Are We on Track
for LEV: Methodology

This document explains the calculations behind the article and its figures. It also traces the quantitative claims that appear in the text without hyperlinks. The projections are benchmarks and thought experiments, not forecasts.

1. Methodology

1.1 Research question

The analysis asks two related questions:

1. How quickly has the best-observed human longevity frontier advanced since 1950?
2. How far is that rate from longevity escape velocity (LEV)?

LEV is operationalized as the point at which expected age at death advances by one year for every calendar year that passes. In the units used in the figures, this is:

LEV threshold = 1 expected-age year per calendar year
              = 10 expected-age years per calendar decade

Exactly one year per year stops remaining life expectancy from shrinking as time passes. A rate above one year per year makes it grow. The article sometimes says “more than ten years per decade”; this is the same condition expressed on a ten-year scale.

This is a simplified, population-level definition. It does not assert biological immortality, and it does not by itself establish that any living cohort has reached LEV.

1.2 Data

The main analysis uses the United Nations Department of Economic and Social Affairs, Population Division, World Population Prospects 2024 (WPP 2024):

• abridged period life tables;
• population by five-year age group and sex;
• Medium variant;
• historical estimates for 1950-2023;
• WPP records classified as Country/Area.

The two source files are:

work/WPP2024_Life_Table_Abridged_Medium_1950-2023.csv.gz
work/WPP2024_Population1JanuaryByAge5GroupSex_Medium.csv.gz

WPP combines observed vital-registration data with demographic estimation. Data quality is stronger in some countries and years than in others, and old-age estimates are especially uncertain. The WPP 2024 methodology report documents the UN’s empirical and model-based procedures.

1.3 Period life tables, not cohort forecasts

All life-expectancy values are period life-table measures. They describe a hypothetical person exposed throughout life to the age-specific mortality rates observed or estimated in one calendar year.

They are not forecasts of the realized lifespan of a person born in that year. When mortality is improving, a real cohort may experience different rates as it ages.

1.4 Expected age at death

Life expectancy at age a, written e(a), is the expected number of remaining years conditional on already reaching age a. To place every age on one intuitive scale, the figures use:

expected age at death at age a = a + e(a)

For example, ten remaining years at age 80 becomes an expected age at death of 90.

The main expected-age figures use starting ages 0, 30, 50, 70, 80 and 90. The rolling-progress figure additionally includes age 60.

1.5 Equal weighting of male and female life tables

For every country or area, year and starting age, male and female outcomes are averaged with equal weight:

sex-neutral expected age at death
  = (male expected age at death + female expected age at death) / 2

The same equal weighting is used for age-specific death rates in the mortality analysis.

This differs from the WPP combined-sex life table. At advanced ages, the combined-sex measure is weighted toward women because more women survive to those ages. Equal weighting was chosen because the analysis is intended to track changes in mortality conditions rather than changes in the sex composition of survivors.

1.6 Eligibility filters

For the expected-age frontier, a country or area is eligible only when:

total population >= 1,000,000

For the age-80 and age-90 expected-age lines, it must also have:

population at or above the starting age >= 25,000

The WPP population files report people in thousands, so the corresponding code thresholds are 1000.0 and 25.0.

These rules reduce instability from very small populations and sparse old-age survivor counts. They do not eliminate measurement error.

1.7 The dynamic top-five frontier

The frontier is recalculated every year. For each year and each starting age:

1. Calculate equal-weight male/female expected age at death for every eligible country or area.
2. Rank the eligible observations from highest to lowest.
3. Select the top five.
4. Report the median of those five values.

Because five is odd, the frontier is the third-ranked observation:

frontier(year, age)
  = median(top five eligible expected ages at death)

The top-five set can therefore change across years and across starting ages. The resulting series represents a moving best-practice frontier, not the history of one fixed group of countries.

Why top five and why the median?

A single-country maximum is highly sensitive to one noisy country-year, particularly at age 90. A broad average of many high-performing countries is more stable but less clearly a frontier. The median of the top five is a compromise: it remains best-in-class while requiring the result to be shared by at least three observations.

The median is preferred to the mean because one extreme estimate can pull the mean upward. With a top-five median, the first- and second-ranked values cannot change the reported frontier unless they also alter the third rank.

1.8 Two related frontier constructions

The article uses two constructions for two different purposes.

Expected-age figures

The historical, projection, rolling-progress and time-to-LEV figures re-rank the top five separately for each starting age. This answers: “What was the best-observed expected age at death among sufficiently large populations for a person who had already reached this age?”

Mortality-schedule figures

The mortality figures need one internally coherent set of age-specific death rates. For each year, countries are therefore ranked by equal-weight expected age at death at birth; the top five countries are selected once; and the median equal-weight death rate is calculated at every age for those same five countries.

