All missions

Motivation

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.

Setting

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.

Formalization targets

All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.

Finite scenario probabilities, losses and positive-part payouts

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`:

$$t\le u\Rightarrow(x-u)_+\le(x-t)_+$$

Increasing retention cannot increase payment for any insured loss.

Finite stop-loss and Rockafellar-Uryasev threshold objective

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`:

$$SL\ge0,\tau>0\Rightarrow K\ge0$$

Nonnegative expected excess and positive tail mass make a valid nonnegative scaling.

Discrete cumulative tails, thresholds and monetary shifts

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`:

$$SL(t)\ge0,\;F_\alpha(t)\ge t,\;F_\alpha^{X+s}(t+s)=F_\alpha^X(t)+s$$

The root joins tail-stop-loss nonnegativity, a valid finite CVaR objective, monetary translation equivalence and the complete lower/upper finite loss partition.

Capstone theorem

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.

$$SL(t)\ge0,\;F_\alpha(t)\ge t,\;F_\alpha^{X+s}(t+s)=F_\alpha^X(t)+s$$

Significance

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.

Difficulty

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.

Formalization scope

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.

Selected references

• R. Tyrrell Rockafellar and Stanislav Uryasev, Optimization of Conditional Value-at-Risk (1999 preprint; Journal of Risk 2000), Equation (4), p. 5. https://sites.math.washington.edu/~rtr/papers/rtr179-CVaR1.pdf
• Rockafellar and Uryasev, Optimization of Conditional Value-at-Risk (2000), Journal of Risk 2, pp. 21–42. https://econpapers.repec.org/RePEc:rsk:journ4:2161159

Motivation

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.

Setting

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.

Formalization targets

All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.

Finite claim-size severity and convolution

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`:

$$f_k\ge0\Rightarrow kf_k\ge0$$

Nonnegative severity weights yield nonnegative weighted claims.

Compound Poisson Panjer recurrence

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`:

$$\lambda,f\ge0\Rightarrow g_k\ge0$$

The uniquely generated compound Poisson aggregate coefficients are nonnegative for nonnegative severity probabilities.

Finite aggregate loss reserve and stop-loss valuation

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`:

$$g=h,\;g_k\ge0,\;G_n+T_n=1,\;SL_n(d)\ge0,\;\lambda\mu_n\ge0$$

The compound aggregate-claims capstone combines genuine uniqueness of forward Panjer probabilities, nonnegative aggregate coefficients, and finite claim-payment cost controls.

Capstone theorem

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.

$$g=h,\;g_k\ge0,\;G_n+T_n=1,\;SL_n(d)\ge0,\;\lambda\mu_n\ge0$$

Significance

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.

Difficulty

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.

Formalization scope

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.

Selected references

• H. H. Panjer, Recursive Evaluation of a Family of Compound Distributions, ASTIN Bulletin 12 (1981), pp. 22–26. https://www.casact.org/sites/default/files/database/astin_vol12no1_22.pdf
• Sundt and Jewell, Further Results on Recursive Evaluation of Compound Distributions, ASTIN Bulletin 12 (1981), pp. 27–39. https://www.casact.org/sites/default/files/database/astin_vol12no1_27.pdf

Motivation

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.

Setting

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.

Formalization targets

All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.

Observed risk sets and interval exits

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`:

$$d/r+c/r=(d+c)/r$$

Death and censoring hazards sum to the total observed exit fraction, without equating censoring to mortality.

Kaplan-Meier product-limit survival

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`:

$$S_{n+1}=S_np_n$$

An actuarial one-step survival update reproduces the exact finite product-limit estimate.

Greenwood uncertainty and finite restricted survival

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`:

$$0\le S_{n+1}\le S_n,\;S_{n+1}+F_{n+1}=1,\;V_{n+1}\ge0,\;RMST_{n+1}\ge0$$

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.

Capstone theorem

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.

$$0\le S_{n+1}\le S_n,\;S_{n+1}+F_{n+1}=1,\;V_{n+1}\ge0,\;RMST_{n+1}\ge0$$

Significance

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.

Difficulty

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.

Formalization scope

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.

Selected references

• Society of Actuaries, Mortality Differences by Handedness, Transactions of the Society of Actuaries 45 (1993), p. 267, on product-limit estimation and Greenwood variance. https://www.soa.org/globalassets/assets/library/research/transactions-of-society-of-actuaries/1990-95/1993/january/tsa93v459.pdf
• Kaplan and Meier, Nonparametric Estimation from Incomplete Observations (1958), Journal of the American Statistical Association 53, pp. 457–481. https://doi.org/10.1080/01621459.1958.10501452

Motivation

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.

Setting

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.

Formalization targets

All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.

Cumulative triangles and selected development factors

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`:

$$N_j\ge0,D_j>0\Rightarrow f_j\ge0$$

Nonnegative claims and positive exposure make the factor nonnegative.

Development to ultimate and outstanding reserve

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`:

$$C+R=U$$

The ultimate estimate is reconciled exactly to paid claims and outstanding reserve.

Prior expected ultimate and Bornhuetter-Ferguson reconciliation

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`:

$$fD=N,\;C+R_{CL}=U_{CL},\;U_{BF}=wU_{CL}+(1-w)E,\;R_{CL},R_{BF}\ge0$$

The substantive capstone reconciles a volume-selected development factor, chain-ladder valuation, and prior-informed Bornhuetter-Ferguson valuation under economically meaningful positivity assumptions.

Capstone theorem

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.

$$fD=N,\;C+R_{CL}=U_{CL},\;U_{BF}=wU_{CL}+(1-w)E,\;R_{CL},R_{BF}\ge0$$

Significance

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.

Difficulty

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.

Formalization scope

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.

Selected references

• Klaus D. Schmidt, Methods and Models of Loss Reserving Based on Run-Off Triangles, CAS Forum (2006), especially section 4.3 and p. 283. https://www.casact.org/sites/default/files/database/forum_06fforum_273.pdf
• Schmidt and Zocher, The Bornhuetter-Ferguson Principle, Casualty Actuarial Society (2008). https://www.casact.org/sites/default/files/2021-07/Bornhuetter-Ferguson-Schmidt-Zocher.pdf

Motivation

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.

Setting

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.

Formalization targets

All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.

Discounted life table functions

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:

$$C_x+(1-v)D_x=D_x-D_{x+1}$$

One-year death and annuity-discount value telescope to the discounted decrement.

Finite commutation functions

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:

$$M_{x:n}+dN_{x:n}+D_{x+n}=D_x$$

The full insurance-annuity-survival decomposition follows by telescoping annual differences.

Assurance and annuity monetary values

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:

$$M+dN+D_{x+n}=D_x,\quad A+d\ddot a+E=1$$

The capstone establishes both raw and conditional actuarial valuation identities and nonnegativity of term assurance and pure endowment for a valid life table.

Capstone theorem

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.

$$M+dN+D_{x+n}=D_x,\quad A+d\ddot a+E=1$$

Significance

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.

