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Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.
Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.
Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.
Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.
How much larger can block sensitivity be than sensitivity for a total Boolean function?
For a total Boolean function , a block of coordinates is sensitive at if flipping every bit in that block changes . Let count sensitive singletons and be the maximum number of pairwise disjoint sensitive blocks. Define and .
This campaign seeks larger exponents for which there exists an infinite family of total Boolean functions , indexed by , satisfying
Here denotes a quantity tending to zero as . Higher is better. The classical exponent is , and Huang's bound gives a ceiling of .
OpenAI's construction achieves an exponent strictly greater than .
One sufficient way to establish an exponent is to find a single total Boolean function with and . Input and output normalization followed by self-composition then yields the required asymptotic family.
Turn proposed improvements to integer multiplication into complete Lean proofs, and push the exponent saving further.
Harvey and van der Hoeven established an algorithm in 2021. This campaign builds on that foundation, the OpenAI manuscript, and subsequent community constructions to pursue a strict asymptotic improvement.
For two -bit integers, the target is
A positive beats asymptotically; larger is better. Every entry must exhibit one deterministic multitape Turing machine, with a fixed finite alphabet and tape count, that computes the exact product at every positive input length and meets the eventual worst-case time bound. The tracked number measures an asymptotic exponent saving.
Classical algorithms solve 3SUM in time. In a 2026 breakthrough, Alman and Vassilevska Williams gave a deterministic algorithm, refuting the integer 3SUM hypothesis. How low can the exponent go?
Building on existing Lean formalizations, this campaign tracks upper bounds for 3SUM on polynomially bounded integers, using a word RAM with -bit words, and pursues smaller exponents.
Classical algorithms solve all-pairs shortest paths in time. In a 2026 breakthrough, Alman and Vassilevska Williams refuted the APSP conjecture with a deterministic algorithm. How low can the exponent go?
Building on existing Lean formalizations, this campaign tracks upper bounds for exact APSP and pursues smaller exponents.
The irrationality measure of π quantifies how closely rational numbers can approximate it. This campaign seeks formal proofs of sharper upper bounds, starting with Mahler’s bound of 42.
The sharp Hlawka inequality for Schatten -norms is a cousin of the triangle inequality: it relates the norms of three matrices to the norms of their pairwise sums and their total sum. For complex diagonal matrices, an exact formula for the best possible comparison constant has been proved in Lean for every real . We conjecture that the same formula holds for all .
What is the smallest cutoff for which this formula holds for every real ?
References:
Is every odd number a sum of primes? This campaign tracks formalized proofs of the smallest that suffices.
Schnirelmann (1930) showed some finite works. Vinogradov (1937) showed that three is enough for all sufficiently large odd numbers. Tao (2012) proved unconditionally. Helfgott (2013) proved that every odd number greater than is a sum of three primes, though the proof is still unrefereed. Ideally, we can formalize this statement here. Note that three is optimal: is neither prime nor + prime.
Schoolbook matrix multiplication takes operations. The exponent is the infimum of all such that two matrices can be multiplied in arithmetic operations; trivially , and is conjectured but open.
Strassen gave the first nontrivial bound, , in 1969, and introduced the laser method in 1986 to reach . Coppersmith and Winograd's 1990 bound of stood for two decades. Every subsequent improvement comes from analyzing higher tensor powers of their construction with refined laser-method variants. That line reached in 2025, and the current record is , from August 2026. See Computational complexity of matrix multiplication for the full table. Can we formalize these results and even improve on them?
Martingales and stopping times link the evolution of an insurer's random capital to finite-horizon ruin probabilities. At issue the insurer holds initial surplus u. Premiums and claims make its later surplus stochastic. A positive exponential transformation of the surplus can become a supermartingale under an appropriate model of premium income, independent claims and an adjustment coefficient. Optional stopping then bounds the probability of negative capital when the contract stops at ruin or at a finite horizon. Promislow develops this connection between probability theory and ruin models in Chapters 18 and 23. This mission states finite probability and stopped-increment hypotheses explicitly. The book is S. David Promislow, Fundamentals of Actuarial Mathematics, Third Edition (2015), chapters 18 and 23. The independently published supporting research reference is available here.
A finite scenario space has nonnegative probability weights p indexed by Fin m and total probability mass one. Net cashflow G(k,ω) is a random realised premium less claim at policy anniversary k, in currency units. The capital process starts at u and accumulates increments up to time k. A scenario-dependent integer stopping time τ is assumed bounded by N. Genuine adaptedness relative to a filtration is mathematically separate and must be checked before asserting that a particular random τ is an admissible stopping time. The stopped process at time k is X(min(k,τ),ω). The exponential transform is exp(-R U(k,ω)), with positive adjustment coefficient R measured in reciprocal currency units. Ruin occurs at strict negative surplus, rather than merely touching zero.
The central actuarial model can be written
The finite probability block defines an actual weighted expectation, a stopped random process, and the expected increments after stopping. Linearity and positivity of expectation are proved over the concrete scenario distribution. A bounded optional stopping theorem follows from summing the stopped increments when each such increment has zero expected value. The supermartingale inequality follows under nonpositive expected stopped increments. Such restrictions are useful explicit finite conditional consequences of genuine martingale models, while their derivation from a more general filtration is a separate mathematical task. The theorem types therefore avoid assuming the desired optional stopping conclusion outright.
The surplus and exponential block defines insurer capital as initial funds plus realised net increments. The surplus recursion is verified path by path. Exponential functions convert additive capital changes into multiplicative factors, and under positive adjustment coefficient the transform decreases as capital increases. The intended later dependence on the claim distribution enters through expected exponential increments. The model does not assert that any arbitrary premium-minus-claim process has the required supermartingale property. This prevents a false blanket application of the Lundberg bound and distinguishes positive initial capital from a possibly negative terminal surplus.
The ruin block introduces an actual event probability, monotonicity of ruin with respect to horizon and capital, and the deficit conditional on negative terminal funds. At a ruined path, exp(-R U)>1 when R>0, so the ruined-path indicator can be bounded by the exponential transform. The nonpositive stopped-increment hypothesis bounds its expectation by the initial exponential reserve. These two independent mathematical steps combine in a finite-horizon exponential bound for genuine ruin events. The initial condition, finite sample weights, and bounded stopping time are all explicit in the capstone.
The proposed root is ActuarialValuation.ruBoundedStoppingRuin_fundamental. Its intended actuarial mathematical relationship is illustrated by
This is a substantive nontrivial model commitment that excludes degenerate substitutes. This formula is stated only under the hypotheses and timing conventions of the actual Lean target. Subordinate theorem declarations provide the named formal prerequisites in the local source inventory. Each supporting milestone requires its own genuine proof rather than a textual assertion that an identity is obvious.
The finite-horizon result is a reusable first formalisation of the reasoning behind adjustment-coefficient solvency bounds. It aligns actuarial capital and stochastic stopping mathematics while making model assumptions auditable. MIT OpenCourseWare's optional stopping notes provide an independent probability reference, and classical actuarial risk theory treats analogous exponential surplus transformations. An ultimate infinite-horizon ruin probability needs additional convergence and stopping-time controls, and the finite bound alone does not establish a limiting probability statement.
The key condition is mathematical rather than syntactic: an arbitrary stopped process need not preserve mean and an exponential of an arbitrary reserve need not be a supermartingale. The finite expected-increment constraints are stated as proof hypotheses; linking them to a filtration and an actuarial claim model is future work. All sums require valid nonnegative weights, and the total weight must equal one to interpret expectations. The proof must use strict negative surplus, R positive, and explicit boundedness of τ. Passing remote declaration elaboration with sorry proves no stochastic result.
The graph includes weighted expectations, stopped processes, a finite optional stopping identity, a discrete actuarial surplus process, exponential transforms, ruin event and deficit definitions, and an exponential upper-bound capstone. Full optional stopping for a measurable filtration, adjusted compound Poisson reserves, deficit-at-ruin distributions and ultimate ruin probabilities remain extensions. The capstone is finite-horizon by construction and carries the assumptions needed to justify it. Detailed dependency graph, exact Lean target types and independent finite probability checks are maintained as distinct artefacts. The staged inventory comprises 13 definitions and 38 open theorem targets. It uses the pinned Mathlib revision and a guarded remote private compiler. The proposed definitions also provide reusable results for future actuarial missions and downstream applications. The planned graph is recorded independently of theorem proof acceptance. The mathematical identities are already established in published actuarial or probability literature; the proposed Lean results remain new unproved formal targets. Every target remains open until an accepted proof is verified, and no owner item is to be confirmed or submitted for moderator review without human assessment.
Discrete arbitrage-free valuation makes insurance and pension investment guarantees amenable to exact replication arguments. A contingent payment can be financed by trading in the underlying risky asset and a risk-free bank account when the market contains a complete two-state one-period set of securities. The corresponding values can be calculated through risk-neutral expectations. This mission develops the explicit share and cash positions and verifies that one strategy replicates both possible obligations. It extends these identities to a two-period option tree and immediate-exercise decisions while keeping market completeness and the appropriate no-arbitrage assumptions visible. The book is S. David Promislow, Fundamentals of Actuarial Mathematics, Third Edition (2015), chapters 20.1–20.11. The independently published supporting research reference is available here.
At time zero, the stock price is s>0. At time one it becomes either su or sd, where u and d are gross stock-return factors. A bank account grows by the strictly positive factor R. A viable two-state frictionless market with strict full-support risk-neutral probabilities requires d<R<u. A European claim pays xu in the up state and xd in the down state, with currency units consistent across both states. The hedge holds Δ shares and a cash bank deposit b from time zero until expiration without any intermediate external cash injection. Replication means both terminal payoffs coincide in the same two-state market. For the multiperiod extension, individual tree nodes have their own exercise and continuation values; immediate exercise is available only at eligible American nodes.
The central actuarial model can be written
First construct the unique risk-neutral probability and its complementary down-state weight using the bank return and the two possible stock returns. Establish strict bounds zero less than q less than one when the bank return lies strictly between the stock outcomes. Risk-neutral valuation is linear in the payoff and positive on nonnegative claims. The discounted expected stock price equals today's stock price, which is the defining one-step martingale restriction. The boundary cases q=0 or q=1 are algebraically valid but no longer correspond to a full-support equivalent martingale measure.
Next construct the actual hedge shares and bank investment. The same positions must deliver the specified payoff in both states, with s, R and u-d explicitly nonzero. The cost of replication then agrees with risk-neutral discounted expected payoff. Theorems handle deterministic payments, zero risky exposure, linearity and one-step put-call parity for a common strike and expiry. The stock/call/put payoff relation is pathwise, including the stock-price-equals-strike boundary; it therefore does not rely on an assumed lognormal distribution or on any incomplete market theorem.
Finally the two-step binomial model calculates the expectation over all four ordered movement paths, preserving separate path-dependent middle cashflows if specified. Backward induction discounts from terminal nodes to time one, then discounts again to issue. An American contract replaces European continuation with the maximum of immediate exercise and expected continuation. The graph includes the fundamental domination of a corresponding European price by an American price when gross bank return is positive and risk-neutral probabilities are valid. Statewise replication at each node is the building block for multistage self-financing hedges.
The proposed root is ActuarialValuation.bnBinomialReplication_fundamental. Its intended actuarial mathematical relationship is illustrated by
This is a substantive nontrivial model commitment that excludes degenerate substitutes. This formula is stated only under the hypotheses and timing conventions of the actual Lean target. Subordinate theorem declarations provide the named formal prerequisites in the local source inventory. Each supporting milestone requires its own genuine proof rather than a textual assertion that an identity is obvious.
The explicit two-asset hedge is useful for formalising investment-linked policy guarantees, embedded options, financial market valuations and replicated hedging of insurer liabilities. The original binomial option framework is published by Cox, Ross and Rubinstein (1979). A separate published Lean 4 formalisation of the broad Fundamental Theorem of Asset Pricing exists: Raphael Coelho (2026), https://arxiv.org/abs/2606.28990. This mission's incremental scope is the explicit binomial hedge, two-period recursion and American exercise boundary rather than a claim to newly establish all finite-market arbitrage theory.
Replication needs a nonsingular two-state payoff matrix and a positive bank numeraire. Different payoff dates cannot be mixed in a single price equation without discounting. Recombining trees can be represented by ordered movement paths without forcing equal middle payoffs for a path-dependent option. American early exercise requires a node-specific immediate payoff and comparison with the discounted continuation; using terminal exercise value prematurely changes the contract. All independent proofs must retain the strict no-arbitrage hypothesis and the role of risk-neutral probabilities. The capstone is a constructive theorem about a particular self-financing portfolio, with actual terminal state equations.
The proposed definitions cover risk-neutral weights, expected payoff, discounted price, share-and-bank hedge, European call and put payoffs, a two-period tree and an American exercise operator. The intended capstone proves existence of the exact replication holdings and equality of hedging cost and risk-neutral valuation. Further research could add a complete finite filtration, dynamic hedging rebalancing, conditional risk-neutral measures and general finite-market completeness, reusing rather than duplicating existing formal work. Every proof target remains open until an authoritative remote compiler accepts an actual Lean proof. The staged inventory comprises 16 definitions and 36 open theorem targets. It uses the pinned Mathlib revision and a guarded remote private compiler. The proposed definitions also provide reusable results for future actuarial missions and downstream applications. The planned graph is recorded independently of theorem proof acceptance. The mathematical identities are already established in published actuarial or probability literature; the proposed Lean results remain new unproved formal targets. Every target remains open until an accepted proof is verified, and no owner item is to be confirmed or submitted for moderator review without human assessment.
