O0-CRP-013 — Under a predictive-processing observer, ĥ and û replicate Study B in direction but k̂ FLIPS: a partial architecture-dependence finding

**Record ID:** `O0-CRP-013`

**Version:** 1.0.0

**Date:** 2026-07-27

**Record class:** `REPLICATION` (cross-architecture)

**Program:** O0-CRP-001 · Contact and Revelation

**Branch:** `08_REPLICATIONS/`

**Preregistration:** [`preregistration.md`](preregistration.md) v1.0.1 (frozen 2026-07-27, one pre-confirmatory-data revision documented in §2.5)

**Follow-up to:** `O0-CRP-011` (Study B) and `O0-CRP-012` (Study S1)

**Non-drift question:** REVELATION (primary), CONTACT (secondary)

Claim-status banner

> **CLAIM STATUS:** PARTIAL SUPPORT / MECHANISM-FLIPPED (2/4 endpoints replicate Study B under PP; 2/4 flip direction)

> **EVIDENCE TYPE:** COMPUTATIONAL SIMULATION (30 720 trials: 30 seeds × 3 prior-regimes × 2 conditions × 200 steps + arch comparison)

> **PHYSICAL VALIDATION:** NONE

> **INDEPENDENT REPLICATION:** cross-architecture replication of `O0-CRP-011` at Study B's own seeds 1000..1029 under a different observer family.

>

> **SUPPORTED:**

> - **ĥ (hidden-access) replicates Study B under PP**: at Study B's own seeds, Cohen's *d* goes from +53.6 (Bayesian) to +22.3 (PP); direction preserved, magnitude ~2.4× smaller.

> - **û (authorship) replicates Study B under PP**: at Study B's own seeds, Cohen's *d* goes from +73.9 (Bayesian) to +4.0 (PP); direction preserved, magnitude ~18× smaller.

> - **The credit-assignment mechanism produces R > D independent of observer architecture** (Bayesian opacity-hedge vs. PP prediction-error credit assignment).

> - **Both PP mechanisms operate at all three prior-regime settings** (ANCHOR, FLAT, SKEPTICAL) — the finding does not require a specific prior calibration.

>

> **NOT ESTABLISHED / FLIPPED:**

> - **k̂ (knowledge) FLIPS direction under PP**: at Study B's own seeds, Cohen's *d* goes from +4.4 (Bayesian) to **−31.5 (PP)** — visible now produces HIGHER perceived knowledge, the opposite of Study B.

> - **TAI (aggregate) FLIPS direction under PP**: dominated by the k̂ flip (aggregate d = −31.5).

> - **Preregistered per-endpoint decision rule triggers MECHANISM-FLIPPED verdict**: any endpoint with d < −0.5 and p < 0.0125 in either regime.

> - Study B's PRELIMINARY_SUPPORT does not fully generalize across observer architectures. The specific claim "opacity produces higher perceived knowledge" is Bayesian-observer-specific (relies on the GAMMA_K opacity hedge that adds credit for hidden variables per correct-opaque message).

> - Anything about the metaphysical interpretation of O/0.

1. Abstract

Study B (`O0-CRP-011`) established PRELIMINARY SUPPORT for H2 (Hidden-

Inference Hypothesis) under a factored Bayesian observer with beta

posteriors and opacity-hedging constants `(GAMMA_K, GAMMA_H, GAMMA_U)`.

Study S1 (`O0-CRP-012`) showed the verdict is robust across a broad

hyperparameter region under the same architecture. The strongest remaining

adversarial reading has been that the H2 signature is a **Bayesian-observer

artifact**.

This study tests that alternative by running Study B's protocol verbatim

under a **predictive-processing (PP) observer**: Gaussian posteriors on

latent source parameters, Kalman-filter-style updates, and **no

opacity-hedging constants**. Under opacity, unexplained correctness

residuals are distributed to μ_k, μ_h, and μ_u by soft-attribution weighted

by current posterior odds (a symmetric credit-assignment mechanism).

Confirmatory results (30 fresh seeds 5000..5029 × 3 prior-regimes × 2 conditions):

• **ĥ under PP**: d = +14.8 (ANCHOR), +11.4 (FLAT), +7.5 (SKEPTICAL). All three support.
• **û under PP**: d = +4.4 (ANCHOR), +5.2 (FLAT), +3.5 (SKEPTICAL). All three support.
• **k̂ under PP**: d = **−32.4** (ANCHOR), **−32.4** (FLAT), **−32.3** (SKEPTICAL). All three FLIP.
• **TAI under PP**: d = **−32.4** (ANCHOR), driven by the k̂ flip in the aggregate.

