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Scaling laws
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How quantities scale, with the exponent, the scope, and the falsifier for each.
68 rows.
| name | relation | exponent | scope | class | source | falsifier |
|---|---|---|---|---|---|---|
| b | b = ≈ 0.74 (Brody, same year: ≈ 0.73) | ≈ 0.74 (Brody, same year: ≈ 0.73) | ~13 spp mammals + birds, 1932 dataset | OBSERVED-CONTESTED | Kleiber (1932), Hilgardia 6:315–353 | See the dispute row below. |
| b_WBE | b_WBE = 3/4 = 0.75 exactly | 3/4 = 0.75 exactly | conditional on: space-filling fractal branching network + size-invariant terminal units + minimised dissipation | MODELED | West, Brown & Enquist (1997), Science 276(5309):122–126, DOI 10.1126/science.276.5309.122 | Exhibit a taxon meeting all three assumptions whose measured b excludes 0.75. Drop any assumption and 3/4 does not follow. |
| b_WS | b_WS = 0.686 ± 0.014 (95% CI); interordinal 0.65 | 0.686 ± 0.014 (95% CI); interordinal 0.65 | 571 mammal spp; BMR normalised to 36.2 °C; excl. Artiodactyla, Macropodidae, Lagomorpha, Soricidae; interspecific ≈0.69, N=619, r²=0.94 | OBSERVED-CONTESTED | White & Seymour (2003), PNAS 100(7):4046–4049, DOI 10.1073/pnas.0436428100; Dodds, Rothman & Weitz (2001), J. Theor. Biol. 209(1):9–27 | Pre-register taxon set + temperature normalisation + regression unit, and show the CI is stable across all three. It currently is not — that is the dispute. |
| b_Kleiber | b_Kleiber = 3/4 (measured slope reported 0.74) | 3/4 (measured slope reported 0.74) | BMR vs mass, 13 data points | OBSERVED-CONTESTED | Kleiber 1932, Hilgardia 6:315–353; n=13 per Kolokotrones et al. 2010, Nature 464:753–756 | See every row below; the dispute IS the falsification record. |
| b_Rubner | b_Rubner = 2/3 | 2/3 | surface-law argument; respiration trials on dogs | OBSERVED-CONTESTED | Rubner 1883 | Data rejecting 2/3 at a stated mass range. |
| b_Savage,binned | b_Savage,binned = 0.737 | 0.737 | mammal BMR, 0.1 log-unit bins | OBSERVED-CONTESTED | Savage et al. 2004, Funct Ecol 18:257–282 (95% CI 0.711–0.762, n=52) | Re-fit with different binning. |
| b_Savage,unbinned | b_Savage,unbinned = 0.712 | 0.712 | mammal BMR, all species — CI excludes 2/3 and 3/4 | OBSERVED-CONTESTED | Savage et al. 2004 (95% CI 0.699–0.724, n=626) | Re-fit; show binning is not what moves the verdict. |
| b_White&Seymour | b_White&Seymour = 0.68 (interspecific); 0.65 (interordinal) — both ≠ 3/4, both = 2/3 within error | 0.68 (interspecific); 0.65 (interordinal) — both ≠ 3/4, both = 2/3 within error | 619 spp. → 469 after excluding Artiodactyla, Lagomorpha, Soricidae, Macropodidae; T_b-corrected to 36.2 °C, Q₁₀=3.0 | OBSERVED-CONTESTED | White & Seymour 2003, PNAS 100(7):4046–9 (CIs not read in this pass) | A dataset with equivalent basal-condition rigour giving 3/4. |
| b_WS,BMR / SMR / RMRt | b_WS,BMR / SMR / RMRt = 0.686±0.014 / 0.675±0.013 / 0.712±0.013 | 0.686±0.014 / 0.675±0.013 / 0.712±0.013 | exponent depends on which rate is measured; the BMR/SMR figures are White & Seymour's own 2003 fits restated in their 2005 review — same authors, same dataset, same regression, so not a replication | OBSERVED-CONTESTED (re-classed: the identical underlying result is OBSERVED-CONTESTED two rows up; one result cannot hold two classes) | White & Seymour 2003, as tabulated in White & Seymour 2005, J Exp Biol 208:1611 | Show the three definitions give one exponent. |
