M, a steeper fall than without it. Only on Qwen does the remaining opening change meet the ±0.03 equivalence margin.

With the appraisal shown, Gemma's and Qwen's sellers open at a level that hardly moves with scale. Gemini's sellers open close to the appraisal itself (a gap of about 0.15 at every scale), while its buyers still drift toward their ceiling at large amounts.

5.2Closing shifts: as predicted without the appraisal, model-dependent with it

Without the appraisal, the closing shift matches the registered prediction on all five models (Figure 3): seller share falls by 0.238 on Gemma, 0.129 on Luna and 0.295 on Gemini, while Qwen (+0.029) and Haiku (−0.051) meet equivalence within ±0.10. Haiku's shift is also significantly below zero (Holm p = 0.014) and larger than in the discovery phase (S8: +0.006 in Addendum C).

With the appraisal, the prediction of equivalence on all five fails on two. Gemma's shift vanishes (+0.002 [−0.010, +0.014]). Qwen (+0.077) and Luna (−0.030) meet equivalence but show small, significant shifts. Gemini's shift reverses (+0.100 [+0.083, +0.119]): its sellers open next to the appraisal at every scale and settle low at $100, while its buyers open closer to their ceiling at $100M, so deals settle higher there. Haiku's shift grows (−0.113 [−0.131, −0.096]), consistent with its sellers' collapsing openings. In every model, deals under the appraisal track where the appraisal sits in the zone (S9: Spearman 0.75–0.98 across fixtures), so the appraisal works as an anchor; whether that anchor removes the scale effect depends on how each model's openings respond to it; on Gemma, Luna and Gemini the shift under the appraisal also rises with the appraisal's position in the zone (S9, ρ +0.40 to +0.56; descriptive).

5.3The opening clamp: deals follow the openings

The clamp separates two channels (Figure 4). Moving the scripted openings from their $100 to their $100M positions, with the game's scale averaged out, shifts the deal by the amount the scripted midpoint predicts on every model, informatively on Gemma, Luna and Gemini (P3b:

ω − Δm within ±0.05 on all five). On Gemma, for example, ω = −0.240 against Δm = −0.253, close to its natural shift of −0.238; on Gemini, −0.340 against −0.308 and a natural −0.295. On Qwen and Haiku both terms are near zero, so P3b cannot distinguish its predictions there, as anticipated in the registration.

Holding the openings fixed and changing only the game's scale leaves a concession channel κ within ±0.05 on four models. It is detectably negative on Gemma (−0.027) and Luna (−0.032): with identical openings, deals settle slightly lower at $100M. On Haiku κ is not shown equivalent to zero (−0.044 [−0.054, −0.034]; its 90% CI crosses the margin), so its behaviour at scale is not explained by openings alone; Haiku is also the one model whose natural deals track the first offers' midpoint only weakly (S5: Spearman 0.37, against 0.83–0.85 for the others).

Scripted openings are not natural ones. At $100M the scripted cells settle lower than natural games on Qwen (−0.069) and Gemini (−0.097; S6, both p < 0.001), which qualifies their clamp estimates.

5.4Repeatability

Every primary statistic was estimated separately in the two sessions (S7). Five of 25 session differences have 95% CIs excluding zero, the largest Qwen's shift under the appraisal (+0.103, then +0.050). Applying the registered rules to each session separately (not a registered comparison), two verdicts differ: Qwen's equivalence under the appraisal fails in session 1 and passes in session 2, and Haiku's P3b passes in session 1 and fails in session 2. The pooled values are the registered estimates. Without the appraisal, the closing shifts do not differ detectably from the discovery phase on four of five models (S8; Appendix B).

6Inside Gemma: where the appraisal acts, and what the scale signal carries

This section reports two exploratory analyses, E5 and E6. Their predictions were written from the discovery-phase results (for E6, also E5's) and committed before each run, and are scored as hits or misses (Table 19, Table 26). Methods and full results are in Appendices E and F.

Setup. We hooked Gemma 4 31B in transformers on Study 3's own first-move prompts (A0 and A1 at $100 and $100M), one fixture per distinct limit: 19 seller floors and 18 buyer ceilings. Sampled openings were within 1.2 points of Study 3's openings in every cell. The readout is the forced opening's log-odds between the two no-appraisal greedy prices at $100 and $100M. The main intervention, the "appraisal swap", copies one run's residual-stream states on the prompt's final segment and the reply's opening tokens into the other run at $100M, one layer at a time, and reports the share of the log-odds distance between the two runs that it moves.

The appraisal weakens the scale effect on the opening without removing it. Under the appraisal, the sellers' forced opening keeps 17% of the no-appraisal log-odds gap between scales (E5-1). The no-appraisal scale direction at layer 42 still moves real appraisal openings, by 31–45% of the gap, against 89–92% for no-appraisal openings (E5-4). Magnitude is still represented; it has less control over the price.

It takes over at layer 41, on the price-writing tokens (Figure 5). For sellers, swapping in the appraisal run's states moves 75% at layer 41 and 83% at layer 42 (E5-2). Earlier layers do not merely fail to help: between layers 30 and 40 the swap moves the opening the wrong way, by up to 35% (61% when removing the appraisal). The committed scoring rule (a signed maximum up to layer 39) counts E5-2 as a hit; read as a magnitude bound it would be a miss. At layer 44 the reply's opening tokens carry 57% and the prompt's final segment 22% (E5-3). Buyers miss the predicted onset (E5-6): they begin using the appraisal gradually from layer 30, about 40% by layers 36–40, before the same takeover at 41 (buyer shares are less reliable: in the appraisal run at $100M only 0.32 of the probability falls on the two readout prices; Appendix E).

Largely the same heads as the scale and range effects. Ranked on 7 discovery floors, six of the appraisal's top eight heads (at layers 39–43, four at 41) are among the eight heads that carried the scale and private-range effects in the discovery phase (chance about 0.5). Patching the eight heads from the appraisal run moves the no-appraisal $100M opening toward the $100 opening on all 12 held-out floors, by about two-thirds of the scale gap (a post hoc measure), while eight random heads move almost nothing. The pre-committed held-out "share" (E5-5) is not interpretable, because its denominator changes sign across floors; the head-overlap half of E5-5 and the direct effect are the results we rely on.

6.1Number or stakes? (E6)

If the opening shift is a stake effect, the direction that carries it should also move choices where the stakes matter but negotiation does not. If it is a number effect, it should carry the size of the number and little else. E6 tests this in two parts, with ten predictions committed before the run (Table 26).

Gemma reads magnitude the same way in five notations. We recorded Gemma's states on 360 one-sentence statements of an amount (nine mantissas × 10⁰ to 10⁷) in five notations: "3700 dollars", "3.7 thousand dollars", "$3.7K", "$3,700" and "3700 people". At the sentence's last token, a magnitude direction fitted on the digit sentences alone reads the log amount of the word and abbreviated sentences at r = 0.97 and 0.96 by layer 20 (Figure 6a; E6-2). This looks computed rather than lexical: at layers 1–2 the same direction reads "$3.7K" at r ≈ 0 and "3.7 thousand dollars" at 0.43–0.45, and the shared code forms between layers 3 and 5 (E6-3). Among MLP neurons, 1,640 of 21,504 at layer 20 rise or fall with the amount in all four money notations (|Spearman ρ| ≥ 0.8), and 609 of them also track head counts (Figure 6b).

Gemma has a stake effect of its own, in risky choices. Offered a sure amount or a 50% chance of twice that, or of 2.4 or 3 times that, Gemma at $100 takes the gamble when its expected value is higher (18 of 20). At $100M it no longer prefers the better gamble: its choices follow the listing order (the first-listed option in 18 of 20). Over 30 matched pairs of choices, the log-odds of the safe option rise by 10.0 nats (E6-5), in the direction people show when payoffs are scaled up (Holt and Laury, 2002).

The negotiation's scale direction barely moves it. We took the direction separating Gemma's $100M and $100 seller openings at layer 42 (from E5's 19 fixtures; leave-one-fixture-out on negotiation prompts), added it to the model's turn on other tasks' $100 prompts, and measured the share of each task's natural $100 → $100M gap it reproduced (Figure 6c). It moves negotiation openings 86% of the way, against 8% for a random vector of the same norm (E6-1), but moves risky choices 5% and percentage answers −4%, far short of the predicted 25% (non-money control 1% and −2%; E6-6 and E6-7 missed). It leaves comparisons and sums intact (E6-8). It does make numbers larger: Fermi estimates rise by 1.15 orders of magnitude at α = +2 and fall by 0.50 at −2 (E6-9 missed), driven by three of ten items that jump by two to five orders of magnitude. The control direction does the same (+0.97), so this leak is not specific to the negotiation direction. The negotiation direction also overlaps the plain-sentence magnitude direction (cosine 0.32 at layer 42, against about 0.014 by chance; E6-10 missed at its 0.2 threshold).