Difficulty

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.

Formalization scope

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.

Selected references

• Institute of Actuaries, International Actuarial Notation and Commutation Functions (1969–1971). https://www.actuaries.org.uk/system/files/documents/pdf/0042-0071.pdf
• David C. M. Dickson, Mary R. Hardy and Howard R. Waters, Actuarial Mathematics for Life Contingent Risks: Annuities (2009). https://www.cambridge.org/core/books/abs/actuarial-mathematics-for-life-contingent-risks/annuities/6E4AF10872F5D04E3437447F025ECA97

Motivation

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.

Setting

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.

Formalization targets

All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.

Prospective reserve and premium difference

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:

$$V(P_2)-V(P_1)=(P_1-P_2)a$$

Change in future premiums shifts the actuarial reserve by the monetised payment difference.

Paid-up insurance and equivalent cover

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:

$$PV(f+g)=PV(f)+PV(g)$$

Paid-up proportional benefit values are additive over cover portions.

Surrender deductions and policy value comparisons

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:

$$V=(P_s-P)a=b_{\rm PU}A_{\rm unit},\quad0\le SV\le V$$

Main policy alteration theorem reconciles prospective reserve, repriced premium difference and reserve-equivalent paid-up cover while bounding surrender cash by the reserve.

Capstone theorem

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.

$$V=(P_s-P)a=b_{\rm PU}A_{\rm unit},\quad0\le SV\le V$$

Significance

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.

Difficulty

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.

Formalization scope

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.

Selected references

• S. David Promislow, Fundamentals of Actuarial Mathematics (2010), Chapter 6. https://onlinelibrary.wiley.com/doi/10.1002/9781119971528.ch6
• David C. M. Dickson, Mary R. Hardy and Howard R. Waters, Actuarial Mathematics for Life Contingent Risks (2009). https://www.cambridge.org/core/books/actuarial-mathematics-for-life-contingent-risks/51C0B04099D4DE1D1DDEF9551024133B/listing

Motivation

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.

Setting

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.

Formalization targets

All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.

Projected service and accrued benefits

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:

$$B_{0,0}=0$$

An empty past and future service ledger has zero pension.

Actuarial liability and normal cost

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:

$$PVFB_{m+1}=PVFB_m+NC_{n+m}$$

Projected PV of benefits increases by the normal cost of the added service year.

Funding shortfall and contribution amortisation

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:

$$PVFB=AL+PVFNC,\quad c^*F=UL,\quad c\ge c^*\Rightarrow S(c)\ge0$$

The pension funding capstone unites projected-unit service attribution with the exact amortisation contribution threshold and surplus monotonicity.

Capstone theorem

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.

$$PVFB=AL+PVFNC,\quad c^*F=UL,\quad c\ge c^*\Rightarrow S(c)\ge0$$

Significance

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.

Difficulty

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.

Formalization scope

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.

Selected references

• American Academy of Actuaries, Fundamentals of Pension Accounting and Funding (2004). https://www.actuary.org/sites/default/files/pdf/pension/fundamentals_0704.pdf
• Institute of Actuaries, GN26 Pension Fund Terminology (2006). https://www.actuaries.org.uk/system/files/documents/pdf/gn26v2-1.pdf
• International Monetary Fund, Conducting Stress Tests of Defined Benefit Pension Plans (2014). https://www.elibrary.imf.org/display/book/9781484368589/ch012.xml

Motivation

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.

Setting

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.

Formalization targets

Cohort probabilities and lifetime laws

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.

Select and fractional-age assumptions

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.

Commutation columns and benefit timing

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.

Principal finite-cohort capstone

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:

$$\ddot a_x=\frac{N_x}{D_x},\qquad A_x=\frac{M_x}{D_x},\qquad A_x+(1-v)\ddot a_x=1.$$

Significance

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.

Difficulty

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.

Formalization scope

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.

Selected references

• S. David Promislow (2015), Fundamentals of Actuarial Mathematics, third edition, Wiley, Chapters 2–5, 7, 9 and 13–14. https://www.wiley.com/en-us/Fundamentals+of+Actuarial+Mathematics%2C+3rd+Edition-p-9781118782521
• David C. M. Dickson, Mary R. Hardy and Howard R. Waters (2020), Actuarial Mathematics for Life Contingent Risks, third edition, Cambridge University Press, Chapters 2–5. https://doi.org/10.1017/9781108784184

Motivation

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".

Setting

Let $x=(x_1,\dots,x_n)\in\mathbb R^n$ be the fluctuations of $n$ thermodynamic quantities from their equilibrium values. For small fluctuations the entropy is quadratic, and Boltzmann's formula gives the Gaussian fluctuation distribution

$$w(x)=\tilde A\,e^{-\frac12\beta_{ik}x_ix_k},$$

where $\beta$ is a positive definite symmetric real $n\times n$ matrix and summation over repeated indices is understood. Write $\langle f\rangle$ for the average of $f(x)$ with respect to $w$.

The thermodynamic conjugate quantities are $X_i=\beta_{ik}x_k$. In the quasi-stationary regime the fluctuations relax according to $\dot x_i=-\lambda_{ik}x_k$ for a real matrix $\lambda$, so that $\dot x_i=-\gamma_{ik}X_k$ with the kinetic coefficients

$$\gamma_{ik}=\lambda_{il}\,(\beta^{-1})_{lk}.$$

The mean value at time $t$ of a fluctuation which takes the value $x$ at $t=0$ is $\xi(t)=e^{-t\lambda}x$, and $\Xi(t)=\beta\xi(t)$. The time correlation is $\langle x_i(t)x_k(0)\rangle=\langle\xi_i(t)x_k\rangle$.

Formalization targets

Goal: Onsager's principle

If fluctuations are symmetric under time reversal,

$$\langle \xi_i(t)\,x_k\rangle=\langle x_i\,\xi_k(t)\rangle\qquad(t\ge0,\ 1\le i,k\le n),$$

then the kinetic coefficients are symmetric:

$$\gamma_{ik}=\gamma_{ki}.$$

Milestones (in the order of the source's argument)

1. $\dot x_i=-\lambda_{ik}x_k=-\gamma_{ik}X_k$ (definition of kinetic coefficients).
2. $\xi(0)=x$ and $\dot\xi_i(t)=-\gamma_{ik}\Xi_k(t)$.
3. $\frac{d}{dt}\langle\xi_i(t)x_k\rangle=-\gamma_{il}\langle\Xi_l(t)x_k\rangle$.
4. $\langle X_ix_k\rangle=\delta_{ik}$.
5. Under time-reversal symmetry, $\gamma_{il}\langle X_lx_k\rangle=\gamma_{kl}\langle X_lx_i\rangle$.

Significance

The symmetry $\gamma_{ik}=\gamma_{ki}$ is the mathematical content of the reciprocal relations: it reduces the number of independent transport coefficients from $n^2$ to $n(n+1)/2$ 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 $\langle X_ix_k\rangle=\delta_{ik}$ and the differentiation of the time-correlation function under the integral sign. No machine-checked formalization of this statement is known to the proposer.

