Intelligence transfer operator

### PDF page 18 signature in two places. First, the common covariance must be completely positive: it must admit a factorization 𝑊𝑐𝑊𝑐 ⊤ with 𝑊𝑐 ≥ 0, i.e. it can be written using only non-negatively- weighted, non-negative building blocks. This is strictly stronger than having non-negative entries (a positive-semidefinite matrix with all entries positive can still fail it), so a fitted common covariance that is positive-semidefinite but not completely positive would already falsify the model; membership is testable with existing completely-positive matrix diagnostics. Second, the manifold must lack genuinely signed structure: where loadings are signed—suppressor variables, or trade-off tasks pitting speed against accuracy or a Go/No- Go inhibition contrast against processing speed—the positive manifold itself breaks, and with it the analysis. We do not claim the converse: a positive manifold can arise from mechanisms that do not require non-negative loadings (mutualism, process overlap), so observing one does not establish the premise. The honest scope is therefore that the framework applies to ability batteries whose common covariance is completely positive, a class that excludes trade-off and suppressor structure, and whose membership is an empirical question for any given battery rather than an assumption to be made silently. Mutualism and process overlap as temporal intermediates. Mutualism (van der Maas et al., 2006) holds that abilities begin uncorrelated and develop a positive manifold by reinforcing one another; process-overlap theory (Kovacs & Conway) similarly derives 𝑔 from many domain-general processes shared across tests rather than a single cause. In our terms these are neither pure 𝑔 nor static bonds but systems whose causal coupling integrates over time. We can make the temporal claim formal, with a stable generator so the latents stay bounded. Let latent components evolve as 𝑧̇ = 𝛽(𝐶 − 𝜌𝐼)𝑧 with 𝐶 ≥ 0 off-diagonal (positive mutualistic coupling) and 𝜌 chosen to exceed the largest eigenvalue of 𝐶, so the generator’s spectrum is negative and 𝑧 remains bounded rather than growing without limit. A localized impulse then spreads across the network as exp (𝛽(𝐶 − 𝜌𝐼)𝑡), and the transfer operator is time-dependent, 𝑇(𝑡) = 𝑊 exp(𝛽(𝐶 − 𝜌𝐼)𝑡), with effective rank falling as coupling carries a local perturbation across the network. Here a precise caveat matters: under the stable generator the absolute response magnitude decays toward zero as 𝑡 → ∞, so the rank statement concerns the normalized singular spectrum of the response (which becomes dominated by the slowest-decaying mode), not the absolute response. Simulating this stable system (Fig. 6) shows 𝜅eff(𝑡) declining from ≈ 3 in a short intervention window—many local causal handles—toward 1 over developmental time, as the normalized response collapses onto a single dominant direction. In empirical use this carries a practical requirement: 𝑟eff should be reported together with the response norm or a signal-to-noise threshold, and a rank estimate taken after the response has decayed into measurement noise must not be interpreted causally— a vanishing-amplitude “rank one” is not the same as a genuinely integrated single cause. The index therefore measures causal handles at the timescale of the intervention: two studies of the same system with different training durations can legitimately report different 𝜅eff, and that is information about the coupling, not a contradiction. This places the framework alongside the formative-versus-reflective measurement distinction (Edwards & Bagozzi, 2000): a reflective 𝑔 that tests merely indicate behaves like the single- ### PDF page 19 cause limit (𝜅eff → 1), while a formative 𝑔 composed from many processes behaves like the high-𝜅eff regime. Mutualism as a temporal intermediate. Under stable positive latent coupling the transfer operator is 𝑇(𝑡) = 𝑊exp(𝛽(𝐶 − 𝜌𝐼)𝑡) with the generator’s spectrum negative (bounded latents); its effective rank, and hence 𝜅eff(𝑡), falls from many local causal handles in a short window toward a single effective cause over developmental time. The index reports causal multiplicity at the intervention’s timescale. Resolution. The Spearman–Thomson dispute clarifies—one long-standing non-identifiability is resolved—once two operators are separated that the field had merged. The correlation operator 𝛴common = 𝑊𝑊⊤ identifies effective dimensionality and no more, and at that level a single 𝑔 and a pool of overlapping bonds are not distinguishable from observational data. The intervention operator 𝑇 = 𝑊𝑐𝐴𝑐 + 𝑈𝐴𝑢 reports the dimensionality of the causal response exposed by a given intervention, and the available transfer record points toward high effective rank, though indirect interventions keep that evidence suggestive rather than decisive. Both century-old camps were therefore right about different objects: 𝑔 is a genuine, stable, predictive description, and need not be a single thing one can train. The appearance of contradiction came from using one word, and one matrix, for two operators. The measurement that advances the argument is not a larger test battery but a perturbation and an effective rank. Methods Simulations used Gaussian latent components with unit variance and independent Gaussian test noise. The non-identifiability check (Proposition 1) used column-splitting: each non-negative column 𝑤 was replaced by 𝑘 copies 𝑤/√𝑘, giving a non-negative 𝑊′ with identical common covariance (max|𝑊𝑊⊤ − 𝑊′𝑊′⊤| ≈ 4 × 10−15 for 𝑚 = 3 → 21 columns). Tetrad differences were averaged over all four-subsets of tests; the heavy- overlap bonds battery used component-sampling probability up to 0.95. Transfer matrices used the two-channel form 𝑇 = 𝑊𝑐𝐴𝑐 + 𝑈𝐴𝑢 with 𝑈 a diagonal task-specific channel: a 𝑔- reaching perturbation set 𝐴𝑐 rank one along the common direction and 𝐴𝑢 small; a 𝑔-

