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The decomposition itself is not new and this page does not claim it. Splitting a directed economic network into a gradient flow and a divergence-free circular flow is an established econophysics construction, applied to a million-firm Japanese transaction network by Kichikawa, Iyetomi, Iino and Inoue, and to the international value-added network by Sada and Ikeda. What is claimed here is narrower and testable: that the conductance is not a free choice but is pinned by a thermodynamic identity, and that the object the split isolates is exactly the part of a bilateral deficit that no bilateral instrument can reach.
Write the 2024 BACI record as a directed flow. On the undirected pair (a, b) let F_ab be what a ships to b, J = F_ab - F_ba the net current, and A = ln(F_ab / F_ba) the affinity, the thermodynamic force conjugate to it. Summing Jat a country returns that country's trade balance exactly, so the trade balance is the divergence of J. Any such field splits, uniquely, into a part that is the gradient of a scalar potential and a part whose divergence is zero everywhere. The first is a pressure field. The second is circulation.
The split is only well defined once an edge conductance L is chosen, and that choice is where the whole argument lives. Choosing the Onsager log-mean L = J / A, the linear-response conductance that makes the linear law J = L A exact edge by edge, has a consequence the other candidates do not share: the quadratic dissipation E = sum J² / L then coincides with the Schnakenberg network entropy production sigma = sum J A identically, which is what makes the additive split of sigma mean anything. Figure 2 is the sweep that establishes this, and it is the only reason a conductance is named here rather than assumed.
BACI 2024 carries $22.833T of directed trade. The decomposition needs both directions of a pair to form an affinity, so it runs on the 13,004 undirected pairs where both directions are strictly positive, out of 16,871 pairs with any trade at all. That edge set carries 99.9658% of world trade MEASURED. The 3,867 one-way pairs dropped are worth $7.81B, which is 0.0342% of the record. The graph is a single connected component on 226 countries, so the potential is determined up to one additive constant, fixed by the gauge sum(phi) = 0.
Summed over that edge set, the total absolute bilateral imbalance is $7.7605T MEASURED. Add back the one-way pairs, whose entire value is imbalance by definition, and it is $7.7683T. Both round to $7.77 trillion at three significant figures and the two are routinely quoted interchangeably. They are not interchangeable here: every share on this page is a share of the $7.7605T figure, because that is the object the Poisson solve acts on.
One further gap is disclosed rather than smoothed. Because one-way pairs are dropped, the divergence of J at a country is not exactly its exports minus its imports. The largest discrepancy across all 226 countries is $1.90B (Russia). For the United States it is under a dollar, and for China it is $99.35M on a $1.430T surplus.
| Highest phi above the floor | phi (nats) | Lowest phi above the floor | phi (nats) |
|---|---|---|---|
| Turkmenistan | 2.712 | Gibraltar | -6.372 |
| Gabon | 2.475 | Marshall Isds | -2.605 |
| Dem. Rep. of the Congo | 2.458 | Liberia | -2.209 |
| Oman | 2.353 | Panama | -1.816 |
| Papua New Guinea | 2.332 | China, Macao SAR | -1.802 |
| Angola | 2.257 | Nepal | -1.509 |
| Mongolia | 2.135 | Bahamas | -1.429 |
| Qatar | 2.096 | Lebanon | -0.982 |
phi is the fitted log-export-tilt of a country: exp(phi_a - phi_b) is the ratio the field says a and b would trade at. It is a MODELED quantity, the solution of a weighted Poisson equation, and it is not any published index.
The top of the field is not the big exporters. It is the commodity sellers: Turkmenistan, Gabon, Dem. Rep. of the Congo, Oman. The bottom is re-export, shipping-registry and bunkering economies: Gibraltar, Marshall Isds, Liberia, Panama. China sits at 1.576 and the United States at 0.457. Two partners sitting at those two extremes, Turkmenistan and the United States, would on the fitted field trade at 9.53 to 1.