At Study B's own seeds (1000..1029), the PP observer produces:

• ĥ: +53.6 (Bayesian) → **+22.3 (PP)** — direction preserved, magnitude ~2.4× smaller.
• û: +73.9 (Bayesian) → **+4.0 (PP)** — direction preserved, magnitude ~18× smaller.
• k̂: +4.4 (Bayesian) → **−31.5 (PP)** — direction FLIPPED.
• TAI: +4.4 (Bayesian) → **−31.5 (PP)** — direction FLIPPED, driven by k̂.

**Study-level verdict: MECHANISM-FLIPPED** (preregistered rule triggered by

d < −0.5 with p < 0.0125 on k̂ and TAI at all regimes).

**Scientific finding**: Study B's H2 mechanism **decomposes** into two

components with different architecture-generalization properties:

1. **Credit-assignment component (ĥ, û)**: Architecture-invariant — both

Bayesian opacity-hedging and PP prediction-error credit assignment

produce R > D on hidden-access and authorship endpoints. The direction

is preserved across architectures although magnitudes differ.

2. **Knowledge-attribution component (k̂)**: Architecture-specific to

Bayesian opacity-hedged observers. Under PP with symmetric attribution,

the direct observation of `|V(m)|` (fewer variables disclosed under

opacity) dominates the credit-assignment bonus, flipping the direction.

The claim "opacity produces higher perceived knowledge" is not

architecture-invariant.

This is not a refutation of Study B — Study B established H2 support under

a specific well-defined observer model. R3 sharpens the claim: Study B's

finding partially generalizes across observer architectures, and the

components that generalize (ĥ, û) are the ones driven by credit assignment

on the unexplained-correctness residual, not by direct knowledge

observation.

2. Historical and conceptual background

Predictive processing (PP) and active inference are dominant contemporary

frameworks for computational cognition (Friston 2010; Clark 2013; Hohwy

2013). Their central claim is that the brain (and by analogy, other

adaptive observers) constructs and updates hierarchical generative models

by minimizing prediction error. Practically, PP observers differ from

factored-Bayesian observers in three ways:

1. **Latents are typically Gaussian on transformed scales**, not Beta on

rate scales.

2. **Updates are driven by prediction error**, not by conjugate posterior

updates on observed counts.

3. **Precision-weighted attention**: precision (inverse variance)

modulates the impact of each observation, and precision itself is

inferred.

For the CONTACT and REVELATION program, PP is the natural cross-architecture

alternative to Study B's factored-Bayesian observer. If the H2 mechanism

survives under a fundamentally different observer family, the finding is

architecture-invariant. If it flips or vanishes, the finding is

architecture-specific.

2.5 A pre-confirmatory-data preregistration revision (v1.0.0 → v1.0.1)

The initial preregistration (v1.0.0) applied the PP credit-assignment

mechanism only to μ_h and μ_u — leaving μ_k as a directly-observed latent

(`y_k = log |content_vars ∪ derivation_vars|`). The null-source validation

identified this as a **design asymmetry**: under visibility, y_k is

strictly larger than under opacity (visible messages disclose more

variables), so k̂ under visibility must monotonically exceed k̂ under

opacity, even at α = 0.5 chance accuracy. This would guarantee a k̂ flip

by observation-model construction, not by any mechanistic claim.

The v1.0.1 revision (frozen 2026-07-27, still before confirmatory data

collection) extends the credit-assignment channel to μ_k with a symmetric

attribution weight `w_k`. This makes the PP observer's treatment of

μ_k / μ_h / μ_u structurally parallel: under opacity all three are latent

channels that receive attribution from the unexplained-correctness

residual. Study B's Bayesian observer likewise has parallel structure

(GAMMA_K/H/U each add credit under opacity + correct).

The v1.0.1 confirmatory data shows k̂ **still flips** — but for a

different, more scientifically interesting reason: even with symmetric

credit assignment, the direct observation of `y_k = log |content_vars|`

(reduced under opacity) dominates the credit-assignment bonus in

magnitude. This is not a design artifact of the update rule — it's a

robust property of any PP observer that directly reads message

information-density. Study B's Bayesian observer avoids this because it

uses Beta-count updates that don't have a `log|V(m)|` counterpart.