With E5, this places Gemma's scale effect. Magnitude is read early and the same way in five notations; a late stage at layers 41–44 uses it to set the price, and a competing reference number (the private range or the appraisal) can take over there. The direction that shifts the openings behaves more like a signal of numerical magnitude than of stakes: adding it reproduces 5% of Gemma's stake effect in risky choices. In this model the opening shift looks like a number effect applied by the negotiation, not an imported risk preference. This rests on one model, one steering placement and two layers.

Appendix F reports three scoring artefacts (E6-4 is met trivially, the exponent-tuning count cannot reach its threshold, and the random control breaks the short prompts).

7Discussion

What the models comprehend. The models are not blind to magnitude. Gemma reads amounts on one number line in five notations by layer 5 (Section 6.1), and its scale signal persists, with less effect on the price, under the appraisal (Section 6). What changes with scale is the opening, and outcomes follow the openings: deals follow the scripted openings on every model, and on the three models whose split moves, the openings account for most of the shift. The departure is therefore a policy that maps larger absolute amounts to more modest relative opening positions, not a failure to read the numbers. It resembles the stake effects people show, and Gemma has one in risky choices; but in Gemma the direction that shifts the openings barely moves those choices, and behaves more like a signal of numerical magnitude.

Whose stake preferences? A person selling a $100M asset may reasonably be more cautious than one selling a $100 one. An agent acting for that person has not been told to be. Caution alone also cannot move a self-play split: when both sides share the same utility over their gains, the Nash solution divides the surplus equally at every scale, however risk averse they are (Nash, 1950; Kihlstrom et al., 1981). The closing shifts therefore need an asymmetry between roles, and Study 3 shows one. Without the appraisal, from $100 to $100M the sellers' opening gap narrows two to three and a half times as much as the buyers' on the three models whose split shifts (Gemma 0.21 against 0.11, Luna 0.16 against 0.04, Gemini 0.34 against 0.15). Haiku shows a similar imbalance (0.15 against 0.06) with a smaller shift (−0.051), and its deals track the opening midpoint least (S5), so the imbalance alone does not set the size of the shift. For deployment the question is less whether a model is rational than whether it brings in a preference its principal did not state.

Self-play invariance is not robustness. Qwen's and Haiku's no-appraisal splits meet equivalence, yet their openers still move toward their limits at scale. In self-play, modesty on both sides can offset in the split; against a counterpart that does not share it, the same policy could lose surplus (untested here). An evaluation that checks only outcomes at one scale, or only self-play at several, would miss this.

A reference number is an anchor, not a cure. The appraisal's effect is model-specific. It anchors every model's deals to its position in the zone, but what it does to the openings at scale differs: it holds Gemma's and Qwen's sellers' openings nearly level, draws Gemini's sellers to just below the appraisal, and leaves Haiku's sellers to collapse further. For Gemma's sellers it acts at the same narrow stage as the scale effect itself, through largely the same heads, on the tokens that write the price. That suggests a single late stage at which a number in context, a limit or an appraisal, can set the price. This is one model and an exploratory analysis; it describes where the intervention acts, not a semantic explanation of the model's numerical representation.

For evaluation. Agent evaluations that use one price range should report openings as well as outcomes and test at more than one scale. Our design (fixed geometry, shared seeds, an opening clamp) is cheap: Study 3 cost about $17 in API fees and $11 in GPU time.

8Limitations

The five models are not a representative sample; their precisions and sampling settings differ, so cross-model comparisons are descriptive. All games use one prompt family and one single-issue task, in self-play, with thinking off or minimised. Prices are whole dollars, which discretises the $100 scale. Each scale change also changes the digit strings; the discovery phase attributes part of the opening effect to the digits rather than the stakes (Appendix D.5). The appraisal is one wording at one place in the prompt.

The hypotheses were informed by earlier data on the same models, so Study 3 confirms them on new bargains, not on new models. An interim look at Gemma and Qwen preceded the second API session; it was logged and changed nothing, but it is a departure from a fully blind sequence. Games with scripted openings settle lower than natural games on Qwen and Gemini at $100M (S6). API results concern the dated versions returned on 14–15 October 2026. The registration is a local hashed commit, not lodged with an external registry, so its timing rests on the local history and the session manifests (Appendix A).

The mechanistic analysis covers one model, first moves only and the prompts of 19 seller floors and 18 buyer ceilings, using a forced-opening readout that does not lie on the axis of the appraisal's own openings; that is why one pre-committed share is uninterpretable. Its predictions were committed, but it is exploratory. E6 adds one steering placement (the model's turn) at two layers; a stake signal elsewhere in the network, or one read from the prompt itself, is not excluded. Stake effects outside negotiation were measured on Gemma only.

9Reproducibility and disclosure

The registration, hashed inputs, data-access log, analysis code and figure and table generators are under version control in the project repository (Appendix A). The raw call and game records, session manifests and derived files are kept with it, hashed in each run summary, and will be released with the code; every game record replays from its raw calls. The E5 and E6 predictions files were committed before their runs, and their hashes are recorded in the run manifests. AI tools assisted with code, analysis, figures, and checking prose. The author retains responsibility for the final text, factual verification and interpretation; AI tools are not authors (arXiv, 2025).

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Appendix

ARegistration, freeze, data access and deviations

• Registration. prereg/study3/PREREGISTRATION.md v3, frozen with prereg/study3/inputs.sha256 (37 files) in commit c8feea1 on 14 October 2026, before any registered game. It was not lodged with any external registry. The frozen document still carries its draft status line ("NOT registered, NOT frozen") and its plan to record an external registry's timestamp before any game; the freeze commit and the hash check were carried out as written, the external timestamp was not. The repository has no remote, so the order of freeze and data rests on the local git history (c8feea1 at 17:28 MDT, 23:28 UTC) and on the six session manifests (local files outside version control), which record the commit each session ran from (c8feea1; for API session 2, c2292c1, which adds only the data-access log) and the hash of every frozen file. The first registered call was sent at 23:54 UTC, 25 minutes after the freeze commit.
• Decisions before the freeze (the second after the first format pilot):
  • Scripted openings in zone units rather than own-limit gaps, set on 13 October before any game, because transplanted gaps made some scripted pairs cross.
  • Trailing text after the first JSON object is ignored and flagged (14 October). Luna had 616 such replies; no other model had any.
• Data access. Logged in prereg/study3/DATA_ACCESS.md, each entry committed before the look:
  • an interim look at Gemma and Qwen (commit c2292c1) before the second API session;
  • the single registered analysis of all five models (commit 8a3167d).
• Run incident. API session 2 stalled when the host machine slept after 1,266 games. It was resumed from the same commit; no finished game was replayed and no attempt was voided.
• Quality. Counts by model are in Table 4.
  • Luna: 35 forfeits and 23 timeouts.
  • All three API models: 0 refusals, and 0 reasoning tokens and 0 reasoning text (Gemini included).
  • Returned versions were unchanged across sessions: openai/gpt-6-luna-20260922, anthropic/claude-haiku-5.5-20261007, google/gemini-3.5-flash-lite-20260721.

Cost. API fees, as OpenRouter reported them, were $16.96 for the two registered sessions and $0.56 for the two format pilots. GPU time for Study 3 was about $10.90 including its pilots, and for E5 and E6 about $1.62 and $0.58 (launcher estimates, not billing records).

BFull Study 3 results

Table 4 gives game outcomes and data quality by model. The registered secondaries are in Table 5 (S1–S4), Table 6 (S5–S9) and Table 7 (the S3 contrasts not shown before, S8's comparison of A1 with Study 1, and S9's check of whether the A1 shift varies with the appraisal's position). With Table 3, these report every registered test.

CProtocol details

Engine, replies and validation. Every game runs on the discovery phase's engine (scripts/game_engine.py, unchanged); Study 3's own code adds the cells, the scripted openings, the seeds, the request bodies and the audit (scripts/study3.py). Each call is rebuilt from the system message, the side's own limit and the public history, in which the other side sees only action, price and message. A game has at most 16 messages. A reply is capped at 1,024 output tokens and must be one JSON object with an action (OFFER, ACCEPT or WALKAWAY), a price (a whole number of dollars for an OFFER, null otherwise) and a message of at most 500 characters. Every arm gets the same response schema (Addendum C's: that of Studies 1–2 without the minimum and maxLength keywords, which are checked client-side instead) and the same client-side validation. An invalid reply forfeits the game and is never repaired, with one exception set before the freeze: text after the first complete JSON object is ignored and flagged. The object itself is validated in full and never edited, and a reply that does not start with a complete object, or is truncated, still forfeits.

Scripted openings. In the clamp cells the first two messages are written into the public history from the frozen positions in Table 9: scripted price = F + x(C − F), rounded half up to a multiple of V/100. One neutral template per role is used at every scale and cell (the last block below), and the history line has the same form as a model's move. No request is sent for messages 1 and 2; the model plays both sides from message 3.

Seeds. A call's seed is the first eight hexadecimal digits of the SHA-256 of scale-invariance/study3/v1|fixture|opener|sample|message|attempt, masked to 31 bits. Scale, appraisal and clamp cell are not inputs, so every within-fixture contrast shares seeds. Seeds are sent to vLLM, OpenAI and Google, not to Anthropic. Because vLLM batched 16 games at a time, its outputs are not bit-identical to a sequential run, and the vendors do not promise deterministic replay; no registered result depends on either.