| b₂ (curvature) | b₂ (curvature) = 0.0322 ± 0.0053 (P = 9.0×10⁻¹⁰); 0.0294 ± 0.0057 with T | 0.0322 ± 0.0053 (P = 9.0×10⁻¹⁰); 0.0294 ± 0.0057 with T | McNab dataset, n=636 (447 with T); unit-scale invariant | OBSERVED-CONTESTED | Kolokotrones et al. 2010, Nature 464:753–756, Table 1 | MacKay 2011 — see contested row; reply: Deeds, Savage & Fontana 2011. |
| b₁ | b₁ = 0.5400 ± 0.0295 (0.5371 ± 0.0305 with T) | 0.5400 ± 0.0295 (0.5371 ± 0.0305 with T) | artifact of M₀ = 1 g; not interpretable alone | INADMISSIBLE as 'the exponent' | Kolokotrones et al. 2010; objection: MacKay 2011, J Theor Biol 280(1):194–6; reply: Deeds, Savage & Fontana 2011, 280:197–8 — concedes the artifact, calls it irrelevant to curvature; the class stands either way | Derive: under M'=kM, b₁' = b₁ − 2b₂log k. |
| local slope | local slope = 0.57 → 0.87 (rises with mass) | 0.57 → 0.87 (rises with mass) | ~3.6 g to ~460 kg; = b₁ + 2b₂log₁₀M | MODELED | Computed in-chapter from Kolokotrones Table 1; matches their stated range | Arithmetic error. |
| M(slope=1) | M(slope=1) = ~7.4 × 10⁷ g ≈ 74 t | ~7.4 × 10⁷ g ≈ 74 t | proposed upper bound on animal size — EXTRAPOLATION: ~2.2 decades beyond the fitted data, which end at ~460 kg (local slope 0.87). Kolokotrones et al.'s own two hedges: the unbounded slope rise 'may be due to the paucity of data for large animals', and the size-limit reading holds only 'If this is correct' | MODELED (extrapolated — rule 1 applies to this row) | Computed in-chapter; Kolokotrones et al. state ~10⁸ g (100 t) | A larger animal; or the slope not reaching 1. |
| k_marsupial | k_marsupial = 0.75 ± 0.01, R² = 0.990 | 0.75 ± 0.01, R² = 0.990 | 70 marsupials (McNab 2008), excluding Tarsipes rostratus + Lasiorhinus latifrons | OBSERVED-CONTESTED | MacKay 2011 | Re-fit with the 2 species retained. |
| WBE assumptions | WBE assumptions = space-filling fractal network; size-invariant terminal unit; energy minimisation | space-filling fractal network; size-invariant terminal unit; energy minimisation | derivation of 3/4 | MODELED | West, Brown & Enquist 1997, Science 276:122–126 | The assumptions are the fence — see rows below. |
| WBE finite-size form | WBE finite-size form = M = c₀B + c₁B^(4/3), both c > 0 → concave; data are convex | M = c₀B + c₁B^(4/3), both c > 0 → concave; data are convex | wrong sign of curvature | MODELED (refuted on this point) | Kolokotrones et al. 2010 | Show c₁ < 0 follows from WBE's own minimisation. |
| b_heart | b_heart = −1/4 | −1/4 | resting mammals; L&H state no n and no mass range for the resting-HR claim — their 34-species / 7 g–500 kg series is maximal heart rate (−0.15) and VO₂max, not this row | OBSERVED-REPLICATED | Lindstedt & Hoppeler 2023, J Exp Biol 226(24):jeb245766 — 'resting heart rate scales as M–1/4' | Modern re-fit with CI outside −0.30 to −0.20. |
| b_lifespan,mammal | b_lifespan,mammal = 0.153 (t_max = 4.88·M^0.153 yr, M in g), R²=0.66 | 0.153 (t_max = 4.88·M^0.153 yr, M in g), R²=0.66 | 856 mammals, cetaceans excluded — not 1/4 | OBSERVED-REPLICATED | de Magalhães, Costa & Church 2007, J Gerontol A 62(2) | Re-fit giving CI containing 0.25. |
| b_lifespan,bird | b_lifespan,bird = 0.218 (t_max = 5.22·M^0.218 yr), R²=0.70 | 0.218 (t_max = 5.22·M^0.218 yr), R²=0.70 | 518 birds | OBSERVED-REPLICATED | de Magalhães et al. 2007 ('body mass explained 70% of the variation in tmax') | As above. |