Difficulty

The algebraic core is short, but the formal argument needs analytic input that the textbook treats as routine: Gaussian integrals over $\mathbb R^n$ for a general positive definite $\beta$ (integrability of polynomial moments against $e^{-\frac12x^{\mathsf T}\beta x}$ 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 $t\ge0$ to an identity between derivatives at the boundary point $t=0$, where only a one-sided derivative is available from the hypothesis.

Formalization scope

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 $\langle f\rangle$ is the ratio $\int f\,w\,dx/\int w\,dx$ of Bochner integrals, so the normalizing constant $\tilde A$ cancels. The mean relaxation is the matrix exponential $\xi(t)=e^{-t\lambda}x$. The time-reversal hypothesis is assumed only for $t\ge0$. The entropy $S$ and Boltzmann's constant are not modelled; $\beta$ is the primitive datum, as in the source's statement of the principle. The hypothesis is non-vacuous (e.g. $\lambda=\beta$ satisfies it), and the conclusion concerns $\gamma=\lambda\beta^{-1}$, not $\lambda$.

All definitions are collected in Definitions.Def_OnsagerReciprocal_basic (namespace OnsagerReciprocal). Reusable Gaussian-moment lemmas for positive definite quadratic forms are welcome contributions.

Selected references

• L. Onsager, Reciprocal Relations in Irreversible Processes. I, Physical Review 37 (1931) 405–426. doi:10.1103/physrev.37.405
• D. G. Miller, Thermodynamics of Irreversible Processes. The Experimental Verification of the Onsager Reciprocal Relations, Chemical Reviews 60 (1960) 15–37. doi:10.1021/cr60203a003
• L. D. Landau, E. M. Lifshitz, Statistical Physics, Part 1, Butterworth-Heinemann (1975).
• Wikipedia, Onsager reciprocal relations, oldid 1355014688.

Motivation

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 $g_{\mu\nu}$ of a space-time metric, $ds^2=\sum_{\mu\nu}g_{\mu\nu}\,dx_\mu dx_\nu$, 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.

• 1912: Einstein's theory of the static gravitational field, with a single scalar potential $c(x)$ (Ann. d. Physik 38).
• 1913: Einstein–Grossmann, Entwurf: metric tensor $g_{\mu\nu}$, energy-momentum law (10), field equations (18), conservation law (19).
• 1915: Einstein, Die Feldgleichungen der Gravitation (Sitzungsber. Preuss. Akad. Wiss.): generally covariant field equations replace (18).

Setting

All objects live in one coordinate chart, $x=(x_1,x_2,x_3,x_4)\in\mathbb R^4$, with $x_4$ the time coordinate. The fundamental tensor $g_{\mu\nu}(x)$ is a symmetric $4\times4$ matrix field with negative determinant $g=\det(g_{\mu\nu})$; $\gamma_{\mu\nu}$ is the reciprocal (contravariant) tensor, the inverse matrix, and $\sqrt{-g}$ is the density factor. All sums run over $1,\dots,4$.

A material process is described by its contravariant stress-energy tensor $\Theta_{\mu\nu}$ (for incoherent dust, $\Theta_{\mu\nu}=\varrho_0\frac{dx_\mu}{ds}\frac{dx_\nu}{ds}$). Its energy-momentum law in the gravitational field is eq. (10):

$$\sum_{\mu\nu}\frac{\partial}{\partial x_\nu}\big(\sqrt{-g}\,g_{\sigma\mu}\Theta_{\mu\nu}\big)-\tfrac12\sum_{\mu\nu}\sqrt{-g}\,\frac{\partial g_{\mu\nu}}{\partial x_\sigma}\Theta_{\mu\nu}=0\qquad(\sigma=1,2,3,4).$$

The second sum is the momentum (for $\sigma=4$, the energy) transferred per unit volume from the gravitational field to matter.

The gravitational field carries its own stress-energy tensor $\vartheta_{\mu\nu}$, eq. (13):

$$-2\kappa\,\vartheta_{\mu\nu}=\sum_{\alpha\beta\tau\rho}\Big(\gamma_{\alpha\mu}\gamma_{\beta\nu}\frac{\partial g_{\tau\rho}}{\partial x_\alpha}\frac{\partial\gamma_{\tau\rho}}{\partial x_\beta}-\tfrac12\gamma_{\mu\nu}\gamma_{\alpha\beta}\frac{\partial g_{\tau\rho}}{\partial x_\alpha}\frac{\partial\gamma_{\tau\rho}}{\partial x_\beta}\Big),$$

and the field equations involve the second-order operator of eq. (15),

$$\Delta_{\mu\nu}(\gamma)=\sum_{\alpha\beta}\frac{1}{\sqrt{-g}}\frac{\partial}{\partial x_\alpha}\Big(\gamma_{\alpha\beta}\sqrt{-g}\,\frac{\partial\gamma_{\mu\nu}}{\partial x_\beta}\Big)-\sum_{\alpha\beta\tau\rho}\gamma_{\alpha\beta}g_{\tau\rho}\frac{\partial\gamma_{\mu\tau}}{\partial x_\alpha}\frac{\partial\gamma_{\nu\rho}}{\partial x_\beta}.$$

The Entwurf field equations (18) read $\Delta_{\mu\nu}(\gamma)=\kappa(\Theta_{\mu\nu}+\vartheta_{\mu\nu})$, with $\kappa$ a constant. Lowering indices with $g$ gives the covariant objects $T_{\mu\nu}=\sum g_{\mu\alpha}g_{\nu\beta}\Theta_{\alpha\beta}$, $t_{\mu\nu}$ (eq. (14)) and the operator $D_{\mu\nu}(g)$ (eq. (16)).

Formalization targets

Goal: eq. (19)

If the energy-momentum law (10) and the field equations (18) hold, then for $\sigma=1,2,3,4$

$$\sum_{\mu\nu}\frac{\partial}{\partial x_\nu}\Big\{\sqrt{-g}\,g_{\sigma\mu}(\Theta_{\mu\nu}+\vartheta_{\mu\nu})\Big\}=0.$$

Milestones

1. The electrostatic model identity of p. 14, which illustrates the method.
2. Identity (12a): $\sum_{\mu\nu}\frac{\partial}{\partial x_\nu}\{\sqrt{-g}\,g_{\sigma\mu}\kappa\vartheta_{\mu\nu}\}=\tfrac12\sum_{\mu\nu}\sqrt{-g}\,\frac{\partial g_{\mu\nu}}{\partial x_\sigma}\{-\Delta_{\mu\nu}(\gamma)+\kappa\vartheta_{\mu\nu}\}$, for every metric.
3. Identity (12b), the covariant counterpart, with $t_{\mu\nu}$ and $D_{\mu\nu}(g)$.
4. Eq. (10) $\Leftrightarrow$ eq. (20), the covariant energy-momentum law.
5. Eq. (18) $\Rightarrow$ eq. (21), $-D_{\mu\nu}(g)=\kappa(t_{\mu\nu}+T_{\mu\nu})$.
6. Eq. (22), the covariant conservation law for matter plus field.