The deterministic claims-development model estimates future claim amounts but does not quantify reserve uncertainty. Thomas Mack's distribution-free standard error framework in ASTIN Bulletin (1993) introduces conditional moments, volume-weighted development, process variability and factor estimation uncertainty. The process uncertainty of future claims and parameter uncertainty from fitted factors must be distinguished. This mission supplies original finite Lean identities corresponding to those actuarial building blocks, without claiming a proof of Mack's full reserve mean-square-error theorem.
The published source supplies the publicly verifiable actuarial context. Original finite Lean formalisation of the corresponding identities remains an open mathematical task; existing actuarial theory is not by itself an accepted formal proof. The input conventions and hypotheses should be compared to the source by an independent human reviewer.
A finite observed paid-claims triangle has common pairs of cumulative claims C(i,j) and C(i,j+1) for origin cohorts i below m. The Mack link factor is f_j=sum C(i,j+1)/sum C(i,j). A predicted next-age amount is C*f_j, and paid-to-ultimate reserve equals predicted amount minus current paid claims. Weighted squared residuals are deviations squared divided by strictly positive cumulative claims. An estimated sigma-squared divides aggregate weighted squared residual by positive degrees of freedom. Under a stated one-step variance model, process variability takes sigma-squared times current claims and estimation variability takes sigma-squared times squared current claims divided by observed development exposure.
All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.
This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.mackProjection_add`:
One common factor distributes over aggregated claims.
This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.mackSigmaSquared_zero`:
No residual variation means zero estimated variance.
This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.mackReserveVariance_fundamental`:
The capstone combines observed factor calibration, separate nonnegative process and estimation variance terms, and claims reserve reconciliation.
The main target is `ActuarialValuation.mackReserveVariance_fundamental`. The capstone combines observed factor calibration, separate nonnegative process and estimation variance terms, and claims reserve reconciliation.
Mack (1993) derives the full distribution-free standard error of chain-ladder reserve estimates from conditional development assumptions. The finite results here are algebraic audit invariants, intended to be reused in a later proof of those conditional-moment claims. The capstone checks that process and factor-estimation error terms are individually nonnegative and reconcile with the selected development factor and reserve. These results support future uncertainty decomposition, diagnostics and coherent reserve modelling.
A Mack residual statistic is not a conditional random variable until a stochastic development model is explicitly established. The factor's denominator must include exactly the cohorts for which the next development observation exists. Strictly positive observed claim amounts are essential for scaled residuals. The process variance and estimated factor variance terms are distinct and must be justified by separate modelling assumptions for a full stochastic theorem. The chosen positive degrees of freedom is an input to the algebraic dispersion estimator; a specific m-minus-one estimator requires enough observed cohorts. The staged capstone does not silently assert unbiasedness or independence.
The original Mack distribution-free method distinguishes model process noise from sampling or parameter estimation uncertainty. The original 1993 paper includes conditional assumptions and a reserve mean-square error derivation; these staged identities prove only algebraic invariants before constructing that complete stochastic theory. For a published actuarial result to be claimed as proved in Lean, corresponding random variables, sigma algebras, conditional expectations and associated independence or uncorrelatedness conditions would all need authoritative formal statements and proofs. This package makes no such claim, and sample residual calculations or a successfully elaborated open theorem do not amount to a stochastic proof.
All staged theorem targets are finite deterministic real-algebraic components motivated by Mack's published reserving model. They do not constitute the full distribution-free stochastic proof, a confidence interval, a tail extrapolation or a quantile estimation result. The Lean source specifies nonzero denominators, finite index sets and valuation units precisely. The nontrivial root assembles calibration, two distinct variance charges and reserve valuation without collapsing either variance term into a tautological special case.
These are original derived finite/analytic Actuarial formalisation targets rather than reproductions of a numbered textbook theorem. Exact Lean binder declarations are authoritative. The complete source uses targeted Mathlib imports and no local Lean compilation. All proof bodies remain genuinely open and do not constitute accepted results. Assumptions about negative rates, survival, policyholder independence, discounting and benefit timing must be preserved exactly.
Stochastic survival distributions provide the probability-theoretic foundation for mortality-based insurance and pension valuation. They describe the actual random duration until failure, its probability distribution and the conditional risk faced by someone still alive at a particular time. Promislow (2015), especially Chapter 14, presents discrete and continuous lifetime distributions, survival functions and hazard rates, building on the continuous mortality material in Chapter 8 and probability foundations in Appendix A.
The established mathematical ideas are available in published literature, and Mathlib has general measure-theoretic infrastructure for probability measures. This mission proposes a bridge between that genuine probability structure and concrete actuarial lifetime calculations. Earlier actuarial work contains life-table and parametric mortality results, but this mission separately constructs a measurable lifetime event model. Every new theorem remains an open Lean target until an authoritative proof is accepted.
Let Ω be the underlying outcome space with a measurable structure, μ a probability measure on that space, and T : Ω → ℝ a measurable lifetime random variable. For a threshold t, define the failure CDF by the probability of T ≤ t and survival by the probability of T > t. This strict-versus-nonstrict distinction is essential when the lifetime law has atoms:
Both quantities are represented by extended nonnegative-real measures. For an economic lifetime model, additional hypotheses specify nonnegative durations and probability normalisation. A lifetime with an atom at issue need not satisfy S_T(0)=1. A lifetime that is only almost surely nonnegative is mathematically sufficient for many distributional identities, although some finite initial targets state stronger pointwise assumptions.
For positive integer k, the discrete failure mass is p_k=P(T=k). The discrete hazard is conditional on surviving beyond k-1:
This ratio is defined operationally only if the at-risk denominator is strictly positive. The underlying Lean division remains total, so the cancellation theorem must explicitly exclude zero and infinite denominators.
The first supporting group formalises survival and CDF monotonicity, event partition, survival before time zero, failure atoms, conditional ratios and residual lifetimes. The residual lifetime after time s is represented by the shifted random variable T-s, while the conditional law of that random variable given T>s is represented separately by a survival ratio. This distinction prevents silently treating an unconditional time shift as conditional survival.
The conditional identity to establish, for a positive survival denominator, is
It applies for nonnegative additional duration t when the numerator event is contained in the conditioning event. The algebraic quotient may also be defined at other real inputs, but its interpretation as a conditional probability requires these event conditions.
A finite probability mass model on durations zero through n supplies explicit sums for expected lifetime and discrete strict-tail survival. The intended tail-sum identity is
This finite identity holds without convergence issues. Probability normalisation and nonnegative mass are additional hypotheses required when interpreting the identity as a lifetime expectation. Discrete term-insurance valuation also appears as a death-year-weighted sum, with death in year k paid at the end of that year. These finite constructions give checkable links from distributional quantities to insurance cashflows.
An exponential lifetime with constant hazard λ has model survival exp(-λt)) and cumulative hazard λt. A **Gompertz** model uses hazard b exp(ct), and a **Makeham** model adds an age-independent force a. Their cumulative hazard formulae are valid under their stated parameter conditions. In particular, the standard integrated Gompertz expression divides by c; when c=0`, its expression must be treated separately as the constant-hazard limit rather than used through division by zero.
General survival and cumulative hazard satisfy, for appropriately regular and nonnegative force of mortality,
This mission stages finite parametric hazard identities and a genuine insurance present-value expectation, rather than claiming the full general theorem without the necessary absolute continuity and integrability formalisation. The expectation of a unit claim payable at the realised lifetime under interest force δ is represented as
The zero-interest and deterministic-lifetime special cases are explicitly targeted.
ActuarialValuation.svSurvivalHazardInsurance_fundamental connects the event-defined CDF of a measurable lifetime with a stipulated exponential survivor law, the corresponding cumulative hazard representation and the genuine measure-integral expected discounted insurance value at zero force of interest. The model-law hypothesis is explicit. The theorem therefore retains stochastic semantics while providing a solvable initial path to a broader general hazard calculus.
Actuarial tables, conditional survivor calculations and life insurance prices can be checked against a consistent random-lifetime model instead of relying solely on formal algebra involving survival symbols. Mathlib's probability-measure library provides a verified environment for event probabilities, with related integration foundations available for subsequent extensions. This builds a reusable formal bridge to mortality projection, multiple-decrement valuation, insurance-loss distributions and the pricing mission LXIV.
The distinction between strict survival and nonstrict failure probability governs boundary atoms. Conditional hazard ratios are meaningful only when the at-risk population has nonzero measure. Continuity, absolute continuity, differentiability and density-based hazard formulae are genuinely additional hypotheses; their absence cannot be repaired by simply assuming exponential survival. General integrated hazard reconstruction will require a later properly regularised theorem. The integral at a nonzero discount rate also needs integration and sign assumptions to justify pricing.
An event-defined random variable has materially different proof obligations from a deterministic model formula. The graph includes nontrivial probability event partitions, monotonicity and conditional mass inequalities, a finite tail-sum expectation identity and a measure-integral valuation. The open declarations are not accepted results.
The staged inventory contains 17 reusable definitions, 36 open theorem targets and 35 supporting milestone specifications. Its finite and continuous-like model families are carefully separated. The capstone is intentionally conditional on a named exponential survival hypothesis and has an actual measure-theoretic loss valuation. Discrete failure times, finite support, residual-life shift, cumulative hazard and time-zero present values are included as individual targets.
Excluded future extensions include general Radon-Nikodym hazard construction, an absolute-continuity theorem for arbitrary lifetime distributions, differentiability of log survival, almost-everywhere equality of densities and insurance PV identities at arbitrary interest forces. These are genuine mathematical next steps, not already established by the initial capstone. Private declaration compilation and any numerical sampling evidence remain separate from accepted proof status.
Fractional-duration life contingencies arise when contractual cashflows are paid more frequently than annually and when valuations occur between policy anniversaries. The frequency of annuity payments, the timing of a death benefit and the assumption used to interpolate survival between integral ages each affect the expected present value. S. David Promislow's Fundamentals of Actuarial Mathematics, third edition (2015), particularly Chapter 7, develops these relationships alongside its treatment of life annuities (Chapter 4) and continuous payments (Chapter 8). This mission constructs independent, explicitly timed Lean 4 definitions and original theorem targets that can be reused in later insurance, pension and reserve formalisation.
The foundational mathematics is already established in the actuarial literature. Its corresponding Lean declarations in this mission are open proof targets. The existing Prove2Me actuarial collection already contains annual annuities, life-table identities and constant-force mortality examples, but these do not by themselves establish the new m-thly cashflow and within-year reserve results. Relevant published material includes Promislow (2015) and Society of Actuaries actuarial work on uniform mortality interpolation.
Time is measured in years after issue. Policy year k is a natural number, payment frequency m is a positive number of instalments per year, and j ranges from zero through m-1. A benefit amount c(k) has currency units per year. A normalised annual payment stream has cash amount c(k)/m at each instalment. Discount D(t) and unconditional survival p(k) are dimensionless. Annual mortality q(k) belongs to the interval from zero to one when interpreted as a valid probability.
An annuity-due payment in subperiod j occurs at k+j/m; an annuity-immediate payment occurs at k+(j+1)/m. The final immediate instalment therefore requires survival to the following policy anniversary. In each policy year, uniform distribution of deaths (UDD) specifies
whereas constant force of mortality specifies
The exponential model assumes a constant force mu over the relevant duration; it is not silently identified with UDD. Constant force of interest delta yields D(t)=exp(-delta*t), but the finite annuity definitions also permit an arbitrary discount function with explicit nonnegativity hypotheses when required.
Construct separate linear and exponential survival models, conditional UDD survival after fraction s, a constant-interest discount factor and the exact indexed instalment time. Establish the zero-duration and full-year boundaries, positivity and monotonicity under justified probability assumptions, conditional-survival multiplication, exponential multiplicativity, and a discount-factor splitting identity. The conditional UDD formula divides by survival at s; a zero-survival endpoint requires an explicit exclusion.
The within-year decomposition is
and the conditional expression divides the left-hand side by S_UDD(s|k) when this is positive.
Define a finite n-year annuity-due as the discounted weighted double sum
Define immediate valuation by changing both the payment time and within-year survival exposure to (j+1)/m. These definitions specify exact UDD actuarial values under the stated unconditional annual survivor weights, rather than using an approximation to annual payments. Establish zero-year and zero-benefit boundaries, linearity, scaling, positivity, and the reduction to annual due valuation for m=1.
For the annual immediate reduction, require the annual survival recurrence p(k+1)=p(k)(1-q(k)). This is necessary because an immediate payment at the annual boundary is conditioned on survival through the full policy year. The case m=0 is retained as an empty finite-sum identity solely to check mathematical totality; it is excluded from economic payment-frequency interpretation.
The equivalence principle premium rate is the expected discounted benefits divided by the present value of a unit-premium annuity. The associated balance is
Here A is a dimensionless instalment-weighted annuity value, B is in currency and pi is currency per year of nominal premium rate. The denominator is required to be nonzero for the funding identity, and positive if the quantity is interpreted as an economically valid payment stream.
For a death benefit payable at the end of the policy year, a retrospective fractional survivor reserve, measured at s after that year's premium has been received, satisfies the conditional balance
where G_{0,s} is accumulation from the anniversary to s, and D_{s,1} discounts an end-year death payment back to s. The reserve is conditioned on survival to s; require 1-sq\ne0. Prove the opening, zero-mortality, end-year and zero-death-benefit boundary identities. These statements do not assert that all deaths occur at an anniversary.
ActuarialValuation.frFractionalValuation_fundamental connects annual reduction for a single due instalment, equivalence-premium funding, exact conditional within-year reserve reconciliation, and positivity of finite-time constant-force survival. The theorem keeps the two mortality assumptions in separate propositions.