| b_VO₂max | b_VO₂max = 0.872 | 0.872 | 34 eutherian species, 7 g – 500 kg — vs basal ~0.70 | OBSERVED-REPLICATED | Lindstedt & Hoppeler 2023 | Show basal and max share an exponent. |
| beats/lifetime scaling | beats/lifetime scaling = ∝ M^(−0.097) → ~4.8× decline over 7 decades of mass | ∝ M^(−0.097) → ~4.8× decline over 7 decades of mass | composition of −0.25 and +0.153 | MODELED (computed in-chapter) | Computed from Lindstedt & Hoppeler 2023 + de Magalhães et al. 2007 | Measure beats/lifetime vs mass directly in one dataset — this composition mixes sources (M22). |
| σ ∝ L | σ ∝ L = stress grows linearly with size at constant shape | stress grows linearly with size at constant shape | F/A ∝ L³/L² | OBSERVED-REPLICATED (geometry) | Galileo 1638, Two New Sciences | Geometric error. |
| elastic similarity | elastic similarity = L ∝ D^(2/3); → D ∝ M^(3/8), L ∝ M^(1/4), S ∝ M^(5/8) | L ∝ D^(2/3); → D ∝ M^(3/8), L ∝ M^(1/4), S ∝ M^(5/8) | McMahon's model | MODELED | McMahon 1973, Science 179:1201–4; cascade computed in-chapter (reproduces his M^(5/8)) | See next row — largely refuted empirically. |
| bone scaling (measured) | bone scaling (measured) = length ∝ M^0.31; diameter ∝ M^0.35 | length ∝ M^0.31; diameter ∝ M^0.35 | 32 mammal spp. (secondary sources say 37 — unresolved here), 0.020–3500 kg — close to geometric similarity, not elastic (0.25 / 0.375) | OBSERVED-REPLICATED (primary not read in this pass; exponents and n via secondary sources) | Alexander et al. 1979, J Zool 189:305–314 | Read the primary; a species count or exponent outside the stated values moves this row. The NEGATIVE verdict rests on the direction (geometric, not elastic), which is corroborated independently, and survives either count. |
| b_colony,metabolic | b_colony,metabolic = 0.81, 95% CI 0.55–1.08 | 0.81, 95% CI 0.55–1.08 | 12 colonies + 391 unitary insects; CI excludes nothing | OBSERVED-CONTESTED | Hou et al. 2010, PNAS 107(8):3634–8 | More colonies; a CI that excludes an alternative. |
| b_colony,production | b_colony,production = 0.74, 95% CI 0.71–0.76 (r²=0.99, combined) | 0.74, 95% CI 0.71–0.76 (r²=0.99, combined) | colonies + unitary organisms — the tight row | OBSERVED-REPLICATED | Hou et al. 2010 | Independent re-fit. |
| b_city,superlinear | b_city,superlinear = cluster 1.07–1.34, not a single value | cluster 1.07–1.34, not a single value | patents 1.27 [1.25–1.29]; R&D empl. 1.34 [1.29–1.39]; GDP 1.15 [1.06–1.23]; wages 1.12 [1.09–1.13]; AIDS 1.23 [1.18–1.29] | OBSERVED-CONTESTED | Bettencourt et al. 2007, PNAS 104(17):7301–6, Table 1 | Leitão et al. 2016 — see next row. |
| b_city,sublinear | b_city,sublinear = gasoline stations 0.77 [0.74–0.81]; road surface 0.83 [0.74–0.92] (n=29); cables 0.87 [0.82–0.92] | gasoline stations 0.77 [0.74–0.81]; road surface 0.83 [0.74–0.92] (n=29); cables 0.87 [0.82–0.92] | Germany/USA 2001–02 | OBSERVED-CONTESTED | Bettencourt et al. 2007 | As above. |
| urban β ≠ 1 | urban β ≠ 1 = model-dependent | model-dependent | 5 models × 15 datasets; depends on fluctuations, their model, and heavy-tailed city sizes | OBSERVED-CONTESTED | Leitão et al. 2016, R Soc Open Sci 3:150649 (arXiv:1604.02872) | A fluctuation model class under which the verdict is stable. |
| Bergmann conformity | Bergmann conformity = 65–71% (mammals); 72–76% (birds) | 65–71% (mammals); 72–76% (birds) | 149 mammals, 94 birds | OBSERVED-CONTESTED | Meiri & Dayan 2003, J Biogeogr 30:331–351 — percentages via Teplitsky & Millien 2014; primary not read (M22) | Read the primary; a value outside these ranges. |