Significance

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 $\Delta_{\mu\nu}(\gamma)$; together with the matter law (10), the field equations (18) and the gravitational stress-energy $\vartheta_{\mu\nu}$ 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 $\gamma_{\mu\nu}$ and $\sqrt{-g}$ in terms of those of $g_{\mu\nu}$, the contractions in (12a) and (12b), and the lowering identities linking $\Delta_{\mu\nu}(\gamma)$ to $D_{\mu\nu}(g)$ and $\vartheta_{\mu\nu}$ to $t_{\mu\nu}$. A search of the platform library found no existing formalization of these statements.

Difficulty

There is no conceptual obstacle, but the identities are long. Identity (12a) involves second derivatives of $\gamma_{\mu\nu}$, the derivative of $\sqrt{-g}$, 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 $\sqrt{-\det g}$ under the standing assumptions), are not written out. A naive expansion of all sums produces thousands of terms.

Formalization scope

• Space-time is $\mathbb R^4$ as one global chart, with points $x:\mathrm{Fin}\,4\to\mathbb R$; two-index fields are matrix-valued functions. Partial derivatives are Fréchet derivatives along coordinate directions.
• Standing assumptions on the metric: entries $C^2$, symmetric, $\det g<0$ everywhere. No signature condition beyond the sign of the determinant; $\gamma$ is the matrix inverse, $\sqrt{-g}$ the real square root of $-\det g$.
• $\kappa\neq0$ is assumed wherever $\vartheta_{\mu\nu}$ or $t_{\mu\nu}$ (defined with the factor $1/(2\kappa)$) appears. Matter tensors are assumed $C^1$, and symmetric where the source's manipulation needs it (eq. (20)).
• In eq. (14) the sum over $\alpha,\beta$ is taken only in the term containing them, following the paper's convention. The printed eq. (22) has an inconsistent index placement; milestone 6 uses the placement of (20) and (12b).
• The goal must not be trivialized: the field equations and the matter law are hypotheses about the same $\Theta_{\mu\nu}$, and the hypotheses are satisfiable (e.g. constant metric, $\Theta=0$) but not degenerate in general.
• Reusable infrastructure: derivative formulas for the inverse and determinant of a $C^k$ matrix field, coordinate tensor calculus on $\mathbb R^4$. Contributions of these as separate lemmas are welcome.

Selected references

• A. Einstein and M. Grossmann, Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation, B. G. Teubner, Leipzig und Berlin, 1913; reprinted in Zeitschrift für Mathematik und Physik 62 (1914). The source of all statements of this mission (Part I, §§4–5, pp. 10–17).
• A. Einstein, Die Feldgleichungen der Gravitation, Sitzungsberichte der Preußischen Akademie der Wissenschaften, 1915, pp. 844–847.

Motivation

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

$$\Delta U = Q - W,$$

where $Q$ is the heat supplied to the system, $W$ is the work done by the system on its surroundings, and $U$ 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 $U$, fixed up to an additive constant. Heat becomes a derived quantity, the residual $Q = \Delta U + W$. This mission formalizes the logical core of the mechanical approach.

Setting

A closed system has a set $\sigma$ of states. A process $p$ has an initial state $p_{\mathrm{start}}$, a final state $p_{\mathrm{finish}}$, a net work $W(p)\in\mathbb R$ 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 $\mathcal P$ of processes stands for the processes that can actually be carried out. It is assumed to satisfy:

1. Composition: if $p,q\in\mathcal P$ and $p_{\mathrm{finish}}=q_{\mathrm{start}}$, then the composite "first $p$, then $q$" lies in $\mathcal P$. Its work is $W(p)+W(q)$, and it is adiabatic exactly when both stages are.
2. Null processes: for every state $A$, leaving the system in state $A$ (with no work and no exchange) is in $\mathcal P$.
3. Path independence of adiabatic work: any two adiabatic processes in $\mathcal P$ with the same initial and final states involve the same work.
4. Adiabatic accessibility: there is a reference state $O$ such that every state $A$ is reached from $O$, or reaches $O$, by some adiabatic process in $\mathcal P$.

A function $U:\sigma\to\mathbb R$ is an internal energy for $\mathcal P$ if $U(p_{\mathrm{finish}})-U(p_{\mathrm{start}})=-W(p)$ for every adiabatic $p\in\mathcal P$. For a given $U$, the heat supplied in a process is the residual

$$Q_U(p)=U(p_{\mathrm{finish}})-U(p_{\mathrm{start}})+W(p).$$

Formalization targets

Goal: existence and uniqueness of internal energy

Under assumptions 1 to 4,

$$\exists\,U \text{ internal energy for } \mathcal P,\qquad\text{and}\qquad U,U' \text{ internal energies} \implies \exists c\in\mathbb R\ \forall A,\ U'(A)=U(A)+c.$$

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, $U$", together with its remark that the internal energy "is defined only up to an arbitrary additive constant of integration".

Milestones

1. Formula (1) and the chain through the reference state. If quasi-static adiabatic work $W_{A\to B}$ satisfies $W_{A\to O}=-W_{O\to A}$ and $W_{A\to B}=W_{A\to O}+W_{O\to B}$, and $U(A)=U(O)-W_{O\to A}$, then $-W_{A\to B}=U(B)-U(A)$.
2. Shifting an internal energy by a constant gives an internal energy.
3. Under adiabatic accessibility, two internal energies differ by a constant.
4. Under assumptions 1 to 4, an internal energy exists (normalized by $U(O)=0$).
5. The Clausius form $\Delta U=Q-W$ holds for the residual heat.
6. Adiabatic processes involve no heat.
7. In an adynamic process (no work), $Q=\Delta U$.
8. Heat does not depend on the additive constant in $U$.
9. In a cyclic process, the net work done equals the net heat taken in.
10. No adiabatic cyclic process produces net work (no perpetual motion machine of the first kind).

Significance

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 $U$ exists, the familiar consequences follow at once: the Clausius equation for every process, the heat-free character of adiabatic processes, $Q=\Delta U$ 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.

Difficulty

The only non-routine step is existence when accessibility holds in just one direction per state. One defines $U(A)=-W(O\to A)$ or $U(A)=W(A\to O)$, depending on which link exists. Showing that this choice is consistent, and that it gives $\Delta U=-W$ for an arbitrary adiabatic process $A\to B$, requires splitting into four cases according to how $A$ and $B$ are linked to $O$. 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.

