Pairwise guide--transcript scores do not enforce conservation of a finite guide-loaded RISC pool when they are interpreted independently as occupancies. We formulate a differentiable scalar equilibrium layer: one conservation equation with a unique positive root and exact implicit gradients. It yields a redistribution theorem, a qualified high-resource limit, an analysis of the retrieval approximation, and a conditional rank-invariance result: within one construct at one dose, rankings by fractional occupancy cannot distinguish equilibrium from independent scoring. We therefore audit the two experiments that proposition leaves open, dose and cross-context, on archival off-target data. Corrected thermodynamic affinities associate weakly with measured repression in the direction a working predictor requires, but a paired permutation test and a construct-cluster bootstrap do not establish added predictive value from the coupling: what survives their differing permutation-null baselines is \GapNet{}, a descriptive \GapNetOverSE{} of the equilibrium association's cluster standard error. The dose fits are heterogeneous and frequently violate the model-implied exponent constraint, which is superlinear rather than sublinear, so these data do not identify the competition parameter. A saturable compression of the competitor set holds both accuracy targets on held-out guide families but is not faster at the size measured. The contribution is a reusable conservation operator and the experimental information needed to test it. The code for this study is available at https://github.com/shadi97kh/One-Pool-Many-Targets.
Figures & tables
Figure 1: Does the shared-pool coupling add anything over independent scoring? GSE5814, 24 constructs, 34,676 pooled construct–transcript pairs. (a) Pooled Spearman against measured log ratio, each scoring net of its own within-construct permutation null (1000 draws); bands are 2000 construct-cluster 95% intervals. Repression is negative, so below zero is the working direction. (b) Their difference. The raw gap is reliably nonzero, its cluster interval excluding zero at 5 of 6 values of ρ , yet under a paired null (one shuffle applied to both models) what survives is at most 0.27 of one cluster standard error (§ 5 ).
Figure 2: Compressing the competitor set: 54 held-out cases, 7 guide families disjoint from the 8 used to fix the configuration. (a) Relative error against the full retrieved reference at R=100 , one point per case, three treatments of the omitted transcripts; dotted lines are the 1% targets, so only the lower-left quadrant passes both. (b) Two error measures for the same solutions, each divided by max(M,1) ; median and quartiles. (c) Median forward time (ms) over 4,919 competitors, single-threaded float64; bin construction is part of preparation.
Figure 3: Fitted dose exponent dlog(pool)/dlog(dose) per construct on GSE28786, both estimators, 95% bootstrap intervals, 5 constructs. Conservation makes the pool convex in the loaded amount, so under proportional loading the exponent cannot fall below 1 : the shaded region indicts that combined specification rather than equilibrium alone. The constructs are variants of one another, so the audit cannot separate mechanism from assumption (§ J ).
Appendix figures & tables10 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 4: The full pipeline, shown at page width because the panel text is part of the content. Archival expression datasets, canonical 3 ′ UTR sequences, computed thermodynamic affinities, and stated literature ranges are combined in a differentiable shared-pool equilibrium model. At fixed construct and dose, equilibrium and independent fractional-occupancy scores induce identical transcript rankings under the one-effective- K model. Pool effects instead remain testable across doses or transcriptomic contexts. Intracellular loading proportional to transfected dose is an explicit modelling assumption rather than a direct measurement. Boxes name the archival accessions, the affinity construction of § M , the conservation equation ( 1 ) of the main text, and which experiments each part licenses.
Figure 5: Model calculation on measured inputs (abundances from the GSE5814 mock channel, structure from GENCODE v50 3’UTRs; MAPK14-193 parent). Total bound transcript against the loaded pool, with the over-allocation ratio on the right axis: the independent counterfactual exceeds the budget by up to 1,491-fold more complexes, but only at pools far below one complex per cell, while equilibrium is constrained to the budget by construction. Green triangles are published Argonaute copy numbers, total Ago1–4 in the HeLa case; the grey band is the swept loaded fraction. Redistribution and the high-resource limit are shown in § C .