| exponent spread | exponent spread = mean 0.738±0.018 but 51% of exponents outside 0.7–0.8; range <0.5 to >1.0 | mean 0.738±0.018 but 51% of exponents outside 0.7–0.8; range <0.5 to >1.0 | 146 relations (Peters 1983), 72% vertebrate | OBSERVED-REPLICATED | Glazier 2005, Biol Rev 80:611–662 | Recount the distribution. |
| intraspecific spread | intraspecific spread = 0.3 to 1.8; mean 0.724, mode 0.667 | 0.3 to 1.8; mean 0.724, mode 0.667 | 220 species (Withers 1992) | OBSERVED-REPLICATED | Glazier 2005 | Recount. |
| MLBH bounds | MLBH bounds = 2/3 (surface-area limits) to 1 (mass/volume power limits) | 2/3 (surface-area limits) to 1 (mass/volume power limits) | metabolic-level boundaries hypothesis | HYPOTHESIZED | Glazier 2005, 2010, Biol Rev 85:111–138 | An exponent stably outside [2/3, 1] with a demonstrated mechanism. |
| Metabolic/rate exponent | Metabolic/rate exponent = 2/3 vs 3/4 — disputed | 2/3 vs 3/4 — disputed | mammals; lineage-dependent | OBSERVED-CONTESTED | 3/4: Kleiber-family. 2/3: White & Seymour (2003) PNAS 100:4046–4049 (619 spp.); Dodds, Rothman & Weitz (2001) JTB; lineage-varying: Capellini, Venditti & Barton (2010) Ecology 91(9) | A phylogenetically-controlled dataset where all lineages converge on one exponent would settle it. |
| t(1 m) | t(1 m) = ~6.2 × 10⁹ (≈195 yr) | ~6.2 × 10⁹ (≈195 yr) | 3D diffusion, D = 27 µm²/s, t = L²/6D | MODELED | Computed in-chapter from BNID 101997 + <r²>=6Dt | Arithmetic error, or a demonstration of 1 m protein transport by diffusion alone. |
| S/V | S/V = 6 vs 0.3 | 6 vs 0.3 | sphere 3/R; R = 0.5 µm vs 10 µm | MODELED | Computed in-chapter (geometry) | Geometric error. |
| δc/c scaling | δc/c scaling = ∝ (D·a·c·T)^(−1/2) | ∝ (D·a·c·T)^(−1/2) | diffusion-limited chemoreception | OBSERVED-REPLICATED (as a scaling) | Berg & Purcell 1977, Biophys J 20:193–219 | A sensor beating the −1/2 exponent. |
| δc/c prefactor | δc/c prefactor = disputed | disputed | B–P vs Bialek–Setayeshgar vs Kaizu: B–S term missing 1/(2(1−n̄)) | OBSERVED-CONTESTED / MODELED | Bialek & Setayeshgar 2005, PNAS 102(29):10040–5; Kaizu et al. 2014, Biophys J 106(4):976–85 | A treatment retaining receptor–ligand correlations that settles the constant. |
| b_prokaryote | b_prokaryote = 1.7 active / 2.0 inactive | 1.7 active / 2.0 inactive | metabolic rate vs body mass; n = 44 / 121 | OBSERVED-REPLICATED | DeLong et al. 2010, PNAS 107(29):12941–5, DOI 10.1073/pnas.1007783107 | Refit with independent data; CI covering 0.75. |
| b_protist | b_protist = 1.0 active / 1.1 inactive | 1.0 active / 1.1 inactive | n = 51 / 52 | OBSERVED-REPLICATED | DeLong et al. 2010 | As above. |
| b_metazoan | b_metazoan = 0.76 active / 0.79 inactive | 0.76 active / 0.79 inactive | n = 71 / 15 | OBSERVED-REPLICATED | DeLong et al. 2010 | As above. |
| Kleiber universality | Kleiber universality = refuted as universal | refuted as universal | 3/4 does not apply across prokaryote/protist/metazoan | OBSERVED-REPLICATED | DeLong et al. 2010 (their explicit conclusion) | A dataset in which one exponent fits all three groups. |
| Metabolic curvature | Metabolic curvature = convex on log-log; quadratic in log-mass required; not a pure power law | convex on log-log; quadratic in log-mass required; not a pure power law | mammals, temperature-corrected | OBSERVED-CONTESTED | Kolokotrones et al. 2010, Nature 464:753–6, DOI 10.1038/nature08920; contested by MacKay 2011, J Theor Biol 280(1):194–196; replied to by Deeds, Savage & Fontana 2011, J Theor Biol 280(1):197–198 — the exchange is live on both sides | Quadratic coefficient CI covering zero on independent data. |