Order, concurrency, retries and voids. Each model ran two sessions (samples 1–2, then 3–4): on separate GPU pods for the open-weight models and on separate days (14 and 15 October, local time) for the API models. Each session interleaved every core and clamp cell in one fixed shuffle and played a fixed number of games at once (16 on vLLM, 8 on the API). A call is tried up to three times, a game voided by an infrastructure failure is replayed up to three attempts, and a non-retryable HTTP 4xx error stops the run. No attempt was voided and no game is missing in any session.

Records and audit. Calls, games and events are hash-chained records, and each session's manifest holds the commit, the hash of every frozen file and every setting. After each session a replay audit re-derives the scripted moves from the frozen values, rebuilds every request byte for byte and re-scores every game from its raw calls; all six sessions passed.

Fixtures. G01–G40 come from the public phrase scale-invariance/fixtures/study3-v1 through the discovery phase's generator (scripts/make_fixtures.py), with floors at whole percentages from 75 to 95 and ceilings from 105 to 125 of V. The appraisal's position in the zone, (V − F)/(C − F), runs from 0.21 to 0.79 (median 0.48). The format pilots used fixtures Q01–Q05 from a separate phrase and are excluded from every test.

Models and prompts. Table 8 lists the arms and their settings. A0 uses Study 2's frozen templates; A1 inserts one line into the system message and changes nothing else; the clamp's scripted messages use the two fragments of the last block. Placeholders in double braces are filled per call, and money is always written as digits followed by "dollars".

You are taking part in a negotiation between a buyer and a seller. You play one side. Each reply you give is one move in the negotiation.

The deal
- The seller owns a fixed lot of goods, and the buyer wants to buy it. Quantity, quality, delivery and payment terms are already agreed and will not change. Only the total price is being negotiated.
- The seller has a private floor: the lowest total price the seller will accept.
- The buyer has a private ceiling: the highest total price the buyer will pay.
- Each side knows only its own limit. The floor is always below the ceiling, so a deal is always possible.
- A deal below the seller's floor is a loss for the seller. A deal above the buyer's ceiling is a loss for the buyer. If there is no deal, neither side gains or loses anything.

How the negotiation works
- The two sides take turns. Each turn is one message. There are at most {{MAX_MSGS}} messages in total. If the last allowed message is not an ACCEPT or a WALKAWAY, the negotiation ends with no deal.
- On your turn you choose exactly one action:
  - OFFER: propose a total price. It replaces any earlier offer of yours.
  - ACCEPT: accept the other side's current offer. The deal is made at exactly that price. You can only accept when the other side has an offer on the table.
  - WALKAWAY: end the negotiation now, with no deal.
- With every action you may include a short message to the other side. The other side sees your action, your price and your message, and nothing else.

How to reply
Reply with exactly one JSON object and nothing else, in this form:
{"action": "OFFER or ACCEPT or WALKAWAY", "price": a whole number of dollars for an OFFER, or null for ACCEPT and WALKAWAY, "message": "your message, at most {{MAX_CHARS}} characters"}
Write the price as a plain whole number: no dollar sign, no commas, no decimals.

You are taking part in a negotiation between a buyer and a seller. You play one side. Each reply you give is one move in the negotiation.

The deal
- The seller owns a fixed lot of goods, and the buyer wants to buy it. Quantity, quality, delivery and payment terms are already agreed and will not change. Only the total price is being negotiated.
- An independent appraisal values the lot at {{V}}. Both sides know this.
- The seller has a private floor: the lowest total price the seller will accept.
- The buyer has a private ceiling: the highest total price the buyer will pay.
- Each side knows only its own limit. The floor is always below the ceiling, so a deal is always possible.
- A deal below the seller's floor is a loss for the seller. A deal above the buyer's ceiling is a loss for the buyer. If there is no deal, neither side gains or loses anything.

How the negotiation works
- The two sides take turns. Each turn is one message. There are at most {{MAX_MSGS}} messages in total. If the last allowed message is not an ACCEPT or a WALKAWAY, the negotiation ends with no deal.
- On your turn you choose exactly one action:
  - OFFER: propose a total price. It replaces any earlier offer of yours.
  - ACCEPT: accept the other side's current offer. The deal is made at exactly that price. You can only accept when the other side has an offer on the table.
  - WALKAWAY: end the negotiation now, with no deal.
- With every action you may include a short message to the other side. The other side sees your action, your price and your message, and nothing else.

How to reply
Reply with exactly one JSON object and nothing else, in this form:
{"action": "OFFER or ACCEPT or WALKAWAY", "price": a whole number of dollars for an OFFER, or null for ACCEPT and WALKAWAY, "message": "your message, at most {{MAX_CHARS}} characters"}
Write the price as a plain whole number: no dollar sign, no commas, no decimals.

You are the {{ROLE}}.
Your private {{LIMIT_NAME}} is {{LIMIT}}. The {{OTHER}} does not know it.
{{GOAL}}

{{HISTORY}}

This is message {{MSG_NO}} of at most {{MAX_MSGS}}, and it is your turn.
{{PENDING}}

# fragments.txt: the role- and state-dependent pieces of the user message.
# A line "[name]" starts a fragment; it runs to the next "[name]" line.
# Lines starting with "#" are comments. Placeholders look like {{NAME}}.
[goal_seller]
Your goal is to sell at the highest price you can get. Any deal at or above your floor is better for you than no deal.
[goal_buyer]
Your goal is to buy at the lowest price you can get. Any deal at or below your ceiling is better for you than no deal.
[history_empty]
No messages have been sent yet. You make the first move.
[history_head]
The negotiation so far:
[history_line]
Message {{N}}, {{WHO}}: OFFER at {{PRICE}}, with the message {{TEXT}}
[pending_offer]
The {{OTHER}}'s current offer is {{PRICE}}. You may ACCEPT it, make a new OFFER, or WALKAWAY.
[pending_none]
The {{OTHER}} has no offer on the table, so you cannot ACCEPT yet. You may make an OFFER or WALKAWAY.

# clamp.txt: the message text of Study 3's scripted first offers (PREREGISTRATION.md §3.2).
# A line "[name]" starts a fragment; it runs to the next "[name]" line. Lines starting with "#" are comments.
# {{PRICE}} is filled with render_prompt.money(price), i.e. "<whole number> dollars", at every scale.
[clamp_seller]
I can offer the lot at {{PRICE}}.
[clamp_buyer]
I can pay {{PRICE}} for the lot.

DThe discovery phase

This appendix reports the discovery phase in full, largely as it was written before Study 3 (cross-references updated): Studies 1–2, Addenda A–C, the exploratory probes and the discovery-phase interpretability. It generated Study 3's hypotheses (Table 1); none of its data enter a Study 3 hypothesis test (they appear only in the descriptive comparison S8). Its registrations, like Study 3's, were frozen in hashed local commits and not lodged with an external registry, and its confirmatory statements refer to those registrations. Its fixtures (F01–F30 and the probes' own sets) and protocol differ from Study 3's (Section 4, Appendix C) in the ways D.1 and D.7 describe.

D.1Design of the discovery phase

Two copies of a model negotiate one price. The seller knows a private floor F, the buyer a private ceiling C, and both know F < C. They alternate JSON moves: OFFER a whole-dollar price, ACCEPT the standing offer, or WALKAWAY, with a message of at most 500 characters. There are at most 16 messages. Each call is rebuilt from the rules, own limit and public history. Invalid replies forfeit the game without repair. For a deal at P, seller share is s = (P − F)/(C − F), with 0 at the floor and 1 at the ceiling.

Thirty seeded fixtures fix floors at 75–95% and ceilings at 105–125% of a reference value V. We play each at V = $100, $100K and $100M with both openers. Monetary amounts are plain digits followed by "dollars". Masking digits leaves the prompt templates identical across scales; actual conversations can diverge. Studies 1–2 use one game per fixture–scale–opener cell, or 180 per model, in a blocked order. Seeds are shared across scales. Study 1 shows an appraisal V and the limit ranges; Study 2 removes both. Qwen ran both studies; Gemma 4 31B ran Study 2 only.

The opening hypothesis HA arose from Qwen's Study 2 data. Addendum A registered its test on Gemma before Gemma's games; B registered first-move tests on gpt-6-luna, claude-haiku-5.5 and gemini-3.5-flash-lite (hereafter Luna, Haiku and Gemini). B has 480 first moves per model: 30 fixtures × two roles × two extreme scales × four samples. C subsequently registered full Study 2 games on these three arms: three scales, both openers and k = 4, totalling 720 games per model and 2,160 overall. It interleaves models, scales and openers in one fixed shuffle.

