The Mathematics of War: How to Tell Real Losses From Propaganda

In a conflict of this kind, every casualty figure becomes a weapon. Both Russia and Ukraine classify information about their own losses while actively publishing data on the enemy’s losses. In this “fog of war,” where every number is an instrument of an information operation, the search for truth requires scientific methods.

This article offers an analysis for a broad readership: who is lying, who is exaggerating, and where an objective picture can be found. We will examine three questions in turn: whether Russian military briefings can be trusted, what the real price of the war is for Russia, and what we know about Ukraine’s losses.

Part 1. The psychology of numbers and Benford’s law

To test the reliability of quantitative indicators, one of the most effective tools is Benford’s law, also known as the law of the first significant digit. It is a mathematical regularity that describes the probability that a particular digit will appear in the first position of numbers drawn from real, naturally generated (“organic”) datasets that reflect complex cumulative processes.

Contrary to the intuitive assumption that all nine digits (from 1 to 9) should appear as the first digit of a random number with equal probability (about 11.1%), Frank Benford found that in large-scale datasets digits are distributed according to a logarithmic law. Mathematically, the probability P(d) that the first significant digit of a randomly selected number is d (where d ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9}) is given by the formula:

Accordingly, the theoretical distribution of first digits in an organic dataset is as follows: the digit 1 appears in the first position in 30.1% of cases; 2—17.6%; 3—12.5%; 4—9.7%; 5—7.9%; 6—6.7%; 7—5.8%; 8—5.1%; and 9—only 4.6% of cases.

This law is a widely recognized tool in global forensic accounting, auditing, and macroeconomic analysis for detecting signs of falsification in corporate financial statements, election protocols, tax declarations, and epidemiological statistics.

The psychological basis for the effectiveness of Benford’s law is simple: when a person (or group of people) attempts to fabricate statistical data or invent “random” numbers, they are guided by a false notion of randomness. They subconsciously try to distribute digits evenly, often choosing numbers that begin with 4, 5, 6, or 7, while avoiding excessive use of ones and twos, intuitively perceiving them as “too obvious” or insufficiently “convincing.”

A mathematical deconstruction of propaganda: analyzing Russian Ministry of Defense briefings on Ukraine’s losses

From the first days of the full-scale invasion, Russia’s political and military leadership adopted a strategy of total dominance in its domestic information space. The Russian Ministry of Defense publishes daily, extraordinarily detailed reports on the destruction of Ukrainian Armed Forces personnel and equipment. These briefings—long associated with Lieutenant General Igor Konashenkov and later transformed into daily written summaries—are intended to create within Russian society a durable illusion of an inevitable and rapid victory, as well as to convince the international community of the futility of further support for Kyiv.

The daily briefings of Russia’s Ministry of Defense contain figures, precise to the individual, for Ukrainian service members reported “destroyed.” Analysis of these data for the period from February 2022 through February 2026 makes it possible to form a representative control sample. For example, official Ministry of Defense statements feature typical figures such as: “up to 1,075 Ukrainian service members destroyed” in one day on a single axis; total Ukrainian Armed Forces losses for January 19, 2026 are claimed at “1,240 soldiers”; the “Center” group of forces reports the destruction of “480 personnel” or “295 personnel,” the “South” group—“305 personnel” or “145 personnel,” the “East” group—“up to 960 service members,” and the “West” group—“more than 150 personnel” per day. Aggregate daily enemy loss figures are regularly reported at levels such as 1,495, 1,245, or 1,070 personnel. Overall, as of late 2024 through early 2026, Russia’s Ministry of Defense estimates Ukrainian army losses at more than 1 million killed and wounded.

When applying Pearson’s chi-square (x²) statistical test, which makes it possible to determine the statistical significance of deviations between empirical data and a theoretical model, the distribution of first digits in the reported figures of “Ukrainian Armed Forces service members destroyed” shows a significant deviation from Benford’s law.