As described in Promislow (2015), fractional valuation provides a common time axis. The graph establishes a common, formally stated time axis for fractional pension and life-insurance cashflows. It distinguishes the recipient's conditional survival at payment from the insurer's future death-benefit payment obligation. The finite indexed formulation supports later derivation of m-thly life-contingent premium conversion, changes in reserving basis and comparison with continuous annuity integrals. The theorem graph can also be composed with existing annual actuarial-valuation work once published theorem identities and exact dependencies have been independently verified.
The main sources of mathematical risk are year-end payment conventions, the survival status of an annuitant exactly at an instalment boundary, the difference between UDD and constant force, and division by the probability of being alive at the valuation duration. The initial model is finite and avoids an unjustified assertion that every m-thly sum converges to a continuous value. Such limiting claims require a further integrability and regularity layer and are reserved for an explicitly justified extension.
The graph includes actual mathematical content beyond zero and cancellation identities: conditional-survival reconciliation, exact change of payment frequency, additivity of survival-discounted cashflows and survivor-weighted reserve funding. The numerical checks and remote declaration checks should be reviewed independently. A successfully elaborated sorry declaration remains unproved.
There are 15 proposed definitions, 38 open theorem targets and 37 subordinate milestone descriptions. The mathematical objects include finite event-weighted cashflow sums, an explicit discount factor, UDD and constant-force survival, and retrospective fractional-duration reserves. The item specifications are the authoritative proposed Lean propositions. Hypotheses on q, s, m, the annual survivor recursion and the premium annuity denominator are present only where required for a proposition.
The mission does not yet include Woolhouse asymptotic expansions, rigorous m\to\infty convergence, a general interest-rate term structure or non-UDD reserve interpolation. Those are identifiable continuation targets rather than presumed established results. No owner item is to be confirmed and no proposal submitted for moderation without human source and mathematical review.
The stochastic approach to insurance pricing treats discounted benefit payments and premium receipts as random variables driven by the policyholder's lifetime and contract cashflow design. This approach makes it possible to study expected insurance losses, second moments, tail probabilities and the adequacy of different premium rates. Promislow (2015), Chapters 15 and 16, develops insurance and annuity random present values, loss distributions, the equivalence principle and percentile premiums.
Earlier Prove2Me actuarial missions have considered expectations, moment identities and reserve recursions. This mission proposes a coherent additional distributional layer using genuinely random, state-dependent present values and finite scenario probabilities. Published mathematical methods already establish the underlying theory, but all of the Lean statements here remain original derived open theorem targets. None is a claim that a remote definition compilation supplies an accepted proof.
The outcome space is a finite set of insured lifetime scenarios indexed by Fin n. Each scenario i has probability mass p(i), discounted benefit B(i) in currency units and discounted unit-premium annuity A(i) in years or in the adopted unit-premium scaling. Valid probabilities satisfy
The index n=0 has no scenarios and cannot support a normalised probability law. Mathematical empty-sum identities remain useful for checking boundary behaviour, while actuarial interpretations require a nonempty normalised scenario space.
At annual premium rate P, the random prospective loss at issue is
Each component is evaluated on the same realised scenario, preserving the correlation induced by common survival time. Whole-life, endowment, and term insurance can be represented through the scenario benefit schedule. A term policy covers specified death years k < N; covered death benefits are paid at k+1 and a death at or after the term boundary receives no term death benefit. A separate endowment benefit may be payable at maturity.
The first group defines benefit PV, premium stream PV, scenario loss, term-cover PV and an endowment event split. The resulting identities relate prospective loss to the benefit and premium cashflows path by path. They include zero-benefit, zero-premium and scaling checks to preserve monetary units and exclude temporal ambiguity.
The key pathwise accounting identity is
While algebraically transparent, this equality determines which realised random variables must subsequently be used when constructing the loss distribution and its expectation.
Define finite expectation and second raw moment by
and define variance as E[L²]-(E[L])². Under nonnegative, normalised scenario masses, the variance theorem must prove nonnegativity by a genuine weighted convexity argument. There is no assumption that loss has only nonnegative support; negative losses correspond to profitable realised outcomes.
The equivalence-principle premium, under nonzero expected premium annuity E[A], is
The denominator condition is essential. Expense loading is explicitly separated as a monetary surcharge or premium margin. An additional group of identities handles finite portfolio expected values and their linearity. An additive variance model for genuinely independent portfolio losses requires independent stochastic assumptions and is not asserted automatically by the finite arithmetic expression.
The loss CDF is constructed from realised values, rather than introduced as an uninterpreted function:
It includes atoms exactly at the threshold. The exact mass at z is the sum over scenarios whose loss equals that value. Separate targets establish CDF nonnegativity, monotonicity, normalisation above the maximum of finite support and vanishing mass below its minimum. Every assertion that a weighted sum represents a probability includes nonnegative mass conditions.
Premium adequacy at annual rate P is
This is the actual probability that realised discounted premium income covers the realised discounted claims obligation. For a required solvency probability α, a percentile premium must satisfy Pr(L_P≤0)≥α. The probability level is prescribed explicitly, with meaningful economic values lying between zero and one. Where scenario annuity values are nonnegative, increasing the premium cannot reduce adequacy probability. Boundary equality is included in the adequately funded event.
For constant unit premium annuity, the premium-adequacy probability is the benefit CDF evaluated at the proposed premium. With variable annuity values, the full loss distribution must be used instead.
The theorem ActuarialValuation.pvRiskPremium_fundamental establishes three linked statements: the equivalence premium produces zero expected prospective loss, actual adequacy probability equals the CDF of realised loss at zero, and a premium increase preserves an explicitly stipulated percentile adequacy threshold. The first statement alone provides no claim that a given high percentile target is met. The capstone requires normalised nonnegative probability weights, nonnegative scenario premium annuity values, a nonzero expected annuity and an ordered comparison of premium rates.
The finite random-loss model provides a reusable way to distinguish pricing by expected cost from pricing by an insurance-loss tail constraint. These are central actuarial techniques documented by Promislow's stochastic insurance chapters. It also supplies a prospective bridge to a more general measurable lifetime and present-value integral model, compound aggregate insurance loss, reserve distributions and market risk allowances. The distinction between expectation, event probability and required percentile is preserved directly in Lean target types.
Positive expected-value pricing can coexist with a substantial probability of loss. For example, a two-scenario benefit distribution with mass two-thirds at zero and one-third at three has expected benefit one under a unit premium annuity. An equivalence premium of one covers claims in only two-thirds of scenarios, well below a ninety per cent adequacy threshold. This is a direct mathematical counterexample to treating an equivalence premium as universally percentile-adequate.
The assumptions that probabilities sum to one, that scenario masses are nonnegative, and that the premium annuity is nonnegative matter to the capstone. A zero denominator invalidates the equivalence funding equation. The CDF includes equality at its threshold, and scenario indices need not identify distinct loss values because separate scenarios may produce the same loss. Variance proof needs the actual normalised distribution.
There are 18 reusable definitions, 40 open theorem targets and 39 supporting milestone descriptions. The mission operates on a finite probability mass model with scenario-dependent discounted benefits and premium annuities. This provides exact finite loss distributions and mathematical endpoint handling without inventing a density for a discrete variable. Term benefits, endowment-event selection, expected benefits, second moments, loadings and percentile adequacy appear in the target graph.
Continuous loss distributions under exponential lifetime, nontrivial portfolio dependence, conditional reserve-loss distributions, general law-invariant risk measures and asymptotic aggregate premium principles require further work. This mission does not claim these general results, and no open sorry stub represents a verified theorem proof. Numerical examples are independent sanity checks, with the remote compiler serving solely as an authoritative declaration-elaboration validator.
Actuaries graduate crude observed mortality rates to remove sampling noise while retaining meaningful age-related mortality structure. Whittaker-Henderson graduation expresses the trade-off between goodness of fit to exposure-weighted crude experience and smoothness of the fitted series through successive finite differences. A 1968 Casualty Actuarial Society account of graduation describes the fit-plus-smoothness criterion. These public actuarial ideas motivate new Lean finite identities for mortality curves, rather than a claim to reprove the original general constrained optimisation theory.
The published source supplies the publicly verifiable actuarial context. Original finite Lean formalisation of the corresponding identities remains an open mathematical task; existing actuarial theory is not by itself an accepted formal proof. The input conventions and hypotheses should be compared to the source by an independent human reviewer.
At ages i the crude empirical rate q(i), graduated rate g(i) and nonnegative reliability weight w(i) define weighted squared error F=sum w(i)(g(i)-q(i))². Adjacent differences are Delta g(i)=g(i+1)-g(i); second differences are Delta²g(i)=g(i+2)-2g(i+1)+g(i). The first and second order roughness measures are finite sums of squared differences. A nonnegative chosen smoothing coefficient lambda yields a composite objective F+lambda S. Uniformly shifting graduated rate levels leaves first and second differences unchanged, though its fit to unchanged crude rates changes.
All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.
This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.whFitPenalty_exact`:
No data fidelity loss arises when graduated values exactly equal crude rates.
This block contributes 12 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.whSecondDifference_linear`:
An affine age trend has exactly zero second finite difference.
This block contributes 10 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.whMortalityGraduation_fundamental`:
The capstone proves a nonnegative exposure-weighted mortality graduation criterion and exact curvature shift invariance under explicit weighting assumptions.
The main target is `ActuarialValuation.whMortalityGraduation_fundamental`. The capstone proves a nonnegative exposure-weighted mortality graduation criterion and exact curvature shift invariance under explicit weighting assumptions.
The Casualty Actuarial Society graduation discussion describes a formal balance between smoothness and fidelity. The staged theorem chain provides reusable mortality graduation quality-control identities: exact finite-difference stencils, nonnegative penalised objective, zero second-difference penalty for affine curves and invariance of smoothing to a uniform level shift. These underlie future actuarial model verification and penalised optimisation work.
The two finite difference orders are not interchangeable. A first difference of an affine age pattern is constant but is not necessarily zero; a second difference of that same pattern vanishes. The fit statistic must account for exposure or credibility weights that cannot be negative when nonnegative loss is asserted. Penalty invariance under adding a constant does not imply that the whole fit-plus-smoothing objective is invariant, because observed crude rates are held fixed. An unconstrained fitted mortality curve may fall outside the unit interval, so future operational graduation must constrain rates appropriately. This mission has no unproved claim of existence or uniqueness of an optimal graduated curve.
The smoothness weight controls the model's balance between observed experience and regularity, and its selection is not proved optimal by the finite arithmetic identities here. An affine age curve has zero second difference but can have a positive first-difference roughness sum. Shifting all estimated rates changes residual fit against unchanged crude rates even as it leaves smoothness unchanged. A realistic actuarial graduation may need constraints on graduated rates, tail-age behaviour and end-condition conventions; these need separate explicit mathematical and administrative specifications. The open theorem set makes these distinctions visible.
Lean formalisation uses finite deterministic age-indexed functions, nonnegative observation weights, squared error and one- or two-step finite differences. The main theorem is the conjunction of exact penalty properties, not a statistical estimator bias theorem or an optimisation convergence theorem. It leaves endpoint boundary conditions and monotonicity constraints for later missions. A genuinely nontrivial result is the simultaneous control of weighted fit and roughness plus invariance of curvature to mortality-level translation.
These are original derived finite/analytic Actuarial formalisation targets rather than reproductions of a numbered textbook theorem. Exact Lean binder declarations are authoritative. The complete source uses targeted Mathlib imports and no local Lean compilation. All proof bodies remain genuinely open and do not constitute accepted results. Assumptions about negative rates, survival, policyholder independence, discounting and benefit timing must be preserved exactly.
Insurance solvency capital requires aggregating standalone module risk charges while acknowledging diversification. The EU Solvency II Directive Article 104 and Annex IV prescribe a correlation aggregation formula; the UK Prudential Regulation Authority's standard formula annexes likewise contain explicit correlation matrices. This mission formalises original exact finite-risk mathematical identities behind the quadratic calculation rather than attempting to certify a particular insurer's regulatory solvency position.
The published source supplies the publicly verifiable actuarial context. Original finite Lean formalisation of the corresponding identities remains an open mathematical task; existing actuarial theory is not by itself an accepted formal proof. The input conventions and hypotheses should be compared to the source by an independent human reviewer.
For insurance risk modules i and j, standalone charges are nonnegative real capital amounts C_i and correlation coefficients rho_ij. The squared correlation-based aggregation is Q=sum over ordered i,j of rho_ij C_i C_j. For exactly two risks and symmetric common pairwise rho, the calculation reduces to C_1 squared plus C_2 squared plus twice rho C_1 C_2. Squared capital before diversification is (C_1+C_2)^2. The difference equals 2(1-rho)C_1 C_2, which is nonnegative when rho is at most one and both standalone charges are nonnegative.
All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.
This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.scrUncorrelatedSquared_nonneg`:
Zero-correlation squared risk capital is nonnegative.
This block contributes 10 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.scrPairShift_zero`:
A zero change in module capital leaves aggregate squared risk unchanged.
This block contributes 10 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.scrSolvencyDiversification_fundamental`:
The capstone proves nonnegative two-module correlation aggregation, exact squared diversification reconciliation and the full-correlation limit.
The main target is `ActuarialValuation.scrSolvencyDiversification_fundamental`. The capstone proves nonnegative two-module correlation aggregation, exact squared diversification reconciliation and the full-correlation limit.
The Solvency II legislative Annex IV gives a source-defined matrix arrangement for module capital aggregation. Exact factor-of-two handling of off-diagonal cross terms is a crucial capital calculation audit. The capstone proves the squared two-risk diversification identity, boundedness under nonnegative coefficients and the perfect positive correlation limit. These results are reusable for portfolio risk controls and independent testing of actuarial capital spreadsheets.
