The Connectome Pipeline

The Caution That Travels with Every d_eff Figure

d_eff is never measured directly. It is computed from the measured critical exponent beta through hyperscaling, d_eff = (2 beta + gamma) / nu, while the other two critical exponents are held at their 3D Ising reference values (gamma = 1.237, nu = 0.63). The number therefore inherits both the fitting choices of the run that produced beta and the assumption that the imported exponents apply. Estimates move accordingly with parcellation, algorithm, and whether the value is a single-resolution measurement or an extrapolation. Metropolis single-spin-flip dynamics on group-averaged Schaefer 400 data return 2.89; Wolff cluster dynamics on 424 individual HCP connectomes parcellated at Lausanne 234 return a mean near 2.37. Both runs are reported below.

The positive connectivity trend is strongest in the reported midrange and cohort summaries; ablation endpoints have reversals. Absolute d_eff remains conditional on imported exponents, parcellation, and fitting choices. Neither a universal ordering nor class membership follows from the conversion alone.

The estimator also has a floor. Holding gamma and nu at their 3D Ising reference values, d_eff = (2 beta + gamma) / nu returns 1.963 at beta = 0, so the formula cannot place any positive beta much below 2 whatever the network is doing. No result computed this way can test whether a system sits below the Mermin-Wagner threshold (the equilibrium continuous-symmetry theorem discussed in Chapter 11, whose hypotheses have not been mapped onto cognition); the arithmetic answers before the physics does.

The Multi-Resolution Finite-Size Scaling Study (Experiment A14)

The simulations used parcellations of increasing resolution from the ENIGMA Toolbox (Schaefer atlases at 100, 200, 300, and 400 regions). At the coarsest resolution (N = 100), the measured critical exponent beta = 0.129, close to the 2D Ising value of 0.125. Finite-size scaling revealed this as an artifact of insufficient resolution. When the parcellation is too coarse, genuine three-dimensional structure is hidden by small system size, the way a photograph taken from too far away makes a mountain range look flat.

At N = 400 the measured beta = 0.238, and extrapolating the four resolutions to infinite system size gives beta = 0.291 +/- 0.031, within 1.2 standard deviations of the 3D Ising value (0.327) and more than five from the 2D value (0.125). The exclusion of two dimensions is the firm result. The match to 3D Ising is softer than the sigma figure suggests, because the extrapolation to infinite size rests on only four parcellation resolutions and the quoted +/- 0.031 is the fit’s internal error, which does not capture the uncertainty in the choice of extrapolation model. Hyperscaling turns that extrapolated exponent into d_eff = 2.89. White matter tracts, by connecting cortical regions that are distant on the sheet but coupled through fiber bundles, contribute enough cross-sheet connectivity to lift the cortex above two effective dimensions. The Binder cumulant, a diagnostic that distinguishes genuine phase transitions from finite-size artifacts, remained consistent across all four resolutions (standard deviation 0.014), confirming the transition is real.¹

The Graded Inter-Hemispheric Ablation (Experiment A14b)

Inter-hemispheric connection weights on the Schaefer 400-region connectome were scaled from zero (complete ablation) to 250% of their natural value, while intra-hemispheric weights were counter-scaled to hold total network weight constant. This isolation ensures that any change in critical behavior reflects topology, not coupling strength.

Eight conditions were run. Beta rises monotonically across the mid-range, from 7.9% through 31.5% inter-hemispheric fraction, then rolls back at the super-physiological endpoint. Across all eight the correlation is r = 0.845:

The two lowest-connectivity conditions do not order cleanly, and their error bars say why. The fully ablated network returns a higher beta than the 3.9% condition, and both carry errors several times larger than any mid-range point: +/- 0.125 on a value of 0.243, +/- 0.058 on a value of 0.076. At that precision neither point constrains anything. The gradient this experiment establishes runs from 7.9% upward.

At the natural connectome’s 15.8% fraction, this run measures beta = 0.313. The multi-resolution study measures beta = 0.238 on the same Schaefer 400 parcellation, a 31% disagreement. The two runs are not independent methods: both are Metropolis single-spin-flip Ising Monte Carlo on the same atlas family. They locate the critical temperature one grid step apart (23.05 here, 21.70 there), and beta is fitted below whatever critical temperature each run locates. The d_eff figures look closer, 2.96 against 2.886, only because the second number is an extrapolation to infinite system size rather than a single-resolution measurement. Read the spread between the two runs as an honest estimate of how tightly beta is pinned at one parcellation, which is less tightly than any single error bar suggests.