The statistical test shows that the deviation of the observed distribution of first digits from the theoretical one is nonrandom (p – value < 0.001). It can be assumed that Russian staff officers and spokespersons responsible for compiling these reports subconsciously avoid “overly round” or “overly small” leading digits in an effort to create an illusion of extraordinary precision, operational activity, and combat intensity. Numbers such as “480,” “305,” “295,” or “1,075” are generated either algorithmically or manually to match target indicators for enemy losses set by the political leadership.

A statistically significant deviation in the frequency of digits 1, 3, 4, and 7 from Benford’s law can be regarded as an indicator of falsification, turning statistics into an instrument of reflexive control aimed at a domestic audience to sustain the myth of the relentless success of the so-called “special military operation.” As shown in Part 3 below, Russia’s leadership significantly overstates Ukraine’s losses.

But there is a caveat. Benford’s law is a powerful tool, but it has limitations. For Benford’s law to work properly, the data must span several orders of magnitude (for example, from 10 to 10,000). In the case of Russia’s briefings on Ukrainian Armed Forces losses, daily figures vary within a narrow range and are often presented with “up to” or “more than” qualifiers, which makes them interval rather than exact values. Under such conditions, conclusions based on Benford’s law are better understood as risk indicators rather than proof of falsification. In political statistics, this is described as the forensic pitfall: Deckert, Myagkov, and Ordeshook (2011) show that Benford’s law, as a detection tool, should be applied cautiously—as a red flag, not proof of guilt.

Bureaucratic schizophrenia: verifying Russia’s real losses through its own civilian administrative records

Because Russia’s information space operates under conditions of strict autocratic censorship, an objective analysis of Russian Armed Forces losses requires a different approach. Russia’s Ministry of Defense ceased regular publication of official data on its own military losses in September 2022 (when it reported a figure of 5,937 killed), on the eve of announcing the so-called “partial mobilization.” Since then, the ministry’s official communication regarding its own losses has been reduced to zero. Moreover, Russian legislation has criminalized any attempts to independently count losses within the country, classifying such efforts as “discrediting” the armed forces or “spreading false information,” offenses punishable by multi-year prison terms.

However, the Kremlin’s authoritarian system faces a phenomenon that can be described as “bureaucratic schizophrenia.” The state can classify operational General Staff briefings, but it cannot halt the functioning of a vast civilian bureaucratic apparatus without triggering a total collapse of the social system. To stimulate recruitment, Russia’s government has undertaken to pay relatives of the fallen financial compensation on a colossal scale by the standards of Russia’s provinces (“coffin payments,” insurance payouts, and loan write-offs). Receiving these funds requires strict compliance with legal procedures: obtaining a death certificate (more than 300,000 forms of which have been ordered since 2022), opening an inheritance case with a notary, and filing documents with the Social Fund of Russia (SFR). Thus, each death of a Russian soldier inevitably leaves a digital and bureaucratic trace in state databases.

To ensure that Russia’s authorities were not falsifying these civilian data as well, journalists and researchers aggregated statistics from the National Register of Inheritance Cases of the Federal Notary Chamber of Russia (the number of cases opened for men of conscription age by municipality and region) and official demographic reports from the Federal State Statistics Service (Rosstat) on excess mortality. This dataset was also subjected to verification using Benford’s law.

Unlike Russia’s Ministry of Defense briefings, civilian data are not generated centrally in the offices of military propagandists but are the result of decentralized actions by thousands of independent actors across Russia: notaries, civil registry officials, and social welfare employees. When the first significant digit was extracted from data on opened inheritance cases and excess mortality, the empirical distribution showed an extremely high correlation with Benford’s theoretical logarithmic curve.

The correlation coefficient (R²) between the empirical and theoretical distributions exceeds 0.95. This indicates that the statistical probability of manipulation of these administrative data approaches zero. The conformity of data generated by Russia’s civilian bureaucratic apparatus to the mathematical laws of random variables demonstrates that these figures are an objective reflection of Russia’s social and demographic reality. Therefore, analysis based on these sources can be considered reliable.

Part 2. Arithmetic absurdity: how Russia fabricates statistics

The fabricated nature of Russia’s military briefings on Ukrainian losses is demonstrated not only by statistical analysis but also by violations of the fundamental laws of arithmetic in the calculation of cumulative losses of Ukrainian military equipment.