A symmetric numerical matrix of pairwise values in [-1,1] is not automatically positive semidefinite. Nonnegative quadratic aggregation of arbitrary signed real vectors requires matrix PSD or a suitable structural hypothesis; the finite portfolio theorem instead explicitly restricts all charges and coefficients to be nonnegative. For two positive risk charges, rho must not exceed one for the usual diversification inequality, and it is only the squared capital that is compared. Cross terms are counted for both ordered pairs. The regulatory total may include operational risk, intangible risk, tax-loss absorption and other adjustments beyond basic quadratic capital, so their signs cannot be invented.
The correlation matrix in a published regulatory formula must be distinguished from an arbitrary symmetric matrix. Two at-most-unit correlated modules with nonnegative standalone risk values enjoy the displayed squared diversification relation; negative or inconsistent correlation matrices can invalidate unsupplemented general positivity assumptions. In actual Solvency II reporting, the calculation depends on rule jurisdiction, date, calibration and other adjustments. The theorem graph is a source-grounded mathematical component, not a regulatory determination or an audit opinion about an individual insurer.
The Lean formalisation is finite real algebra for risk charges and selected correlation weights, with a two-module squared capital capstone. It does not assert unconditional positivity for a non-PSD arbitrary coefficient matrix or identify the square root with regulatory capital without establishing nonnegative Q. Published correlation matrices and applicable UK or EU dates must be checked independently when the formula is deployed operationally. The theorem statements concern a nontrivial parametric diversification identity, not a preselected numerical example.
These are original derived finite/analytic Actuarial formalisation targets rather than reproductions of a numbered textbook theorem. Exact Lean binder declarations are authoritative. The complete source uses targeted Mathlib imports and no local Lean compilation. All proof bodies remain genuinely open and do not constitute accepted results. Assumptions about negative rates, survival, policyholder independence, discounting and benefit timing must be preserved exactly.
Lee-Carter mortality projection is a standard demographic approach for age-specific death rates. The original paper by Ronald Lee and Lawrence Carter represents mortality's logarithm by age effects and a common period index. The Social Security Administration working paper explains this decomposition and the identifiability of its components. Existing actuarial literature establishes the parametric modelling framework; these are original formal finite-period Lean identities, not claims that the statistical model has been estimated from historical data.
The published reference identifies the actuarial basis for this work. Finite source-specific formalisation in Lean remains open, while the motivating mathematical identities and technical conventions have been established in the literature. No compiler check or numerical example is an accepted proof.
At attained age x and calendar year t, the central death rate is m(x,t)=exp(a_x+b_xk_t), with a_x a real age intercept, b_x real age sensitivity and k_t a real period-mortality index. A deterministic index drift evolves k_(t+n)=k_t+ndelta. The ratio of death rates at the same age across two periods equals exp(b_x*(k_2-k_1)). With a nonnegative constant annual mortality force m, one-year survival is exp(-m) and the one-year death probability equals one minus survival. A cohort receives a possibly age- and calendar-dependent sequence of valid annual death probabilities q(t).
All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.
This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.lcRate_zero_period`:
Zero index gives the exponential age baseline.
This block contributes 12 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.lcDeathProb_partition`:
One-year death and survival are complementary under a constant-force basis.
This block contributes 12 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.lcLeeCarterCohort_fundamental`:
The comprehensive mortality capstone links Lee-Carter rate ratios with a valid bounded cohort survival law and the nonnegative one-year insured death outgo.
The main target is `ActuarialValuation.lcLeeCarterCohort_fundamental`. The comprehensive mortality capstone links Lee-Carter rate ratios with a valid bounded cohort survival law and the nonnegative one-year insured death outgo.
The original Lee-Carter work provides a tractable time trend at each age. The proposed mathematical capstone relates positive death-rate projections and exact age-specific rate ratios to bounded multi-year cohort survival and nonnegative expected death benefit cost. These are reusable model-validation invariants for actuarial mortality improvement projections. Unlike a fitted SVD or time-series forecast, a deterministic rate ratio and finite survival-product identity can be verified without probabilistic estimation assumptions.
A central death rate is not automatically an annual death probability. Converting an annual nonnegative force of mortality to a death probability is a modelling assumption that holds with constant force during the year. The one-year mortality intensities for ages x+n along a cohort differ from fixed-attained-age period ratios. The survival product must use each cohort-year probability exactly once, with half-open endpoints. Negative annual force values would yield death probabilities outside the unit interval and are ruled out where probabilities are asserted. The Lee-Carter age and period parameters have a scale identifiability ambiguity unless normalisation constraints are imposed; the current mission asserts algebraic properties at fixed parameters, not unique statistical identification.
An age-period mortality rate for the same age in two calendar years is not the same as the sequence of annual death probabilities experienced by a cohort ageing through consecutive calendar years. The first is represented as a ratio of positive central mortality intensities; the second is a product of annual survival complements. The conversion from a rate to an annual probability assumes a constant mortality force within the relevant year. That is distinct from mere exponential positivity and must be explained in review. A parameter-normalisation condition would be necessary for identification from experience data, but fixed-parameter algebraic rate-ratio statements do not require identifying the historical fit.
The nontrivial root combines exact exponential rate-ratio reconstruction with independent multi-year cohort survival bounds and expected assurance claims, rather than using degenerate zero mortality or zero benefits. It does not prove statistical estimation, covariance forecasts, stochastic longevity risk, or maximum likelihood fit. The formalisation uses Lean real arithmetic, exp, finite products and model-specific definitions, preserving positivity assumptions. Future missions can reuse the established identities for age-calendar time series and annuity valuation of projected cohorts.
These are original derived finite/analytic Actuarial formalisation targets rather than reproductions of a numbered textbook theorem. Exact Lean binder declarations are authoritative. The complete source uses targeted Mathlib imports and no local Lean compilation. All proof bodies remain genuinely open and do not constitute accepted results. Assumptions about negative rates, survival, policyholder independence, discounting and benefit timing must be preserved exactly.
Actuarial insurance ratemaking links expected claims to a measurable risk exposure base. The Actuarial Standards Board's adopted ASOP No. 53 discusses prospective risk-transfer cost estimation, exposure data and risk classification. Pure premium is cost per earned exposure and class relativity describes relative risk classification in a common basis. The already established actuarial pricing framework is conceptual; the new Lean statements are independent original finite proofs.
The published reference identifies the actuarial basis for this work. Finite source-specific formalisation in Lean remains open, while the motivating mathematical identities and technical conventions have been established in the literature. No compiler check or numerical example is an accepted proof.
A finite set of insured risk classes has real earned exposures e(i), loss costs L(i), and selected dimensionless relativity r(i). A common base pure rate b charges technical class premium e(i)br(i), summed over n disjoint classes. The relativity-weighted exposure W equals the sum of e(i)r(i). A projected aggregate claim cost T is financed by the uniquely calibrated b=T/W, provided W is strictly positive. Historical observed pure premium L/E is distinct from the future indicated base price.
All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.
This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.ratClassIndicated_zero`:
No expected frequency leads to zero prospective claim cost.
This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.ratPortfolioPremium_factor`:
The class-by-class technical premium sum factors as base rate times weighted exposure.
This block contributes 12 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.ratExposureRatemaking_fundamental`:
The substantive pricing capstone establishes unique expected-cost calibration and technical premium adequacy across arbitrary finite risk-class exposures and relativities.
The main target is `ActuarialValuation.ratExposureRatemaking_fundamental`. The substantive pricing capstone establishes unique expected-cost calibration and technical premium adequacy across arbitrary finite risk-class exposures and relativities.
The ASOP No. 53 actuarial source distinguishes the estimate of future insured losses from an actual insurance selling price. This formal capstone proves a finite technical income reconciliation and a nonnegative adequacy margin at every higher base rate. These equalities form reusable verification invariants for actuarial pricing spreadsheets and downstream portfolio studies. It does not assert legally required rates, premium tax calculations, competitive prices or expense allocation.
Class exposures must be measured on consistent earned-exposure units. Multiplying a base rate by class relativity differs from taking an unweighted arithmetic average of class premiums. If weighted exposure vanishes, a positive target cannot be financed by the formula T/W. Positivity of weighted exposure is needed for uniqueness and increasing-rate adequacy; aggregate exposure alone may be positive even when all relativity weights are zero. A risk class's exposure may be zero without invalidating the rest of the portfolio. A zero-cost portfolio is a legitimate boundary, but an all-zero exposure scenario cannot establish the general theorem.
A pure indicated base premium in this finite model covers only the agreed future claim-cost total. It is not a retail premium including expense, profit, risk or market pricing provisions. The same class relativity and earned exposure basis must be applied to the projected loss cost and to the pricing denominator. An independent actuarial reviewer can identify a rate-mix error through the exact aggregate reconstruction identity. Positive aggregate unweighted exposure alone does not ensure a viable technical rate when all class multipliers vanish. The central root is therefore a financial reconciliation theorem with an explicit strictly positive relativity-weighted exposure base.
The formal verification uses finite indexed class data and real-valued expected costs, independent of claim arrival distributions or stochastic credibility assumptions. The results prove exact identities over a fixed prospective insurance coverage basis. A loading for acquisition costs, handling expenses, reinsurance and capital is not automatically contained in the technical pure premium. The nontrivial root quantifies over every alternative higher rate and proves genuine fiscal sufficiency, rather than only simplifying an illustrative numeric case.
These are original derived finite/analytic Actuarial formalisation targets rather than reproductions of a numbered textbook theorem. Exact Lean binder declarations are authoritative. The complete source uses targeted Mathlib imports and no local Lean compilation. All proof bodies remain genuinely open and do not constitute accepted results. Assumptions about negative rates, survival, policyholder independence, discounting and benefit timing must be preserved exactly.
Insurers price retention layers and assess high-loss capital needs using distributions of losses rather than only expected claim frequency. The Rockafellar and Uryasev paper introduces an auxiliary loss-threshold objective for conditional value-at-risk (CVaR). Its central mathematical construction uses expected positive-part loss in excess of a threshold. The existing core relationship is established in published research; this mission develops an original exact finite-scenario Lean formalisation and its necessary boundary lemmas.
The published reference identifies the actuarial basis for this work. Finite source-specific formalisation in Lean remains open, while the motivating mathematical identities and technical conventions have been established in the literature. No compiler check or numerical example is an accepted proof.
A finite lattice has loss outcomes 0 through n with nonnegative scenario weights p(k); it represents a genuine probability distribution only when their total mass equals one. The scenario-weighted excess above deductible t is SL_n(t)=sum p(k) max(k-t,0), which is an expected excess under unit total probability mass. For a conventional confidence alpha in the interval from zero to one, excluding one, the tail fraction 1-alpha is positive and the Rockafellar-Uryasev objective equals F_alpha(t)=t+SL_n(t)/(1-alpha). The lower cumulative weight includes outcomes less than or equal to an integer threshold, while upper mass includes only strictly larger losses. Both parts partition the original finite weight. The Lean identities hold more generally for nonnegative weights without asserting that unnormalised weights define a probability distribution or that a fixed threshold minimises the CVaR objective.
All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.
This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.tailExcess_antitone_threshold`:
Increasing retention cannot increase payment for any insured loss.
This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.tailScaledExcess_nonneg`:
Nonnegative expected excess and positive tail mass make a valid nonnegative scaling.
This block contributes 10 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.tailTailRisk_fundamental`:
The root joins tail-stop-loss nonnegativity, a valid finite CVaR objective, monetary translation equivalence and the complete lower/upper finite loss partition.
The main target is `ActuarialValuation.tailTailRisk_fundamental`. The root joins tail-stop-loss nonnegativity, a valid finite CVaR objective, monetary translation equivalence and the complete lower/upper finite loss partition.
A scenario version of the original threshold-objective formula is suitable for rigorous deterministic actuarial software tests. Exact stop-loss monotonicity and positive-part partition results are reusable in reinsurance, capital modelling and future optimisation research. The root proves meaningful financial invariants: nonnegative ceded loss, a lower bound on the objective, and translation equivariance. It does not claim the objective at an arbitrary threshold equals optimised CVaR: that requires minimising over all thresholds.
The confidence parameter alpha must remain below one, otherwise dividing by 1-alpha is meaningless and positivity assertions fail. An insurance loss scenario at the deductible does not generate ceded payout. Discrete distributions may have atoms at a quantile: the conditional tail mean needs partial quantile mass, not a careless conditional expectation over all values greater than or equal to VaR. The auxiliary objective is defined for all real thresholds, whereas finite lower/upper tails at integer thresholds have disjoint endpoints. Zero-weight degenerate examples cannot establish the universal finite-scenario results.
The Rockafellar-Uryasev expression represents an auxiliary objective at each fixed threshold, with the conditional value-at-risk corresponding to its minimum across thresholds. This proposal does not assume that any arbitrary deductible minimises the objective. It distinguishes a discrete lattice distribution from a continuous loss density and preserves the mass at the chosen deductible in the lower tail. Taking a retention greater than every possible loss yields zero ceded premium but does not prove a zero capital risk measure. Translation equivariance follows from shifting the distribution of losses and the monetary threshold together, so the excess layer itself remains unchanged.
This is finite-scenario algebraic formal verification of a classic CVaR optimisation building block, not a continuous-distribution conditional expectation theorem or an assertion of existence/uniqueness of the minimising quantile. No distributional normalisation is inferred from truncated mass. The meaningful root relates insurance payouts, risk objective and monetary translation across arbitrary finite nonnegative-weight scenarios, preserving explicit denominator restrictions. This supports independent future research on convexity, optimum threshold and subadditive coherent measures.