Doubling the inter-hemispheric fraction from 15.8% to 31.5% pushes d_eff from 2.96 to 3.36. The same absolute increase in intra-hemispheric weight, running along the cortical sheet, would barely register. At 39.4% inter-hemispheric fraction (2.5 times the natural value), beta rolls back from its peak of 0.439 to 0.395: excess cross-sheet connectivity begins to homogenize the network, washing out the geometric structure that supports a clean phase transition. The real connectome sits near the bottom of the optimal band (roughly 15-30% inter-hemispheric), where each additional cross-sheet connection yields the largest marginal increase in d_eff.

Full callosal ablation separates two N = 200 hemispheres and yields the reported Binder cumulant 0.35. This reduction does not prove loss of a common critical point: for two independent identical zero-mean subsystems, the Binder cumulant of their averaged magnetization is half the single-system value. Identical disconnected systems can retain the same Tc and β. Estimator floors and the absence of multi-resolution scaling limit the ablation’s dimensional inference.

Sex-Stratified Topology: Cohorts, Tertiles, Quintiles (Experiments A14c-d)

Inter-hemispheric fraction is positively associated with estimated d_eff in the reported comparisons: r = 0.845 for Schaefer 400 ablation, 0.995 for OASIS-3 tertiles, and 0.976 for HCP tertiles.² The ablation has reversals at its low and high ends, so the trend is neither all-level monotonicity nor near-perfect correlation in every dataset. Its midrange rise remains the more limited observation.

Individual-level Ising MC on all 424 HCP subjects resolves what group averaging obscured.³ Across individual connectomes, inter-hemispheric fraction correlates with d_eff at r = 0.51 (p < 0.001, n = 424). The correlation survives controlling for network density (partial r = 0.45): inter-hemispheric topology predicts d_eff independently of how many total connections a brain has. Sex predicts d_eff at Cohen’s d = 0.32 (p = 0.006): female mean d_eff = 2.374, male mean d_eff = 2.356. The effect is real, small, and entirely mediated by inter-hemispheric fraction.

Real single-subject connectomes average around 2.37, well below the 2.89 the group-averaged Schaefer 400 data returns, which is the method dependence flagged at the top of this annex. The gap is a reason to read the ordering across subjects rather than the level.

A quintile split reveals the gradient’s shape and the sex composition at each level:

The gradient saturates: d_eff rises steeply from Q1 to Q2, then flattens through Q3-Q5. The same saturation appeared in the Schaefer ablation at high inter-hemispheric fraction. The quintile correlation (r = 0.83) is lower than the tertile correlations (r = 0.976, 0.995) because finer binning exposes the nonlinearity.

The highest-connectivity quintile is 70% female; the lowest is 54%. Females cluster toward higher inter-hemispheric connectivity, consistent with Ingalhalikar’s findings.⁴ Joel et al.’s mosaic point holds quantitatively: 65.6% of males fall above the female 25th percentile for inter-hemispheric fraction, and 63.8% of females fall below the male 75th percentile.⁵ The overlap is massive. Sex is a weak filter on a continuous distribution.

The decisive test: within each quintile, does sex still predict d_eff? If inter-hemispheric topology is the causal variable, the sex effect should vanish once topology is controlled. It does. The female-minus-male d_eff gap shrinks from +0.038 in Q1 to +0.006 in Q4, and reverses to -0.013 in Q5 (males higher). Sex adds nothing beyond what inter-hemispheric topology already explains.

HCP corpus callosum volumetrics are consistent with the anatomical basis: after adjusting for intracranial volume, the CC/ICV ratio runs higher in females. The direction matches recent HCP work finding the corpus callosum genu larger in women once brain size is controlled, though the classic meta-analysis (Bishop and Wahlsten, 1997) found no significant CC sex difference after brain-size correction. The structural difference, where it appears, is small relative to individual variation.

In-Silico Confirmation: Bilateral Bridges

Two language models (Qwen 2.5 1.5B, one instruct-tuned, one fine-tuned on natural language inference) were connected by cross-attention bridges at 5% bandwidth. The bridge is the artificial counterpart of a callosal fiber: it lets each model attend to a narrow slice of the other’s internal state as it runs, 5% of the available channels.

Jointly trained activations show the reported 18% participation-ratio increase relative to either separate perturbation. PR = (Σλ)^2 / Σλ^2 measures the spread of activation variance across spectral directions. Connectome d_eff here is inferred from Ising critical exponents and imported hyperscaling parameters. They are different estimators; a PR increase does not validate the Ising dimension prediction with the same mathematics.