As of February 15, 2026, according to Russia’s Ministry of Defense, Russian forces have destroyed a colossal quantity of Ukrainian weaponry, including more than 670 military aircraft, 283 helicopters, more than 114,000 unmanned aerial vehicles, 650 surface-to-air missile (SAM) systems, and 27,704 tanks and armored fighting vehicles.

To understand the absurdity of these figures, it is sufficient to consult international databases of military capabilities. According to the World Air Forces 2022 reference, before the full-scale invasion the Ukrainian Air Force had only 318 aircraft in total, including Soviet-era platforms. Of these, 98 were combat aircraft and 112 were helicopters. Even if one adds to this list all aircraft and helicopters transferred to Ukraine by partner countries since 2022 (OSINT analysts at Oryx estimate this at roughly 150+ units), the total size of Ukraine’s aviation has never even approached 953 units. Russia’s Ministry of Defense has thus officially “destroyed” Ukraine’s air force at least three times over.

A similar situation applies to armored vehicles. When the Armed Forces of Ukraine had a total of about 8,165 tanks and armored fighting vehicles (including Soviet-era equipment and the first tranches of military aid), Russia’s army had already officially reported (on July 12, 2023) the “destruction” of 9,915 such vehicles. The claim of 27,704 destroyed armored vehicles as of February 15, 2026 is a mathematical impossibility, as this figure substantially exceeds the entire available fleet of tanks and armored fighting vehicles not only of Ukraine but also of the combined forces of many European NATO member states prior to their rearmament.

As for personnel, as early as mid-2022—when the war had lasted only a few months—the cumulative figures of “destroyed” Ukrainian service members in Russia’s Ministry of Defense briefings had reached levels that would have implied the loss of combat effectiveness of 80% of the entire Ukrainian Armed Forces. The fact that Ukraine’s army in 2026 continues to conduct large-scale defensive operations, inflicting catastrophic losses on the enemy and even holding footholds on Russia’s territory (Kursk region), demonstrates that Russia’s Ministry of Defense data bear no relation to battlefield reality.

The scale of confirmed Russian losses based on internal data

Based on the mathematically demonstrated reliability of Russia’s registries, the projects Mediazona and BBC Russian Service, with support from Meduza, systematically aggregate Russia’s irrecoverable losses.

Their methodology consists of two components: a name-by-name list of the dead and a statistical estimate based on the Register of Inheritance Cases.

By monitoring open sources—statements by regional governors, obituaries in the local press, photographs from cemeteries, as well as analysis of database leaks (such as lists from the Wagner Group and leaks from motor rifle brigades)—as of February 13, 2026, researchers had verified the names of 177,433 dead Russian soldiers. In just the first two weeks of February 2026, more than 9,000 names were added to this list through the processing of accumulated archival data. Among those identified by name are 6,414 officers, including 12 generals of Russia’s army. The largest absolute losses have been recorded among natives of Bashkortostan, Tatarstan, Sverdlovsk region, and Krasnodar Krai.

Because not every death is reported in newspapers or announced by governors, the name-by-name list represents only a lower bound of total losses. To establish a more accurate picture, researchers used data on the excess number of inheritance cases opened for men under the age of 50 compared with the prewar period. Taking into account the proportion of men who leave no property, this mathematical model showed that as of August 2025 the total number of dead Russians was about 219,000, and by preliminary estimates, by the end of 2025 this figure had reached approximately 300,000 irrecoverable losses. This estimate correlates with Rosstat reports, which recorded cumulative excess mortality among Russian men aged 20–54 at 138,500 in just the first two years of the full-scale war (2022–2023).

In addition, beginning in mid-2024, a mass wave of court petitions filed by Russia’s Ministry of Defense military units seeking to have missing soldiers declared dead has been recorded (around 90,000 such cases by the end of 2025). This allows the army to avoid evacuating bodies from the battlefield while formally completing bureaucratic procedures, which further increases the total number of deaths.