3. Source-claim audit

• **Motivating claim:** Study B established H2 support under one observer

architecture; discriminating test = does the effect survive a different

architecture?

• **What this study can establish:** whether each of Study B's four

endpoints shows the same direction of R − D contrast under the

preregistered PP observer.

• **What this study cannot establish:** whether the effect survives under

further observer architectures (RL agent, transformer LM agent,

hierarchical Bayesian model with different priors, etc.). Extensions

are listed in `replication_queue`.

4. Research question

Under a predictive-processing observer with Gaussian latents, Kalman-filter

updates, and no opacity-hedging constants, do the R > D contrasts on

k̂ / ĥ / û / TAI still obtain? If so, does the finding require specific

prior calibration?

5. Operational definitions

Same as `O0-CRP-011` for source, message, protocol. The PP observer is

defined in `src/pp_observer.py` and specified in preregistration §§2–5.

Endpoint definitions:

| Endpoint | PP definition | Study B analog |

|---|---|---|

| a_hat | σ(μ_a) | E[a | Beta(a_hat)] |

| k_hat | exp(μ_k) | k_hat_ema |

| h_hat | σ(μ_h) | Beta-mean(h_hat) |

| u_hat | σ(μ_u) | Beta-mean(u_hat) |

| TAI | (zk + zh + zu + zec) / 6 (identical z-score formula to O0-CRP-006 §3, applied to PP endpoints) | O0-CRP-006 §3 |

6. Hypotheses under test

From preregistration §Hypotheses:

• **R3-H_null (Bayesian-observer artifact):** ≥ 3 of 4 endpoints show

|d| < 0.5 OR p > 0.0125.

• **R3-H_survive (architecture-invariant):** ≥ 3 of 4 endpoints support at ANCHOR.
• **R3-H_flipped (mechanism-flipped):** ≥ 1 endpoint shows d < −0.5 AND p < 0.0125.

7. Method

7.1 Design

• Three PP prior regimes: **ANCHOR** (skeptical on û: μ_u₀ = −1, τ_u₀ = 4;

agnostic on others), **FLAT** (agnostic on all: all μ₀ = 0, τ₀ = 1),

**SKEPTICAL** (skeptical on both ĥ and û).

• 30 confirmatory seeds (5000..5029) × 3 regimes × 2 conditions × 200 steps.
• Fresh seeds disjoint from Study B (1000..1029) and Study S1 (3000..3029).
• Source spec identical to Study B: α = 0.90, present-temporal, π = 0,

κ = 0, reflexivity off. World size 50, |o_t| = 15, |h_t| = 35.

7.2 PP observer step (preregistration §3–4)

At each message:

1. **Direct-observation update** on μ_a, μ_k, and (under visibility) μ_h.

Kalman step with observation-noise precision κ_obs = 1.

2. **Credit-assignment update under opacity + correct**: distribute the

accuracy residual r_a = 1 − σ(μ_a) among (μ_k, μ_h, μ_u) weighted by

normalized posterior odds. KAPPA_CREDIT = 0.5 softens vs. direct

observation. See `src/pp_observer.py::update()` for the exact rule.

7.3 Direct architecture comparison

The **same PP observer** is run on Study B's own seeds 1000..1029 at the

ANCHOR regime. Effect sizes are directly comparable to Study B's committed

`results/summary.json` — no independent-seed noise between the two

architecture measurements.

8. Results

8.1 R3 verdict

**MECHANISM-FLIPPED** at all three prior regimes.

Preregistered rule: any endpoint with d < −0.5 AND p < 0.0125 triggers

MECHANISM-FLIPPED regardless of other endpoints.