The from-scratch comparison reports PR 8.8 for unlike-seed streams versus 6.4 for identically initialized streams, about 38% higher. The welfare-probe annex also reports a single-stream baseline of 9.7, above either paired condition. These activation-space results motivate studying initialization and coupling, without establishing a general cross-substrate critical mechanism or guaranteed dimensional gain over a single stream.⁶

The Prediction Ledger

A developmental increase in connectivity could be tested against the connectome estimator. Interpreting any resulting dimension change as a transition from binary to graded cognition additionally requires a model satisfying the continuous-symmetry theorem and validated cognitive observables. Measuring all critical exponents would improve the network inference, yet would not alone provide that bridge.

Musical training. Professional musicians show enlarged corpus callosum, particularly those who began training before age seven.⁷ Musicians’ connectomes should therefore have higher d_eff than matched non-musicians, and this increase should correlate with the expanded cognitive repertoire musicians demonstrate in tasks requiring coordination across modalities (sight-reading integrates visual, motor, auditory, and temporal processing simultaneously).

Bilingualism. Bilinguals show increased white matter density in inter-hemispheric tracts, particularly in the anterior corpus callosum.⁸ Bilingual connectomes should show elevated d_eff relative to matched monolinguals. Managing two language systems requires continuous coordination between competing representations, the kind of graded, both/and processing that the Mermin-Wagner framework associates with d_eff above 2.

Neurodegeneration. Multiple sclerosis destroys myelin sheaths; small vessel disease damages white matter through chronic ischemia. Both reduce long-range connectivity. d_eff should decrease as white matter degrades. Clinically, patients should shift from graded to binary coordination, a phenomenon clinicians describe as cognitive rigidity, black-and-white thinking, and loss of nuance. The prediction is quantitative: d_eff computed from patient connectomes should correlate with measures of cognitive flexibility, and the correlation should be mediated by the Mermin-Wagner threshold.

Traumatic brain injury. Diffuse axonal injury, the most common pathology in moderate-to-severe traumatic brain injury, shears long-range white matter tracts. Decreased d_eff should follow injury, with the magnitude of the decrease correlating with the severity of cognitive inflexibility in recovery. Patients with preserved inter-hemispheric connectivity (higher post-injury d_eff) should retain more graded coordination than patients with equivalent total lesion volume concentrated in inter-hemispheric tracts.

The Comorbidity Test Record

Six falsifiable predictions from the coordination-class mismatch framework were tested against publicly available connectome data.

The autism d_eff prediction is a clean null (d = -0.093, p = 0.895, n = 154), despite the inter-hemispheric fraction mechanism confirming at r = +0.709 on the same dataset.⁹ The ADHD network segregation prediction was wrong in direction: the framework predicted elevated segregation, the data showed reduced segregation (d = -0.559, p = 0.0018 across four sites), and the framework was reinterpreted post hoc to accommodate that reversal.¹⁰ The d_eff effect-size predictions, then, did not confirm: one was a null, the other reversed. What holds is the mechanism-level correlation (r = +0.709) and the double dissociation between the two conditions’ topological signatures (autism: reduced inter-hemispheric connectivity; ADHD: reduced network segregation with normal inter-hemispheric connectivity). That dissociation is suggestive, not decisive.

Pipeline sensitivity is a live caveat. Anatomically constrained tractography (dipy ACT) recovers the inter-hemispheric-fraction-to-d_eff mechanism (r = +0.709); coercive label-assignment methods invert it. Any replication must therefore report its tractography method alongside its d_eff values; the sign of the mechanism correlation is not robust to that choice.

Remaining predictions (EDS white matter, comorbidity-d_eff correlation, masking effort, within-autism gender diversity) await population-specific data.

The cluster extends beyond autism, ADHD, and EDS. Anorexia,¹¹ dissociative identity disorder,¹² functional neurological disorders,¹³ and depersonalization¹⁴ show similar integration-substrate signatures, with atypical interoception as the common functional thread.¹⁵ Every member of the extended cluster shows disrupted cross-hemispheric integration manifesting as disrupted interoceptive integration, regardless of phenotype. The full cross-condition evidence base, including formal falsification criteria and direct empirical tests on ABIDE-II tractography data, is developed in the author’s companion paper on disrupted bilateral integration as a transdiagnostic substrate (in preparation).