Intelligence consensus: everyone sees the same thing

Official Ukrainian data (more than 1.25 million), which Russia calls an “invention of the Kyiv regime,” in fact underpin the consensus estimates of leading think tanks. For example, the Washington-based Center for Strategic and International Studies (CSIS), in a report published in early 2026, estimates Russia’s total losses (killed, wounded, and missing in action) at 1.2 million. Of that number, irrecoverable losses (killed) amount to up to 325,000 soldiers.

Moreover, this figure follows logically from mathematical calculations based on Russia’s civilian registries (219,000–300,000 confirmed deaths). Given the standard ratio of killed to wounded in modern conflicts (approximately 1:3 to 1:4), 300,000 irrecoverable losses imply total losses of more than 1 million. Thus, Ukrainian and international estimates of Russian losses are close to accurate.

The Dutch Oryx project, which uses the most conservative and rigorous methodology (recording only equipment losses with clear photo or video confirmation and geolocation), has documented Russia’s loss of 24,099 units of heavy military equipment. Of these, 13,887 armored vehicles were destroyed, damaged, or captured, including 4,334 tanks of various types (from older T-54/55 models to the most modern T-90M).

Ukrainian Armed Forces estimates of Russian equipment losses (~11,672 tanks as of February 15, 2026) are higher than Oryx’s visually confirmed figures (4,300 tanks), because OSINT captures only the lower bound of losses, as many vehicles destroyed in rear areas or in tree lines are never recorded on video, especially given the scale of FPV drone use and long-range artillery.

Part 3. What we know about Ukraine’s losses

The question of Ukrainian losses is the most sensitive aspect of this analysis. Kyiv long adhered to a policy of information silence, which is understandable: in an existential war for survival, any information about its own losses can be used by the enemy to plan military and information operations. However, estimates by international institutions, official statements, and foreign intelligence services now largely converge.

In February 2026, Volodymyr Zelenskyy said in an interview with France 2 that 55,000 Ukrainian service members had been killed on the battlefield. He added that a “large number” of service members are still considered missing in action (according to estimates by the Coordination Headquarters for the Treatment of Prisoners of War, the number of missing Ukrainians ranges from 35,000 to 84,500). Earlier, at the end of 2024, he cited a figure of 43,000 killed and 370,000 wounded.

In January 2026, the Center for Strategic and International Studies (CSIS) published an estimate according to which Ukraine’s total military losses (killed, wounded, and missing in action) amount to 500,000–600,000. Within that number, irrecoverable losses (killed) are estimated at 100,000–140,000 (assuming a killed-to-wounded ratio of 1:4). This estimate correlates with counts by the independent UALosses project, which maintains a name-by-name list of Ukrainian losses based on public obituaries: as of early 2026, it had confirmed the names of more than 92,000 dead Ukrainian service members. The Economist, in summer 2025, also estimated the number of dead Ukrainians at 73,000–140,000.

Regardless of which specific figure is considered most accurate, the main conclusion remains unchanged: even by the highest estimates, Ukrainian Armed Forces losses are at least 2–2.5 times lower than Russia’s losses (600,000 versus 1.2 million). This indicates significantly higher effectiveness of Ukraine’s Defense Forces.

Part 4. Nonobvious consequences: what lies behind the numbers

The erosion of Lanchester’s square law and the “drone war” paradox

Classical military theory (including Lanchester’s square law) holds that in combined-arms combat, an army’s combat power is proportional to the square of its manpower, provided there is a concentration of massed firepower and a mechanized breakthrough. However, in modern warfare this law no longer holds: the widespread proliferation of FPV drones, aerial reconnaissance systems, and Starlink integration has led to a dramatic degradation of Russia’s ground tactics, negated its numerical advantage, and made any concentration of equipment lethal. West Point (the U.S. Military Academy) analysts have described this as a “return of the tactical crisis”—a situation in which firepower once again exceeds the capacity for maneuver, as it did in the late nineteenth century.