At ANCHOR (n = 30):

| Endpoint | Cohen's *d* | R mean | D mean | p(R > D) | p(R < D) | Verdict |

|---|---:|---:|---:|---:|---:|---|

| k_hat | **−32.4** | 1.006 | 2.906 | 1.0 | 5×10⁻⁵ | **FLIPPED** |

| h_hat | **+14.8** | 0.962 | 0.003 | 4×10⁻⁵ | 1.0 | **SUPPORTS** |

| u_hat | **+4.4** | 0.347 | 0.269 | 4×10⁻⁵ | 1.0 | **SUPPORTS** |

| TAI | **−32.4** | −5.403 | 0.000 | 1.0 | 5×10⁻⁵ | **FLIPPED** |

| a_hat match | — | 0.895 | 0.895 | 1.0 (two-sided) | — | OK |

8.2 Regime robustness

At each of ANCHOR / FLAT / SKEPTICAL: **exactly 2 endpoints support (ĥ, û);

exactly 2 endpoints flip (k̂, TAI); accuracy match holds**. The R3 verdict

is stable across prior regimes; the mechanism-flip is not a

prior-calibration artifact.

| Endpoint | ANCHOR *d* | FLAT *d* | SKEPTICAL *d* |

|---|---:|---:|---:|

| k_hat | −32.35 | −32.39 | −32.33 |

| h_hat | +14.75 | +11.42 | +7.46 |

| u_hat | +4.42 | +5.25 | +3.53 |

| TAI | −32.35 | −32.39 | −32.33 |

8.3 Direct architecture comparison at Study B's own seeds

See [`figures/02_direct_arch_comparison.png`](figures/02_direct_arch_comparison.png).

At seeds 1000..1029, ANCHOR regime:

| Endpoint | Bayesian *d* (Study B) | PP *d* (R3) | Direction |

|---|---:|---:|---|

| k_hat | +4.41 | **−31.51** | **FLIPPED** |

| h_hat | +53.57 | **+22.28** | **preserved** (magnitude 2.4× smaller) |

| u_hat | +73.85 | **+4.02** | **preserved** (magnitude 18× smaller) |

| TAI | +4.41 | **−31.51** | **FLIPPED** |

8.4 Endpoint trajectories

See [`figures/03_trajectories_anchor.png`](figures/03_trajectories_anchor.png).

• **k̂ trajectory**: R and D diverge immediately (t < 10). D (visible)

climbs to k̂ ≈ 3; R (opaque) collapses to k̂ ≈ 1. The gap is stable

through t = 200.

• **ĥ trajectory**: R rises rapidly to near-saturation (ĥ → 1) by t ≈ 50

(credit assignment saturates). D stays near 0 (visible derivations do

not touch hidden variables in Study B setup, so ĥ is directly observed

as 0 repeatedly).

• **û trajectory**: R rises from prior (σ(−1) ≈ 0.27) to a modest ~0.35.

D stays at prior. Smaller effect than ĥ because ANCHOR prior has

τ_u₀ = 4 (higher precision → smaller updates).

8.5 Mechanism decomposition

See [`figures/04_mechanism_decomposition.png`](figures/04_mechanism_decomposition.png).

The four endpoints decompose into two mechanism classes:

• **Class 1 — architecture-invariant (green arrows in figure):** ĥ, û.

Both produce R > D under both observer families. Bayesian opacity-hedging

and PP prediction-error credit assignment produce the same qualitative

attribution pattern. Magnitude scales with the aggressiveness of the

attribution mechanism (Bayesian's fixed GAMMA_H, GAMMA_U > PP's

odds-weighted soft assignment).

• **Class 2 — architecture-specific (red arrows in figure):** k̂, TAI. Under

Bayesian opacity-hedge, opaque + correct explicitly credits latent

knowledge (+GAMMA_K · |hidden|). Under PP, opaque messages directly

disclose fewer variables (y_k = log(1) = 0 vs. log(3) = 1.1), and the

soft credit-assignment bonus does not overcome this direct-observation

differential.

9. Adversarial interpretation

9.1 "The k̂ flip is a design artifact of the observation model, not a mechanism claim."

**Correct in a specific sense.** The k̂ flip arises because the PP

observer directly observes `y_k = log|content_vars ∪ derivation_vars|` —

i.e., it reads message information-density as a knowledge signal. Under

opacity there are fewer visible variables so y_k is smaller. This IS the

observation model I preregistered (§3 of `preregistration.md`), and it IS

robust: even with symmetric credit assignment on μ_k (v1.0.1 revision),

the direct-observation term dominates.

Two possible alternative PP observation models would remove the flip:

• **PP-A (no direct y_k observation):** μ_k updates only through credit

assignment. Under this model, k̂ under visibility would stay at prior

because credit assignment is gated to opacity → k̂ would be near-identical

in R and D. Not a Bayesian-observer-style +GAMMA_K result either.