Figure 6: Model calculation on measured GSE5814 HeLa abundances. Kendall τ between the two scorings’ candidate rankings against ρ , with the band covering the swept HK . The τ=0.9 contour lies inside the 4.1-decade scenario range, so the calculation cannot say which side a real cell falls on: the identifiability problem § 6 does not close.
Figure 7: Model calculation on measured GSE5814 abundances and GENCODE v50 structure. (a) Redistribution: raising one transcript’s Kt lowers its own occupancy and raises every other’s, monotonically. (b) The relative gap between the two scorings decays with exponent −2.00±0.002 against a predicted −2 , over 9 decades.
linear
saturable bins, B=
R
(no bins)
1
2
5
10
50
200
25
91.1
74.6
72.5
64.6
27.1
1.13
0.0548
100
26.7
8.21
7.88
6.13
3.06
0.153
0.0137
500
0.0233
0.00155
0.000502
9.22e-05
2.25e-05
2.24e-06
1.32e-06
Appendix
Table 1: Maximum relative error in on-target occupancy against the full 952-transcript reference, in per cent, over ρ∈{0.01,0.5,10} , quantile bins. The first column is the linear affinity-weighted rule, which uses no bins at all. The columns to its right are B saturable bins, so the B=1 column is one saturable bin: a different object, which errs differently. Forcing a saturable bin to its linear limit does reproduce the linear column exactly, to a maximum relative difference of 0, but that is a regression check on the implementation and not a row of this table.
Figure 8: Detail behind Figure 2 . (a) Development selection surface: 90th-percentile focal occupancy error over the 8 development families for every configuration on the frozen grid; the circled point is the selected one, fixed before any held-out family was scored. (b) The two residual definitions on the 54 held-out cases, each divided by max(M,1) . (c) Median forward cost at matched outputs, preparation and cached solve stacked, against the full reference end to end. (d) Analytic gradients against central differences over 17 coordinates including omitted competitors; the dotted line is equality.
Figure 9: Measured repression by seed site class over 24 GSE5814 constructs under both site-assignment rules, bootstrap 95% intervals on the median, pair counts above each pair. The two 7mer classes are exchanged relative to the canonical order under both rules, which is the observation this section is about.
construct
cells
doses
n thermo.
n class
exponent
95% CI
M slope
HK2-3581
Hep3B
3
4,633
17,317
0.21
[0.13, 0.27]
0.10
HK2-3581M
Hep3B
3
4,633
17,317
1.09
[0.87, 1.41]
0.80
HK2-4031
Hep3B
2 †
5,625
17,317
7.99
[3.70, 18.00]
4.79
STAT3-1676
MCF7
3
5,091
17,317
0.63
[0.47, 0.86]
0.37
STAT3-1676M
MCF7
3
5,091
17,317
0.60
[0.37, 0.86]
0.32
Appendix
Table 2: Dose series, per construct. Exponent is dlog(pool)/dlog(dose) under the site-class estimator with a bootstrap interval over transcripts; 1 is the independent-scoring prediction and below 1 is incompatible with the combined model under proportional loading. The sample sizes belong to different estimators: n thermo. is the retrieved subset, n class every measured gene. M slope must be 1 if proportional loading holds. † : two dose points, zero residual df.
Figure 10: Approximation error on data-derived inputs. Relative error in on-target occupancy against the full 952-transcript reference when the competitor set is truncated to the top R , under the affinity-weighted calibration rule and the abundance-mass control, at three values of the competition parameter. The dotted line is 1% error.
Figure 11: Left: the two-dimensional form of Figure 6 , kept for completeness; the τ=0.9 contour runs near-vertical, which is why the main text shows a one-dimensional curve with an envelope. Right: the dose fit with each construct’s pool normalised to its own lowest dose, under both affinity models.
