Formula developed by long-dead British polymath shows one side nearing defeat in Ukraine

The iron law of warfighting mathematics shows the Ukraine war is only a few months away from a tipping point beyond which the losses of one side will snowball, leading inexorably to defeat.

The Ukraine War has now passed its 50th month — over four years in old money. Who’s winning? Both the mainstream and alternative media obsess over this question. Whether in genteel opinion columns, febrile social media arguments, or the intellectual foreign policy press, endless words are written to explain to us that one side or the other is leading the race to victory.

We monitoring-the-situation bros read these opinions every day. Most feel like propaganda, or else nakedly biased and laced with emotion and personal investment. Some, we must concede, are interesting — and a few are even useful. But can we do better? Yes. Yes we can. To understand how, we must travel over a century back in time to the great abattoir of iron and human flesh that was the First World War. We must also learn a bit of maths (but don’t worry, it’s easy to grasp).

Before 1914, wars were characterised by manoeuvre and the decisive battle. Skilled generals would wage fast, fluid campaigns and chase a single crushing victory. By the eve of the First World War, this concept had developed into the so-called Cult of the Offensive. The military planners of the European Great Powers believed that wars would be fought through bold attacks, rapid movement, and sheer élan — and thus decided quickly, in a single summer campaign. The study of Napoleon and Moltke was de rigeur.

As we all now know, the machine gun, barbed wire and the trench changed all that. They made manoeuvre impossible and bold attacks suicidal. For year after bloody year, the generals struggled to escape this grand charnel house. The solution was finally demonstrated between 8 and 11 August 1918 at the Battle of Amiens: combined arms warfare.

At Amiens, the British, under General Sir Henry Rawlinson, deployed military innovations, such as the creeping artillery barrage, tanks and aircraft, and perfectly coordinated them against a single point to punch a hole in the front, through which followup forces could flood. On the first day alone, they advanced 7-8 miles (more than 11km) — a shocking distance on the Western Front — captured 29,000 prisoners and 338 guns, and liberated dozens of villages. German resistance collapsed in places; whole units surrendered en masse. The German commander, Erich Ludendorff, famously called 8 August “the black day of the German Army.”

British Mark V tank in action at Lamotte-en-Santerre, 8 August 1918

Manoeuvre had returned to warfighting. And it stayed. Combined arms warfare reached its zenith in the theory of Blitzkrieg (or the Soviet version, ‘Deep Operation’) and became the template of warfare through the 20th Century and into the 21st. Certainly no great power wished a return to the slaughterhouse of 1914-18. Whether in the Ardennes in 1940, the breakout from Stalingrad in 1942, the Manchuria Campaign in 1945, or the Gulf War in 1991, the formula would be something like that at Amiens. Forces would be concentrate. A mass of tanks would combine with artillery and air power to punch through the front lines. Mobile infantry would follow, coordinate with, and support, the armour to exploit the gap. Manoeuvre deep in the enemy rear — in a fluid, rapidly developing battle — would decide the war.

In Ukraine, technology has now stymied manoeuvre again. Modern intelligence, reconnaissance and surveillance equipment has made the battlefield transparent. This makes tactical (or even strategic) surprise almost impossible. The democratisation and diffusion of precision strike, either through missiles or various types of drones, has transformed the concentration of forces from a necessary prerequisite for battle into a deadly error. Much as trenches, barbed wire and the machine gun rendered fronts immovable, so these new technological developments have turned Ukraine into a grinding and bloody battle of position.

Modern ISR has made the battlefield in Ukraine transparent to both sides

Yet while the First World War’s state of attrition was defined by two trench networks facing each other across a no-man’s land that might be as little as a hundred yards deep, such concentration is impossible in Ukraine. Instead, the front line is characterised by interlocking strong points and captured tree lines, and might extend for as many as 25 miles (or more than 40km) deep.

Different as the physical structure may be, the zone of interlocking positions is as much of a killing field and as difficult to penetrate as the no-man’s land of the First World War. The Ukraine War, then, is, like the First World War, a war of attrition. Such wars are characterised not by attempts to capture capital cities or key logistics nodes, or by decisive battles. Instead, each side seeks to pay in smashed machinery and broken men to drain the other’s combat effectiveness. And a revolution in our understanding and assessment of warfare developed during the attrition of First World War can help us predict the outcome of this one.