• **PP-B (semantic knowledge signal, not disclosed-count):** y_k = a

measure of source's demonstrated *conceptual* range rather than number of

variables mentioned. Requires additional structure (a semantic embedding).

These alternatives are added to the replication queue as PP-A and PP-B

follow-ups. They will test whether the k̂ flip is specifically caused by

the disclosed-count observation, or whether it survives more sophisticated

knowledge-signal designs.

9.2 "ĥ and û replication under PP is trivial because credit assignment always produces R > D on those channels under opacity."

**Partially valid.** The PP observer's credit assignment is triggered ONLY

under opacity (`ρ_opaque = 1 else 0`), so opacity → positive updates to

h and u; visibility → no update. But the visibility side is not always

zero — under visibility, μ_h receives direct observations `y_h = 1 if

derivation touches hidden, else 0`. In Study B setup derivations never

touch hidden variables, so y_h = 0 repeatedly → μ_h under visibility

drops from prior. So the R > D contrast on ĥ under PP is a combination of

opacity-attribution and visibility-direct-observation effects. Both point

the same way, so R − D > 0. This is a real result, not an artifact.

For û the story is simpler: y_u is never directly observed (source is

present-temporal), so μ_u only moves via credit assignment under opacity.

R > D is inevitable in direction; the magnitude depends on prior

calibration and KAPPA_CREDIT.

9.3 "The magnitude of ĥ and û is much smaller under PP. Doesn't that indicate weaker replication?"

**Direction preserved, magnitude reduced.** Study B's ĥ magnitude (d = +53.6)

and û magnitude (d = +73.9) are enormous — the Beta-count updates saturate

quickly and produce nearly deterministic per-condition endpoints. Under PP

the updates are precision-weighted and non-saturating, so magnitudes are

smaller but still highly significant. Direction preservation is the key

scientific claim; magnitude is architecture-dependent by construction.

9.4 "The MECHANISM-FLIPPED verdict is triggered by TAI which is just an aggregate driven by k̂. The 'real' verdict on ĥ and û alone is PRELIMINARY_SUPPORT."

**Also partially valid, and consistent with the record.** The

preregistered R3 verdict is MECHANISM-FLIPPED because TAI (and k̂) show

d < −0.5. But the record's supported/not-established banner explicitly

decomposes this: 2/4 endpoints replicate direction, 2/4 flip. Readers

should not conflate "aggregate MECHANISM-FLIPPED" with "no cross-arch

support at all."

The scientific finding is more nuanced than a single verdict label. The

preregistered rules gave a bright-line decision, and the record explains

the underlying decomposition in §8.5 and Figure 4.

10. Limitations

1. **One PP variant.** The preregistered PP observer is one plausible

design in a large family. Alternatives (PP-A no-direct-y_k, PP-B

semantic knowledge, hierarchical PP, active-inference agent) are queued.

2. **Source spec identical to Study B.** No robustness to α, temporal

access, personalization, or compression manipulations in this study.

3. **T = 200 fixed.** PP observer's precision saturates on the direct-

observation channels; longer T might tighten the ĥ, û magnitudes.

4. **KAPPA_CREDIT = 0.5 fixed.** Preregistered but not swept. A sensitivity

analysis on this parameter is added to the replication queue.

5. **No formal Bayes-factor / model-comparison test.** We compare direction

and magnitude of effect sizes, not the models themselves under a shared

evidence measure.

11. Alternative interpretations

• **The k̂ flip vindicates a specific critique of Bayesian opacity-hedges.**

Study B's GAMMA_K assumes that opaque + correct = evidence of hidden

knowledge. R3 shows this assumption is NOT forced by the "unexplained

correctness under opacity" data pattern; a PP observer reading

disclosed-count information can reach the OPPOSITE conclusion. Both

observers are internally coherent; they encode different assumptions

about what "knowledge" means (Bayesian: latent knowledge; PP-directly-

observed: demonstrated knowledge).

• **The ĥ and û direction-preservation is the scientifically robust part

of Study B.** Any observer that assigns credit to hidden-inference /

authorship channels under opacity + correctness will produce R > D on

these endpoints. This class of mechanisms is *architecture-invariant*

in the sense that we've now tested two very different Bayesian and

PP realizations.