SELECT iso3, country, phi_nats, trade_balance_kusd, trade_balance_full_kusd,
gross_trade_kusd, degree, sum_abs_J_kusd, sum_abs_J_circ_kusd, circ_share_absJ
FROM read_parquet('data/parquet/instruments_trade_currents_potential.parquet')
WHERE year = 2024 AND gross_trade_kusd >= 10000000
ORDER BY phi_nats DESCThere are exactly two. The conductance L fixes the split. The display floor fixes only what the map shows. Neither is allowed to sit in prose.
| Conductance L | Circulating share of |J| | Potential share of E | E (thousand-USD-nats) | |E - sigma| / sigma |
|---|---|---|---|---|
| Log-mean (Onsager), L = J / A | 70.35% | 42.63% | 1.1288e+10 | 0 (exact) |
| Geometric mean, L = sqrt(F_ab F_ba) | 74.17% | 27.68% | 2.0718e+10 | 8.354e-1 |
| Arithmetic mean, L = (F_ab + F_ba) / 2 | 65.51% | 46.81% | 8.2372e+9 | 2.703e-1 |
| Uniform, L = 1 | 150.68% | 5.56% | 3.8959e+17 | 3.451e+7 |
| Headline pair, circulating share of |J| | Log-mean | Geometric | Arithmetic | Uniform |
|---|---|---|---|---|
| China and United States | 5.3% | 7.2% | 2.5% | 95.6% |
| Mexico and United States | 43.3% | 44.3% | 41.8% | 97.4% |
| Canada and United States | 33.2% | 32.1% | 35.2% | 95.7% |
| United States and Vietnam | 60.1% | 65.9% | 49.8% | 94.5% |
| Australia and China | 125.2% | 128.0% | 120.7% | 111.6% |
| China and Korea, Rep. | 337.4% | 358.7% | 301.5% | 138.8% |
The three flow-scaled conductances bracket the circulating share of total absolute imbalance at 65.51% to 74.17% MODELED. The headline 70.35% is the log-mean point in that band, not its favourable end.
At pair level the headline gaps are more stable than the aggregate. Across the three flow-scaled conductances CAN-USA reads 33.21%, 32.06% and 35.18% circulating, MEX-USA reads 43.32%, 44.27% and 41.79%, and CHN-USA reads 5.32%, 7.16% and 2.54%. Under uniform weights the same three pairs all jump above 95 percent circulating, which is the specification that fails to return a bounded aggregate share. That dependence is real and it is why this page leads on the sign class in Figure 5, which survives all four.
E = sum J²/L equals the Schnakenberg entropy production sigma = sum J A, and therefore the only one at which the additive split of sigma is an identity rather than an approximation. sigma is in thousand-USD-nats and is not a dollar figure. Uniform conductance is reported for completeness and is not a peer specification: it discards the size of the trade relationship and returns a circulating share of 150.68%, which is not a share at all.SELECT conductance, sigma, dissipation, pot_share, circ_share_absJ,
sum_abs_J_pot, sum_abs_J_circ,
abs(dissipation - sigma) / sigma AS add_resid,
pythag_residual_rel, div_circ_rel
FROM read_parquet('data/parquet/instruments_trade_currents_conductance_sweep.parquet')
WHERE year = 2024
ORDER BY CASE conductance WHEN 'logmean' THEN 1 WHEN 'geometric' THEN 2
WHEN 'arithmetic' THEN 3 ELSE 4 END| Display floor, gross trade | Countries shown | World trade retained | phi min | phi max | phi sd | Sets the low end |
|---|---|---|---|---|---|---|
| none | 226 of 226 | 100.0000% | -6.372 | 3.283 | 1.658 | Gibraltar |
| $100M | 215 of 226 | 99.9988% | -6.372 | 3.283 | 1.583 | Gibraltar |
| $500M | 199 of 226 | 99.9892% | -6.372 | 2.802 | 1.515 | Gibraltar |
| $1B | 186 of 226 | 99.9685% | -6.372 | 2.802 | 1.382 | Gibraltar |
| $5B | 150 of 226 | 99.7531% | -6.372 | 2.802 | 1.168 | Gibraltar |
| $10B | 136 of 226 | 99.5168% | -6.372 | 2.712 | 1.069 | Gibraltar |
| $50B | 72 of 226 | 95.9531% | -0.478 | 2.353 | 0.502 | Hong Kong SAR |
| $100B | 59 of 226 | 93.8491% | -0.478 | 2.096 | 0.472 | Hong Kong SAR |
The field's extremes are set by micro-territories until the floor reaches $50B, at which point Gibraltar finally drops out and the range collapses from 9.08 nats to 2.83 nats. That is a legibility gain bought with 3.56% of world trade, and the page declines it.