Frederick Lanchester was born in London in 1868. He did not distinguish himself at school, despite the gift of a private education. He more than made up for this in later life, developing Britain’s first four-wheeled motorcar, several innovations in internal combustion engines, an epicyclic gearbox, cantilever suspension, worm-drive transmission, disc brakes, wire wheels, pressure-fed lubrication, and even early ideas toward power steering and four-wheel drive. Mr Lanchester was also a pioneer in aviation, developing a circulation theory of lift and a vortex theory explaining how wings generate lift and drag.

Yet he is perhaps best known for the mathematical laws he developed in 1916 to determine the relative strengths of armies. Mr Lanchester’s crucial insight was related to the nature of modern, mechanised warfare. He realised that while in pre-industrial warfare a single soldier could only engage one enemy combatant at a time (with his sword, spear or bow), a modern soldier (with a machine gun or aeroplane) could engage many.

Mr Lanchester was the first to understand that this meant, in pre-modern war, the relation between the size of an army and the combat power of that army followed a linear law. In other words, if you doubled the army, you got twice the combat strength: 3+3=6.

In modern combat, however, a superior force would not only would kill more of the enemy, but would lose fewer men than the enemy. Over time, this process would multiply the initial superiority, making the kills and losses even more lopsided, further accentuating the superiority. The increase in combat power would therefore have a multiplying effect (kills and losses), not a linear effect. In turn, this meant that increasing the size of an army would follow a square law: 3x3=9.

Mr Lanchester developed mathematical equations to show how this square law applied to war. Modified versions are still used by military operational planners today. They are especially useful for applying to attritional wars, because, as aforementioned, in attritional wars, the degradation of combat power is the aim in and of itself, rather than victory in a decisive battle or the capture of land or specific key points, like cities. Therefore, as one side loses power, the terrible logic of the square law takes hold: it doesn’t lose a linear amount of relative power, but the square root.

Graphs, per Lanchester’s Law, which show the outcomes of battles between differing sized forces, with depletion rates on the vertical axis and time on the horizontal.

Dr Warwick Powell is Adjunct Professor at Queensland University. He usually works on issues related to China, digital technologies, supply chains, financial flows and global political economy and governance. He has an excellent Substack that covers many of these topics in an accessible, interesting way. Recently, however, he has turned his econometrics and financial expertise to the Ukraine War. What he has found, using a modified version of Lanchester’s Law to create a power model of the war, is that one side is rapidly approaching a tipping point beyond which its losses will start snowballing — where the ruthless logic of the square law will start taking hold and begin a precipitous fall toward ultimate defeat. With his permission, we share these findings with you today¹.

The inputs to the model.

Dr Powell first creates a unit of combat effectiveness. To do this, he aggregates “lethal units” for comparability. Each personnel member counts as 1 lethal unit. Armoured vehicles count as as 10 units (reflecting firepower). Munitions as 0.01 unit per shell (approximating strike equivalence). This process — called ‘normalisation’ — allows Dr Powell to model the war as a system of differential equations, where Ukraine’s force U(t) changes as net replenishment minus losses proportional to Russia’s force R(t), and vice versa.

Dr Powell takes the data needed to count these ‘lethal units’ from a diverse range of open sources as of February 2026. The ranges of discrete inputs to his model reflect the divergences between different sources: Western estimates (e.g., the Ukrainian General Staff, CSIS or the Kiel Institute) tend to report lower losses for Ukraine and higher levels of expected international aid (and thus expected replenishment). Pro-Russian sources (the Russian Ministry of Defence, Rybar, and pro-Russian ‘mil-blogs’) report, of course, higher levels of damage inflicted on the Ukrainian military.

The aggregates Dr Powell builds from these sources focus on theatre (frontline) capacities: while global stocks are of course larger, they are delivery-constrained. Here, those readers less interested in the minutiae and mathematics of the model can skip to the section, below, headlined “Caveats”. Those who are interested, though, the numbers are as follows.

Manpower (Theatre Active Strength):

• Ukraine: 450,000-550,000 (Western: ~500,000 frontline, rotations strained; Russian: ~450,000 effective, implying higher cumulative depletion). Cumulative losses since 2022: 400,000-1,200,000 (Western ~500,000; Russian ~1.2M).

• Russia: 600,000-700,000 (consistent across sources; total committed ~1.2M). Cumulative: 800,000-1,200,000 (Western ~1.16M; Russian lower, ~600,000).

Daily casualties: Ukraine 500-1,800 personnel (Western 500-700; Russian 1,200-1,800); Russia 900-1,200 (Western higher; Russian ~1,000).

Recruitment: Ukraine 5,000-10,000/month (net ~0 due to losses); Russia 25,000-30,000/month (net +700-900/day).