| Exponent-by-subset | Exponent-by-subset = small-dominated → ~2/3; large-dominated → ~3/4 | small-dominated → ~2/3; large-dominated → ~3/4 | mammals; artefact of fitting a line to a curve | OBSERVED-CONTESTED (with the row above) | Kolokotrones et al. 2010 | Both subsets return the same slope. |
| α ∝ f² | α ∝ f² = exponent 2 | exponent 2 | classical (Stokes–Kirchhoff) term only | MODELED | classical acoustics | Not the observed exponent below 10 kHz — see next rows. |
| Rayleigh | Rayleigh = ∝ 1/λ⁴ | ∝ 1/λ⁴ | scatterers ≪ λ | OBSERVED-REPLICATED | Strutt [Rayleigh] 1871, Phil. Mag. 41:107–120, 274–279 | A small-particle scattering exponent ≠ 4. |
| c ∝ √T | c ∝ √T = exponent ½ | exponent ½ | ideal gas; independent of pressure | MODELED | Computed in-chapter from √(γRT/M) | Demonstrate a pressure dependence of c at fixed T. |
| W/S ∝ m^(1/3); v ∝ m^(1/6) | W/S ∝ m^(1/3); v ∝ m^(1/6) = exponents ⅓, ⅙ | exponents ⅓, ⅙ | isometric scaling of a flyer | MODELED | Computed in-chapter | Show non-isometric wing-area scaling that breaks it. |
| P_req ∝ m^(7/6) | P_req ∝ m^(7/6) = exponent 7/6 | exponent 7/6 | induced power, isometric | MODELED | Computed in-chapter; = Pennycuick's 7/6 law | Arithmetic; or measured power scaling ≠ 7/6. |
| P_avail ∝ m^(2/3) | P_avail ∝ m^(2/3) = exponent 2/3 | exponent 2/3 | muscle mass ∝ m, wingbeat frequency falls with size | MODELED | Computed in-chapter; the second curve in the crossing | Measured power-available scaling ≠ 2/3. |
| max flapping mass (theory) | max flapping mass (theory) = ~12 | ~12 | aerobically powered continuous flapping flight; m^(7/6) vs m^(2/3) crossing | MODELED (NOT-SOURCED — via secondary reports, not the primary) | Pennycuick's aerodynamic theory; secondary reports also give 'largest extant flying species ≈12–14 kg' (Pennycuick 1989) | Fetch the primary; or a measured power-available exponent ≠ 2/3. |
| ceiling vs census | ceiling vs census = exceedance, not agreement | exceedance, not agreement | heaviest bustards sit above the ~12 kg theoretical ceiling | OBSERVED-CONTESTED | this chapter; the two rows above, each in its own scope | Show a >16 kg bird sustaining continuous aerobic flapping → refutes the ceiling. Show bustard flight is burst-only → the ceiling's scope condition holds and the exceedance is not one. |
| α (MLR) | α (MLR) = 2.028 / 4.572 / 5.743 / 4.329 / 3.967 / 2.865 | 2.028 / 4.572 / 5.743 / 4.329 / 3.967 / 2.865 | six pieces over 0.179–31 M☉; 509 stars, detached eclipsing binaries | OBSERVED-REPLICATED | Eker et al. 2018, MNRAS 479(4):5491–5511, doi:10.1093/mnras/sty1834 (Table 4) | An independent DEB sample with a single exponent, or different break points. |
| α = 3.5 | α = 3.5 = not found in any piece | not found in any piece | the textbook value | SUPERSEDED | contradicted by Eker et al. 2018 | Show a mass range where 3.5 is the calibrated fit. |
| t_MS(30 M☉) | t_MS(30 M☉) = 2 × 10⁶ (α=3.5) vs 1.8 × 10⁷ (α=2.865); robust range 10⁶–10⁷ | 2 × 10⁶ (α=3.5) vs 1.8 × 10⁷ (α=2.865); robust range 10⁶–10⁷ | t ∝ M^(1−α), anchored on the Sun | MODELED | Computed in-chapter from Eker et al. 2018 + t ∝ M/L | A detailed evolutionary track for 30 M☉ outside 10⁶–10⁷ yr. |