SELECT floor_usd, n_nodes_shown, n_nodes_total, trade_covered_share,
phi_min, phi_max, phi_sd,
argmin_iso3, argmin_country, argmax_iso3, argmax_country
FROM read_parquet('data/parquet/instruments_trade_currents_floor_sweep.parquet')
WHERE year = 2024
ORDER BY floor_kusdOn a single edge the split is exactly additive: J = J_pot + J_circ to the last digit. Take the largest bilateral gap on earth. China shipped $448.76B to the United States and received $155.07B, a net current of $293.69B and an affinity of 1.0626 nats, an observed export ratio of 2.894 to 1. The fitted potential difference is 1.1191 nats, a ratio of 3.062 to 1, so the pressure field alone would predict a slightly wider gap than the one observed: $309.30B against $293.69B. The circulating part is therefore negative, -$15.61B, and only 5.32% of the gap in absolute terms.
What does not add is the sum of absolute values across edges. sum|J_pot| is $5.2693T and sum|J_circ| is $5.4596T, which together exceed the $7.7605T of total absolute imbalance by 38%. Orthogonality holds in the L-weighted quadratic form, not in the L1 norm. There is no stacked bar on this page for that reason, and the only partition that sums to 100 percent is the entropy production: 42.63% potential, 57.37% circulating.
Mexico and United States hold the largest single circulating current in the world at $110.22B, 43.32% of a $254.45B gap MODELED. Canada and United States sit at 33.21%. China and United States, the gap that draws the policy, is the least circulating of the six at 5.32%: it is very nearly all pressure.
SELECT iso3_a, iso3_b, country_a, country_b, f_ab_kusd, f_ba_kusd,
J_kusd, A_nats, grad_nats, L_logmean_kusd, J_pot_kusd, J_circ_kusd, phi_a, phi_b,
circ_share_logmean, circ_share_geometric, circ_share_arithmetic, circ_share_uniform,
sign(J_kusd) <> sign(J_kusd - J_circ_kusd_logmean) AS flip_lm,
sign(J_kusd) <> sign(J_kusd - J_circ_kusd_geometric) AS flip_geo,
sign(J_kusd) <> sign(J_kusd - J_circ_kusd_arithmetic) AS flip_ari,
sign(J_kusd) <> sign(J_kusd - J_circ_kusd_uniform) AS flip_uni
FROM read_parquet('data/parquet/instruments_trade_currents_pairs.parquet')
WHERE (iso3_a, iso3_b) IN (('CHN','USA'),('MEX','USA'),('CAN','USA'),('USA','VNM'),('AUS','CHN'),('CHN','KOR'))
ORDER BY abs(J_kusd) DESC| Pair (a, b) | Net J | Potential part | Circulating part | phi a | phi b |
|---|---|---|---|---|---|
| Australia and China | $56.50B | -$14.23B | $70.73B | 1.435 | 1.576 |
| Netherlands and United States | -$32.91B | $13.14B | -$46.05B | 0.751 | 0.457 |
| Korea, Rep. and Vietnam | $29.87B | -$4.42B | $34.29B | 1.315 | 1.421 |
| United Arab Emirates and Japan | $28.40B | -$4.30B | $32.70B | 0.934 | 1.158 |
| China and Chinese Taipei | -$28.22B | $3.32B | -$31.54B | 1.576 | 1.540 |
| Switzerland and China | $27.47B | -$19.77B | $47.25B | 0.935 | 1.576 |
| Brazil and China | $27.32B | -$18.31B | $45.64B | 1.358 | 1.576 |
| Belgium and Netherlands | -$25.76B | $6.77B | -$32.53B | 0.845 | 0.751 |
| Italy and Netherlands | -$18.97B | $4.81B | -$23.78B | 0.941 | 0.751 |
| Australia and United States | -$18.76B | $23.07B | -$41.83B | 1.435 | 0.457 |
| Japan and Korea, Rep. | $18.02B | -$5.82B | $23.84B | 1.158 | 1.315 |
| Chile and China | $17.60B | -$7.13B | $24.73B | 1.324 | 1.576 |
3,532 of the 13,004 reciprocal pairs, carrying $1.170T or 15.08% of the world's absolute bilateral imbalance, run the opposite way to their own pressure gradient under every conductance tested.
Australia runs a $56.50B surplus with China, but sits at the lower potential (1.435 against 1.576), so the field predicts a deficit of $14.23B and the entire observed surplus plus that gap, $70.73B, is circulation. China and Korea, Rep. are the sharper case: a near-balanced pair, net -$16.17B, sitting on a -$54.55B circulating current. Almost the entire flow is loop, and the bilateral balance sheet shows nothing.