In 2023 this coherent mortality frontier contains Hong Kong, Japan, the Republic of Korea, Switzerland and Australia.

Re-ranking countries independently at every age would splice death rates from different country sets into one schedule. Such a curve could not be interpreted as a life table, and calculations of expected age at death from it would be internally inconsistent.

The coherent mortality schedule uses the total-population threshold of one million. The additional age-80/90 population filter belongs to the age-specific expected-age frontier and is not applied to this birth-ranked schedule.

1.9 Figure 1: historical frontier, current trend and LEV ramp

For each starting age, an ordinary least-squares (OLS) line is fitted to the annual frontier values from 1950 through 2023:

frontier expected age at death = intercept + slope * year

The slope is multiplied by ten and reported as years gained per decade.

The multiple is a ratio of rates:

multiple needed to reach LEV = 10 / fitted years gained per decade

It is not a multiplier of lifespan, survival probability, research funding, personnel or therapeutic efficacy.

Current-trend projection

The thin straight line extends the 1950-2023 fitted rate after 2023:

V_current(t) = V_2023 + r0 * (t - 2023)

where r0 is the historical slope in years per calendar year.

LEV-ramp scenario

The dashed LEV path is illustrative. It assumes that the annual rate of progress rises linearly from the historical rate in 2023 to one expected-age year per year in 2050:

V_LEV(t) = V_2023
         + r0 * (t - 2023)
         + 0.5 * ((1 - r0) / (2050 - 2023)) * (t - 2023)^2

The curve reaches the LEV rate in 2050. It does not forecast that medicine will actually do so.

1.10 The article’s age-90 illustration

The long-run age-90 slope is about 0.370 years gained per decade. Extending that rate mechanically from 2023 to 2078 gives:

0.370 * (2078 - 2023) / 10 = 2.03 additional expected-age years

The 2023 frontier expected age at death at 90 is 95.686, or 5.686 remaining years. Adding approximately 2.03 produces about 7.72 remaining years, rounded in the article to “roughly eight rather than six.” This is a period-trend illustration, not a cohort forecast for today’s 38-year-olds.

1.11 Figure 2: rolling progress and acceleration

For every endpoint year t, progress is estimated with OLS over the 11 annual observations from t-10 through t:

rolling progress(t)
  = 10 * OLS slope of frontier values from t-10 through t

The window contains 11 observations but spans exactly ten calendar years. Therefore:

first endpoint = 1960, using 1950-1960
last endpoint  = 2023, using 2013-2023

The rolling values are expressed in years gained per decade.

“Acceleration” is a second OLS fit to those rolling progress rates:

rolling progress(t) = intercept + beta * t
fitted acceleration = 10 * beta

Its unit is years gained per decade, per calendar decade. For example, an acceleration of +0.13 y/dec per decade means that the fitted pace of progress rises by 0.13 years gained per decade every ten calendar years.

All rates in the table are in years gained per decade; acceleration columns are per calendar decade. The fitted 2023 pace differs from the actual 2013-2023 pace because the former is the endpoint of a regression line fitted through every rolling window.

Adjacent rolling windows share ten of their 11 observations. The fitted acceleration is therefore descriptive, not a causal estimate with independent annual observations.

The primary figure includes 2020-2023 and the COVID shock. A 2009-2019 fit is used only as a pre-COVID sensitivity in the mortality figure.

1.12 Convexity and the shape of the expected-age curve

A sustained positive acceleration means the fitted slope rises over calendar time. Under that fitted model, the expected-age-at-death curve is convex: successive future calendar decades add progressively more expected-age years.

Positive acceleration alone does not imply LEV. LEV is reached only if the pace itself eventually reaches ten years gained per decade. A small positive acceleration can remain far below that threshold for centuries.

1.13 Figure 3: mechanical time to LEV

The time-to-LEV figure asks a deliberately narrow question: if the fitted 1960-2023 acceleration continued linearly without limit, when would the fitted pace cross ten years per decade?

For a positive acceleration:

crossing year
  = 2023 + (10 - fitted pace in 2023)
           / (acceleration per calendar year)

The stored acceleration is per calendar decade, so it is divided by ten before solving for calendar years.

These dates are not forecasts. They are scale indicators showing that the observed historical acceleration is much too small to imply near-term LEV. The earliest mechanical crossing in the current specification is 2459 at age 30.

1.14 Figures 4 and 5: mortality schedules and the Gompertz fit

For the coherent mortality frontier, the analysis uses the WPP central death rate mx at each age. Male and female mx values are averaged within each selected country, then the median across the five selected countries is taken at each age.

Adult mortality is summarized with the Gompertz model:

mu(x) = alpha * exp(beta * x)
ln(mu(x)) = ln(alpha) + beta * x

An OLS line is fitted to ln(mx) against the midpoint of each five-year age group from 35-39 through 85-89. The line is displayed through age 95 as a visual out-of-sample check.