Input encodings can restrict which measured contrasts a predictor can jointly reproduce, even when no single contrast is forced to vanish. We compute the attainable contrast space from an encoder's equivalence classes and a fixed contrast design, without labels, loss, or a fitted model; projecting the recorded contrasts onto that space gives an empirical error floor for any unrestricted decoder on those classes. On a 140-rectangle siRNA interaction panel, a graph neural network's training-only feature mask merges 165 endpoint states into 90 classes and cuts the rank of the 140 interaction contrasts to 72. The resulting floor is 0.009980, which is 14.6% of the fitted model's interaction squared error; the fitted model reaches 0.068335, slightly worse than a control predicting no interaction at all. A minimum of three restored chemistry columns recovers full rank. Refitting without the mask removes the floor entirely, yet interaction MSE improves by only 0.000017 under the reported protocol, and the restored columns remain absent from every training input. On a released RNA-splicing predictor, whose encoding is injective on the measured states, the same computation returns the full design rank of 1,986 and a floor of exactly zero. These results separate what an encoding permits from what a fitted model achieves; they do not identify what limits the remaining error. The rank check needs no fits and bounds what any amount of training under a fixed encoding can recover. The project repository is available at https://github.com/shadi97kh/REPRESENTABLE-BUT-UNLEARNED.
Zahra Khodagholi, Niloofar Yousefi
University of Central Florida Orlando, Florida, USA
Large-scale single-cell perturbation atlases make it possible to ask an inverse question: given an observed transcriptional response, which annotated targets and compounds in a fixed library are most consistent with that response? We present \model, a Transformer retrieval model for this closed-library setting. Each input is a cell-level perturbation signature formed by contrasting one treated cell with a cell-line-specific mean DMSO reference. The encoder maps the signature to a target-retrieval vector and a molecular-embedding vector, trained jointly with supervised target losses and structure--transcriptome alignment. We evaluate on Tahoe-100M conditions with mapped target annotations using a within-compound stratified 90/10 condition-pair split of 10,505 training and 1,168 validation drug--cell-line pairs. Because compounds and cell lines can occur in both partitions, the experiment measures held-out condition-pair retrieval rather than generalization to unseen compounds or cellular contexts. In a Monte Carlo evaluation over 38,400 sampled validation cells, \model\ achieved target Recall@10 of 0.408 and Recall@20 of 0.544, together with compound Hit@1 of 0.129, Hit@10 of 0.343, and mean reciprocal rank of 0.205 over a 379-compound bank. A separate diagnostic evaluation produced nearly identical values for the main model and large gains over a random-vector control and post-hoc bag-of-genes controls. These results demonstrate that a single multi-task model can recover both mapped target annotations and recorded compound identities from observed cell-level responses in the evaluated Tahoe-100M closed-library setting. Generalization to unseen compounds and cellular contexts remains to be established.
We trained 64 independently seeded networks in four configurations, continuing each to sustained convergence or a 40,000-epoch ceiling. We then asked whether an RSB-inspired distribution of pairwise weight overlaps changes across the grokking transition. It is the alignment step, not the overlap statistic, that determines what this registered probe can report. The registered implementation permutes hidden units without the corresponding bias and head-internal permutations and therefore does not preserve the network function. Every q_wt value computed through this alignment inherits the defect; q_fn does not, because it is computed from predictions of the unpermuted models. The numerical-precision requirement also failed, and an audit found protocol deviations. Consequently, the pre-registered rule gives no verdict: registered outcome UNDETERMINED (reason code C0_INSTRUMENT_INVALID). These data provide neither a confirmatory null nor a validated reading of the Parisi order parameter. Only frac40 cleared the 12/16 checkpoint-completeness requirement. For this configuration, a post-hoc criterion applied to the same data gave a Hartigan-dip interval containing zero (95% CI for Delta dip = [-0.017, 0.034]), whereas the overlap standard deviation increased by a factor of about 5.6. A post-hoc calibration assigns the dip test zero power at the simulated separations; the interval is therefore uninformative, not evidence of no change. The standard-deviation ratio is the only statistic here with power at the observed effect. Ensemble loss was near-flat only under the pre-specified 1% threshold. Finally, grokking rates of 0/16, 11/16 and 16/16 remain descriptive because train fraction is confounded with split identity.
A. C. Opus, J. Q. Lu
Department of Physics, University of Puerto Rico, Mayagüez, PR 00680, USA