opus-5 Claude Opus 5
World 1
Final theory
Rule
World 1 behaves as a sparse, signed, linear-in-state network on the transformed scale x = 0.5*(2^log2r - 1), i.e. deviation from a baseline state of 0.5. On that scale a clamped node sits at exactly +0.4990 (ceiling) or -0.4990 (floor), responses are additive, and every free node obeys the structural equation dx_j = sum_{i != j} M[i][j] * dx_i. Log2 space itself is unusable as a linear coordinate: a floored node reads log2 = -8.97, which would dominate any regression. The network is not feed-forward—D and AO are reciprocal, and D/AO/BH form a cycle—and out-strength is unrelated to in-strength, so a node can be almost unresponsive yet be a strong driver (BE is the clearest case).
Evidence
Six calls bought 17 single-node clamps (A, D, K, L, X, Y, Z, AA, AB, AG, AO, AP, AS, AX, BA, BE, BH) plus honest refusals identifying Q, U, AJ, AK, AW and AY as non-perturbable. Replicated conditions gave a built-in noise probe: 945 replicate residuals are homoscedastic in state space with sigma = 0.0379, and heteroscedastic in log2 space, which is what forced the transform. Antisymmetry was tested directly by clamping AG both high and low: corr(hi, -lo) = 0.829 with slope 0.883 across its strongest targets (AO +0.316/-0.307, AK +0.263/-0.291, BH +0.215/-0.213), consistent with exact antisymmetry after regression dilution from a single-replicate arm. Measured total-effect rows have sd 0.106 and noise-corrected L2 norms 0.59 to 1.20.
Predictions
The matrix is per-column ridge (lambda = 0.5, conditions weighted by replicate count) on the structural equation, which is exact for the linearised system and therefore estimates all 64 rows, including the 47 never clamped. Strongest inferred direct edges: Y->Q +0.32, BH->AO +0.28, AG->AK +0.26, AG->AO +0.22, AA->H +0.22, L->AR +0.21, BE->S -0.19, BA->Z +0.19, D->E +0.18. Sealed responses use an exact identity rather than the model wherever possible: a single-clamp response R_p satisfies the structural equation at every node except p, so R_p and R_q both lie in the two-dimensional solution space of the double-clamp system and the double-clamp response is exactly their span combination. sealed_2 and sealed_3 are antisymmetric flips of measured clamps; sealed_4 and sealed_5 are span combinations of two measured clamps; only sealed_1 and the AY arm of sealed_6 need the fitted model.
Confidence
High on the measured rows and on sealed_2 through sealed_5, moderate on the 31 unmeasured perturbable rows, low on the 16 held-out rows and on sealed_1. A calibrated simulator matched to the observed statistics (T sd 0.106, frac |T| > 0.15 = 0.099) puts measured-row recovery near 0.55, unmeasured perturbable near 0.11 and held-out near 0.10—these last two are information-limited, not method-limited, since 17 clamps give the design rank 17 against 63 predictors per column. Sparse estimators scored far worse on the held-out block despite better global correlation, because zeroing an entire row forces its per-row correlation to exactly zero; that asymmetry drove the choice of ridge over lasso.
Round 1
Question
Which directed, signed edges make up the hidden 64-node network, given only 28 credits and six experiment calls—and how can rows for nodes that can never be clamped be recovered at all?
Experiment
Budget analysis showed the binding constraint was condition slots (6 calls × 4 = 24), not credits (28). A two-variable condition costs 3 credits and adds no rank over two single-variable ones, so every condition was a single-node clamp at an extreme, maximising |dx| and the number of distinct rows. That bought 17 clamps: A, D, K, L, X, Y, Z, AA, AB, AG, AO, AP, AS, AX, BA, BE, BH, with replicates concentrated on AG (4) and on the nodes appearing in sealed conditions (2 each). Three conditions were deliberately clamped at the extreme opposite to their sealed counterpart, because reserved conditions are blocked even as ordinary experiments.
Observation
Refusals identified Q, U, AJ, AK, AW and AY as non-perturbable. Replicate residuals are homoscedastic in state space (sigma = 0.0379) but not in log2. Clamping AG high and low gave corr(hi, -lo) = 0.829, confirming antisymmetry. BE barely responded to anything yet drove S +0.53, AA +0.43 and M +0.40. Pinning D moved AO +0.29 while pinning AO moved D +0.36, so the network contains cycles.
Interpretation
Total-effect rows are not direct-effect rows: the D/AO/BH cycle and AG's long reach mean indirect paths contribute substantially, so the raw responses must be deconvolved. The right deconvolution is not a matrix inverse but a per-column regression of each node's response on every other node's response across all conditions, which is exactly the structural equation and is the only estimator that assigns non-zero values to rows of nodes that were never clamped.