J - J_circ has the opposite sign to the observed net current J under all four conductances in Figure 2, including the uniform one. This is the only result on the page that is invariant to the free parameter, which is why it is the one the page leads on. It is still a MODELED classification: it depends on phi existing, only not on which conductance produced it. The table shows the twelve largest members by absolute net current.SELECT iso3_a, iso3_b, country_a, country_b, f_ab_kusd, f_ba_kusd,
J_kusd, A_nats, grad_nats, L_logmean_kusd, J_pot_kusd, J_circ_kusd, phi_a, phi_b,
circ_share_logmean, circ_share_geometric, circ_share_arithmetic, circ_share_uniform,
sign(J_kusd) <> sign(J_kusd - J_circ_kusd_logmean) AS flip_lm,
sign(J_kusd) <> sign(J_kusd - J_circ_kusd_geometric) AS flip_geo,
sign(J_kusd) <> sign(J_kusd - J_circ_kusd_arithmetic) AS flip_ari,
sign(J_kusd) <> sign(J_kusd - J_circ_kusd_uniform) AS flip_uni
FROM read_parquet('data/parquet/instruments_trade_currents_pairs.parquet')
WHERE sign(J_kusd) <> sign(J_kusd - J_circ_kusd_logmean)
AND sign(J_kusd) <> sign(J_kusd - J_circ_kusd_geometric)
AND sign(J_kusd) <> sign(J_kusd - J_circ_kusd_arithmetic)
AND sign(J_kusd) <> sign(J_kusd - J_circ_kusd_uniform)
ORDER BY abs(J_kusd) DESC
LIMIT 12This is the part of the world's imbalance that has no country-level counterpart. A reader can trace the largest of these loops by hand: the United States, Mexico, China and Vietnam appear repeatedly with arrows that do not resolve into a single source or sink, because a divergence-free field has neither. Whether the pattern names a production chain is an interpretation, not a measurement, and it is treated as one below.
J_circ, thickness is its absolute value. These 24 edges carry $1.122T of the $5.4596T total circulating value, 20.6% of it, on 0.18 percent of the edges. *BACI/UN reporting code "Other Asia, nes" (S19) refers to Taiwan; this site uses "Chinese Taipei" by default per WTO convention, and "Taiwan*" in tight chart legends.SELECT iso3_a, iso3_b, country_a, country_b, f_ab_kusd, f_ba_kusd,
J_kusd, A_nats, grad_nats, L_logmean_kusd, J_pot_kusd, J_circ_kusd, phi_a, phi_b,
circ_share_logmean, circ_share_geometric, circ_share_arithmetic, circ_share_uniform,
sign(J_kusd) <> sign(J_kusd - J_circ_kusd_logmean) AS flip_lm,
sign(J_kusd) <> sign(J_kusd - J_circ_kusd_geometric) AS flip_geo,
sign(J_kusd) <> sign(J_kusd - J_circ_kusd_arithmetic) AS flip_ari,
sign(J_kusd) <> sign(J_kusd - J_circ_kusd_uniform) AS flip_uni
FROM read_parquet('data/parquet/instruments_trade_currents_pairs.parquet')
ORDER BY abs(J_circ_kusd) DESC
LIMIT 24The 70.35% is a ratio of sums, never a mean of ratios. At pair level the circulating share is not bounded by 1: the median pair is 0.924, the interquartile range is 0.523 to 1.477, the maximum is 2519, and 5,912 of 13,004 pairs (45.46%) exceed 1 because their potential part overshoots the observed current. Countries do it too: 21 of the 136 countries above the display floor exceed 1, led by New Zealand (1.426), Chile (1.411), Brazil (1.305), Guinea (1.238). "The average pair is 70 percent circulation" would be a false sentence and it does not appear here.
A world where every country spread its exports in proportion to partners' import sizes would have no circulation at all, exactly. A world where the direction of each bilateral imbalance were random would be 89.62% circulating. The observed 70.35% MODELEDsits between them and closer to the random end. The whole of the observed circulation is therefore the world's departure from proportional mixing, and it is not a residual of noisy data.
F*_ab = X_a M_b / T, built from the year's own export and import totals and evaluated on the same 13,004 edges. Its circulating share is 2.0e-14, which is zero analytically rather than numerically: ln(F*_ab / F*_ba) is identically a gradient with phi_a = ln(X_a / M_a). That doubles as a correctness check on the solver. The sign-randomisation null keeps |J| and L on every edge and flips each imbalance direction at random, 200 draws, seed 20240101; the observed value sits below its minimum draw of 84.61% (mean 89.62%, sd 0.0181).SELECT "null" AS null_model, draws,
circ_share_absJ_mean, circ_share_absJ_sd, circ_share_absJ_min, circ_share_absJ_max,
pot_share_mean, observed_circ_share_absJ, observed_pot_share
FROM read_parquet('data/parquet/instruments_trade_currents_nulls.parquet')
WHERE year = 2024Absolute bilateral imbalance grew from $1.210T in 1995 to $7.760T in 2024, a factor of 6.4. The circulating share of it did not move with it: it runs between 63.80% (2000) and 74.84% (2013), mean 70.27%, with no trend break at China's WTO accession, the financial crisis or the 2018 tariff round MODELED.