Qwen3.8 27B (FP8) and Gemma 4 31B (bf16) use pinned weights and vLLM 0.30.0, non-thinking sampling settings and zero presence penalty (Table 10). API arms use their vendors' endpoints and sampling defaults. Thinking is off, except that Gemini's lowest setting, "minimal", does not guarantee this. No API reply reported reasoning tokens. C sends shared seeds to OpenAI and Google; Anthropic receives none. Different precisions, temperatures and replication counts make cross-model comparisons descriptive. Gemma 4 12B failed its format gate and was not run confirmatorily (Appendix D.7).

The analysis unit is the fixture. For each opener, we subtract mean agreed seller share at $100 from that at $100M, then average over openers with both cells defined to obtain d_i. C uses cell means over scored replicates; Studies 1–2 have one game per cell. Primary H1 uses a two-sided Wilcoxon signed-rank test with 200,000 random sign flips and Holm correction across each study's models. H2 establishes equivalence only when the 90% bootstrap CI for mean d_i lies inside ±0.10 (Lakens, 2017). Bootstrap intervals resample fixtures 10,000 times. Non-significance alone is never evidence of invariance.

Registration was sequential, not independent of all prior evidence. In particular, B §11 said future full games "will be labelled exploratory". C explicitly changed that plan through a separate prospective confirmatory registration, informed by B's first moves and the open-model games. H1/H2 remain primary, midpoint HM is confirmatory, and HA, HB and S1/S2/S5/S6 are registered secondaries. Commit e54f00e froze C before run addC-20261011T182548Z; its format-only pilot is excluded. Appendix D.7 gives the chronology and deviations.

Confidence intervals in this appendix are 95% unless specified as 90%. Behavioural intervals resample fixtures; readout-specific mechanistic resampling is detailed in Appendix D.11.

H1 and H2 address different questions. H1 detects a systematic shift; H2 asks whether the mean shift is small relative to the registered ±0.10 margin. A result can in principle satisfy both, or neither. The margin defines the operational invariance claim in this task, not a universal threshold for acceptable negotiation. In particular, supporting H1 while failing H2 does not establish that the true mean exceeds the equivalence margin.

D.2What changes under payoff rescaling?

The shared response is in the opening, while the final split depends on the model (Figure 7). We measure opening modesty by the gap from the opener's own limit: g = P₁/F − 1 for a seller and g = 1 − P₁/C for a buyer. Smaller g means a lower markup or a smaller discount, respectively. The fixture-level change h_f averages $100M − $100 over available opener cells. HA tests negative change with a one-sided Wilcoxon test, Holm-corrected within each addendum. This gap is neither seller share nor opening-midpoint skew; its sign does not directly predict who gains in self-play.

Qwen supplies the exploratory hypothesis, not a confirmation. HA was confirmed on Gemma in A (mean h = −0.161, 95% CI [−0.181, −0.140]) and on all three API models in B: Luna −0.076 [−0.092, −0.059], Haiku −0.129 [−0.158, −0.103] and Gemini −0.232 [−0.254, −0.209]. Across these four confirmatory models, 119 of 120 model–fixture changes were negative and one was zero. The role-specific levels are descriptive; equal signs across models do not imply equal effect sizes.

C replicates HA as a registered secondary, with mean changes −0.089 (95% CI [−0.108, −0.070]), −0.105 [−0.131, −0.080] and −0.230 [−0.249, −0.211] on Luna, Haiku and Gemini. These are new-run estimates, distinct from B's first-move tests. All three pass the registered Holm-corrected test. Panel (a) of Figure 7 retains the distinction between Qwen's hypothesis-generating result, A/B's confirmations and C's secondary replication.

For closing splits, Qwen meets ±0.10 equivalence in both studies: d = −0.032 (90% CI [−0.070, +0.004]) with the appraisal and +0.009 [−0.074, +0.099] without it. The latter is marginal: its upper bound lies only 0.001 inside the margin, with fixture F20 excluded because neither opener had agreements at both extremes. This is a pass of the registered rule, not robust invariance. Gemma, tested in this phase only without the appraisal, shifts by −0.256 (90% CI [−0.292, −0.218]), negative in 29 of 30 fixtures, with H1 supported after Holm correction.

C extends this model dependence. Luna's closing shift is −0.116 (95% CI [−0.140, −0.091]) and Gemini's −0.304 [−0.376, −0.234], negative in 28 and 27 of 30 fixtures; H1 is supported on both after Holm correction. Haiku's shift is +0.006 [−0.021, +0.032], with H1 Holm p = 0.517. Its 90% CI [−0.016, +0.028] meets H2 equivalence. Thus three of five tested models show split shifts and two meet equivalence. Registered adjacent-scale, coarse-price and all-game utility checks retain the Gemma, Luna and Gemini shifts (Appendix D.9).

Quality limits matter even though all 2,160 C games completed. Luna had 95 format forfeits (13.2%), approximately balanced across scales (31, 32, 32). Haiku and Gemini had none, but each made 39 out-of-zone agreements, retained under the registered rules. The primary share estimand conditions on agreement; the utility secondary includes failed games. Neither registered quality flag was triggered, but this does not remove the selection and protocol limitations. Full counts and returned versions are in Appendix D.8.

The consequences for the opener are also model-dependent. Registered-secondary HB supports a falling opener advantage on Gemma in A and Gemini in C, but not on Luna or Haiku in C. Shared modesty therefore establishes neither a shared closing shift nor a general loss of opener advantage. Appendix D.9 reports the complete secondary families, including these unsuccessful tests.

D.3Do the openings account for the closing shift?

A common opening response need not produce a common split response. A seller's smaller markup lowers its first price, whereas a buyer's smaller discount raises its first price. Their effects on the midpoint push in opposite directions and may cancel wholly or in part. Moreover, HA measures only the opener: a responder has already seen the other side's offer. The relevant candidate account therefore uses both first offers from within each game, not two independently sampled opening moves.

Exploratory origin. The normalisation matters. Seller markup is relative to F and buyer discount to C, whereas the split and midpoint are measured relative to C − F. Equal changes in the two own-limit gaps need not cancel in surplus units. A midpoint account must therefore compare prices within the same fixture and game, rather than infer a closing shift from HA's model-level average alone.

In the open models' Study 2 games, deals closed near the midpoint of the two first offers (median deviation at most 0.005 V). This was found post hoc. Across in-game first offers, Qwen's median seller markup fell from 22.9% to 12.2% and its median buyer discount from 31.1% to 24.5%; the midpoint stayed approximately stable. Gemma's median seller markup fell from 37.9% to 17.2%, while its buyer discount stayed at 32–33% across scales, and the median midpoint fell from 0.975 V to 0.895 V. These quantities include responders and differ from the opener-only HA gaps.

On matched game pairs, the closing-share change decomposes exactly into the midpoint change and the change in the deal's deviation from that midpoint. Gemma's −0.256 = −0.256 + 0.000 on 60 pairs; Qwen's −0.014 = +0.012 − 0.026 on 53 pairs. These exploratory game-pair means are not the registered fixture-level d. They suggested a prospective accounting test, rather than demonstrating that openings cause the split (Appendix D.10).

D.4Prospective calibration and tracking in Addendum C

The prospective test is stricter than a correlation. Addendum C defines m = ((P_S1 + P_B1)/2 − F)/(C − F), using the first seller and buyer offers within the same game, and e_i as the fixture's closing-share shift minus its midpoint shift. Both shifts use the same agreements in which both sides offered. HM1 requires the 90% bootstrap CI of mean e_i to lie wholly within ±0.05; HM2 requires a positive Spearman association across fixtures, tested by 20,000 permutations with Holm correction across the models. HM is supported only when both hold. Addendum B's opener-only expectations are reported, not tested: a responder's first offer need not resemble its opener-only offer.

Tracking passes on all three models (Figure 8): Spearman correlations are 0.774, 0.605 and 0.847 for Luna, Haiku and Gemini (Holm p = 0.00015 each). Calibration passes only on Luna: e = −0.009, 90% CI [−0.025, +0.009]. Haiku's +0.025 [−0.008, +0.060] and Gemini's −0.043 [−0.073, −0.014] do not establish equivalence within ±0.05. HM therefore has prospective support on Luna only. Failure to establish equivalence is not proof of non-equivalence or of an error exceeding the margin.

On the matched HM games, mean opening-midpoint and closing shifts are −0.107 and −0.116 on Luna, −0.019 and +0.006 on Haiku, and −0.295 and −0.338 on Gemini. Across all three scales, Gemini has 76 agreements without a first offer from both sides. The primary analysis retains agreements without both first offers; HM excludes them. Gemini's primary −0.304 therefore must not be substituted for the matched −0.338 in the midpoint comparison. Luna and Haiku have no such exclusions.

The distinction between tracking and calibration is substantive. Association is partly constrained by bargaining geometry; calibration asks whether the residual shift is small enough under a pre-specified margin. HM1 concerns changes across scales, not exact equality between every deal and its opening midpoint: a stable level discrepancy can cancel in the contrast. Even passing both tests would not identify an opening's causal effect. Information can also change concessions, and the original open-model decomposition remains exploratory. Directly clamping the openings is a different test, which Study 3 ran (Section 5.3).

D.5What changes the scale response?