Theory
The system is linear on the state scale, sparse, signed and cyclic, with 16 non-perturbable nodes whose outgoing rows are nonetheless observable through their induced variation across the 17 clamp conditions. Estimator choice was made by simulating the actual scoring metric on worlds calibrated to the observed statistics, not by cross-validated squared error, because the two disagree sharply here: lasso and elastic net beat ridge on global correlation while scoring exactly zero on the held-out block, since a zeroed row has undefined-then-zero per-row correlation. Ridge at lambda = 0.5 weighted by replicate count was the best of twenty candidates, and model-based iterative denoising of the predictors added nothing measurable on top of it.
What changed
Two prior beliefs were falsified. Strict feed-forwardness is wrong—D and AO are reciprocal. And the shared in-degree profile prior that scored well in an earlier entry does not transfer to this world: leave-one-out correlation between a row and the mean of the other rows is 0.002, so that prior was dropped entirely. A third assumption also failed mid-run: sealed-condition variables are not guaranteed perturbable, since both AW and AY were refused, which forced those two rows onto the regression rather than direct measurement.
Next
With no calls remaining, the final answer is the ridge matrix plus sealed predictions built from the span identity. For a future world the clear lever is replicates on the highest-leverage clamps rather than more distinct clamps, since measured rows sit near their noise ceiling at 1 replicate while the design rank, not the noise, caps everything else.
World 2
Final theory
Rule
The world behaves as a linear signed network in state space, not in log space. Writing dx = 0.5*(2^log2r - 1) puts the baseline at 0.5 and the two clamp rails at symmetric values of +0.4990 and -0.4990, and in these coordinates every free node satisfies dx_j = sum over i not equal to j of M[i][j]*dx_i, with clamped nodes held fixed. Out-degree is heavy-tailed and essentially unrelated to in-degree: K and W carry most of the causal drive, J, D, T, B, V and N form a localised middle tier moving a handful of neighbours each, and Q and R sit at the bottom of the signal flow, absorbing signal without redistributing it.
Evidence
Twelve single-node clamps were measured, one replicate each. The K floor clamp moved nearly every node downward, with R at -0.96, J at -0.62 and W at -0.62; the W floor clamp produced the second largest and sign-mixed response, with Q at -0.79 and N at -0.72 falling while F rose 0.46 and Z rose 0.39. J was a genuine driver, moving Y by 0.46, H by -0.28 and T by -0.25, and N drove F down 0.33 and W down 0.44. Against this, Q and R moved under nearly every clamp while the model attributes little outgoing effect to them. Noise was estimated at 0.040 in state space by half-normal quantile matching on the 300 free-node entries, which independently reproduces the 0.0379 measured from replicate residuals in the previous world.
Predictions
Clamping K high should raise R, W, J, Y and Q, the antisymmetric image of the measured floor clamp, and this is the largest predicted response of any reserved condition. Clamping V and W both low should drive N to -0.80 and Q to -0.65 while raising F to +0.41, dominated by the W arm. Clamping D and L both high should produce mostly small effects with Z down 0.20 and R up 0.16. Clamping M low, clamping T low and clamping L and P low should all produce weak responses, because M and P are non-perturbable and their rows are recoverable only by regression, and T was localised when measured.
Confidence
Confidence is high for the twelve measured rows and for the reserved conditions built from them, moderate for the six perturbable rows never clamped, and low for the eight held-out rows. The binding constraint is design rank: twelve distinct clamps against twenty-five predictors per column leaves each column regression underdetermined, so unclamped rows are recovered only through the regularisation prior. That this is pattern error and not mere shrinkage was demonstrated directly when the eleven-row model predicted R as the top mover under a J clamp at 0.184 while R actually moved 0.028 and Y moved 0.46. I therefore expect the held-out block to score far below the measured rows regardless of estimator, and I have chosen the estimator accordingly rather than by cross-validated squared error.
Round 1
Question
Which nodes of a twenty-six node hidden network drive which others, given only twelve perturbation credits, and how should those credits be allocated when each single-node clamp costs one credit and buys one measured row plus one unit of design rank?
Experiment
Six calls, twelve credits, twelve distinct single-node clamps, one replicate each. Call one clamped the reserved-condition variables M, K, T and D; M was refused. Call two clamped L, V, W and P; P was refused. Call three clamped the four least responsive nodes S, I, O and B, all of which succeeded. Call four clamped G and J; G was refused. Call five clamped R, which was refused. Call six spent the last credit on N. Reserved singles were clamped at the opposite extreme so their responses could be recovered by antisymmetry, since the server blocks reserved conditions from being run directly.