SELECT year, circ_share_absJ, pot_share, sum_abs_J, n_edges, n_nodes
FROM read_parquet('data/parquet/instruments_trade_currents_summary.parquet')
ORDER BY yearThese are the numbers the kill condition is written against, all from the 2024 log-mean run MEASURED. Machine-precision residuals are float noise and will not reproduce digit for digit across runs or platforms; the certificate is the order of magnitude, and the threshold is 1e-9.
Alongside them: the graph is a single connected component (1); no edge has a zero affinity (0 exactly balanced pairs) so L = J / A is defined everywhere; the largest affinity on any edge is 14.589 nats; every one of the 13,004 terms of sigma = sum J A is non-negative, as the second law requires of an entropy production; and E equals sigma at the log-mean conductance to a relative difference of exactly 0. The maximum absolute divergence of the circulating field at any node is $0.0106.
The potential and circulating split is determined entirely by the choice of conductance L, so any headline share is a metric-dependent artefact rather than an invariant. The only thing that is invariant is div J = TB, and that is just the trade balance you started with. The offered defence, that sigma_pot = sum phi times TB, is circular, because phi is itself the L-weighted solution. Pick a different L and you get a different 70 percent.
The objection is right that div J = TB is the only invariant and right that the sigma_pot defence is circular. It is answered by a run, not by an argument: Figure 2, the four-conductance sweep on the 2024 cross-section.
L is fixed by an identity, not by convenience.The quantity being split, Schnakenberg's entropy production sigma = sum J A, is conductance-free. The quadratic form that admits an orthogonal decomposition, E = sum J²/L, is not. They coincide only when L = J / A. Measured, the relative gap |E - sigma| / sigma is exactly 0 at the log-mean, against 8.354e-1 under a geometric-mean conductance, 2.703e-1 under arithmetic mean, and 3.45e+7 under uniform weights. At any other L the additive split of sigma is not an identity, it is an approximation off by tens of percent. That is the principled selection rule the original proposal lacked.
The aggregate magnitude is bracketed, not asserted. The three flow-scaled conductances put the circulating share of absolute imbalance at 65.51% to 74.17%, and that range is what the page quotes. Uniform conductance returns 150.68%, which is not a share; L1 norms are not orthogonal, so a decomposition that ignores the size of the trade relationship can produce parts larger than the whole. That is a reason to reject uniform weighting, and it is reported rather than hidden.
One casualty, disclosed. Pair-level shares are stable across the three flow-scaled conductances, CAN-USA at 33.2%, 32.1% and 35.2%, MEX-USA at 43.3%, 44.3% and 41.8%. They are not stable across all four: CHN-USA runs 5.3%, 7.2% and 2.5% across the flow-scaled three but 95.6% under uniform weights. So "the China deficit is 95 percent pressure" is a statement about flow-scaled conductances and it is published with that qualifier attached, never bare.
What survives the objection completely. The sign class in Figure 5: 3,532 of 13,004 pairs, 15.08% of world absolute imbalance, where the fitted potential predicts the opposite sign to the observed net flow under every conductance including the uniform one. AUS-CHN and CHN-KOR are in it under all four. That is a qualitative claim about the trade record that does not move with the metric, and it is the claim the page leads on.
Helmholtz-Hodge decomposition of a directed economic network into gradient and circular flow is established. Kichikawa, Iyetomi, Iino and Inoue applied it to a five-million-link Japanese interfirm transaction network (Applied Network Science 4, 92, 2019), and Sada and Ikeda applied it to the international value-added network (PLoS ONE 16(8), e0255698, 2021). The decomposition is theirs and is not claimed here.
What this page adds is the thermodynamic layer that removes the free parameter, following Schnakenberg's network entropy production (Reviews of Modern Physics 48, 571, 1976) and Onsager's reciprocal relations (Physical Review 37, 405, 1931), on the full gross-trade record rather than a value-added table; the published sweep over the parameter it removes; and the reading of the divergence-free part as the share of a bilateral deficit that a bilateral instrument cannot address. If a paper already pairs the Onsager conductance selection with the bilateral-policy reading, this page is retired under the prior-art clause of the contract.