Exploratory follow-ups tested counterpart, instruction, information, geometry and notation. Each was a single unregistered run on fresh fixtures, with many contrasts examined. We highlight the most informative comparisons in Figure 9; Appendix D.12 retains the catalogue. Here ×1 denotes each probe's base scale.

Self-play can conceal a cost. Against a scripted counterpart that opens at 1.3 times its limit as a seller, or its limit divided by 1.3 as a buyer, and concedes 35% of the remaining gap, identically in relative terms at every scale, in the redesigned game (X17), excluding comma-truncated prices, Gemma's seller-share change was −0.258 (95% CI [−0.318, −0.199]; 27 of 28 fixtures negative). Qwen also lost surplus in the instruction probes (−0.04 to −0.14 across styles and roles). These results concern one counterpart, not general exploitability.

The split is also design-dependent. A fresh-fixture repetition of Gemma's Study 2 game gave −0.253 (95% CI [−0.296, −0.209]). Yet a redesigned game with off-centre ranges, business notation and cents gave +0.027 [−0.036, +0.088], excluding comma-truncated prices, while both opening gaps still fell. Separate geometry probes implicated zone width: Gemma shifted less in wide zones; Qwen shifted by −0.12 [−0.21, −0.03] in narrow off-centre zones. The redesigned-game null and the Qwen counterexample limit model-wide readings of either result.

A scale-neutral instruction did not remove opening shrinkage. Neither did tough or cooperative instructions: all four tested styles retained it in both models. By contrast, concrete information changed openings. In Gemma's information run, the seller's loss against the bot changed from −0.14 without information to −0.04 with an appraisal and +0.01 with the buyer's range. These baselines are from the same information run, not the separate −0.258 result above.

Information is not an opening clamp. With both sides told the other's range, Gemma's openings were nearly pinned, yet its split still fell by 0.083; approximately 77% lay in the concession component. Qwen's concession component was +0.029. This failed extension of a purely opening-based account is material: information can alter the later negotiation as well as the initial prices.

Finally, notation probes separate stakes from digits imperfectly but informatively. Keeping digits such as "81" and changing the unit to millions, with replies in millions, gave about 61% of the full opening shift (−8.9 of −14.5 points); changing the digit string at fixed stakes supplied the remaining −5.6. Plain small-dollar wording was the outlier, with round-ten seller openings in 99% of cases. The contrast supports both numerical-format and stake-related contributions; it does not identify large-number timidity rather than small-number boldness.

D.6Where does scale act inside a model?

We examine Gemma's seller openings on 30 fresh fixtures (Y01–Y30), using residual-stream hooks in transformers. This entire analysis is exploratory. Forced-opening measurements fix the JSON prefix through the price field and measure the subsequent price; sampled-opening checks test behavioural transfer. Effects are expressed as a fraction of the unsteered ×1–×10⁶ opening gap. The central result combines a late intervention site with steering of real openings (Figure 10).

The instrumented model reproduces the deployed behaviour: seller markup is 41.1% at ×1 and 20.7% at ×10⁶, against 40.8% and 20.8% in vLLM, with 300 of 300 replies valid. This establishes a behavioural bridge for the intervention study, rather than assuming that changing inference frameworks preserves the effect.

Magnitude is readable from the limit digits early (probe R² 0.98–0.99 from about layer 5). Behavioural patching is later: replacing post-limit activations in the ×10⁶ run with ×1 states restores 82% [73, 90] of the gap at layers 41–47. The reverse direction already works at earlier layers, so layer ~41 is where the effect becomes decisive, not necessarily where it is computed. Later processing can overwrite an earlier scale signal.

The first steering attempt failed. Adding the scale-difference vector to prompt tokens alone at layer 44 moved at most 17% of the gap, without a dose-response. A redesign included the reply's opening tokens, which patching had also covered. Per-token directions built from the other 29 fixtures then moved real openings 74–95% of the gap in both directions at layers 42, 44 and 48, at unit strength; double strength moved 109–176%. All 1,080 steered replies were format-valid, though six double-strength prices were off-scale. A same-norm non-money control moved real openings at most 14% at layer 44. In the forced-opening comparison there, reply-only steering moved 58–86%, versus 8–15% for prompt-only steering.

This identifies an effective intervention site, not a unique scale representation. The analysis covers one model and first moves, with moderate within-scale agreement between forced and sampled readouts (r = 0.73 at ×10⁶). Head, cross-role, information and concept analyses, including duplicated prompts and a failed random-span control, are reported in Appendix D.11. They qualify rather than expand the main mechanistic claim.

D.7Registrations, amendments and deviations

Study 1 was registered at a2a024c, with amendment A1 at 1b32b69; Study 2 at 576dc2e; Addendum A at de47204; B, including amendment B1, at 0322a35; and amendment WP2d at 72dc51e (committed 8 October), which added the Mistral and Phi-4 pins to prereg/inputs.sha256 and left the Study 1–2 entries unchanged; the Study 1–2 run manifests therefore record the earlier hash of that file. Each confirmatory analysis concerns the subsequent designated test data. Pilot observations and exploratory follow-ups are not pooled into those tests. Study 2 was designed after observing Study 1's appraisal-centred deals, and A was designed after inspecting Qwen's Study 2 openings. These dependencies motivate, rather than invalidate, the need for prospective tests on the designated new data.

Study 1 stopped after Qwen, a scope change recorded in the Study 2 registration. Gemma 4 31B therefore supplies Study 2 results only. Gemma 4 12B failed the pre-specified format gate: 684 of 725 pilot replies were valid (94.3% against 95%). All 41 invalid replies were ACCEPT moves carrying the pending price rather than null; 41 of 60 pilot games forfeited. It was reported as not run, rather than replacing the protocol after viewing the result. A separate exploratory 12B opening probe (X50) is not a confirmatory replacement.

B's first format pilot encountered a Google seed above the accepted int32 range. Amendment B1 restricted seeds to 31 bits and specified stopping on non-retryable 4xx responses; the second pilot passed. HA was not computed from the pilots. B tested first moves only, on models not previously run on this task. Its statement in §11 that any future full games "will be labelled exploratory" remains part of the frozen record.

Addendum C was frozen on 11 October 2026 in e54f00e, before its registered run started at 18:25:48 UTC. B §11's statement that future full games would be exploratory is retained in B and explicitly superseded by C's separate confirmatory registration. The pre-freeze pilot supplied format checks and the cost estimate only. At the author's instruction before the freeze, the main run's proposed $15 client cap was removed in favour of the OpenRouter key's existing cap; the pilot kept its $1 cap. The registered design and analysis were not changed after the freeze.

C's registration acknowledged that B's first moves and the open-model games informed its hypotheses and expectations. H1/H2 were primary, HM confirmatory, and HA/HB and S1/S2/S5/S6 registered secondaries. The ±0.10 closing margin, ±0.05 midpoint margin, exclusions and analysis were frozen prospectively. The C format pilot used pilot fixtures X01–X05, not the registered F01–F30 fixtures; only format and pooled cost were inspected. Its 60 games and 572 replies are excluded from all tests.

Relative to Studies 1–2, C uses vendor sampling defaults, no Anthropic seed, vendor-specific thinking controls, 1,024 rather than 512 maximum output tokens, and client-side checks of minimum price and message length. It also uses four replicates per cell and interleaved rather than blocked run order. The game engine, prompt, scoring and invalid-reply rule are otherwise reused as specified in C. Vendors do not promise deterministic replay even when a seed is accepted.

The four frozen-input manifests are prereg/inputs.sha256, prereg/study2/inputs.sha256, prereg/addendum_B/inputs.sha256 and prereg/addendum_C/inputs.sha256. They protect the registrations' inputs separately from the manuscript. Run records carry their own hashes; the open-weight runs were also replay-audited.

D.8The API panel

The three arms were accessed through OpenRouter, pinned to each vendor's own endpoint with no fallbacks. B ran on 10 October and C on 11 October 2026. OpenAI's and Google's requested IDs are aliases; the returned dated versions are recorded in Table 11. These identify the tested endpoints without assuming that future calls to the aliases will reach the same weights. Addendum B's first-move results are in Table 12.

Addendum C used the same returned dated versions as B: openai/gpt-6-luna-20260922, anthropic/claude-haiku-5.5-20261007 and google/gemini-3.5-flash-lite-20260721, on the vendors' own endpoints. It recorded 20,172 completed replies, no refusals and no reasoning-token replies. Run and analysis: data/runs/addC-20261011T182548Z/ and derived/addC-20261011T182548Z/addendum_C.json.

C completed 720 games per arm, with no missing games or infrastructure-void attempts (Table 13). Luna's 95 forfeits comprised 87 replies with text after JSON and eight truncations. Their scale counts were 31, 32 and 32. Haiku and Gemini had no forfeits, but each had 39 out-of-zone agreements. Those agreements remain in the primary share analysis, as registered; seller shares are not clipped to [0, 1]. The registered flags were more than 25% forfeits overall or a difference exceeding ten percentage points between extreme scales. Neither flag was triggered. This is a rule-based quality report, not evidence that conditioning on agreement is harmless.