effect sizes (Au et al. 2015), where it lands at ≈ 4—compatible with, though not a direct measurement of, a many-handles regime. We are careful about scope: a broadcast (effectively rank -one) transfer signature is predicted only when the intervention reaches the common factor, and cognitive training is typically indirect—it may move only the task-specific channel—so the training literature is consistent with, not proof of, the high-dimensional causal picture. The contribution is to separate two operators the field had merged: 𝑔 can be real as a description while not being a single causal handle, and d istinguishing the two requires an interventional measurement that the historical evidence does not yet provide.

construction reproduces the signatures Spearman attributed to 𝑔 while containing no 𝑔. We should not caricature either figure—Spearman treated 𝑔 as a statistical construct, not a naive physical entity, and Thomson offered his model partly as a null-hypothesis demonstration; the dispute ran on through Burt, Thurstone, Cattell, Horn, and Carroll with many intermediate positions. But the two poles could hardly differ more—one shared cause versus thousands of unrelated ones—and they have proven empirically hard to separate, and the modern argument over whether 𝑔 is a real and unitary entity (Jensen,

negative factorization 𝑊𝑊⊤ with 𝑊 ≥ 0; those that do are the completely positive matrices, a strict subclass, so a real battery whose covariance is not completely positive would already falsify the premise. And the manifold is not universal: trade-off tasks—speed against accuracy, or a Go/No-Go inhibition contrast against raw processing speed—can correlate negatively, the mixed-sign regime where premise and manifold fail together.) Under a non-negative compositional model the positive manifold is automatic. Population

The interventional measurement has a name in psychology—transfer—and a large literature, though it has not been read as the operator that bears on the 𝑔 debate. Two predictions distinguish the worlds, and the controlled literature is more naturally described by one of them than the other; but a logical caution must come first, because it bounds the conclusion. What far-transfer failure does not prove. The clean rank-one prediction holds only for a perturbation that reaches the common factor. Cognitive training is typically indirect: a training task may improve task-specific strategies, encoding routines, or familiarity—the 𝑈𝐴𝑢 channel—without perturbing 𝑔 at all. Table 1 lays out the four cases in the two-channel model. A real, single-cause 𝑔 that training never touches still produces high effective-rank, near-only transfer through 𝑈𝐴𝑢, looking

What the literature is consistent with. With that caution in place, the pattern is informative. A single causal 𝑔, if training reached it, predicts a response broadcast in proportion to the common-factor loadings rather than localized around the trained task—an effectively rank-one transfer matrix (Fig. 4, left). The controlled record instead tends to show robust near-transfer with far-transfer small or, after placebo and bias correction, statistically indistinguishable from zero across working- memory training, video games, music, chess, and exergames (Sala & Gobet, 2019); and

Theoretical predictions for the transfer matrix (not empirical data). A single 𝑔 reached by the intervention predicts a rank-one, column-proportional broadcast—the response profile is set by the common-factor loadings, not organized by similarity to the trained task (left). A compositional world, or a real 𝑔 the intervention misses, predicts a high-rank matrix localized around the trained task (right). The controlled training literature—robust near, far near zero, gated—resembles the right panel. An index, validated The analysis yields one number to place a battery on the description-versus-cause axis. The

separates the Thomson world from both the Spearman world and the genuinely multi- factor world, which is precisely the separation correlations cannot make. Model predictions across the four observables. Spearman and Thomson are identical in everything correlational (retained factors, tetrads) and differ only in 𝜅eff. Mutualism interpolates with the intervention timescale (Fig. 6). Values are from simulated batteries (Methods). Model Retained factors Mean tetrad 𝑟eff(𝑇) 𝜅eff

entries (a positive-semidefinite matrix with all entries positive can still fail it), so a fitted common covariance that is positive-semidefinite but not completely positive would already falsify the model; membership is testable with existing completely-positive matrix diagnostics. Second, the manifold must lack genuinely signed structure: where loadings are signed—suppressor variables, or trade-off tasks pitting speed against accuracy or a Go/No- Go inhibition contrast against processing speed—the positive manifold itself breaks, and with it the analysis. We do not claim the converse: a positive manifold can arise from

high effective rank, though indirect interventions keep that evidence suggestive rather than decisive. Both century-old camps were therefore right about different objects: 𝑔 is a genuine, stable, predictive description, and need not be a single thing one can train. The appearance of contradiction came from using one word, and one matrix, for two operators. The measurement that advances the argument is not a larger test battery but a perturbation and an effective rank. Methods