Formalization scope

• States form an arbitrary type $\sigma$. Work is real-valued and is work done by the system (Clausius convention), so that $\Delta U=-W$ for adiabatic processes and $\Delta U=Q-W$ in general.
• "Adiabatic" is a primitive label on processes, as adiabatic walls are primitive in the mechanical approach. Heat is not primitive: it is defined as the residual $Q_U(p)=\Delta U+W$.
• The physically realizable processes form a set $\mathcal P$. All hypotheses are stated about $\mathcal P$, never about every conceivable process.
• The goal is not vacuous: assumptions 1 to 4 can all hold at once (for example, states $\mathbb R$, with adiabatic work from $a$ to $b$ equal to $a-b$).
• Open systems, matter transfer, the extensivity statement $U_0=U_1+U_2$, and the differential form $dU=\delta Q-\delta W$ are out of scope for this first mission.

Selected references

• Wikipedia, First law of thermodynamics, revision oldid=1366986255. https://en.wikipedia.org/w/index.php?title=First_law_of_thermodynamics&oldid=1366986255
• C. Carathéodory, Untersuchungen über die Grundlagen der Thermodynamik, Mathematische Annalen 67 (1909), 355–386. https://ui.adsabs.harvard.edu/abs/1909MatAn..67..355C
• M. Born, Kritische Betrachtungen zur traditionellen Darstellung der Thermodynamik, Physikalische Zeitschrift 22 (1921), 218–224.

Motivation

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.

Setting

Fix a strike $K>0$, an expiry $T$, a constant risk-free rate $r\in\mathbb R$ and a constant volatility $\sigma>0$. At time $t<T$ the time to maturity is $\tau=T-t>0$ and the spot price of the (non-dividend-paying) stock is $S>0$. Let

$$N(x)=\frac1{\sqrt{2\pi}}\int_{-\infty}^x e^{-z^2/2}\,dz,\qquad N'(x)=\frac1{\sqrt{2\pi}}e^{-x^2/2}$$

be the standard normal distribution and density functions, and set

$$d_+=\frac{1}{\sigma\sqrt{\tau}}\Big[\ln\frac SK+\Big(r+\frac{\sigma^2}2\Big)\tau\Big],\qquad d_-=d_+-\sigma\sqrt\tau .$$

The Black–Scholes call and put prices are

$$C(S,t)=N(d_+)S-N(d_-)Ke^{-r\tau},\qquad P(S,t)=N(-d_-)Ke^{-r\tau}-N(-d_+)S .$$

In the alternative (Black '76) parametrisation one uses the discount factor $D=e^{-r\tau}$ and the forward price $F=e^{r\tau}S$, with $d_\pm=\frac1{\sigma\sqrt\tau}\big[\ln\frac FK\pm\frac12\sigma^2\tau\big]$, and $C=D[N(d_+)F-N(d_-)K]$. Under the risk-neutral measure the terminal price is $S_T=S\exp\big((r-\tfrac12\sigma^2)\tau+\sigma\sqrt\tau\,Z\big)$ with $Z$ standard normal.

Formalization targets

Goal: the formula solves the Black–Scholes equation

On the domain $\{S>0,\ t<T\}$ the call price $C$ is $C^\infty$ and satisfies

$$\frac{\partial C}{\partial t}+\frac12\sigma^2S^2\frac{\partial^2C}{\partial S^2}+rS\frac{\partial C}{\partial S}-rC=0,$$

together with the boundary and terminal conditions $C(S,t)\to0$ as $S\to0^+$, $C(S,t)-(S-Ke^{-r(T-t)})\to0$ as $S\to\infty$, and $C(S,t)\to\max\{S-K,0\}$ as $t\to T^-$.

Milestones

1. The density $N'$ is the derivative of $N$.
2. The alternative (forward) formulation of the call and put prices.
3. Put–call parity, $P=Ke^{-r\tau}-S+C$, and its forward form $C-P=D(F-K)$.
4. The Greeks table: Delta, Gamma, Vega, Theta and Rho for calls and puts.
5. Risk-neutral pricing: $C=e^{-r\tau}\,\mathbb E[\max(S_T-K,0)]$.
6. Binary options: cash-or-nothing values $e^{-r\tau}N(\pm d_-)$ and asset-or-nothing values $SN(\pm d_+)$.

Significance

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.

Difficulty

Each identity is elementary on paper, but the calculus is long: every Greek requires differentiating $N\circ d_\pm$ through a logarithm, a square root and a quotient, and the simplifications rest on the identity $SN'(d_+)=Ke^{-r\tau}N'(d_-)$. The limits in the goal need Gaussian tail estimates (e.g. $SN(-d_+)\to0$ as $S\to\infty$, where $S$ grows while $N(-d_+)$ 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.

Formalization scope

All objects are real-valued and deterministic: $N$ is defined by the integral formula above (a Lebesgue integral over $(-\infty,x]$), $N'$ by its closed form, and prices by the formulas, as functions of $(S,t)$ with parameters $(K,r,\sigma,T)$. 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 $C^\infty$ clause on the open domain. Each theorem carries the standing assumptions $K>0$, $\sigma>0$, $S>0$, $t<T$ where they are needed; the parity identities hold without them and are stated unconditionally. At $S=0$ and $t=T$ 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 $q=0$.

Contributions of reusable lemmas (derivative of $N$, symmetry $N(x)+N(-x)=1$, limits of $N$ at $\pm\infty$, Gaussian half-line integrals) are welcome.

Selected references

• F. Black, M. Scholes, The Pricing of Options and Corporate Liabilities, Journal of Political Economy 81(3), 1973. https://doi.org/10.1086/260062
• R. C. Merton, Theory of Rational Option Pricing, Bell Journal of Economics and Management Science 4(1), 1973. https://doi.org/10.2307/3003143
• J. C. Hull, Options, Futures, and Other Derivatives, Pearson.
• Wikipedia, Black–Scholes model. https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model

Motivation

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 $x^2 - dy^2 = \pm 1$. 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 $\mathbb R$ and $\mathbb C$. 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).

Setting

A number field $K$ is a field containing $\mathbb Q$ with finite dimension $n = [K:\mathbb Q]$ as a $\mathbb Q$-vector space. Its ring of integers $\mathcal O_K$ consists of the elements of $K$ that are roots of monic polynomials with integer coefficients, and its unit group $\mathcal O_K^\times$ is the group of invertible elements of $\mathcal O_K$.

The field $K$ has exactly $n$ field embeddings $\sigma : K \to \mathbb C$. Let $r_1$ be the number of embeddings with image in $\mathbb R$, and $r_2$ the number of pairs of complex-conjugate non-real embeddings, so that $n = r_1 + 2r_2$. Equivalently, $r_1$ and $r_2$ count the real and complex infinite places $w$ of $K$, i.e. the absolute values $|x|_w = |\sigma(x)|$. Write $N_w = 1$ for a real place and $N_w = 2$ for a complex place.

The rank of an abelian group is the largest number of elements that are independent over $\mathbb Z$. For $\mathcal O_K^\times$ this is the largest number of multiplicatively independent units. Units of finite order are exactly the roots of unity in $K$.