These are original derived finite/analytic Actuarial formalisation targets rather than reproductions of a numbered textbook theorem. Exact Lean binder declarations are authoritative. The complete source uses targeted Mathlib imports and no local Lean compilation. All proof bodies remain genuinely open and do not constitute accepted results. Assumptions about negative rates, survival, policyholder independence, discounting and benefit timing must be preserved exactly.
Actuarial collective risk models represent total insured losses as a random sum of claim amounts. Panjer's 1981 recursive method provides a tractable recurrence for aggregate probabilities when claim number frequencies belong to a specified recursive family. This mission uses the compound Poisson special case and formalises the positive integer severity setting, its unique recursive aggregate law and finite premium and stop-loss valuations.
Finite actuarial valuation follows published literature, including the principal source. Definitions of finite event counts, valuation factors and cashflow conventions must be treated as precise mathematical inputs. Established actuarial formulae motivate the work; the resulting Lean statements are newly proposed, open proof obligations. The formal verification of these identities is distinct from the existing published mathematical development. No numerical tests or private declaration compilation constitute accepted proofs.
For intensity lambda at least zero, individual claims take strictly positive integer sizes with nonnegative severity weights f(j) and f(0)=0. The Poisson zero-claims probability is exp(-lambda). The aggregate loss lattice masses g(k) satisfy, for positive k, kg(k)=lambdasum from j=1 to k of j*f(j)*g(k-j). A finite probability prefix through n gives G(n), and 1-G(n) denotes its modelled tail complement. Finite expected aggregate loss and stop-loss premiums use sums over outcomes 0 through n.
All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.
This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.panjerWeightedSeverity_nonneg`:
Nonnegative severity weights yield nonnegative weighted claims.
This block contributes 12 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.panjerRecurrence_nonneg`:
The uniquely generated compound Poisson aggregate coefficients are nonnegative for nonnegative severity probabilities.
This block contributes 12 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.panjerAggregateClaims_fundamental`:
The compound aggregate-claims capstone combines genuine uniqueness of forward Panjer probabilities, nonnegative aggregate coefficients, and finite claim-payment cost controls.
The main target is `ActuarialValuation.panjerAggregateClaims_fundamental`. The compound aggregate-claims capstone combines genuine uniqueness of forward Panjer probabilities, nonnegative aggregate coefficients, and finite claim-payment cost controls.
The core mathematical target proves that the forward recurrence with initial mass exp(-lambda) uniquely fixes all aggregate coefficients, and nonnegative intensity and severities imply all generated probabilities are nonnegative. It reconciles finite cumulative mass, a tail complement, expected aggregate costs and a ceded stop-loss value in one theorem. The recursion is independent of the earlier negative-binomial frequency moment mission and leaves full normalisation to separate infinite-series proofs.
The importance of the formal result is the availability of reusable, source-explicit lemmas for actuarial software audit and subsequent independent mathematical work, as described in the published literature. It goes beyond an elementary reformulation of a single definition: the capstone simultaneously connects different construction stages with monetary or probability reconciliation and precise boundary hypotheses. A vacuous zero-data specialisation cannot establish these relationships across arbitrary finite schedules.
The Panjer recurrence requires positive severity weights indexed by j at least one: allowing mass at zero without changing the denominator and initial atom would make its stated form incorrect. For each target k only lower aggregate coefficients occur, permitting ordinary induction. Natural index subtraction is safe within the finite sum. Intensity positivity, exp(-lambda), natural-to-real index coercions and finite sums need careful attention. This mission deliberately does not claim that a truncated mass automatically sums to one or that a sequence satisfying the recurrence is normalised without additional assumptions.
The recurrence's fundamental mathematical content is the coefficient-by-coefficient relationship: the coefficient at aggregate loss k depends only on previously calculated coefficients for aggregate losses strictly below k. This well-founded structure supports a strong induction proof of uniqueness and a separate positivity induction. The initial mass exp(-lambda) applies only when each individual claim has strictly positive loss. A severity law admitting zero-sized claims would require a corrected initial atom and a denominator involving zero-severity mass. A finite prefix of recursive probabilities is not, by itself, an exhaustive law. Further work can add total normalisation and full expectation formulas.
The model is the lattice compound Poisson special case, not the full (a,b,0) family. It distinguishes algebraic consistency of a recursively specified sequence from measure-theoretic construction of an independent compound random sum. Stop-loss results are finite truncation values and do not by themselves control omitted large losses. All theorems are original derived proof obligations.
These are original derived finite/analytic Actuarial formalisation targets rather than reproductions of a numbered textbook theorem. Exact Lean binder declarations are authoritative. The complete source uses targeted Mathlib imports and no local Lean compilation. All proof bodies remain genuinely open and do not constitute accepted results. Assumptions about negative rates, survival, policyholder independence, discounting and benefit timing must be preserved exactly.
Insurance mortality investigations must estimate survival probabilities from observed death times when some policyholders leave the observation period alive or are otherwise censored. Kaplan-Meier product-limit estimation accounts for changing risk sets without misclassifying censoring as mortality. Greenwood's formula provides the standard plug-in variance. This mission formalises a finite death-event calculation, risk-set conservation, survival probability bounds, and variance bookkeeping.
Finite actuarial valuation follows published literature, including the principal source. Definitions of finite event counts, valuation factors and cashflow conventions must be treated as precise mathematical inputs. Established actuarial formulae motivate the work; the resulting Lean statements are newly proposed, open proof obligations. The formal verification of these identities is distinct from the existing published mathematical development. No numerical tests or private declaration compilation constitute accepted proofs.
At event index t, the observed risk set r_t is positive immediately before deaths d_t and censoring exits c_t. Deaths and censors are disjoint exits, so the next risk set is r_t-d_t-c_t under a closed cohort. The estimated event-time mortality fraction is d_t/r_t, and Kaplan-Meier survival through n events is the product of 1-d_t/r_t. The complement gives cumulative failure. With 0<=d_t<r_t, the Greenwood term is d_t/[r_t(r_t-d_t)] and the plug-in variance equals squared Kaplan-Meier survival times the accumulated terms. Restricted expected lifetime is a finite sum of discrete event-time survival values.
All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.
This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.kmRiskExit_reconciliation`:
Death and censoring hazards sum to the total observed exit fraction, without equating censoring to mortality.
This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.kmSurvivalStep_exact`:
An actuarial one-step survival update reproduces the exact finite product-limit estimate.
This block contributes 12 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.kmSurvivalGreenwood_fundamental`:
The survival-estimation capstone proves the complete finite product-limit probability bounds, failure partition, nonnegative Greenwood plug-in variance and nonnegative restricted expected lifetime under strict risk-set conditions.
The main target is `ActuarialValuation.kmSurvivalGreenwood_fundamental`. The survival-estimation capstone proves the complete finite product-limit probability bounds, failure partition, nonnegative Greenwood plug-in variance and nonnegative restricted expected lifetime under strict risk-set conditions.
The capstone formally guarantees nonnegative and nonincreasing estimated survival, a complementary failure estimate, a nonnegative Greenwood plug-in variance and a nonnegative finite restricted survival integral under actual risk-set constraints. These are foundational actuarial quality-control properties and provide reusable results for experience investigations. Censoring changes future exposures but never directly counts as a death.
The importance of the formal result is the availability of reusable, source-explicit lemmas for actuarial software audit and subsequent independent mathematical work, as described in the published literature. It goes beyond an elementary reformulation of a single definition: the capstone simultaneously connects different construction stages with monetary or probability reconciliation and precise boundary hypotheses. A vacuous zero-data specialisation cannot establish these relationships across arbitrary finite schedules.
The denominator must be the individuals under observation immediately before the death time. Event times with r_t=0 cannot identify a mortality fraction. The Greenwood variance denominator requires r_t>d_t, so a complete mortality exhaustion creates a boundary where the usual finite formula is undefined. Event-time product-limit survival does not imply unbiasedness under informative censoring; external independent/non-informative censoring assumptions are needed for statistical interpretations. A finite restricted sum is not an unrestricted life expectancy.
The survival estimator is indexed by actual ordered death-event intervals. People censored without dying should remain at risk up to the censoring time and leave the risk set before later events. At a death time the product decreases by a factor determined by deaths divided by lives currently exposed. At an interval containing no death the factor is one. The standard Greenwood approximation is only defined when positive risk remains after each included set of deaths. At complete exhaustion the finite variance expression must not divide by zero, and the usual delta-method approximation should not be silently extrapolated. Results are deterministic algebra conditional on observed counts, not a theorem of statistical consistency.
The formal model is a finite discrete event-time product-limit calculation, not an asymptotic proof of the Kaplan-Meier central limit theorem. Greenwood's plug-in formula is defined explicitly; the proof obligations concern its algebra, positivity and consistency under stated boundaries, not an estimator-unbiasedness claim. Definitions and milestones are original derived Lean results based on published survival-method principles.
These are original derived finite/analytic Actuarial formalisation targets rather than reproductions of a numbered textbook theorem. Exact Lean binder declarations are authoritative. The complete source uses targeted Mathlib imports and no local Lean compilation. All proof bodies remain genuinely open and do not constitute accepted results. Assumptions about negative rates, survival, policyholder independence, discounting and benefit timing must be preserved exactly.
Paid claims triangles are essential to general insurance reserving. A volume-weighted chain-ladder link factor projects mature cohorts and less developed cohorts to a common ultimate basis, while the Bornhuetter-Ferguson approach apportions expected remaining development to an externally selected prior ultimate. This research mission formalises their finite algebra and reconciliation without asserting stochastic unbiasedness or any distributional variance result.
Finite actuarial valuation follows published literature, including the principal source. Definitions of finite event counts, valuation factors and cashflow conventions must be treated as precise mathematical inputs. Established actuarial formulae motivate the work; the resulting Lean statements are newly proposed, open proof obligations. The formal verification of these identities is distinct from the existing published mathematical development. No numerical tests or private declaration compilation constitute accepted proofs.
For origin cohort i and development period j, C(i,j) denotes cumulative paid claims and C(i,j+1)-C(i,j) the incremental payment. A selected rectangular subset with m fully observed adjacent development points gives volume-weighted factor f_j=Sum_i C(i,j+1)/Sum_i C(i,j), defined when denominator nonzero. The remaining development factor F is the finite product of selected nonnegative age-to-age factors. Latest cumulative paid claims C yield chain-ladder ultimate CF and reserve C(F-1). With a priori ultimate E, the Bornhuetter-Ferguson unpaid reserve is E(1-1/F), and ultimate is C plus this reserve.
All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.
This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.clVolumeFactor_nonneg`:
Nonnegative claims and positive exposure make the factor nonnegative.
This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.clUltimate_reserve_balance`:
The ultimate estimate is reconciled exactly to paid claims and outstanding reserve.
This block contributes 14 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem `ActuarialValuation.clClaimsDevelopment_fundamental`:
The substantive capstone reconciles a volume-selected development factor, chain-ladder valuation, and prior-informed Bornhuetter-Ferguson valuation under economically meaningful positivity assumptions.
The main target is `ActuarialValuation.clClaimsDevelopment_fundamental`. The substantive capstone reconciles a volume-selected development factor, chain-ladder valuation, and prior-informed Bornhuetter-Ferguson valuation under economically meaningful positivity assumptions.
The main formal result reconciles numerator and denominator on the same observed cohorts, the paid-plus-reserve valuation identity, and the exact BF weighted-average identity. It distinguishes development-based estimation from prior-informed estimation. These results establish algebraic control checks for finite claims triangles and reserve reconciliation, while reserving model predictive uncertainty for subsequent work.
The importance of the formal result is the availability of reusable, source-explicit lemmas for actuarial software audit and subsequent independent mathematical work, as described in the published literature. It goes beyond an elementary reformulation of a single definition: the capstone simultaneously connects different construction stages with monetary or probability reconciliation and precise boundary hypotheses. A vacuous zero-data specialisation cannot establish these relationships across arbitrary finite schedules.
Carefully distinguish incremental and cumulative claims, the latest paid amount and ultimate, and an age-to-age factor from the full age-to-ultimate product. The volume-weighted ratio must use identical origin-cohort exposure in numerator and denominator. A zero denominator does not identify a meaningful observed development factor. The BF interpretation requires strictly positive F, and nonnegative future emergence requires F at least one. Deterministic algebra is not an assertion of unbiased predictive reserves without statistical assumptions.
An age-to-age ratio represents an explicitly selected rectangular cohort of observed cumulative values. It is not interchangeable with the arithmetic average of individual ratios, nor with an extrapolated tail factor after the final observed maturity. Where paid and incurred development patterns differ, separate sets of factors must be selected. In the Bornhuetter-Ferguson setting the chosen external expectation must correspond to the same basis as the observed cumulative paid claims: mixing paid claims and reported case estimates invalidates the interpretation. The final theorem therefore tests linked monetary identities rather than asserting an objective optimal reserve.
The task is finite deterministic paid-claims reserving in a single currency and valuation date, with nonnegative selected cumulative development under hypotheses of each theorem. It formalises no future-tail extrapolation or reserve variability, and no stochastic independence or Mack conditional unbiasedness. Every theorem is an original derived Lean goal requiring an accepted proof.
These are original derived finite/analytic Actuarial formalisation targets rather than reproductions of a numbered textbook theorem. Exact Lean binder declarations are authoritative. The complete source uses targeted Mathlib imports and no local Lean compilation. All proof bodies remain genuinely open and do not constitute accepted results. Assumptions about negative rates, survival, policyholder independence, discounting and benefit timing must be preserved exactly.