| b_geom | b_geom = 1.0 | 1.0 | predicted length-vs-circumference exponent, geometric similarity | MODELED | Kilbourne & Makovicky 2010, J Anat, Table 8 | Derivation error. |
| b_elastic | b_elastic = 0.67 | 0.67 | elastic similarity (L ∝ D^(2/3)); attributed to McMahon 1975a | MODELED | Kilbourne & Makovicky 2010, Table 8 | Derivation error. |
| b_stress | b_stress = 0.5 | 0.5 | static stress similarity | MODELED | Kilbourne & Makovicky 2010, Table 8 | Derivation error. |
| b_fem,Trex | b_fem,Trex = 0.5341 — 95% CI 0.04159–0.9718 | 0.5341 — 95% CI 0.04159–0.9718 | T. rex, femur, ontogenetic, RMA log L vs log C — CI contains static-stress (0.5), elastic (0.67) and nears geometric (1.0): discriminates among NONE; carries no contrast with any other taxon | OBSERVED-SINGLE (one growth series, one study; CI spans the model space — downgraded from OBSERVED-REPLICATED) | Kilbourne & Makovicky 2010, J Anat, Table 3 | Re-measure the growth series; RMA slope or CI outside the published 0.04159–0.9718. |
| b_fem,Allo | b_fem,Allo = 0.82 | 0.82 | Allosaurus fragilis, femur, ontogenetic | OBSERVED-REPLICATED | Kilbourne & Makovicky 2010 | As above. |
| b_fem,sauropodomorph | b_fem,sauropodomorph = ~1.0 (Massospondylus 0.81) | ~1.0 (Massospondylus 0.81) | sauropodomorphs, femur, ontogenetic | OBSERVED-REPLICATED | Kilbourne & Makovicky 2010 | As above. |
| b_fem,hadrosaur | b_fem,hadrosaur = 1.05 (Maiasaura); 1.094 (Hypacrosaurus, 95% CI 1.072–1.113) | 1.05 (Maiasaura); 1.094 (Hypacrosaurus, 95% CI 1.072–1.113) | hadrosaurids, femur, ontogenetic — tight CI excludes every standard model from below; this is the section's real signal, and it stands alone without the T. rex row | OBSERVED-REPLICATED | Kilbourne & Makovicky 2010, Table 3 | Re-measure; slope or CI overlapping 1.0. |
| b_fem,interspecific | b_fem,interspecific = 0.83 (tibia 0.78; MT III 0.80) | 0.83 (tibia 0.78; MT III 0.80) | interspecific, non-avian dinosaurs — do not conflate with ontogenetic | OBSERVED-REPLICATED | Carrano, as reported by Kilbourne & Makovicky 2010 | Fetch Carrano primary; femoral exponent outside ~0.83 ± 0.05. |
| EE_odontocete | EE_odontocete = decreases with body mass | decreases with body mass | energy captured ÷ energy expended | OBSERVED-REPLICATED | Goldbogen et al. 2019, Science 366:1367–1372 | Tag data showing EE rising with size in odontocetes. |
| EE_rorqual | EE_rorqual = increases with body mass | increases with body mass | lunge filter feeders on krill | OBSERVED-REPLICATED | Goldbogen et al. 2019, Science | As above, inverted. |
| v ∝ d | v ∝ d = linear | linear | conduction velocity vs outer diameter, myelinated | OBSERVED-REPLICATED | Hursh 1939, Am J Physiol 127(1):131–139; Gasser & Grundfest 1939 | A myelinated preparation with non-linear v(d). |
| W ∝ G^(4/3) | W ∝ G^(4/3) = 4/3 (less a small cortical-thickness correction) | 4/3 (less a small cortical-thickness correction) | white vs grey matter, mammals, several orders of magnitude | OBSERVED-REPLICATED | Zhang & Sejnowski 2000, PNAS 97:5621–5626 | A mammalian dataset fitting a materially different exponent. |
| V_wire ∝ R⁶ | V_wire ∝ R⁶ = 6 | 6 | crude model: wiring volume to hold τ fixed as R grows; assumes fixed fraction of long-range neurons, linear v(d), spherical brain | MODELED | Computed in-chapter; cf. Zhang & Sejnowski's more careful 4/3 derivation | The assumptions — chiefly fixed fraction of long-range neurons, which modular brains violate by design. |
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