Put $r = r_1 + r_2 - 1$. If $u_1, \dots, u_r$ generate $\mathcal O_K^\times$ modulo roots of unity, the $r \times (r+1)$ matrix $\big(N_w \log |u_i|_w\big)$ has every row summing to $0$. The absolute value $R$ of the determinant of the square matrix left after deleting any one column is the regulator of $K$.

Formalization targets

Goal: Dirichlet's unit theorem

$$\mathcal O_K^\times \text{ is finitely generated and } \operatorname{rank}\mathcal O_K^\times = r_1 + r_2 - 1.$$

Milestones

1. The signature identity $n = r_1 + 2r_2$. If $K/\mathbb Q$ is Galois, then $r_1 = 0$ or $r_2 = 0$.
2. Two other ways to compute $(r_1, r_2)$: real and non-real roots of the minimal polynomial of a primitive element, and $\mathbb R \otimes_{\mathbb Q} K \cong \mathbb R^{r_1} \times \mathbb C^{r_2}$.
3. Quadratic fields have unit rank $1$ when real and $0$ when imaginary. The rank is positive except for $\mathbb Q$ and imaginary quadratic fields.
4. Torsion: the torsion units are the roots of unity and form a finite cyclic group. When $K$ has a real embedding they are $\pm 1$.
5. Rank equality in a proper extension $L/K$ holds exactly when $K$ is totally real and $L/K$ is a totally complex quadratic extension.
6. The unit theorem for an arbitrary order $\mathcal O \subseteq \mathcal O_K$.
7. The logarithmic picture: $\sum_w N_w \log |u|_w = 0$ for every unit, and the vectors $\big(N_w \log |u|_w\big)_w$ form a lattice in the hyperplane of coordinate sum $0$.
8. The regulator does not depend on the generators or on the deleted column, and it is positive.
9. Examples: $\operatorname{Reg} = 1$ for $\mathbb Q$ and imaginary quadratic fields. $\operatorname{Reg}(\mathbb Q(\sqrt5)) = \log\frac{1+\sqrt5}{2}$. For the cyclic cubic field defined by $x^3 + x^2 - 2x - 1$, the fundamental units are $\alpha^2 + \alpha - 1$ and $2 - \alpha^2$, and $\operatorname{Reg} \approx 0.5255$.

Significance

The result itself. The unit theorem gives $\mathcal O_K^\times \cong \mu(K) \times \mathbb Z^{r_1 + r_2 - 1}$. 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 $S$-unit problems and the structure of solutions of Pell's equation. It is the model for Hasse and Chevalley's $S$-unit theorem and for Leopoldt's conjecture on $p$-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.

Difficulty

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 $r_1 + r_2 - 1$ 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 $\mathcal O^\times$ and $\mathcal O_K^\times$ through a finite-index argument.

Formalization scope

Statements are in Lean 4 over Mathlib, in the namespace DirichletUnitTheorem. They use Mathlib's notions directly: NumberField, 𝓞 K, InfinitePlace, nrRealPlaces/nrComplexPlaces for $r_1$/$r_2$, InfinitePlace.mult for $N_w$, 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 $\mathcal O_K$ 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 $r_1 + r_2 - 1$, because $r_1 + r_2 \ge 1$ 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.

Selected references

• Wikipedia contributors, Dirichlet's unit theorem, Wikipedia. https://en.wikipedia.org/wiki/Dirichlet%27s_unit_theorem
• J. Neukirch, Algebraic Number Theory, Grundlehren 322, Springer, 1999, Chapter I §7. https://doi.org/10.1007/978-3-662-03983-0
• H. Cohen, A Course in Computational Algebraic Number Theory, GTM 138, Springer, 1993, Table B.4. https://doi.org/10.1007/978-3-662-02945-9

Motivation

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".

Setting

Prices and quantities are real numbers. A supply curve is a function $S:\mathbb R\to\mathbb R$ giving the quantity supplied at each price, and a demand curve is a function $D:\mathbb R\to\mathbb R$ giving the quantity demanded. A set $I\subseteq\mathbb R$ of admissible prices is fixed (for example all reals, or the positive reals). Supply is upward-sloping: $S$ is strictly increasing on $I$. Demand is downward-sloping (the law of demand): $D$ is strictly decreasing on $I$.

A pair $(p,q)$ is a market equilibrium of $(S,D)$, written $\mathrm{IsEquilibrium}(S,D,p,q)$, when

$$D(p)=q \quad\text{and}\quad S(p)=q,$$

i.e. the market clears at price $p$ with quantity $q$ transacted.

An increase in demand (a right shift of the demand curve) from $D_1$ to $D_2$ means $D_1(p)<D_2(p)$ for every admissible price $p$; an increase in supply from $S_1$ to $S_2$ means $S_1(p)<S_2(p)$ for every admissible price.

The market curve of finitely many individual curves $f_i$, $i\in s$, is their horizontal sum $p\mapsto\sum_{i\in s}f_i(p)$. The point elasticity of a curve $Q$ at $P$ is $Q'(P)\,P/Q(P)$. The article's example curves are $Q(P)=3P-6$ and $Q(P)=5P^{0.5}$ (supply) and $Q(P)=32-2P$ and $Q(P)=3P^{-2}$ (demand).

Formalization targets

Goal: comparative statics of a curve shift

For upward-sloping supply and downward-sloping demand on $I$, with equilibria at admissible prices:

1. if demand increases from $D_1$ to $D_2$ (supply $S$ fixed), equilibria $(p_1,q_1)$ of $(S,D_1)$ and $(p_2,q_2)$ of $(S,D_2)$ satisfy

$$p_1<p_2\quad\text{and}\quad q_1<q_2;$$

2. if supply increases from $S_1$ to $S_2$ (demand $D$ fixed), equilibria $(p_1,q_1)$ of $(S_1,D)$ and $(p_2,q_2)$ of $(S_2,D)$ satisfy

$$p_2<p_1\quad\text{and}\quad q_1<q_2.$$

The corresponding statements for decreases (left shifts) are the same statements with the indices $1,2$ exchanged.

Milestones

• uniqueness of the equilibrium for strictly monotone curves;
• the demand-shift and supply-shift laws separately;
• the Ibn Taymiyyah law for simultaneous shifts: demand up and supply down raises the price, and demand down and supply up lowers it;
• horizontal aggregation preserves the slope of supply and demand curves;
• the article's example curves have the stated slopes, the isoelastic curves have the stated log-linear forms, and their point elasticities are the constants $0.5$ and $-2$.

Significance

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.

Difficulty

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 $P<2$, and the isoelastic curves are only meaningful for $P>0$); 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.

Formalization scope

All curves are total functions $\mathbb R\to\mathbb R$; slope assumptions are imposed only on the admissible set $I$, an arbitrary set of reals, and every equilibrium price is assumed to lie in $I$. 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 $P^{0.5}$ and $P^{-2}$ are Mathlib's Real.rpow, and statements about them are restricted to $P>0$. 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.