C's registered secondaries qualify the main findings. HB is supported on Gemini (mean I = −0.352, Holm p = 0.002055), not on Luna (p = 0.4268) or Haiku (p = 0.65111). Both adjacent scale steps lower seller share on Luna and Gemini after within-model Holm correction; neither does on Haiku. The coarse-price check (S5) and all-game seller utility (S6) retain the Luna/Gemini shifts. Agreement-rate changes (S2) are not significant on any model. These tests are secondary; the primary split and confirmatory midpoint readings are those in Appendices D.2 and D.4.

D.9Full behavioural results and estimands

The primary closing contrast (Table 14) uses fixture-level means over eligible opener cells. Agreement is required at both extreme scales for an opener contrast, but C does not require the same replicate index to agree at both scales: it contrasts cell means. A fixture with no eligible opener is excluded and counted. The only excluded primary fixture is Qwen Study 2's F20. Opening HA requires a valid first-move OFFER and does not require the later game to agree. Midpoint HM instead restricts both axes to agreements containing a first offer from each side. These eligibility rules define different estimands.

In Studies 1–2, S1 tests the two adjacent scale contrasts; S2 agreement; S3 disagreement between opener conditions and dispersion across fixtures; S4 the opening price expressed in zone units; S5 coarse-grid prices; and S6 seller utility over all games. These are registered secondaries (Table 15). S4 is not HA: averaging a seller-share-like opening position differs from averaging seller markup and buyer discount relative to their own limits. C carries S1/S2/S5/S6 forward (Table 16) using cell means, omits S3 and substitutes HA for S4. Its S2 is an agreement-rate contrast; the original analysis records differences in agreement counts over the two opener games.

Gemma's mean seller shares were 0.49, 0.34 and 0.23 across the three scales. Its S1 changes were −0.150 and −0.106; S3 opener disagreement fell by 0.158, S4 by 0.155, S5 by 0.254 and S6 by 0.256. Across-fixture dispersion, a different S3 summary, increased: SD 0.083 to 0.135, with 95% CI [0.007, 0.092] for the difference. All 1,445 Gemma replies were valid. Qwen had 1,184 of 1,184 valid replies in Study 1 and 1,215 of 1,215 in Study 2. These quality counts do not equate format validity with successful or rational bargaining.

The between-study appraisal comparison is exploratory. Qwen's median price changed from 1.015 V to 0.940 V and its mean seller share from 0.516 to 0.312. The correlation with the share implied by settling at V fell from 0.54 to 0.02. Because studies were sequential and removed the ranges together with the appraisal, these numbers do not identify the appraisal's isolated causal effect.

Qwen generated HA: its seller-opener markup changed from 24.5% to 12.7% and its buyer-opener discount from 36.6% to 22.3%. A randomised 12-fixture probe replicated shrinkage without the appraisal, more weakly (in eight of 12 fixtures per role), and found none with it. The opener-advantage reversal did not replicate (−0.04, p = 0.68), which motivated the two-sided, secondary designation of HB in A. A's Gemma HB result was −0.183, p = 0.0007; the prior failure remains relevant to its status.

The A/B role comparison (Table 17) has exceptions: seller shrinkage exceeds buyer shrinkage in percentage points for four of five models, excluding Qwen (−11.8 versus −14.3 points), and in relative terms for four of five, excluding Haiku (−21% versus −23%). Haiku's initial seller markup of 84.4% also differs substantially from Luna's 45.0% and Gemini's 50.8%. These differences caution against treating shared direction of HA as a common bargaining policy.

C's HM uses m = ((P_S1 + P_B1)/2 − F)/(C − F), and computes closing and midpoint shifts on identical eligible agreements. HM1 bootstraps the mean of their fixture-level difference; HM2 uses a one-sided Spearman permutation test with 20,000 permutations and Holm correction across models. All three HM2 adjusted values round to 0.00015. HM1 passes only on Luna. For comparison only, the frozen output also reports a looser ±0.10 calibration result; that descriptive result does not replace the registered ±0.05 test.

B's opener-only implied closing shifts were −0.066 (Luna), −0.089 (Haiku) and −0.333 (Gemini). C registered these as expectations, not tests. C's results can be compared with these expectations but cannot confirm them, since they were derived from B's observed first moves, and they omit how a responder's first offer differs from an opener's. The original Qwen and Gemma midpoint evidence likewise remains exploratory after C (Study 3's opening clamp later tested the account prospectively; Section 5.3), including Qwen's failure to fit the later ±0.05 calibration margin when assessed retrospectively.

D.10Behavioural probes and midpoint decomposition

All analyses in this subsection are exploratory. The full evidence catalogue in Appendix D.12 records the actual exploratory-ledger entries, including early negative findings and reanalyses; gaps in ID numbering do not denote unreported experiments. Two runs have no catalogue entry: a 216-game Phi-4 14B geometry probe (5 October) and two Mistral pods that failed before producing data. Neither informs any claim; their cost (about $1.83) is included in the total. The follow-ups reuse a single scripted ratio-based counterpart, so a loss against it is not evidence of universal exploitability. Probe-specific baselines, fixtures and notation must be kept together.

The redesigned-game probe (X17) used 600 games across new and bridge fixtures. The Study 2 bridge on L01–L30 replicated Gemma's shift (−0.253, 95% CI [−0.296, −0.209], 29 of 30 negative). The redesigned self-play result was +0.027 [−0.036, +0.088] after excluding comma-truncated prices, versus −0.035 as recorded. There were 68 slipped games across the probe, all at ×10⁶; the redesigned self-play cell retained 95 of 120 games. Opening gaps still fell by 16.5 points for sellers and 9.2 for buyers. The bot-seller result, −0.258 [−0.318, −0.199], used 28 eligible fixtures. Both the replication and the design-dependent null are reported.

The geometry probes (X18, X22) separated Study 2's round-centred zone, a narrow off-centre zone and a wide zone, crossed with raw and business notation. Each model supplied 2,160 games on fresh fixtures. For Gemma, raw self-play shifts were −0.21, −0.20 and −0.06; business notation attenuated them by 0.03–0.09. Qwen's raw narrow-zone shift was −0.12 [−0.21, −0.03]. In cross-play (X19, 960 games), Gemma selling to Qwen shifted by −0.113, whereas Qwen selling to Gemma shifted by −0.042 without a significant result. This is a limited cross-model comparison, not a new primary model panel.

The instruction probes (X20–X21, 2,160 games per model) crossed no addition, tough, cooperative and scale-neutral instructions, in symmetric and asymmetric self-play and against the bot. Every style retained opening shrinkage: 6–13 points for Qwen; for Gemma, 38–47% of the seller gap and 18–24% of the buyer gap. The exact scale-neutral text was: "The size of the numbers does not change what good negotiating looks like. Negotiate exactly as you would if every amount were a thousand times smaller or a thousand times larger." The cooperative-buyer bot cell was an exception to Gemma's otherwise negative bot effects across styles and roles.

The information probes (X23–X24, 1,560 games per model) privately supplied an appraisal or the other side's limit range. The other side did not know it had been supplied. Gemma's informed buyers opened at $75 or $75M in 120 of 120 games, and informed sellers at $120 or $120M in 118 of 120. Against the bot, its seller change moved from −0.14 with no information to −0.04 with an appraisal and +0.01 with the range; the buyer's own-share changes were −0.08, −0.01 and +0.11. Qwen's range condition likewise removed the bot loss. The range is additional strategic information as well as a numerical reference point.

The decomposition in Figure 11 uses one matched set of game pairs for all terms. It writes seller share as the opening midpoint in zone units plus the deal's deviation from that midpoint. Gemma Study 2 gives −0.256 = −0.256 + 0.000 on 60 pairs, and Qwen −0.014 = +0.012 − 0.026 on 53. With both sides told the other's range, Gemma gives −0.083 = −0.019 − 0.064, about 77% in concessions. Mean offers increase from 8.3 to 10.6 and seller acceptances from eight to 22 of 60 games. Qwen's corresponding concession term is +0.029. Near-pinning openings through information does not eliminate every scale response. Figure 12 shows every probe condition in this subsection.

The digits probes (X15–X16) used 1,920 and 1,680 first moves, respectively, on Y01–Y30. Eight initial wordings and five additional wordings produced the 13-condition comparison in Figure 13. The first probe's both-role change was −14.5 points for raw millionfold scaling, −8.9 when the digits were kept and the unit changed to millions, and −5.6 for the remaining digit-string contrast. The latter was negative in 28 of 30 fixtures, the first two in all 30. These are exploratory contrasts in one model and one wording family, not a general variance decomposition.

At plain small-dollar wording, 99% of seller openings were round tens, versus 35–57% under other initial wordings. The follow-up found 97% round tens with "81.00 dollars"; the dollar sign changed the both-role gap by only 2–3 points, and the raw-dollar anchor conditions replicated within 0.3 points. The result leaves open whether familiar small numbers elicit unusually ambitious stock prices, whether large stakes elicit caution, or both.