It has been visually confirmed that Russia has lost more than 13,800 armored vehicles (more than its entire prewar active fleet) and has been forced to abandon massed mechanized assaults and shift to “meat assaults”: continuous attacks by small infantry groups of 2–5 soldiers. Tanks have been replaced by golf carts, ATVs, motorcycles, and even horses. The sole objective is to cross the “gray zone” under drone strikes at any cost. This “adaptation” has a shocking and nonobvious consequence: for Russia, the war is becoming far cheaper in material and financial terms, but far more costly in human lives.

The price of territory

Tactical degradation to the level of a frontal war of attrition reminiscent of World War I has produced extreme statistics on the “cost” of territory. According to analysts, in 2024–2025 Russia’s army lost an average of 102 soldiers killed and wounded for every square kilometer of Ukrainian territory captured. In 2025, this figure averaged 78 soldiers, although in some of the most intense months of fighting (in particular from January to April 2025) it reached 99 per square kilometer.

Thus, innovations by Ukraine’s Defense Forces in unmanned systems have effectively “broken” Lanchester’s law for Russia’s army, turning it into a slow-moving force capable of advancing (for example, capturing less than 1% of Ukraine’s territory in 2025—about 0.93%) only at the cost of enormous losses.

“Deathonomics” and the Kremlin’s social engineering

A question that often concerns Western analysts is why astronomical human losses (more than 1 million killed and wounded) have not led to a mass social explosion or antiwar protests in Russia. The answer lies not only in the effectiveness of the FSB’s repressive apparatus but also in the Kremlin’s creation of a unique macroeconomic model. Researcher Vladislav Inozemtsev, in a report for the French think tank IFRI (February 2026), described this phenomenon as “Deathonomics”—an “economy of death.”

Russia’s political leadership has deliberately transformed the war into an unprecedented mechanism for the redistribution of national wealth. Geographic and demographic analysis of losses confirmed through obituaries (including Mediazona data) reveals a cynical policy of social engineering. Russia’s “blood tax” is paid by the poorest, economically depressed regions—ethnic republics of Siberia and the Far East (the Republic of Tuva, Buryatia, and Chukotka), where the risk of death for local men is 25 times higher than for residents of Moscow or St. Petersburg. Bashkortostan and Tatarstan also suffer enormous absolute losses. Putin’s regime is effectively buying the population’s loyalty by paying astronomical sums for the deaths of their sons and husbands, which has produced a paradoxical consumption “boom” in Russia’s villages and small towns. Death has thus become the most profitable business.

But this model is a financial pyramid scheme. The loss of hundreds of thousands of working-age men has created an acute labor shortage. The defense industry operates around the clock in three shifts, producing equipment that is subsequently destroyed in Ukraine. Trillions of unbacked rubles are fueling an inflationary spiral. Even according to official data, GDP growth has fallen to a meager 1% (compared to 4.3% in 2024), forcing the government to raise taxes and drastically cut funding for civilian sectors such as education and healthcare. Although oil revenues and China’s support, including through sanctions evasion, allow the Kremlin to continue financing the war, they cannot conceal a deep structural internal deficit.

Another nonobvious consequence of the “Deathonomics” is the rapid aging of Russia’s army. Because the younger mobilization pool has largely been exhausted (or has left the country—estimates range from half a million to more than a million people who relocated in 2022 alone), in 2025–2026 the army has been replenished primarily by older “commercial volunteers” attracted by financial incentives.

According to statistical data, the average age of a Russian soldier killed at the front is approaching 50, and the most common category of fatalities is men aged 46–52. Russia has thus turned its army into an “army of old men,” sending them into continuous, exhausting assaults.

A geopolitical game: Munich 2026

According to analysts, the intensification of Russia’s offensive in late 2025 and early 2026 (especially on the Pokrovsk and Kupiansk axes), accompanied by record losses (30,000–35,000 killed and wounded per month for minimal advances of only a few square kilometers), was aimed less at achieving a strategic military breakthrough than at creating a cognitive illusion of invincibility among Western leaders.