Selected references

• Wikipedia contributors, "Supply and demand", revision 1378800284, https://en.wikipedia.org/w/index.php?title=Supply_and_demand&oldid=1378800284
• A. A. Cournot, Recherches sur les principes mathématiques de la théorie des richesses, 1838.
• F. Jenkin, "The Graphical Representation of the Laws of Supply and Demand, and their Application to Labour", in Recess Studies, 1870, https://books.google.com/books?id=NC5BAAAAIAAJ
• A. Marshall, Principles of Economics, Macmillan, 1890.
• T. M. Humphrey, "Marshallian Cross Diagrams and their Uses before Alfred Marshall", Economic Review (Federal Reserve Bank of Richmond), Mar/Apr 1992, 3–23.

Motivation

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:

• 1776 — Adam Smith, The Wealth of Nations: trade based on absolute advantage.
• 1814–1815 — anonymous pamphlet Considerations on the Importation of Foreign Corn; Torrens, Essay on the External Corn Trade: first formulations of comparative advantage.
• 1817 — Ricardo's England/Portugal example with cloth and wine.
• 1833 — William Whewell publishes the first mathematical Ricardian model.
• 1930 — Gottfried Haberler restates comparative advantage in terms of opportunity costs and production possibility curves.

Setting

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 $L>0$; producing one unit of cloth takes $a_{LC}>0$ units of labour and one unit of wine takes $a_{LW}>0$. Foreign's data are written with a prime: $L'$, $a'_{LC}$, $a'_{LW}$.

Home has a comparative advantage in cloth when its opportunity cost of cloth in terms of wine is lower than Foreign's:

$$\frac{a_{LC}}{a_{LW}}<\frac{a'_{LC}}{a'_{LW}},$$

which is the same as saying that Home is relatively more productive in cloth, $a_{LC}/a'_{LC}<a_{LW}/a'_{LW}$. Nothing is assumed about absolute productivity.

In autarky a country can produce and consume any bundle $(Q_C,Q_W)\ge 0$ with $a_{LC}Q_C+a_{LW}Q_W\le L$. Under free trade both goods sell at world prices $P_C,P_W>0$; the wage in the cloth sector is $P_C/a_{LC}$ and in the wine sector $P_W/a_{LW}$, and workers move to the sector with the higher wage. A country that produces only cloth and trades at the world relative price $P_C/P_W$ can consume any bundle with $a_{LC}Q_C+a_{LC}(P_W/P_C)Q_W\le L$; a country that produces only wine can consume any bundle with $a_{LW}(P_C/P_W)Q_C+a_{LW}Q_W\le L$.

Formalization targets

Goal: gains from trade

If Home has a comparative advantage in cloth and the world relative price lies between the two autarky prices,

$$\frac{a_{LC}}{a_{LW}}\le\frac{P_C}{P_W}\le\frac{a'_{LC}}{a'_{LW}},$$

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.

Milestones

1. The two forms of the comparative-advantage assumption are equivalent.
2. The five-case description of world relative supply: who specializes in what at each relative price.
3. As long as relative demand is finite, the equilibrium relative price lies between the two autarky prices.
4. The autarky production possibility frontier $Q_C=L/a_{LC}-(a_{LW}/a_{LC})Q_W$.
5. The free-trade consumption possibility frontier $Q_C=L/a_{LC}-(P_W/P_C)Q_W$ lies on or above the autarky frontier.
6. Terms of trade: specializing in a good leaves a country no worse off exactly when the world price lies on the right side of its own opportunity cost.
7. Ricardo's numerical example: absolute and comparative advantage, the gain in world output from specialization, and terms of trade between $5/6$ and $9/8$ units of wine per unit of cloth.

Significance

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.

Difficulty

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.

Formalization scope

All quantities are real numbers. A country is a structure carrying $L$, $a_{LC}$, $a_{LW}$ and proofs that all three are positive, so no unit labour requirement can be zero. Prices $P_C,P_W$ are explicit arguments with positivity hypotheses wherever they are divided by. Bundles are pairs $(Q_C,Q_W)\in\mathbb R^2$. A competitive output is a bundle with full employment ($a_{LC}Q_C+a_{LW}Q_W=L$, $Q_C,Q_W\ge 0$) 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.

Selected references

• Wikipedia contributors, Comparative advantage, Wikipedia (PDF export supplied with the proposal). https://en.wikipedia.org/wiki/Comparative_advantage
• D. Ricardo, On the Principles of Political Economy and Taxation, John Murray, London, 1817. https://www.econlib.org/library/Ricardo/ricP.html
• P. Krugman, M. Obstfeld, M. Melitz, International Economics: Theory and Policy, Pearson (chapter on the Ricardian model).

Motivation

Zipf's law is the empirical observation that when a set of measured values is sorted in decreasing order, the value of the $n$-th entry is often approximately inversely proportional to $n$. Its best-known instance is word frequency in natural language: in the Brown Corpus the most frequent word, "the", accounts for nearly $7\%$ of all word occurrences, the second, "of", for slightly over $3.5\%$, and so on. The same rank–size relation has been reported for city populations (Auerbach 1913), firm sizes (Axtell 2001), and many other data sets; it is named after the linguist George Kingsley Zipf (1932, 1935, 1949) and remains a central concept of quantitative linguistics (Piantadosi 2014).

In mathematical statistics the law is formalized as the Zipf distribution, a family of discrete probability distributions whose rank–frequency relation is an inverse power law, together with its relatives: the zeta distribution (Lotka's law), the Zipf–Mandelbrot law and the Yule–Simon distribution. Several generative explanations — random typing (Miller 1957, Li 1992) and preferential attachment — produce power laws by provable mechanisms.

Timeline of the results formalized here:

• 1913–1932 — Auerbach (city sizes), Estoup (1916), Dewey (1923), Condon (1928) and Zipf (1932) observe the inverse rank–frequency relation.
• 1957 — Miller: a power law emerges in randomly generated texts (doi:10.2307/1419346).
• 1992 — Li: in random texts with equiprobable letters, word frequency follows Zipf's macro-trend (doi:10.1109/18.165464).

Setting

For a real exponent $s$ and an integer $N \ge 1$, the generalized harmonic number is

$$H_{N,s} = \sum_{k=1}^{N}\frac{1}{k^s}, \qquad H_N = H_{N,1}.$$

The Zipf distribution on $N$ elements with exponent $s$ assigns to the element of rank $k \in\{1,\dots,N\}$ the probability

$$f(k;N,s) = \frac{1}{H_{N,s}}\,\frac{1}{k^{s}} .$$

The classical Zipf law is $s=1$. The source lists the parameter range $s\ge0$, $N\in\{1,2,3,\dots\}$.