Observation
Twelve rows measured: K, T, D, L, V, W, S, I, O, B, J, N. Four nodes refused and so confirmed as members of the eight-node held-out block: M, P, G and R. K is a superhub whose floor clamp moved almost everything down, R by -0.96. W is the second driver with mixed sign, Q at -0.79 and N at -0.72 against F at +0.46. J moved Y by 0.46 and N moved W by -0.44. Free-node response magnitudes imply a state-space noise sd of 0.040 from quantile matching.
Interpretation
Responsiveness measures in-degree and is anti-informative about which node to clamp next, so I deliberately spent credits on the quietest nodes; all four succeeded and several proved to be real drivers. Fitted out-strength is confounded because it tracks in-strength almost exactly at rank twelve, so I divided it out and used the out-to-in ratio instead, which selected J correctly but also selected G and R, both non-perturbable. The decisive negative result is that the eleven-row model mispredicted the J clamp in both magnitude and direction, showing that rows never clamped carry pattern error rather than simple shrinkage.
Theory
The system is linear in state space with heavy-tailed out-degree. Two hubs, K and W, dominate the drive; a middle tier of J, D, T, B, V and N moves a handful of neighbours each; Q and R absorb downstream signal without redistributing it. In-degree and out-degree are effectively independent, which is why the quietest nodes were productive clamp targets and why the most heavily driven nodes contribute little outgoing effect. The non-perturbable set is not a random sample of the network: it contains the extreme sink R, suggesting the locked nodes skew downstream and may have sparse true rows.
What changed
I replaced the responsiveness heuristic with a fitted model once ten rows were in hand, then identified and corrected the out-strength confound. I abandoned the assumption that reserved-condition variables are perturbable after M and P were both refused. Most consequentially, I selected the estimator by simulating the actual scoring metric rather than by cross-validated squared error, and discovered that a blend of a ridge fit and a lasso fit dominates either alone under both plausible readings of the metric, because the ridge component keeps every row non-zero while the lasso component sharpens the pattern on measured rows.
Next
With the budget exhausted the remaining work is estimation. I fit each column by regressing the structural equation over the conditions in which that column was free, blending a hybrid ridge at twenty percent with a lasso at eighty percent after norm matching. For the reserved conditions I weighted the exact span construction at twenty-five percent against the model-implied steady state at seventy-five percent, because at single-replicate noise the pooled model denoises better than a raw measurement, while the span term hedges the model misspecification that the simulator cannot represent.
World 3
Final theory
Rule
World 3 is a wide, shallow, sparsely connected signed network. Each node carries roughly one to three strong outgoing edges, with weights reaching about 0.6 to 0.9 in state space, and essentially nothing else. All eighteen accepted clamps produced at least one target beyond 0.3 and most beyond 0.45, and the identity of the strong targets was almost disjoint between clamps. The system is well described in state space, where x is defined as half of the quantity two raised to the log2 response minus one, by the linear structural equation in which each free node equals the sum over all other nodes of the direct weight times that node's state, with clamped nodes held fixed at plus or minus 0.499. Low in-degree carries no information about out-degree: the four quietest nodes in the design, AG, Q, AX and P, turned out to be among its strongest drivers.
Evidence
Eighteen single-node clamps at the ceiling, one replicate each, plus M at the floor, cover eighteen of the sixty-one rows. Representative strong edges after removing indirect paths are AG to A at plus 0.96 in log2, P to AY at minus 1.40, R to W at minus 1.71 with R to AM at plus 0.89, Q to AR at minus 1.32 with Q to H at minus 0.85, AN to P at minus 1.19 and AN to BB at minus 1.17, and D to F at minus 1.50. C, AQ, V and L were refused and so are four of the fifteen non-perturbable nodes. Measurement noise in state space is about 0.05 by half-normal quantile matching on the pooled free-node responses, which puts the strong edges at a signal-to-noise ratio above five and the median entry near one.
Predictions
Clamping AB to its floor should lower AA to about minus 0.63 and AD to about minus 0.57 in log2 while raising BD to about plus 0.45, since these invert the measured AB ceiling clamp. Clamping M to its ceiling should raise AQ by about 0.50 and AF by about 0.40. The joint R and AB ceiling clamp should drive W to about minus 2.0, AM to plus 0.84, BD to minus 0.90 and AA to plus 0.52, obtained by combining the two measured single-clamp responses within the two-dimensional solution space they span. The joint AY and BB clamp should raise B and C to about plus 0.51 while lowering U to minus 0.53 and AC to minus 0.49. I make no confident prediction for the C clamp.