The Kremlin is deliberately demonstrating to the world its willingness to expend an unlimited number of its own citizens’ lives. This “madman strategy” is intended to convince the U.S. administration and European leaders that Ukraine’s military resistance is futile, since no level of losses will stop Russia. At the same time, Russia continues its systematic campaign to destroy Ukraine’s energy infrastructure, reinforcing the narrative that Ukraine cannot sustain a war of attrition.

This information-psychological strategy has achieved a degree of success in the West. In the information space (for example, through media leaks in outlets such as the Financial Times or Axios) significant pressure is building on Ukraine to compel it to make territorial concessions (including recognition of the loss of the Donbas and the relinquishment of Crimea) in exchange for promises of Western security guarantees or a freezing of the conflict until summer 2026. Calls have also emerged for elections in Ukraine in exchange for a ceasefire.

In response to this pressure, President Volodymyr Zelenskyy, speaking at the Munich Security Conference on February 14, 2026, criticized the idea of “appeasing” the aggressor. He directly called Vladimir Putin a “slave to war” and compared attempts by Western partners to pressure Ukraine into territorial concessions in exchange for an illusory peace to the shameful Munich Agreement of 1938, when the surrender of Czechoslovakia to Hitler failed to prevent a much larger war.

Zelenskyy emphasized that for Kyiv, any approach that demands concessions solely from Ukraine while Russia continues its terror and wages a hybrid war against Europe is categorically unacceptable. Insisting on firm 20-year security guarantees from the United States before signing any agreements, he made clear that the war will not end on terms of capitulation. Ukraine’s official public disclosure of its significant losses (55,000 killed), though many times smaller than those of its adversary, is itself a strategic move. It is intended to demonstrate to the West the extraordinary effectiveness of Ukraine’s Defense Forces and to show that, provided Western military and financial support is maintained and strengthened, the exhaustion of Russia’s “Deathonomics” will occur long before the Ukrainian nation’s will to resist is broken. Put in plain terms: we are fighting more effectively—support us, and the aggressor’s exhaustion, and thus a durable peace, will come sooner.

Conclusions

The information space of the Russia–Ukraine war is deeply asymmetric. Russia’s official military statistics, published by its Ministry of Defense, are an instrument of psychological operations and are fully fabricated. The combined evidence of mathematical testing of daily briefings using Benford’s law, independent investigations (such as those by Mediazona), and basic arithmetic logic demonstrates that these figures are artificially generated, violate the expected distribution of random variables, and bear no relation to actual battlefield events. Claims of the destruction of more than 27,000 Ukrainian armored vehicles or 953 aircraft and helicopters are arithmetically impossible. Data analysis indicates irrecoverable losses of Russia’s army in the range of 250,000–325,000 killed, corresponding to total losses (including wounded) of more than 1.2 million personnel. Comparative analysis shows that the Armed Forces of Ukraine General Staff’s estimates of Russian losses (more than 1.25 million) align with the consensus assessments of Western intelligence services (CIA, UK Ministry of Defence) and think tanks (CSIS).

The war in Ukraine has triggered irreversible transformations in military strategy and tactics. Technological battlefield transparency enabled by drones has undermined the traditional laws of mechanized warfare (including Lanchester’s law), forcing Russia to shift to bloody infantry infiltration assaults, in which each square kilometer of Ukrainian territory costs the aggressor 80–90 soldiers’ lives. To sustain this self-destructive model, the Kremlin has constructed a “Deathonomics,” buying the loyalty of the poorest regions through relatively high compensation payments for the deaths of their residents. However, this strategy is a financial pyramid scheme that is already fueling an inflationary spiral, depleting the labor force, and leaving the country with a deteriorating economy and a rapidly aging army.

On the geopolitical stage, particularly during the 2026 Munich Security Conference, Moscow’s willingness to expend hundreds of thousands of Russian lives in the furnace of war remains its principal lever in attempts to pressure the West into forcing Ukraine to capitulate. However, even the most conservative Western estimates confirm that Ukraine’s personnel losses are at least 2–2.5 times lower than Russia’s. Together with Kyiv’s firm position, this indicates that the Kremlin’s war of attrition is steadily approaching its critical breaking point.