Actuarial valuation of these claims follows published actuarial mathematics (https://www.actuaries.org.uk/system/files/documents/pdf/0042-0071.pdf). Actuarial commutation functions provide a classical exact connection between life-table survival and financial discounting. The Institute of Actuaries' International Actuarial Notation (1969–1971) records the discounted survivor and death columns and their accumulated insurance and annuity values. Dickson, Hardy and Waters' Actuarial Mathematics for Life Contingent Risks (2009), Chapter 5, explains valuation relationships between assurance and annuity payments. The objective is a connected proof of the finite-horizon assurance, annuity-due and pure endowment identity, rather than a cataloguing of conventional symbols.
Existing known actuarial results from the reference sources motivate these new original-derived theorem obligations. The actuarial formulae in published literature provide the mathematical background, while the proposed Lean theorem declarations are original derived results and remain unproved. A genuine nonzero mortality, service or insurance cashflow model is required rather than a vacuous zero-value special case. Accepted results will support more advanced actuarial valuation tasks.
An integer-age life table l(x) gives survivors at age x and decreases over the covered term. An annual discount factor v is strictly positive. The commutation functions are D_x=v^x l_x and C_x=v^(x+1)(l_x−l_(x+1)), where the second discounts a death claim payable at the end of the year. The finite sums N_(x:n)=sum_(k<n)D_(x+k) and M_(x:n)=sum_(k<n)C_(x+k) measure the temporary annuity-due and term assurance numerators. Terminal survival is D_(x+n), and the financial effective discount rate is d=1−v. Dividing the numerator sums by D_x gives the corresponding conditional actuarial present values, when D_x is nonzero.
All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.
This block contributes 10 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem ActuarialValuation.commLocalDeathIdentity:
One-year death and annuity-discount value telescope to the discounted decrement.
This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem ActuarialValuation.commGrossReconciliation_identity:
The full insurance-annuity-survival decomposition follows by telescoping annual differences.
This block contributes 12 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem ActuarialValuation.commAssuranceAnnuity_fundamental:
The capstone establishes both raw and conditional actuarial valuation identities and nonnegativity of term assurance and pure endowment for a valid life table.
The main target is ActuarialValuation.commAssuranceAnnuity_fundamental. The capstone establishes both raw and conditional actuarial valuation identities and nonnegativity of term assurance and pure endowment for a valid life table.
The resulting formal identities will be reusable by future downstream pensions and insurance valuation missions. Formal actuarial modelling of these obligations follows the published valuation literature (https://www.actuaries.org.uk/system/files/documents/pdf/0042-0071.pdf). The one-year identity C_x+(1−v)D_x=D_x−D_(x+1) telescopes over successive payment years. The resulting finite equation M_(x:n)+dN_(x:n)+D_(x+n)=D_x reconciles three disjoint financially timed payment streams. Dividing by D_x yields A_(x:n)+d a_due_(x:n)+E_(x:n)=1, connecting term assurance, annuity due and survival to term expiration. This provides a powerful actuarial valuation consistency check: omitting a terminal survival payment or discounting deaths at the beginning rather than end of the year breaks the identity.
The annual life-table decrement l_x−l_(x+1) must not be confused with the financial discount d=1−v. The annuity-due includes the initial survivor payment at age x whereas the insurance claim is payable at the end of the death year. Each exponent and half-open finite sum endpoint must be exact. The main telescope holds without assumptions on real survivor counts, but financial probability interpretation requires nonnegative remaining lives and a nonincreasing table, plus positive valuation-age survivors. The conditional ratios require a nonzero D_x. A general theorem must not assume the result by defining the numerator balance identically equal to D_x.
The model is a finite integer-age life table and a common annual interest basis with unit annuity payments, year-end unit death claims and a unit pure endowment at the terminal payment date. It makes no independence assumption about additional lives and introduces no continuous-time death benefits. The capstone proves the raw commutation identity, the actuarial present value identity and nonnegativity of its insurance and terminal survival components. All declarations are original Lean formalisation targets and should remain open until genuine accepted remote proofs exist.
These are original derived finite/analytic Actuarial formalisation targets rather than reproductions of a numbered textbook theorem. Exact Lean binder declarations are authoritative. The complete source uses targeted Mathlib imports and no local Lean compilation. All proof bodies remain genuinely open and do not constitute accepted results. Assumptions about negative rates, survival, policyholder independence, discounting and benefit timing must be preserved exactly.
Actuarial valuation of these claims follows published actuarial mathematics (https://onlinelibrary.wiley.com/doi/10.1002/9781119971528.ch6). Life insurance policy alterations require a mathematically coherent link between reserves, remaining premium payments, reduced paid-up cover and cash surrender settlements. In Fundamentals of Actuarial Mathematics, S. David Promislow develops the premium difference and paid-up formulae in Chapter 6, Section 6.8 (2010/2015). Dickson, Hardy and Waters cover policy values and alterations within life-contingent reserve theory. The proposal creates a connected sequence of algebraic valuation results for these choices, without treating a surrender value as if it were always the prospective reserve.
Existing known actuarial results from the reference sources motivate these new original-derived theorem obligations. The actuarial formulae in published literature provide the mathematical background, while the proposed Lean theorem declarations are original derived results and remain unproved. A genuine nonzero mortality, service or insurance cashflow model is required rather than a vacuous zero-value special case. Accepted results will support more advanced actuarial valuation tasks.
At the policy alteration date, B denotes the complete present value of remaining insured benefits and a denotes the value of future unit premium payments. A level premium P leaves prospective policy reserve V=B−Pa. An equivalent newly repriced premium Ps=B/a is meaningful for nonzero a. A reserve-equivalent paid-up benefit amount b is V/Aunit for an explicitly positive unit insurance present value Aunit. A proportionally scaled paid-up policy uses z=V/B under nonzero B. Cash surrender value is the simplified payment SV=V−c, with a separate nonnegative surrender deduction c not exceeding V when positive surrender cash is required. All values are compared at the same time and on one consistent actuarial valuation basis.
All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.
This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem ActuarialValuation.alterationRevisedReserve_delta:
Change in future premiums shifts the actuarial reserve by the monetised payment difference.
This block contributes 10 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem ActuarialValuation.alterationPaidUpScaledValue_add:
Paid-up proportional benefit values are additive over cover portions.
This block contributes 9 theorem targets and 3 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem ActuarialValuation.alterationReservePaidUpSurrender_fundamental:
Main policy alteration theorem reconciles prospective reserve, repriced premium difference and reserve-equivalent paid-up cover while bounding surrender cash by the reserve.
The main target is ActuarialValuation.alterationReservePaidUpSurrender_fundamental. Main policy alteration theorem reconciles prospective reserve, repriced premium difference and reserve-equivalent paid-up cover while bounding surrender cash by the reserve.
Formal actuarial modelling of these obligations follows the published valuation literature (https://onlinelibrary.wiley.com/doi/10.1002/9781119971528.ch6). The premium difference theorem identifies reserve as the present value of the deficiency of original premiums relative to the equivalent premium currently required for remaining benefits. Paid-up insurance conversion substitutes a benefit without future premiums whose expected present value equals the original reserve. A surrender amount can differ from reserve when contractual charges apply. These three concepts reconcile actuarially while remaining different legal and financial benefit forms. The final theorem combines premium-difference equivalence, paid-up monetary equivalence, and the inequality between surrender value and reserve under explicit reserve and deduction constraints.
Divisions in reprice and paid-up formulae require nonzero insurance and premium annuity denominators, unlike Lean's totalised real division. A prospective reserve can be negative if future premiums exceed remaining insurance values, so nonnegative paid-up cover and cash surrender cannot be claimed without additional assumptions. An actual policy might use tax, expenses, guaranteed surrender values, bonus distributions or other legal terms not represented by a simple deduction. The model distinguishes a change in premium value from a change in benefit value, and the capstone cannot be established through a degenerate zero-cover simplification.
The mathematical scope is a finite real-valued prospective valuation model supplied with consistent actuarial present-value factors. Benefit discounting, mortality basis and benefit dates are represented by B, a and Aunit rather than silently re-derived. Paid-up and surrender transformations are idealised actuarial alternatives and are not universal contractual entitlements. Asset share is a separate retrospective concept and is not asserted equal to prospective reserve without further matching assumptions. Every theorem remains an original derived obligation, suitable for subsequent extensions with expense-adjusted reserves and policyholder nonforfeiture terms.
These are original derived finite/analytic Actuarial formalisation targets rather than reproductions of a numbered textbook theorem. Exact Lean binder declarations are authoritative. The complete source uses targeted Mathlib imports and no local Lean compilation. All proof bodies remain genuinely open and do not constitute accepted results. Assumptions about negative rates, survival, policyholder independence, discounting and benefit timing must be preserved exactly.
Actuarial valuation of these claims follows published actuarial mathematics (https://www.actuary.org/sites/default/files/pdf/pension/fundamentals_0704.pdf). Projected unit credit funding is a standard actuarial cost method that allocates pension retirement benefits to completed and future years of service. The American Academy of Actuaries describes actuarial accrued liability and normal cost under this method in its Fundamentals of Pension Accounting and Funding (2004). The Institute of Actuaries' GN26 Pension Fund Terminology (2006) likewise describes projected unit cost, standard contribution rates and amortisation. This formal research mission unites the benefit attribution and funding calculations in a single theorem graph; it goes beyond the separate salary-linked CARE/DC terminal equivalence already treated in the preceding mission.
The actuarial formulae in published literature provide the mathematical background, while the proposed Lean theorem declarations are original derived results and remain unproved. A genuine nonzero mortality, service or insurance cashflow model is required rather than a vacuous zero-value special case. Accepted results will support more advanced actuarial valuation tasks.
A finite salary-linked accrued annual pension is expressed as a sum of projected retirement benefit units b(t). Completed service years occupy the half-open index range zero through n−1, and m future years occupy n through n+m−1. An actuarial retirement factor f is a survival probability times an appropriate discount and pension annuity package. The accrued liability is AL=f sum of completed service units, and projected future normal costs equal f times the future-service units. Assets A are measured at the same valuation date. An unfunded liability UL=AL−A is financed by a constant contribution rate c applied to a strictly positive present value F of pensionable payroll, and S=A+cF−AL is the resulting simplified funding surplus.
All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.
This block contributes 10 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem ActuarialValuation.pucTotalBenefit_zero:
An empty past and future service ledger has zero pension.
This block contributes 11 theorem targets and 5 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem ActuarialValuation.pucProjectedBenefitPV_add_future:
Projected PV of benefits increases by the normal cost of the added service year.
This block contributes 11 theorem targets and 3 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.
Representative theorem ActuarialValuation.pucPensionFunding_fundamental:
The pension funding capstone unites projected-unit service attribution with the exact amortisation contribution threshold and surplus monotonicity.
The main target is ActuarialValuation.pucPensionFunding_fundamental. The pension funding capstone unites projected-unit service attribution with the exact amortisation contribution threshold and surplus monotonicity.
Formal actuarial modelling of these obligations follows the published valuation literature (https://www.actuary.org/sites/default/files/pdf/pension/fundamentals_0704.pdf). A sound pension valuation must reconcile the projected retirement benefit with both its accrued-service obligation and the cost of further service. The formal capstone proves that PVFB=AL+PVFNC under the same projected benefit assumptions. It additionally derives the unique amortisation percentage c*=UL/F required to extinguish a nonnegative valuation-date deficit and proves that every larger contribution percentage produces nonnegative surplus. This establishes a financial and actuarial accounting invariant across valuation and funding. It can support later analyses of demographic experience, recovery periods, employer contributions and aggregate scheme funding, but does not automatically impose any scheme-specific contribution rule.
The benefit units represent projected retirement benefits, not past salary amounts at historical ages. Past and future service must be disjoint and exhaustion of the projection horizon must be exact. The pension factor uses the same assumed retirement date, survival and payment schedule throughout each monetary reconciliation; changes in that basis are outside the elementary attribution equation. Zero service and zero accrued liability are meaningful. A nonzero payroll base is necessary to define the funding rate, and positive payroll PV is essential to monotonicity. Deficits and surpluses are financially different: the chosen capstone assumes a nonnegative unfunded deficit rather than assuming the same payment solves an overfunded scheme under all regulations.
The formal results treat an individual or coherently aggregated finite service schedule under one actuarial pension valuation basis and a simplified fixed-rate contribution financing model. They do not assert that the selected funding contribution must be paid by law or accounting standards, nor do they value ancillary expenses, deferred pension increases or new entrants automatically. The main theorem combines independently defined actuarial benefit allocation, pension liability and funding surplus. Existing pension funding textbooks provide the conceptual derivations, while every proposed Lean statement is an original finite formalisation requiring a real proof.
These are original derived finite/analytic Actuarial formalisation targets rather than reproductions of a numbered textbook theorem. Exact Lean binder declarations are authoritative. The complete source uses targeted Mathlib imports and no local Lean compilation. All proof bodies remain genuinely open and do not constitute accepted results. Assumptions about negative rates, survival, policyholder independence, discounting and benefit timing must be preserved exactly.
A life table summarises the survival of a cohort as ages advance, as developed in Promislow's life-table treatment. It provides an essential common language for life assurance, retirement pensions, annuities and reserves. A credible mathematical foundation must make precise when cohort counts define conditional probabilities, how death counts exhaust an assumed closed population, and why commutation-function ratios give the same values as discounted contractual payments. The classical notation is widely understood by actuaries but can conceal assumptions at terminal ages, when there are no remaining lives, and at fractional-year payment times. This formalisation specifies those assumptions so that later results can safely reuse them.