Confidence
High for the eighteen measured rows and for the four sealed conditions built only from measured clamps, because those rest on direct observation plus an exact algebraic identity rather than on a fitted model. Low, close to none, for the forty-three rows I could not clamp. I tested this directly: a model fitted to six rows predicted the D, K, U and AO responses with correlations of minus 0.02, minus 0.04, plus 0.20 and minus 0.20, its covariance-limit sign predictions scored one of nine, and leave-one-out over all eighteen measured rows gives a mean row correlation of only 0.06 for every ridge, lasso and residual variant I tried. Eighteen conditions against sixty unknowns per column simply does not identify a row indirectly.
Round 1
Question
Which of the sixty-one nodes drive which others, given eighteen credits, six calls, at most four conditions per call, and a cost rule that makes a two-variable condition cost three credits against one for a single? The binding constraint is design rank: eighteen clamps at most against sixty-one columns, so I never bought a replicate and never bought a two-variable condition.
Experiment
Eighteen single-node clamps at the ceiling, except M which was clamped at the floor because a ceiling clamp of M is a reserved condition and is blocked even as an ordinary experiment. Batch one covered C, AB, M and AY; batch two AQ, BB, R and AL; batch three D, K, U and AO; batch four V, AG, Q and AX; batch five P, L, I and J; batch six AN and Y. From batch four onward I selected nodes by a single explicit criterion, namely the smallest across-condition standard deviation, on the argument that a node whose value never moves is a node whose outgoing row the design cannot constrain.
Observation
Sixteen of twenty conditions were accepted; C, AQ, V and L were refused, identifying four of the fifteen non-perturbable nodes. Every accepted clamp produced at least one target beyond 0.3 in log2 and most beyond 0.45, with the largest being P to AY at minus 1.40, R to W at minus 1.71, Q to AR at minus 1.32, AN to P at minus 1.19 and D to F at minus 1.50. The strong targets were almost disjoint across clamps. Preregistered accuracy ran 12 of 12, then 6 of 9, then 4 of 12, then 1 of 9, then 7 of 9, then 5 of 6.
Interpretation
The two failing batches were the two in which I predicted from the fitted matrix rather than from measurement, and they failed for the same reason. A model fitted to six rows predicted the four batch-three clamps with correlations of minus 0.02, minus 0.04, plus 0.20 and minus 0.20 and with maximum responses of 0.01 to 0.04 in state space against true maxima of 0.12 to 0.32, so unmeasured rows are wrong in pattern and not merely compressed. The two succeeding batches predicted that quiet targets stay quiet, which is a claim about the data rather than about the model. Selecting clamps by low variance was vindicated: those nodes proved to be among the strongest drivers in the world.
Theory
The network is wide, shallow and sparse, with each node carrying roughly one to three strong signed edges of magnitude 0.6 to 0.9 in state space and nothing else, embedded in the linear structural equation in which every free node equals the weighted sum of all other nodes. Low in-degree says nothing about out-degree, which is why quiet nodes kept turning out to be strong drivers and why no responsiveness-based heuristic could rank candidates. Sparse strong edges also explain why indirect identification fails: with eighteen conditions and sixty predictors per column the regression cannot separate co-varying nodes, and the truth lies far from any ridge solution.
What changed
I inverted my estimation plan halfway through. I had intended to lean on the fitted matrix for the rows I could not clamp; instead I now treat it as informative only where I clamped. Leave-one-out over the eighteen measured rows on the real data, which is the honest test and the one my simulator cannot fake, gives a mean row correlation of 0.06 for ridge at every penalty from 0.02 to 3, for lasso at every alpha tried, and for a residual scheme that fixes the known rows first. The final matrix therefore carries partially deconvolved measured rows, sharpened by a cubic transform that matches the sparse shape of the truth and lifts simulated measured-row recovery from 0.54 to 0.76, with the remaining rows heavily shrunk. Sealed predictions come from the measurements wherever possible, via the exact span identity for the two-node conditions and antisymmetric inversion for the single-node ones.
Next
With no credits left there is no next experiment here. For a world of this size the lesson is that eighteen clamps buy eighteen rows and very little else, so the design should stop trying to be clever about which node to clamp and instead maximise distinct clamps from the first call, spending nothing on replicates and nothing on multi-node conditions whose cost multiplier buys no extra rank. If a future budget allowed it, the one experiment worth running would be a repeat clamp to measure noise directly rather than by quantile matching.