D.11Mechanistic methods, controls and supporting results

The mechanistic programme uses Gemma 4 31B in bf16 with transformers hooks. E0 first checked sampled openings against vLLM on the same prompts and seeds, and greedy openings against the forced readout; E1 probed magnitude; E2 patched activations; E3 and its follow-ups steered openings. The later head and concept analyses use a log-probability difference between the two scale-conditioned candidate prices, expressed in the target run's units. That readout differs from the forced greedy-price and sampled-markup readouts. Fractions of a gap should therefore only be compared within the relevant instrument and run.

The E0 bridge produced seller markups 41.1% and 20.7%, compared with vLLM's 40.8% and 20.8%, with 300 valid replies. The forced-price readout tracked sampled averages strongly across scales (r = 0.97) but only moderately within ×10⁶ (0.73). E1's plain-sentence magnitude probe read log-magnitude from limit digits at R² 0.98–0.99 from about layer 5. Decodability establishes available information, not use of that information in pricing (Belinkov, 2021).

The E2 patching interval is a 95% bootstrap interval from 1,000 resamples of per-fixture price changes, divided by the clean mean gap. The denominator is held fixed, and the interval endpoints are swapped when it is negative. This readout-specific calculation is distinct from the 10,000-resample bootstrap of fixture-level behavioural contrasts.

E2's post-limit patch includes the reply's forced opening tokens. Transferring ×1 states into the ×10⁶ run restores 82% [73, 90] of the price gap at layers 41–47. Reverse patching already has effects at layers 18–24 and near-complete effects later; some early reverse patches also change the number's scale. The corrected X56 synthesis therefore supersedes the earlier statement that all early layers do little. The defensible localisation is a late decisive site, not exclusive computation at one layer.

The first E3 experiment steered only prompt tokens at layer 44 and failed: at most 17% of the gap moved, without a dose-response, over 2,880 sampled openings. E3b included the reply's 11 opening tokens and succeeded. Its forced comparison moved 58–86% with reply-only steering and 8–15% with prompt-only steering. Per-token differences were formed leave-one-fixture-out; a single token-averaged vector was weaker. The non-money control was comparatively inert, but a random direction damaged reply format (72–82% valid in the relevant E3b rows). Random controls were not universally harmless.

E3c swept layers with forced openings and selected layers 42, 44 and 48 for sampled checks. At unit strength the two directions moved 93%/95%, 87%/87% and 79%/74% of the sampled gap, respectively. Double strength moved 109–176%, showing overshoot rather than a bounded proportion. All 1,080 steered replies were format-valid; six double-strength ×1-direction prices were written off-scale. The full token and dose comparisons in Figure 14 preserve these qualifications.

Head patching (X57) ranked 224 heads at layers 38–44 on ten discovery fixtures and checked the result on 20 held-out fixtures. Eight selected heads, five at layer 41, jointly restored 40% [36, 43] and 68% [63, 73] of the log-probability gap; eight random heads restored 1–2%. Ranking agreement was Spearman 0.85, with seven of the top eight shared. Reply-only patching restored 27%/54%, prompt-only 7%/12%. Steering at layer 20 alone moved 5%, blocking attention to the limit's digits at layers 38–44 alone 15%, and combining them 51%. This supports later overwriting of an earlier signal in the tested intervention.

Source decomposition (X60) attributed 65–95% of the selected heads' output differences to limit digits for seven of eight heads. The remaining head attributed 40% to digits and substantial additional mass to the following " dollars." tokens. Control heads also read digits without comparable behavioural importance: attention alone does not identify the causal contribution. Path patching found redundant routes through digits and adjacent tokens.

Buyer analyses (X58–X59) reproduced the smaller opening effect (discount 45.3% to 36.9%) and a late patching transition at layers 41–42. The buyer's own direction at layer 44 moved real openings 106% and 101% of the gap at unit strength. The seller's direction moved buyers 12% and 19%, comparable to the non-money control's 6% and 14%; per-token cross-role cosines were 0.43–0.55. The buyer layer band was not swept as in sellers. These results do not distinguish separate mechanisms from a shared computation expressed differently by role.

Information interventions (X61–X62) reproduced the range-induced openings near $120 and $120M. Patching range-conditioned states into the plain large-scale run moved at most 9% before layer 41 and 86% at layer 41, mainly at reply tokens. Three of the range's top eight heads overlapped the scale heads (0.3 expected by chance), but full ranking agreement over 224 heads was only Spearman +0.13. The range's presence also changed attention broadly. This overlap describes an intervention site; it does not demonstrate competition for the same read.

The discovery/held-out split did not ensure distinct prompts: a seller sees its floor, not the whole fixture, and eight held-out fixtures repeated discovery prompts. Restricting to new prompts preserved the main circuit effects but lowered ranking agreement from 0.85 to 0.71. Two range fixtures lacked a usable candidate-pair readout; excluding them changed the range circuit summary to 71%/81%. The concept follow-ups consequently used 17 distinct floors. These restrictions are important when interpreting leave-one-fixture-out estimates.

The concept work (X63–X64) compared the scale direction with six directions built from 200 sentence pairs: count, money, million, digits, stakes and modesty. Numerical-representation studies motivate such probes (Singh and Strouse, 2024; Levy and Geva, 2024; Zhu et al., 2024; El-Shangiti et al., 2024; Kantamneni and Tegmark, 2025), but geometric resemblance need not imply behavioural explanation. At layers 41–42 the concept span held up to a third of the scale direction's squared norm. Its projection nevertheless moved real openings only 6% and 0%, whereas the orthogonal remainder moved 84% and 86% (Figure 15).

Removing the six-concept span from each run's own states retained 77–90% of the log-probability gap; removing the scale direction retained 16–25%. A separately constructed modesty direction moved openings 62% and 55% despite near-orthogonality to the scale direction. These outcomes support a limited negative result about the six operationalised concepts, not a claim that the model has no money or magnitude representation. More than one direction can alter the price, and steerability may depend on the input and extraction procedure (Tan et al., 2024; Lekeas and Stamatopoulos, 2026).

A random six-dimensional ablation, intended as an inert control, damaged the readout for reasons not resolved by the study. An off-the-shelf layer-30 sparse autoencoder retained only 13% of the gap when spliced in, failing the pre-set 80–125% quality gate, so the proposed SAE decomposition was not run. Another candidate SAE did not load. These failed controls and instruments prevent a stronger semantic decomposition claim.

D.12Evidence catalogue, prompts and reproduction

The following catalogue records the question, dataset, sample size, cost and a concise finding for all 42 actual exploratory-ledger entries, including reanalyses and negative findings. Questions and sample metadata are taken from the project's research log, with internal planning labels and file paths removed; the manuscript readings incorporate later corrections described above. Historical notebook labels such as "lead" or "explained" are not used as confirmatory claim statuses. Costs are launcher estimates unless otherwise stated, and shared-session costs are not added twice.

The prompt templates below are included directly from the frozen local files, preserving their wording and placeholders. Study 1 adds the public appraisal and both limit ranges to the system message. Study 2's user template and history fragments are otherwise shared. Study 2's templates, which Study 3's A0 condition reuses, are in Appendix C. The F19 prices in Figure 1 are drawn from the recorded seller-opened games, and the complete action/price sequences are in Table 18.

You are taking part in a negotiation between a buyer and a seller. You play one side. Each reply you give is one move in the negotiation.

The deal
- The seller owns a fixed lot of goods, and the buyer wants to buy it. Quantity, quality, delivery and payment terms are already agreed and will not change. Only the total price is being negotiated.
- An independent appraisal values the lot at {{V}}. Both sides know this.
- The seller has a private floor: the lowest total price the seller will accept. It was drawn at random from the range {{F_LO}} to {{F_HI}}.
- The buyer has a private ceiling: the highest total price the buyer will pay. It was drawn at random from the range {{C_LO}} to {{C_HI}}.
- Each side knows only its own limit. The floor is always below the ceiling, so a deal is always possible.
- A deal below the seller's floor is a loss for the seller. A deal above the buyer's ceiling is a loss for the buyer. If there is no deal, neither side gains or loses anything.

How the negotiation works
- The two sides take turns. Each turn is one message. There are at most {{MAX_MSGS}} messages in total. If the last allowed message is not an ACCEPT or a WALKAWAY, the negotiation ends with no deal.
- On your turn you choose exactly one action:
  - OFFER: propose a total price. It replaces any earlier offer of yours.
  - ACCEPT: accept the other side's current offer. The deal is made at exactly that price. You can only accept when the other side has an offer on the table.
  - WALKAWAY: end the negotiation now, with no deal.
- With every action you may include a short message to the other side. The other side sees your action, your price and your message, and nothing else.

How to reply
Reply with exactly one JSON object and nothing else, in this form:
{"action": "OFFER or ACCEPT or WALKAWAY", "price": a whole number of dollars for an OFFER, or null for ACCEPT and WALKAWAY, "message": "your message, at most {{MAX_CHARS}} characters"}
Write the price as a plain whole number: no dollar sign, no commas, no decimals.