The Riemann zeta function is $\zeta(s)=\sum_{k\ge1}k^{-s}$ for $s>1$. The Hurwitz zeta function is $\zeta(s,a)=\sum_{n\ge0}(n+a)^{-s}$. The Zipf–Mandelbrot law on ranks $k=1,2,\dots$ with shift $q$ and exponent $s$ is $f(k;q,s)=\frac{1}{C}\frac{1}{(k+q)^s}$ with $C=\zeta(s,q+1)$. The Yule–Simon distribution with parameter $\rho>0$ is $f(k;\rho)=\rho\,B(k,\rho+1)=\rho\,\Gamma(k)\Gamma(\rho+1)/\Gamma(k+\rho+1)$, $k = 1, 2, \dots$.

Formalization targets

Goal — the infinite-item extension

For $s \ge 0$:

$$\sum_{k\ge1}\frac1{k^s}<\infty \iff s>1;\qquad s>1 \implies H_{N,s}\xrightarrow[N\to\infty]{}\zeta(s);\qquad s\le1 \implies H_{N,s}\xrightarrow[N\to\infty]{}\infty .$$

That is, the generalized Zipf distribution extends to infinitely many items exactly when $s>1$, in which case the normalizing constant becomes Riemann's zeta function.

Milestones

1. Normalization: $\sum_{k=1}^N f(k;N,s)=1$.
2. CDF: $\sum_{k=1}^n f(k;N,s)=H_{n,s}/H_{N,s}$ for $n\le N$.
3. Mean: $\sum_{k} k\,f(k;N,s)=H_{N,s-1}/H_{N,s}$.
4. Variance: $\frac{H_{N,s-2}}{H_{N,s}}-\frac{H_{N,s-1}^2}{H_{N,s}^2}$.
5. Entropy: $\frac{s}{H_{N,s}}\sum_{k=1}^N\frac{\ln k}{k^s}+\ln H_{N,s}$.
6. Mode: rank $1$ is a most probable element (for $s \ge 0$).
7. Ratio: $f(1;N,s)/f(n;N,s)=n^s$ — "the most common word occurs about $n$ times the $n$-th most common one" when $s = 1$.
8. Log–log linearity: $\ln f(k;N,s) = -s\ln k-\ln H_{N,s}$, an affine function of $\ln k$ with slope $-s$.
9. Zipf–Mandelbrot: with $C=\zeta(s,q+1)$ the law is a probability distribution ($s>1$, $q\ge0$).
10. Yule–Simon tail: $f(k;\rho)\sim C/k^{\rho+1}$ for some constant $C>0$.
11. Random typing (Li): word probability is bounded above and below by a power of the rank.
12. Benford connection: the leading digits of a Zipf ($s=1$) random rank follow Benford's law as $N\to\infty$.

Significance

The goal pins down exactly when "Zipf's law with infinitely many items" makes sense, and identifies the resulting distribution (the zeta distribution) with Riemann's zeta function. The milestones are the infobox facts that statisticians use when fitting Zipf models (CDF, mean, variance, entropy, mode), the log–log diagnostic used to test data against the law, two of the related laws the source lists (Zipf–Mandelbrot, Yule–Simon), one of the generative explanations (random typing), and the stated link to Benford's law.

The results are classical; the work is formalizing them. Most milestones are finite algebra about $H_{N,s}$; the goal, the Yule–Simon tail, the random-typing bound and the Benford connection need analysis.

Difficulty

The finite identities are routine once the support conventions are fixed. The analytic content is concentrated in four places. The goal needs the $p$-series test and the identification of the limit with Mathlib's complex riemannZeta, which is defined by analytic continuation rather than by the series. The Yule–Simon tail needs the asymptotics $\Gamma(k)/\Gamma(k+a)\sim k^{-a}$. The random-typing bound requires controlling the rank of a word through the number of strictly shorter words, uniformly over every rank order compatible with the probabilities. The Benford connection requires comparing the harmonic sums over each block of integers with a fixed leading digit with $\ln(1+1/d)$, with an error that is negligible against $H_N\sim\ln N$.

Formalization scope

• $H_{N,s}$ uses a real exponent (Real.rpow); $H_{0,s}=0$.
• zipfPMF N s k is defined for every natural $k$ and is $0$ outside the support $\{1,\dots,N\}$. Statements that need a nonempty support assume $N\ge1$.
• The goal states convergence to the complex number riemannZeta s of the real partial sums viewed as complex numbers, and divergence as Tendsto … atTop atTop. The source's parameter range $s\ge0$ is kept as a hypothesis.
• The Hurwitz zeta function is defined by its series (value $0$ when the series diverges); the Zipf–Mandelbrot law is the infinite-item version on ranks $k\ge1$, with $q\ge0$.
• The Yule–Simon distribution is defined through the Gamma function; the tail statement asserts existence of a positive constant, matching the source's "[constant]".
• Random typing is formalized in Li's equiprobable model: $m\ge2$ letters each typed with probability $q>0$, a space with probability $1-mq>0$, and a word $w$ has probability $q^{|w|}(1-mq)$. "Sorted by rank" means any bijection $e$ from $\mathbb N$ onto the nonempty words along which the probability is non-increasing; the conclusion is a two-sided power-law bound with exponent $\log_m(1/q)$.
• The Benford connection uses the leading decimal digit of the rank $k$, with $k$ drawn from the Zipf distribution with $s=1$.
• Not formalized: the empirical claims (word, city, income data), the atlas-model result, Belevitch's Taylor-series argument, the parabolic fractal distribution and the Pareto exchange of variables; their statements in the source are not precise enough to formalize faithfully.

Selected references

• G. K. Zipf, Human Behavior and the Principle of Least Effort, Addison-Wesley, 1949. https://archive.org/details/in.ernet.dli.2015.90211
• E. U. Condon, Statistics of vocabulary, Science 67 (1928), 300. https://doi.org/10.1126/science.67.1733.300
• G. A. Miller, Some effects of intermittent silence, American Journal of Psychology 70 (1957), 311–314. https://doi.org/10.2307/1419346
• W. Li, Random texts exhibit Zipf's-law-like word frequency distribution, IEEE Trans. Inform. Theory 38 (1992), 1842–1845. https://doi.org/10.1109/18.165464
• B. Conrad, M. Mitzenmacher, Power laws for monkeys typing randomly: the case of unequal probabilities, IEEE Trans. Inform. Theory 50 (2004), 1403–1414. https://doi.org/10.1109/TIT.2004.830752
• R. L. Axtell, Zipf distribution of U.S. firm sizes, Science 293 (2001), 1818–1820. https://doi.org/10.1126/science.1062081
• M. E. J. Newman, Power laws, Pareto distributions and Zipf's law, Contemporary Physics 46 (2005), 323–351. https://arxiv.org/abs/cond-mat/0412004
• S. T. Piantadosi, Zipf's word frequency law in natural language: a critical review and future directions, Psychon. Bull. Rev. 21 (2014), 1112–1130. https://doi.org/10.3758/s13423-014-0585-6
• Wikipedia, Zipf's law. https://en.wikipedia.org/wiki/Zipf%27s_law