Consider an integer-age cohort with a finite terminal age, nonnegative numbers alive and nonincreasing numbers alive at successive ages. At the terminal age the remaining cohort is zero. Annual deaths are the change in cohort size over each year. Conditional survival starts only from an age with positive numbers alive. The model also has a strictly positive annual discount factor; when the corresponding effective interest rate is nonnegative, that factor is at most one. A due annuity pays on the integer anniversaries at which the insured life is alive. An annual death assurance pays at the end of the year of death. All financial amounts in the principal identities are per unit benefit, valued at the attained age of the insured life.
The opening group establishes that successive survivor and decrement counts conserve population mass. Multi-year survival, conditional and deferred mortality must be derived from original counts and remain coherent under time shifts. A zero survivor count is an explicit boundary of the probability model. Curtate lifetime moments are obtained from the annual death distribution and its tail probabilities. Continuous survival and hazard relationships form an additional interface requiring a genuine measurable lifetime law. They cannot be inferred from the table's integer-age observations without an interpolation assumption.
The next group distinguishes age at selection from time since selection and attained age. Ultimate mortality applies only after the selection period. Within a year, uniform distribution of deaths, constant force and Balducci interpolation share the annual survival endpoints but generally disagree between those endpoints. Each conditional fractional formula needs the corresponding survival denominator to be positive. These are alternative models, so the choice of interpolation must remain explicit.
The commutation columns are defined from the same original survivor counts and a fixed annual discount factor. Discounted survivors and year-end deaths create the first-order columns. Their finite forward sums create the second-order columns, and sums of those forward sums create the increasing-benefit columns. In particular, for survivors the discounted payments begin at the valuation anniversary, whereas for deaths the benefit is payable on the next anniversary. The finite terminal convention is essential for the last year's assurance payment and the last available annuity instalment.
The principal goal is the finite-cohort valuation reconciliation. Starting independently from annual numbers alive and numbers dying, it shows that total deaths exhaust the living cohort, actual survival-contingent payments agree with the annuity commutation ratio, actual annual death payments agree with the assurance commutation ratio, and the commutation sums satisfy their backward recursions. It also establishes the capital identity expressing unit present value as annual death assurance plus the discount loss on an annuity due. No commutation recurrence or present-value equivalence is assumed as an input.
At a valid attained age, the annuity and assurance are independently valued from the original cohort:
As in the classical actuarial notation, the resulting interfaces separate cohort mass from conditional probability and separate mortality from the economic discounting basis. This allows a later actuarial mission to combine them only where the required regularity and positivity assumptions are available. The commutation functions supply a transparent independent valuation route alongside direct finite cashflow sums. Their ratios remain useful as a reference calculation even when production models value very large pension populations by member-level cashflow projection. Higher-order commutation functions are also the natural algebraic representation of benefits that increase by one unit at each subsequent anniversary.
This foundation is reusable across future mortality and valuation missions. A foundational namespace is particularly important where related existing missions already use different local definitions. The present work provides a compatibility design and a proposed canonical public vocabulary. Publishing an older equivalent definition does not automatically establish a safe imported interface: exact source, revision, visibility, and declaration signatures must match. The programme therefore retains existing mission objects and develops explicit crosswalks, rather than silently replacing earlier results.
The mathematical difficulty centres on conditioning, index shifts and independent routes to the same financial result. At an age with no survivors, a formal programming language may return a totalised value for division by zero even though no conditional mortality probability exists. A common off-by-one error values a death at the beginning of its year or includes a due annuity payment after the terminal age. Another error confuses discounting to time zero with discounting to the current valuation age. Select mortality requires careful handling of attained age at the exact selection-duration boundary. Continuous lifetime hazard and moment formulas additionally need probability, measurability and integrability assumptions that finite cohort counts alone cannot supply.
The double-sum commutation identities require a finite discrete summation argument and must use published survivor and assurance conventions consistently. The goal compares separate original definitions of payment cashflows, the population exit distribution, and commutation columns under one valid finite life table. The capstone is nonvacuous: a positive starting cohort and positive discount factor exclude empty-cohort substitutions and make its conditional cashflows meaningful. Its assumptions impose cohort validity and positive discounting; they do not contain the desired present-value identities as hypotheses.
The initial formalisation supplies a canonical finite discrete cohort and annual cashflow basis. The continuous-law, hazard, select and fractional-age results are separately specified mathematical interfaces and are admitted into the formal theorem inventory only when remote declaration checks establish valid Lean signatures. For nonnegative effective interest the annual discount is in the interval from zero excluded to one included. More general deterministic term structures and stochastic discount factors require separate cashflow definitions. Joint-life and competing-decrement extensions must identify any independence or mutually exclusive decrement assumptions explicitly.
No assertion is made about mortality being stationary across calendar years or about the validity of a specific mortality table for a real population. The terminal-age assumption imposes certain death before the finite horizon. It is unsuitable without modification for a lifetime distribution with unbounded support. A lifetime with an atom at an integer boundary requires separate weak and strict survival events; interpreting a one-year death count as the hazard at the death instant is unjustified without further assumptions. Open formal statements remain proof obligations, and passing a declaration check would establish only well-formed Lean, not the mathematical truth of the intended result.
The finite-cohort valuation targets also include independently defined one-year and multi-year mortality probabilities, temporary and deferred annuities, term and endowment assurances, pure endowments, annuities in arrears and increasing finite-term cashflows. The expanded candidate graph has 26 definitions and 48 open theorem obligations in a coherent reconciliation programme. New results require independently derived cashflow-to-commutation identities.
Related proposals LII (commutation functions and finite-term benefits) and LXII (fractional-duration valuation) are in review, not public importable dependencies. Existing published life-table theorems in XLV must be reused where signatures match. The fractional-age and continuous-law interfaces require separate hypotheses. All research targets remain provisional until their exact published-name overlap checks, authoritative declaration elaboration and dependency links are validated. No private diagnostic is counted as a proved theorem.
The increasing-insurance commutation column requires the relative-duration or absolute-age convention to be declared explicitly. Annual UDD includes certain year-end death, but constant-force and Balducci interpolation require annual death probability strictly below one.
Near thermodynamic equilibrium, flows of extensive quantities (heat, matter, charge) depend linearly on the thermodynamic forces (gradients of the conjugate intensive variables). Some of the coefficients in these linear laws describe cross effects: a temperature gradient driving an electric current (Seebeck effect), a voltage driving a heat flow (Peltier effect), a temperature gradient driving diffusion (Soret effect) and its converse (Dufour effect). In 1931 Lars Onsager showed that, as a consequence of the time-reversibility of microscopic dynamics, the matrix of these coefficients is symmetric (Onsager 1931). These reciprocal relations were confirmed experimentally across many classes of irreversible processes (Miller 1960), and Onsager received the 1968 Nobel Prize in Chemistry for them.
This mission formalizes the abstract, finite-dimensional version of Onsager's principle as presented by Landau and Lifshitz (Statistical Physics, Part 1, 1975) and reproduced in the Wikipedia article Onsager reciprocal relations, sections "Abstract formulation" and "Proof".
Let be the fluctuations of thermodynamic quantities from their equilibrium values. For small fluctuations the entropy is quadratic, and Boltzmann's formula gives the Gaussian fluctuation distribution
where is a positive definite symmetric real matrix and summation over repeated indices is understood. Write for the average of with respect to .
The thermodynamic conjugate quantities are . In the quasi-stationary regime the fluctuations relax according to for a real matrix , so that with the kinetic coefficients
The mean value at time of a fluctuation which takes the value at is , and . The time correlation is .
If fluctuations are symmetric under time reversal,
then the kinetic coefficients are symmetric:
The symmetry is the mathematical content of the reciprocal relations: it reduces the number of independent transport coefficients from to and yields the equality of measurable cross-coefficients such as the Peltier and Seebeck coefficients. Together with the positive semi-definiteness of the coefficient matrix (from non-negative entropy production), it is the basic structural input of linear irreversible thermodynamics.
The result is classical and its proof is standard in physics textbooks; this mission produces a machine-checked version of the finite-dimensional argument, including the Gaussian moment identity and the differentiation of the time-correlation function under the integral sign. No machine-checked formalization of this statement is known to the proposer.
The algebraic core is short, but the formal argument needs analytic input that the textbook treats as routine: Gaussian integrals over for a general positive definite (integrability of polynomial moments against and the value of the second moments), differentiation of a parameter-dependent integral involving a matrix exponential, and the passage from an identity valid for to an identity between derivatives at the boundary point , where only a one-sided derivative is available from the hypothesis.
Vectors are functions Fin n → ℝ with Lebesgue measure; matrices are Matrix (Fin n) (Fin n) ℝ, and β.PosDef (which includes symmetry) is assumed in every theorem. The average is the ratio of Bochner integrals, so the normalizing constant cancels. The mean relaxation is the matrix exponential . The time-reversal hypothesis is assumed only for . The entropy and Boltzmann's constant are not modelled; is the primitive datum, as in the source's statement of the principle. The hypothesis is non-vacuous (e.g. satisfies it), and the conclusion concerns , not .
All definitions are collected in Definitions.Def_OnsagerReciprocal_basic (namespace OnsagerReciprocal). Reusable Gaussian-moment lemmas for positive definite quadratic forms are welcome contributions.
In 1913 Albert Einstein and the mathematician Marcel Grossmann published the Entwurf ("outline") of a generalized theory of relativity and a theory of gravitation. It is the first theory in which the gravitational field is described by the ten components of a space-time metric, , and in which Ricci and Levi-Civita's absolute differential calculus (Grossmann's mathematical part) is brought into physics. Its gravitational field equations were superseded by the generally covariant field equations of November 1915, but the framework of the Entwurf — metric as gravitational potential, the energy-momentum law of matter in a gravitational field, a stress-energy expression for the gravitational field itself — carried over to general relativity.
The physical part (Part I, by Einstein) derives the field equations from one requirement: the energy-momentum law must hold for matter and gravitational field together. The derivation runs through an explicit differential identity, eq. (12), whose rewritten form (12a) yields the field equations (18) and the total conservation law (19). This mission formalizes that chain of statements.
Timeline.
All objects live in one coordinate chart, , with the time coordinate. The fundamental tensor is a symmetric matrix field with negative determinant ; is the reciprocal (contravariant) tensor, the inverse matrix, and is the density factor. All sums run over .
A material process is described by its contravariant stress-energy tensor (for incoherent dust, ). Its energy-momentum law in the gravitational field is eq. (10):
The second sum is the momentum (for , the energy) transferred per unit volume from the gravitational field to matter.
The gravitational field carries its own stress-energy tensor , eq. (13):
and the field equations involve the second-order operator of eq. (15),
The Entwurf field equations (18) read , with a constant. Lowering indices with gives the covariant objects , (eq. (14)) and the operator (eq. (16)).
If the energy-momentum law (10) and the field equations (18) hold, then for
The result itself. Eq. (19) is the property Einstein required of any relativistic theory of gravitation: the energy of the gravitational field acts as a source in the same way as every other form of energy, and matter together with field obeys conservation laws of divergence form. Identity (12a) is what makes this work for the Entwurf operator ; together with the matter law (10), the field equations (18) and the gravitational stress-energy are fixed by it.
Formalizing it. These statements are proved in the source by explicit computation, and the steps (12) → (12a) and the covariant forms (20)–(22) are given without the intermediate algebra. A machine-checked development provides the full index computations: derivatives of and in terms of those of , the contractions in (12a) and (12b), and the lowering identities linking to and to . A search of the platform library found no existing formalization of these statements.
There is no conceptual obstacle, but the identities are long. Identity (12a) involves second derivatives of , the derivative of , and contractions over up to six indices. The source gives only the final form of (12) and says the identity is obtained "nach dem angedeuteten Verfahren"; the intermediate algebra, and the analytic facts behind it (differentiability of matrix inversion and of under the standing assumptions), are not written out. A naive expansion of all sums produces thousands of terms.
The first law of thermodynamics is the form the law of conservation of energy takes for thermodynamic processes. For a closed system (no transfer of matter) it is usually written, in the sign convention of Clausius, as
where is the heat supplied to the system, is the work done by the system on its surroundings, and is the internal energy. Heat and work describe a process and depend on the path; the internal energy is a property of the state, and its change depends only on the initial and final states.
The Wikipedia article First law of thermodynamics (revision oldid=1366986255) describes two ways of reaching this statement. In the original "thermodynamic approach" (Clausius 1850, Rankine), heat, defined by calorimetry, and work are primitive notions. In the "mechanical approach" (Bryan 1907, Carathéodory 1909, Born 1921), only adiabatic work is primitive. The law then asserts that adiabatic work is independent of the path, and that this work defines a state function , fixed up to an additive constant. Heat becomes a derived quantity, the residual . This mission formalizes the logical core of the mechanical approach.
A closed system has a set of states. A process has an initial state , a final state , a net work done by the system on its surroundings, and a label recording whether it is adiabatic (carried out inside an adiabatic enclosure, so that energy is transferred only as work). A set of processes stands for the processes that can actually be carried out. It is assumed to satisfy:
A function is an internal energy for if for every adiabatic . For a given , the heat supplied in a process is the residual
Under assumptions 1 to 4,
This is the article's statement: "For all adiabatic processes between two specified states of a closed system of any nature, the net work done is the same regardless the details of the process, and determines a state function called internal energy, ", together with its remark that the internal energy "is defined only up to an arbitrary additive constant of integration".
The goal makes precise the step in the mechanical approach where internal energy is defined from adiabatic work instead of being postulated. It shows which properties of the set of realizable processes are needed for the definition to make sense. Once exists, the familiar consequences follow at once: the Clausius equation for every process, the heat-free character of adiabatic processes, for adynamic processes, the equality of work and heat over a cycle, and the impossibility of perpetual motion of the first kind. Milestones 5 to 10 record these.