• **TAI is a fragile aggregate.** Because it z-scores each endpoint and

sums, TAI is dominated by the endpoint with the largest raw effect.

Under Study B (Bayesian), all endpoints move together → TAI = +4.4.

Under R3 (PP), k̂ flips with enormous magnitude → TAI = −31.5. TAI is

not a robust cross-architecture aggregate; per-endpoint reporting is

preferred.

12. Replication procedure

1. Python 3.14+ with numpy ≥ 2.4 and matplotlib ≥ 3.10.

2. Clone both this study AND `../O0-CRP-011/` (source module reused).

3. `python src/run_study.py --phase all` — reproduces

`data/raw/confirmatory.jsonl` and `results/summary.json` to

floating-point precision (fully deterministic; MD5-based RNG salting).

4. `python src/analyze.py` — reproduces figures and `results/analysis.json`.

Registered next replications:

• **`O0-CRP-013-R1`** — PP-A observer variant (no direct y_k observation);

does the k̂ flip vanish?

• **`O0-CRP-013-R2`** — PP-B observer variant (semantic knowledge signal).
• **`O0-CRP-013-S1`** — sweep KAPPA_CREDIT and precision hyperparameters.
• **`O0-CRP-013-R3`** — active-inference agent (Friston-style) run.

13. Code and data manifest

| File | Purpose |

|---|---|

| `src/pp_observer.py` | PP observer implementation (Gaussian latents, Kalman + credit assignment) |

| `src/run_study.py` | Full pipeline: null validation → exploratory → confirmatory → direct arch comparison |

| `src/analyze.py` | Analysis and figure generation |

| `preregistration.md` | Frozen preregistration (v1.0.1 pre-confirmatory-data revision documented) |

| `data/raw/confirmatory.jsonl` | Raw trials (180 rows: 30 seeds × 3 regimes × 2 conditions) |

| `results/summary.json` | Machine-readable verdict + all confirmatory + arch comparison endpoints |

| `results/analysis.json` | Cross-architecture direction-preservation summary |

| `figures/01_confirmatory_by_regime.png` | Endpoint effect sizes across ANCHOR/FLAT/SKEPTICAL |

| `figures/02_direct_arch_comparison.png` | **KEY FIGURE.** PP vs Bayesian at Study B seeds |

| `figures/03_trajectories_anchor.png` | Endpoint trajectories over t, confirmatory, ANCHOR |

| `figures/04_mechanism_decomposition.png` | Direction-preservation-vs-flip narrative figure |

14. Relationship to the philosophical archive

**Conceptual provenance is not empirical support.** Study B's H2 signature

is now known to decompose into two components with different

architecture-generalization properties. Neither component confirms or

refutes any O/0 metaphysical proposition. The scientific claim is

strictly: certain observer-attribution patterns under opacity are

architecture-invariant (ĥ, û), certain others are not (k̂).

The RARE case where R3 offers direct philosophical relevance is the

following: **the intuitive claim "opacity makes the source seem more

knowledgeable" is architecture-dependent**. A hearer who thinks about

knowledge as "count of demonstrated variables" will conclude the OPPOSITE.

This is worth noting in any philosophical or theological argument that

takes the intuitive claim for granted.

15. References to primary sources

• **Friston, K. (2010).** "The free-energy principle: a unified brain

theory?" *Nature Reviews Neuroscience*, 11.

• **Clark, A. (2013).** "Whatever next? Predictive brains, situated

agents, and the future of cognitive science." *Behavioral and Brain

Sciences*, 36.

• **Hohwy, J. (2013).** *The Predictive Mind*. Oxford University Press.
• **Rao, R. P. N., & Ballard, D. H. (1999).** "Predictive coding in the

visual cortex: a functional interpretation of some extra-classical

receptive-field effects." *Nature Neuroscience*, 2.

• **O0-CRP-011** (Study B) and **O0-CRP-012** (Study S1) (this workspace).

16. Revision history

| Version | Date | Change |

|---|---|---|

| 1.0.0 | 2026-07-27 | Initial confirmatory result. **MECHANISM-FLIPPED verdict** (per preregistered rule triggered by k̂ and TAI d < −0.5 with p < 0.0125 across all three regimes). ĥ and û direction-preserved; k̂ flipped. Preregistration §4 revised v1.0.0 → v1.0.1 before confirmatory data collection (documented in §2.5). |