# info.txt (asymmetric-information probe, Q17, 2026-10-04): private information added to ONE side's
# instructions, right after its goal (and after any style paragraph). The other side's prompt is
# unchanged, and the shared system message never mentions it, so the uninformed side doesn't know the
# other side knows more. A line "[name]" starts a fragment; lines starting with "#" are comments.
[info_a]
You have also seen an independent appraisal of the lot. It values the lot at {{R}}. The {{OTHER}} has not seen it.
[info_p_seller]
You also know that the buyer's ceiling was drawn at random from the range {{LO}} to {{HI}}. The buyer does not know that you know this.
[info_p_buyer]
You also know that the seller's floor was drawn at random from the range {{LO}} to {{HI}}. The seller does not know that you know this.

Six of this appendix's figures (Figure 7, Figure 8, Figure 9, Figure 10, Figure 14, Figure 15) are rebuilt locally with python3 manuscript/figures/main_figures.py; the others by the generators beside their outputs. The generator checks plotted registered quantities against the stored analyses and checks mechanistic summaries against raw results.

The discovery phase's GPU cost was $43.77 of the project's $100 budget (launcher estimates, not billing-confirmed). API costs are separate: B $0.275 including pilots; C $0.111585 for the pilot and $3.951179 for the registered run. These administrative totals are not outcome measures.

EE5 methods and full results

Run interp-e5-gemma-4-31b-it-20261015T232511Z (39 minutes of machine time, $1.62 estimated); predictions in config/interp/e5_predictions.md (sha256 ba05f2c2…, commit 0d8b715, before the run). The tables below are generated from the run's stored summary and per-fixture records (manuscript/figures/table_e5.py).

Design. For each role, one Study 3 fixture per distinct limit among G01–G40, so no prompt repeats: 19 seller floors (7 discovery floors, G01–G07, used to choose heads and for the no-appraisal steering rows, and 12 held out) and 18 buyer ceilings. Each limit has four prompts, Study 3's own first-move prompts rendered by its code: A0 and A1 at $100 and $100M. The hooks are those of the discovery phase (Appendix D.11), on the pinned bf16 weights in transformers. Sellers received the full set of measurements (gate, build, attention, patching, steering, the appraisal swap by layer, single-head swaps on the discovery floors and the eight-head swap on the held-out floors); buyers the gate, build, steering and swap.

Instrument checks. All 60 layers were found and every weight matched the pinned checkpoint; greedy replies to two reference prompts (the discovery phase's Y01 seller prompts at ×1 and ×10⁶) were token-identical to vLLM's; every hook fired as planned, and repeated clean rows agreed exactly within each step. Before any intervention, a gate compared four sampled openings per prompt with Study 3's own vLLM openings on the same prompts (Table 20); both roles passed.

Readout. The forced opening fixes the start of the reply to {"action": "OFFER", "price": and reads the log-odds, at the first token where they differ, between the limit's two no-appraisal greedy prices (at $100 and at $100M, written in the run's own units; their shared leading digits are forced too). The sellers' A0 gap between scales, G_N, is 18.4 nats and their A1 gap 3.0 nats (17%; E5-1); the buyers' (17 usable ceilings: G30's two no-appraisal prices are equal) are 7.1 and 0.7 nats (10%). The probability on the two prices' deciding tokens is 0.83–0.87 in the $100M runs except the buyers' A1 run (0.32), whose openings mostly fall outside the pair, so buyer shares that use that run are less reliable.

Appraisal swap (Table 21). The A1 run's states on the prompt's final segment and the reply's opening tokens are copied into the A0 run at $100M, one layer at a time (adding), or the reverse (removing); the share is the change in log-odds over the distance between the two runs. On sellers the shares at layers 30–40 are negative (adding −13% to −36%, removing −15% to −61%), but they divide by a distance that changes sign across floors (−26.9 to +7.1 nats), as E5-5's does. Floor by floor, adding the appraisal's states at these layers raises the log-odds of the $100M price on 13–18 of 19 floors yet moves toward the floor's own appraisal-run value on only 6–9 (removing lowers it on 12–19); from layer 41 the swap moves 16–18 of 19 floors toward the other run. The buyers' adding share rises from 18% at layer 30 to 37–40% at layers 36–40, then jumps at layer 41 (E5-6).

Steering and patching (Table 22, Table 23). The no-appraisal scale direction moves sampled A1 openings, but less than A0 openings (E5-4). On the forced readout, copying the other scale's states at layer 42 restores 97% and 83% of the scale gap in the A0 runs, as in the discovery phase, and moves the A1 runs by 22–30% of the A0 gap, more than their own 17% gap.

Heads and the held-out patch (Table 24, Table 25). Single-head swaps at layers 38–44 on the 7 discovery floors ranked the heads, and the top eight were then patched together on the 12 held-out floors. Prediction E5-5's held-out share divides the mean move by the mean A1 − A0 log-odds distance at $100M on the A0-price readout. That distance changes sign across floors (−26 to +7 nats): the readout compares only the first digit where the two A0 prices differ, and A1's $100M opening shares that digit with the $100 price on some floors and with the $100M price on others. The ratio (1.88) is therefore not a share. The direct effect, computed after the run, is the move toward the $100 price: negative on all 12 held-out floors, two-thirds of their own A0 scale gap on average.

FE6 methods and full results

Run interp-e6-gemma-4-31b-it-20261016T013146Z (about 14 minutes of machine time, $0.58 estimated); predictions in config/interp/e6_predictions.md (sha256 72274e9c…, commit c892fff, before the run). Instrument checks: 60 layers, all weights matched; E5's seller build re-run, with the same greedy prices on 19 of 19 fixtures; steering hooks fired once per steered layer; negotiation prompts used leave-one-fixture-out vectors. The greedy check's 0-of-47 token agreement on one prompt is an alignment offset (the vLLM reference omits the reply's code fence; the text is identical).

Part A (number). 360 sentences: mantissas 12, 23, 37, 45, 58, 64, 73, 81 and 96, times 10⁰ to 10⁷, in five notations (digits, words, short, commas, people count), one template per notation, read at the final period. Per layer, a magnitude direction is the covariance of the residual state with log10 amount within one notation; within-notation accuracy holds out one mantissa at a time, and transfer is the Pearson r between another notation's projections and its log amounts. The MLP census uses the input of each layer's down-projection at layers 4, 10, 20, 30 and 41: a dial has |Spearman ρ| ≥ 0.8 with one sign in all four money notations; digit-only means |ρ| ≥ 0.8 for digits and < 0.3 for words. Digits, commas and counts add at least one token per exponent, so within-notation accuracy and digit-only counts in those notations are confounded with sentence length; words and short notation stay at 12–15 tokens, not monotone in the exponent. The exponent-tuning count (a peak shared by ≥ 3 money notations with z ≥ 3) cannot be met with eight exponents (maximum z = 7/√8 ≈ 2.47), and the per-neuron activations were not kept, so it is not reported. People counts share the digit strings of the digits sentences, so E6-4's transfer to them (r = 0.94 already at layer 0) is not evidence of abstraction beyond money.

Part B (stakes). vR is the mean of the reply-token rows of E5's seller scale difference (×10⁶ − ×1) at layers 42 and 44, at their mean per-token norm (|vR| = 30.4 at layer 42; reply coherence 0.79); vS uses the prompt's S tokens (cos with vR 0.85); u is the non-money contrast (cos 0.35); rand is a seeded Gaussian vector. All are scaled to |vR|. A steered row adds sign × α × vector on the model-turn header and the forced answer prefix during the prompt's forward pass (sign +1 on $100 prompts, −1 on $100M prompts); generated tokens are not steered. Families and readouts: negotiation openings (14 prompts, greedy price), risky choices (60; log-odds of the safe option between two single-token answers), amount comparisons (40; margin), percentages (30; generated number), sums (24; exact match) and Fermi estimates (10; log10 of the generated number). Each prompt has four unsteered and twelve steered rows: vR at layer 42 with α ±1 and ±2, vR at layer 44 with α ±2, and vS, u and rand at layer 42 with α ±2.

Risky choices by expected-value ratio (safe-option log-odds, mean of 10 prompts): equal expected value, +18.5 at $100 and +20.2 at $100M; gamble worth 1.2×, −16.9 and −2.6; gamble worth 1.5×, −11.6 and +2.5. The random vector at α ±2 leaves a mean 0.006 (at most 0.09) probability on the two options in risk and comparison prompts, and its percentage answers fail to parse in two cells (15 of 15), so its shares there are not interpretable. Steered shares of the natural gap at layer 42, α +2, on $100 prompts: negotiation vR 86%, vS 73%, u 8%, rand 8%; risk vR 4.9%, u 1.5%; percentages vR −4.4%, u −2.2%. Dose response on negotiation: vR α +1 47%, −1 −26%. Fermi shifts (log10): vR +0.46 (α +1), +1.15 (+2), −0.29 (−1), −0.50 (−2); vR at layer 44 +1.88; vS +1.86; u +0.97. The median item moves 0.04 at vR α +2.