Machinery (Operational Tanks/AFV/Artillery):

• Ukraine: 2,000-2,500 (losses ~10,000 cumulative; aid ~2,000 delivered). Refurbishment: 50-100/month (net negative from attrition).

• Russia: 7,000-8,500 (losses ~30,000-35,000 cumulative, but ~1,000/month refurbished from stocks).

Daily equipment losses: Ukraine 20-50 (Western lower); Russia 30-60.

Munitions (Artillery Shells, Missiles):

• Ukraine: Stock 500,000-1,000,000 (low; daily use 2,000-3,000). Production/Delivery: 20,000-40,000/month (US/EU ramp to 100,000 delayed; PURL adds ~50,000/year but queued). Air defence: Patriot stocks ~200-300 interceptors (covers 20-30 salvos vs. 450 threats/month); production diversion via PURL delays replenishment.

• Russia: Stock 4M-6M; production 4M-5M/year (~12,000-15,000/day). Fire ratio: 5:1-10:1 advantage.

Recent constraints: US inventories at 25% (1,000-1,500 Patriots total; production 740/year, 75% to Ukraine via Europe). Germany: €20B+ exhausted, no more direct transfers; contingent ~35 PAC-3 (~1 week’s defence).

Net Replenishment and Effectiveness:

• Ukraine r_U: -100 to +100 units/day (pre-February; now -100 to 0 from aid cuts). β (Russian effectiveness): 0.0012-0.0025 (up 10% from air gaps).

• Russia r_R: +700-900/day. α (Ukrainian effectiveness): 0.0018-0.0020.

How the model works

Dr Powell then puts his lethal unit numbers, and depletion and replenishment rates, into a modified Lanchester model to simulate combat dynamics. At its heart, his model tracks two variables: each side’s effective force level over time, influenced by replenishment rates and the opponent’s inflicted losses.

The model assumes linear attrition until a threshold, then introduces the non-linearity of the square law. The point at which this occurs is based on operational research for the scale of losses after which force integrity starts to yield non-linear results, 73%. This, in turn, leads to collapse after 50%.

Replenishment (r) includes recruitment, repairs and aid inflows minus decay (e.g., equipment wear). Effectiveness coefficients (α for Ukraine’s impact on Russia, β vice versa) embed doctrinal factors: Russia’s mass artillery yields higher β (0.0012-0.0025 losses per Russian unit-day), while Ukraine’s precision strikes give α around 0.0018-0.0020. Initial conditions are set from theater estimates (U_0 ≈ 550,000 in November 2025, adjusted downward by observed attrition).

Numerical integration — discretising time in daily steps — projects forward:

U_{t+1} = U_t + r_U - β R_t, iterated until U hits θ U_0.
θ ≈ 0.73, based on historical density for coherent defence.

Post-threshold, a multiplier γ (2.5) amplifies losses, simulating breakthroughs. This yields time to tipping (t*) and full collapse (to 50% force, proxy for operational failure).

Caveats

Dr Powell concedes that caveats abound, and that these mean his model offers an estimation, not a firm prediction. For instance, there are biases in his data: Western sources might, for instance, underreport Ukrainian losses (500-700/day) so that Western governments can sustain domestic political support for the rather significant costs of the current policy (often underplayed and excluding the substantial economic costs of maintaining sanctions). On the other hand, Russian claims (1,200-1,800) might be inflated for propaganda purposes. Dr Powell believes that the reality is likely somewhere in the middle; however, readers should be aware that the fog of war exists here.

He is also careful to note that other variables are not modelled. Examples of such unmodelled variables include morale (potentially accelerating or holding off collapse), weather (winter slows the cadences of the war, so a longer or harsher winter has an effect), or black swans such as drone surges or some kind of successful third party mediation or intervention. Aid is also volatile and difficult to predict. For instance, Europe’s Prioritised Ukraine Requirements List programme (best known in policy circles as PURL) could ramp up substantially. If it did so, it would change the outcome significantly. Yet a complete end of all international aid would do likewise in the other direction. Points between these extremes would obviously have their own, lesser effects.

Furthermore, Dr Powell notes that his model assumes constant parameters. Yet adaptations (e.g., Ukrainian drones flipping local α) could localise deviations. Threshold θ is empirical, drawn from past phases (e.g., Pokrovsk encirclements at ~70% density), but varies by terrain. Finally, aggregation into lethal units simplifies where reality can somewhat defy such reduction. For example, a tank’s value is not in reality fixed at ‘10’ to an infantryman’s ‘1’, and munition efficacy depends on targeting.