For the 2023 frontier:

alpha = 8.334e-6
beta = 0.10451 per year
R-squared = 0.9966
mortality-rate doubling time = ln(2) / beta = 6.63 years

For 1950, beta = 0.09691 and the doubling time is 7.15 years. The estimated level parameter alpha fell by about fivefold from 1950 to 2023, while the age slope did not flatten. This is the basis for the article’s description of the mortality curve moving down without materially “tilting” toward slower age-related growth.

The figure stops at age 95. WPP’s age-100 interval is open-ended and largely model-dependent; plotting it as if it were an ordinary five-year point can create a false plateau.

1.15 Survival from age 100

The article’s extreme-old-age examples use official national life tables rather than the open WPP 100+ interval.

Japan

In Japan’s 23rd complete life table for women, 8,153 of the synthetic cohort are alive at exact age 100 and 47 at exact age 110:

P(survive 100 to 110) = 47 / 8,153 = 0.005765
inverse probability   = 173.5

Rounded: about one in 173.

United States

Using the female one-year death probabilities in the US Social Security Administration’s 2023 period life table:

P(survive 100 to 110) = product of (1 - qx), x = 100,...,109
                      = 0.002617
inverse probability   = 382.1

Rounded: about one in 382.

Continuing the published probabilities through age 119 gives:

P(survive 100 to 120) = 7.405e-10
inverse probability   = 1.350 billion

In the article’s counterfactual plateau, the published US probabilities are retained through age 104 and the annual probability of death is set to 50% from age 105 onward:

P(survive 100 to 120) = observed survival through 104 * 0.5^15
                      = 2.833e-6
inverse probability   = 353,016

Rounded: about one in 350,000. This plateau is a hypothetical illustration, not an empirical claim.

1.16 Figure 6: mortality decline equivalent to LEV

The mortality-decline figure translates expected-age progress into a proportional change in the entire age-specific mortality schedule.

Expected age from a mortality schedule

WPP provides abridged death rates at ages 0, 1, 5, 10, ..., 100. Hazards are treated as constant within each interval: width one at age 0, width four at age 1 and width five thereafter. The age-100 interval is treated as open.

For interval mortality m, width w and survival entering the interval S, expected person-years contributed by the interval are:

S * (1 - exp(-m*w)) / m

Survival then becomes:

S * exp(-m*w)

The open age-100 interval contributes S/m.

Converting an expected-age trend into a mortality decline

For each comparison period, an OLS trend is first fitted to the coherent frontier’s expected age at death at birth:

For each row, every age-specific death rate in the period-end schedule is multiplied by the same factor exp(-r). A binary search finds the r that produces the same expected-age gain when the life table is recalculated.

The LEV benchmark applies the same procedure to the 2023 schedule and finds the common decline that increases expected age at death by exactly one year. This calculation uses the complete abridged mortality schedule; it does not assume that mortality follows a Gompertz curve:

annual log mortality decline required for LEV = 10.9277%
equivalent ordinary annual reduction          = 10.3518%

An earlier version used the 2023 fitted Gompertz age slope, beta = 0.10451, as a shortcut and obtained 10.45%. Under a pure Gompertz model, a parallel mortality shift of beta moves expected age at death by approximately one year. The shortcut is close, but the exact life-table result is preferable because the observed schedule is not purely Gompertz at young and extreme ages. The final mortality-target figure therefore uses the full-life-table value, not the Gompertz approximation.

The upper age-specific lines in the figure are diagnostics. At each age, they show -slope(ln(mx)) over 2013-2023, 2009-2019 and 1950-2023. They are not separate LEV targets for each age.

Important percentage convention

The chart reports continuously compounded log-rate declines because the age-specific diagnostics are estimated as changes in ln(mx). The computed 10.9277% log-rate is equivalent to the following ordinary one-year proportional reduction:

1 - exp(-0.109277) = 10.3518%

Under the implemented log-rate convention, compounding for 27 years from 2023 to 2050 leaves:

exp(-0.109277 * 27) = 5.23% of the 2023 mortality-rate baseline

or a 94.77% reduction. For general readers, the clearest wording is: age-specific mortality rates would need to fall by about 10.4% each year. Treating 10.9% as an ordinary discrete percentage and calculating (1 - 0.109)^27 leaves 4.40%, but that mixes two conventions and is not the calculation implemented in the figure.

The phrase “10.9% fewer people dying each year” is only an intuition. The calculation concerns age-specific period mortality rates, not the crude number of deaths, which also depends on population size and age structure.

1.17 Verification

The expected-age frontier and acceleration results were independently recomputed directly from the raw WPP files. The audit found:

• 2,590 top-five membership rows matched;
• zero membership-identity mismatches;
• zero candidate-count mismatches;
• zero population-filter failures;
• maximum rolling-rate difference of 0.00005 y/dec;
• maximum mechanical-crossing difference of 0.08 year in the current specification.