The mathematics is elementary and the results are classical. The value of the formalization is that it states exactly the hypotheses under which "path-independent adiabatic work defines a state function" holds. The article itself notes the gaps: experimental evidence for independence from the order of stages is thin (Bailyn), and Münster observes that one cannot always go adiabatically in both directions between two states. Assumption 4 therefore asks only for one direction.
The only non-routine step is existence when accessibility holds in just one direction per state. One defines or , depending on which link exists. Showing that this choice is consistent, and that it gives for an arbitrary adiabatic process , requires splitting into four cases according to how and are linked to . In some of these cases the composite process is an adiabatic cycle, which must be compared with a null process. Without null processes, or without closure under composition, the naive definition is not well-defined.
The Black–Scholes model (Black and Scholes, 1973; Merton, 1973) is the reference model for pricing European options. Its closed-form price, the Black–Scholes formula, is quoted daily by options markets, its partial derivatives (the Greeks) are the standard risk measures that trading desks limit and hedge, and the Black–Scholes equation is the prototype of the parabolic pricing PDEs that are solved numerically when no closed form exists. Merton and Scholes received the 1997 Nobel Memorial Prize in Economic Sciences for this work.
Timeline. Bachelier (1900) first applied Brownian motion to option pricing. In the 1960s Sprenkle, Boness, Samuelson and Merton refined the theory. Black and Scholes found the risk-neutral hedging argument in 1968 and published the formula in The Pricing of Options and Corporate Liabilities (Journal of Political Economy, 1973); Merton's paper of the same year extended the mathematical framework and coined the name.
Fix a strike , an expiry , a constant risk-free rate and a constant volatility . At time the time to maturity is and the spot price of the (non-dividend-paying) stock is . Let
be the standard normal distribution and density functions, and set
The Black–Scholes call and put prices are
In the alternative (Black '76) parametrisation one uses the discount factor and the forward price , with , and . Under the risk-neutral measure the terminal price is with standard normal.
On the domain the call price is and satisfies
together with the boundary and terminal conditions as , as , and as .
The result. The goal is the statement that ties the model's two faces together: the closed-form price is a classical solution of the pricing PDE with the payoff as terminal datum. The Greeks are what hedging strategies are built from (delta hedging is the core of the model's replication argument), put–call parity is the basic no-arbitrage relation, and the risk-neutral and binary-option identities are the probabilistic reading of the two terms in the formula.
Formalizing it. All results here are classical and proved in textbooks (e.g. Hull, Options, Futures, and Other Derivatives). The work is formalization: building a reusable, machine-checked library for the normal distribution function, the Black–Scholes price and its sensitivities, and Gaussian expectations of log-normal payoffs, on top of Mathlib's real analysis and its Gaussian measure gaussianReal.
Each identity is elementary on paper, but the calculus is long: every Greek requires differentiating through a logarithm, a square root and a quotient, and the simplifications rest on the identity . The limits in the goal need Gaussian tail estimates (e.g. as , where grows while decays). The risk-neutral identities require computing Gaussian integrals over half-lines with an exponential weight, i.e. a change of variables shifting the mean of the normal distribution.
All objects are real-valued and deterministic: is defined by the integral formula above (a Lebesgue integral over ), by its closed form, and prices by the formulas, as functions of with parameters . Partial derivatives are stated with HasDerivAt (genuine differentiability plus the value), except in the goal's PDE, where deriv is used together with an explicit clause on the open domain. Each theorem carries the standing assumptions , , , where they are needed; the parity identities hold without them and are stated unconditionally. At and the formula is undefined in the source, so the boundary and terminal conditions are stated as one-sided limits rather than values. Risk-neutral expectations are integrals against the standard Gaussian gaussianReal 0 1 of an explicit log-normal terminal price; no stochastic calculus (Itô integrals, SDEs, Girsanov, Feynman–Kac) is required or formalized. The dividend-paying extensions, American options and the perpetual put are out of scope; binary options are formalized for the non-dividend case .
Contributions of reusable lemmas (derivative of , symmetry , limits of at , Gaussian half-line integrals) are welcome.
The units of a ring of algebraic integers control much of the arithmetic of a number field. They decide when two generators of the same principal ideal differ, they enter the analytic class number formula through the regulator, and for real quadratic fields they are the solutions of Pell's equation . Dirichlet's unit theorem (Dirichlet, 1846) determines the structure of this unit group completely: it is finitely generated, and its rank is computed from the way the field embeds into and . With the finiteness of the class group, it is one of the two basic finiteness theorems of algebraic number theory and appears in every introductory text on the subject.
This mission follows the account of the theorem, the regulator and its standard examples given in the Wikipedia article Dirichlet's unit theorem (https://en.wikipedia.org/wiki/Dirichlet%27s_unit_theorem).
A number field is a field containing with finite dimension as a -vector space. Its ring of integers consists of the elements of that are roots of monic polynomials with integer coefficients, and its unit group is the group of invertible elements of .
The field has exactly field embeddings . Let be the number of embeddings with image in , and the number of pairs of complex-conjugate non-real embeddings, so that . Equivalently, and count the real and complex infinite places of , i.e. the absolute values . Write for a real place and for a complex place.
The rank of an abelian group is the largest number of elements that are independent over . For this is the largest number of multiplicatively independent units. Units of finite order are exactly the roots of unity in .
Put . If generate modulo roots of unity, the matrix has every row summing to . The absolute value of the determinant of the square matrix left after deleting any one column is the regulator of .
The result itself. The unit theorem gives . This is the input for the analytic class number formula, where the regulator appears next to the class number. It also yields the finiteness of solutions of norm-form and -unit problems and the structure of solutions of Pell's equation. It is the model for Hasse and Chevalley's -unit theorem and for Leopoldt's conjecture on -adic regulators.
Formalizing it. The theorem itself is classical and has already been formalized: Mathlib contains the unit group's rank, a system of fundamental units and a regulator in its NumberField.Units development. This mission adds the surrounding statements of the source article as explicit, independently checkable targets. These are the alternative descriptions of the signature, the rank in quadratic fields and in extensions, the unit theorem for non-maximal orders, the well-definedness of the regulator in the generator-free form given in the article, and the worked examples. The examples involve explicit numerical regulator values that are not in the library.
The upper bound on the rank follows by showing that the logarithmic image of the units is discrete. The hard part is the lower bound: one must produce independent units. This requires constructing units of prescribed size at the infinite places. The obvious approach of writing down units explicitly does not work in general. The standard route uses Minkowski's geometry of numbers. The examples carry their own difficulty: showing that given units are fundamental and pinning a regulator to four decimal places takes certified bounds, not just exhibiting units. The statement for orders requires passing between and through a finite-index argument.
Statements are in Lean 4 over Mathlib, in the namespace DirichletUnitTheorem. They use Mathlib's notions directly: NumberField, 𝓞 K, InfinitePlace, nrRealPlaces/nrComplexPlaces for /, InfinitePlace.mult for , IsTotallyReal/IsTotallyComplex for real/imaginary quadratic fields, and NumberField.Units.regulator for the regulator in the example milestones. The rank is Module.rank ℤ of the additive version of the unit group. This is a cardinal, so finite generation and the rank are stated separately. An order is a subring of of finite additive index. Approximate numerical values from the source are encoded as open intervals.
Units of finite order are written with IsOfFinOrder, so no statement depends on a particular torsion API. The goal is not trivialized by the truncated subtraction in , because always holds.
Welcome contributions include proofs that bridge to the existing Mathlib development, certified numerical bounds for regulators, and the order and extension statements, which are reusable beyond this mission.
Supply and demand is the basic model of price determination in a perfectly competitive market: holding everything else fixed, the price of a good adjusts until the quantity buyers want equals the quantity sellers offer. The model is the standard first chapter of microeconomics, it underlies partial-equilibrium policy analysis (taxes, subsidies, price controls), and it is applied to labour markets (with the wage as the price) and money markets (with the interest rate as the price). See the Wikipedia article, which is the source of this mission.
Timeline. Demand curves were first drawn by Cournot (1838); supply curves were added by Fleeming Jenkin (1870), who also drew the first comparative-statics diagrams of a curve shift; Marshall's Principles of Economics (1890) popularised the "Marshallian cross". Qualitative versions of the comparative-statics law are much older: Ibn Taymiyyah (14th century) wrote that "if desire for goods increases while its availability decreases, its price rises".
Prices and quantities are real numbers. A supply curve is a function giving the quantity supplied at each price, and a demand curve is a function giving the quantity demanded. A set of admissible prices is fixed (for example all reals, or the positive reals). Supply is upward-sloping: is strictly increasing on . Demand is downward-sloping (the law of demand): is strictly decreasing on .
A pair is a market equilibrium of , written , when
i.e. the market clears at price with quantity transacted.
An increase in demand (a right shift of the demand curve) from to means for every admissible price ; an increase in supply from to means for every admissible price.
The market curve of finitely many individual curves , , is their horizontal sum . The point elasticity of a curve at is . The article's example curves are and (supply) and and (demand).
For upward-sloping supply and downward-sloping demand on , with equilibria at admissible prices:
The corresponding statements for decreases (left shifts) are the same statements with the indices exchanged.
The comparative-statics laws are the qualitative predictions that make the supply-and-demand model usable: they say in which direction price and quantity move after a change in a non-price determinant (tastes, incomes, input costs, technology). The simultaneous-shift law is the precise form of the qualitative statement quoted above, and it also records what the model does not predict (the direction of the quantity change is left open).
These results are classical and elementary. The point of this mission is a clean, reusable formal vocabulary for partial-equilibrium market models (equilibrium, curve shifts, market curves, elasticity) on which later missions about taxes, surplus, and elasticity can build.
The mathematics is elementary; the care is in the modelling. Two traps: assuming the equilibria exist and lie in the admissible price set is necessary (the article's linear supply curve is negative for , and the isoelastic curves are only meaningful for ); and the law of demand or of supply cannot be dropped, since for an upward-sloping demand curve (Veblen or Giffen goods, discussed in the article) the comparative-statics conclusions can fail.
All curves are total functions ; slope assumptions are imposed only on the admissible set , an arbitrary set of reals, and every equilibrium price is assumed to lie in . Strict monotonicity is used for both supply and demand, as in the article's figures and its law of demand. Existence of an equilibrium is not asserted; the statements are about given equilibria. Real powers and are Mathlib's Real.rpow, and statements about them are restricted to . The point elasticity uses Mathlib's deriv.
The definitions live in a single definition file, SupplyDemand_Model, in the namespace SupplyDemand, shared by all statements.
Comparative advantage is the principle that two parties can both gain from trade even when one of them is more productive at producing every good. It is usually credited to David Ricardo's On the Principles of Political Economy and Taxation (1817), with earlier formulations in an anonymous 1814 pamphlet and in Robert Torrens' Essay on the External Corn Trade (1815). It is widely described as one of the most powerful yet counter-intuitive insights in economics, and it underlies the standard argument that trade patterns are driven by relative, not absolute, productivity.
A short timeline of the two-country, two-good theory:
The Ricardian model has two countries, Home and Foreign, and two goods, cloth and wine. Labour is the only factor of production; it moves freely between sectors inside a country but not between countries. Home has a labour force ; producing one unit of cloth takes units of labour and one unit of wine takes . Foreign's data are written with a prime: , , .
Home has a comparative advantage in cloth when its opportunity cost of cloth in terms of wine is lower than Foreign's:
which is the same as saying that Home is relatively more productive in cloth, . Nothing is assumed about absolute productivity.
In autarky a country can produce and consume any bundle with . Under free trade both goods sell at world prices ; the wage in the cloth sector is and in the wine sector , and workers move to the sector with the higher wage. A country that produces only cloth and trades at the world relative price can consume any bundle with ; a country that produces only wine can consume any bundle with .
If Home has a comparative advantage in cloth and the world relative price lies between the two autarky prices,
then Home's autarky set is contained in its consumption set when it specializes in cloth, and Foreign's autarky set is contained in its consumption set when it specializes in wine. When the corresponding inequality is strict, the inclusion is strict: the country can consume bundles it could not produce itself.
The result is the textbook form of the classical argument for specialization and trade: the gain does not depend on either country being absolutely more productive, only on a difference in relative labour costs. The price-bound milestone says where world prices can settle, and the terms-of-trade milestone says that both countries gain exactly at prices between their opportunity costs. Ricardo's numbers make the general statements concrete.
These are classical results with elementary proofs. The value of the mission is a reusable, precisely stated Ricardian model — definitions of countries, autarky sets, trade consumption sets and competitive outputs — on which later missions in this series (many goods, many countries, the Dornbusch–Fischer–Samuelson continuum, Haberler's production possibility curves) can build.
There is no deep obstacle; the work is in stating the model faithfully. The obvious shortcut is to compare autarky and trade only at the two corner bundles, or to drop the wage-based description of who produces what and simply assume each country specializes. The milestones instead derive specialization from the wage comparison, so the five-case relative-supply statement has to cover the boundary prices where workers are indifferent and output is not unique.
All quantities are real numbers. A country is a structure carrying , , and proofs that all three are positive, so no unit labour requirement can be zero. Prices are explicit arguments with positivity hypotheses wherever they are divided by. Bundles are pairs . A competitive output is a bundle with full employment (, ) in which a sector produces only if its wage is at least the other sector's wage. Relative demand is not modelled as a function; the price bound is stated for any competitive outputs whose world cloth and world wine supplies are both strictly positive (finite, nonzero relative supply, as market clearing with a finite positive relative demand requires). The comparative-advantage hypothesis is kept in the goal as the section's standing assumption, although the price-bound hypotheses imply its non-strict form.
Contributions welcome: proofs of the milestones and the goal, and follow-up missions extending the model to many goods or many countries.