These limitations mean that trajectories are probabilistic ranges, not fixed paths. The value of the model lies in its sensitivity: by tweaking r_U by +100 units/day (e.g., via Japanese Patriot backfill of the Ukrainain anti-air magazine) delays t* by 30-60 days. This shows how data parameters inform adjustments to war cadences.

All these caveats boil down to the fact the model enables a reasoned estimation, revealing the war’s underlying arithmetic, but does not claim omniscience.

Conclusion

Dr Powell’s model suggests that the war is rapidly nearing a tipping point. Ukraine is approaching the fulcrum beyond which the weight of the war tips its forces into rapid depletion, allowing the Russian military to start achieving much more significant territorial gains. In turn, this would lead to accelerated Ukrainian losses through the ruthless reality of the square law. This road heads toward an eventual Ukrainian collapse.

Ukraine’s effective combat power (a composite of manpower, machinery and munitions) is depleting, the model shows, at a net rate that outpaces its replenishment, while Russia’s holds steady or grows marginally. This imbalance, compounded by recent reductions in Western support, suggests a tipping point where Ukrainian force density thins below viability, triggering rapid territorial losses and operational collapse.

Based on his integrated projections from an updated model (as of 15 May), Dr Powell estimates window for this tipping point is 3-6 months from now (July-September 2026), followed by a 3-4 month cascade to functional exhaustion. Overall, this yields a 6-9 month horizon to “floodgates opening,” where advances accelerate from the current 0.3-1 km/day to 5-10 km/day, as seen in historical breakthroughs like the 2022 Kherson retreat. This non-linear result is the manifestation of the pitiless square law which Frederick Lanchester first codified into a usable model for military operations in 1916. But it boils down to this: Dr Powell’s model estimates Ukrainian collapse by early Spring next year (nine months from mid-May 2026) at the outside.

Ukrainian infantry fighting vehicles, smashed on the battlefield, 2023

Two factors support Dr Powell’s conclusion. The first is that Anusar Farooqui, the hedge fund CEO and international relations scholar, and friend of Multipolarity, has managed to replicate Dr Powell’s results using the model. The second is that Peter Turchin, emeritus professor at the University of Connecticut in the departments of ecology and evolutionary biology and mathematics, has produced his own model, which reaches a similar conclusion: Ukraine is heading toward a tipping point beyond which its position becomes irretrievable.

A key understanding here is that it matters not who is attacking and who is defending; who is capturing territory and who is losing it. The losses of men, munitions and weapons platforms carry on regardless, and in an attrition war, it is exactly that which counts. Absent a black swan event, Ukraine continues heading toward the tipping point whether it is successfully counter-attacking or doggedly defending.

It must be noted that the tipping point Dr Powell’s model suggests would not be the end of the war. It would simply be the point at which the war of attrition finally makes Ukraine’s conventional military position irretrievable, with Russian territorial gains accelerating from that moment. That escalation phase itself would last months, per Dr Powell — and potentially even longer.

There are also many examples in history in which a military has passed this point and fought on for a lengthy period. In the American Civil War, the Confederacy passed the tipping point after the battles of Gettysburg and Vicksburg in summer 1863, but it was not until the Spring of 1864 that defeat was finally conceded. Most estimates put the number of Confederate KIAs at north of 100,000 after Vicksburg. The Nazis on the Eastern Front in World War II passed the tipping point after the Battle of Kursk in 1943, yet were not finally defeated for another two years, despite the Eastern Front turning into a series of rolling catastrophes for the Wehrmacht and Waffen-SS. Insurgency or terrorist war could also be fought long after a conventional military defeat, especially with European support.

Yet Dr Powell has reached what should be a stunning conclusion for western policymakers. Absent a black swan event, or a vastly underestimated influence of drones, Ukraine is fast approaching the point of no-return, perhaps as early as this summer, and no later than the beginning of winter of this year. After that, Dr Powell estimates at most another four months to the point of collapse. This would mean substantially accelerated territorial gains for Russia and ultimate victory.

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Finally, Multipolarity wishes to thank Dr Powell for his kind permission to use the results of his model in this essay. We encourage readers to subscribe to his brilliant Substack, here.

1. Much that appears hereafter is taken directly from Dr Warwick Powell’s original and supplementary papers on this research project. It has not been put in quotation marks, and some minor changes have been made for reasons of clarity and order, but everything that appears hereafter must be considered Dr Powell’s work and property, not Multipolarity’s. The original paper can be viewed here. The update can be viewed here.