The remaining differences are display rounding.

1.18 Limitations

1. Period versus cohort: period life tables are snapshots, not realized cohort forecasts.
2. Frontier switching: the top-five membership changes over time and, in the expected-age figures, by starting age.
3. WPP estimation: many country-years, particularly at advanced ages, include modeled mortality rather than complete direct observation.
4. Old-age uncertainty: small survivor counts, age misstatement and the open 100+ interval limit inference at extreme ages.
5. Overlapping rolling windows: adjacent rolling-progress estimates are strongly dependent.
6. Linear fits: OLS summaries compress nonlinear historical changes into one rate.
7. COVID: the current 2013-2023 endpoint includes the pandemic and its aftermath; 2009-2019 is shown as a sensitivity.
8. Mechanical extrapolation: the LEV ramp and crossing years are scenarios, not forecasts.
9. No causal attribution: the analysis measures outcomes but does not estimate which technologies, policies or expenditures caused them.
10. No input-output production function: a 7-27x outcome-rate gap does not imply 7-27x more funding or personnel. Biomedical inputs may translate into outcomes sublinearly, superlinearly or only after long lags.

1.19 Future validation with the Human Mortality Database

The Human Mortality Database (HMD) could be used in a future sensitivity analysis to:

• validate ages 80 and 90 with harmonized death and exposure data;
• test whether surprising old-age frontier members persist in a higher-quality-data sample;
• compare the dynamic frontier with a fixed-country panel;
• quantify sensitivity to age misstatement and old-age population thresholds.

HMD has narrower and uneven country-year coverage. It should therefore be reported as a separate robustness analysis, not spliced into the WPP series.

2. Sources and claim audit

2.1 Primary demographic and actuarial sources

1. UN DESA, World Population Prospects 2024, abridged life tables and population by age and sex, Medium variant.
2. WPP 2024 methodology report.
3. Japan Ministry of Health, Labour and Welfare, 23rd Complete Life Tables (2020).
4. US Social Security Administration, 2023 Period Life Table.

2.2 Article claims derived from this analysis

2.3 Biomedical progress comparison claims

The comparisons are context, not inputs to the LEV calculations. “Today” means the latest year used when the article was drafted; where 2000 was unavailable, the article used 2001 or 2002.

2.4 Interpretation notes for the biomedical comparisons

1. Mortality percentage convention: the exact target is 10.9277% on the chart's continuous log-rate scale, equivalent to an ordinary annual reduction of 10.3518%. The reader-facing prose should use approximately 10.4%, with 5.2% of the 2023 mortality-rate baseline remaining in 2050.
2. Biotech venture capital: the corrected comparison is $3.9B -> $18.4B, or 4.7x nominal and approximately 2.7x in 2023 CPI-adjusted dollars. The values come from different data regimes, so they establish direction and scale but are not a clean like-for-like growth series.
3. Therapeutic modalities: the numerical 5-6 -> 20-21 comparison has been removed. The first-approval milestones communicate the expansion without relying on an arbitrary taxonomy.
4. INDs: the comparison now uses the FDA's 2025 total of 1,960.
5. Global pharma R&D: the nominal comparison is sourced to Evaluate. The 2.2x real estimate should be described as approximate because it applies the US BRDPI to a global company-spending series.
6. NIH funding: the comparison uses program level, not budget authority, obligations or outlays. ARPA-H and emergency supplemental funding are excluded.

2.5 Reproducibility files

Core expected-age outputs:

outputs/top5_longevity_frontier_values_1950_2023.csv
outputs/top5_longevity_frontier_slopes_1950_2023.csv
outputs/top5_longevity_frontier_members_long_1950_2023.csv
outputs/top5_longevity_frontier_lev_vs_current_rate_1950_2050_summary.csv

Rolling progress and time-to-LEV:

outputs/top5_equal_weight_frontier_acceleration_1960_2023_and_2000_2023.csv
outputs/top5_equal_weight_frontier_time_to_lev_1960_2023.csv
outputs/top5_equal_weight_time_to_lev_raw_wpp_audit.csv

Mortality and Gompertz analysis:

outputs/gompertz_frontier_results.json
outputs/gompertz_charts_ABGM_styled_checks.json
outputs/gompertz_chart_G_corrected_checks.json
outputs/gompertz_chart_G_original_vs_corrected_values.csv

Principal calculation scripts:

work/draw_top5_longevity_frontier_1950_design.py
work/render_wpp_acceleration_and_lev_timing_verified.py
work/gompertz_frontier_analysis.py
work/render_chart_g_corrected_comparison.py

The raw-WPP audit, rather than visual agreement between charts, is the final check on the expected-age and acceleration values.