Motivation

Sampling an entire finite-state execution is a different computational task from recognizing a supplied word. A membership circuit must answer correctly on its given input. A sampler instead chooses the input together with its associated output, while preserving their joint distribution. This distinction is relevant when finite-state models describe allowable executions and when parallel generation should avoid a sequential pass through every transition. The source paper establishes constant-depth sampling results for regular languages and finite Markov chains in Appendix E (Zhang et al., 2026).

The mission concerns that appendix's finite-state sampling results. It formalizes all six numbered results E.1–E.6, together with the seed bounds stated in the surrounding text. Each milestone points to a published theorem with a separately checked proof. The objects to sample are complete words, word–acceptance pairs, and complete state trajectories. Sampling only the final state would lose the joint temporal information required by the targets.

Finite-state models and fair-bit circuits

A binary deterministic finite automaton has finitely many states, a designated start state, a transition for each state and bit, and a set of accepting states. Starting from its initial state, a binary word of length $n$ determines a run. Let $L$ be the set of words whose final state is accepting. Put $L_n=L\cap\{0,1\}^n$. An arbitrary such automaton represents an arbitrary binary regular language, with no assumption that membership belongs to $\mathrm{AC}^0$.

A finite Markov chain consists of a nonempty finite state set $Q$, an initial probability vector $\mu$, and a row-stochastic transition matrix $P$. Entries are nonnegative real numbers, not restricted to rational numbers. For $\gamma=(x_0,\ldots,x_n)$, its probability is $p_M(\gamma)=\mu(x_0)\prod_{i=1}^{n}P(x_{i-1},x_i)$. The distribution of all $n+1$ states is denoted $\mathrm{Path}_n(M)$. For $F\subseteq Q$, conditioning is allowed when the probability that $X_n\in F$ is positive.

A randomized $\mathrm{AC}^0$ sampler is a deterministic Boolean circuit supplied with independent, unbiased random bits. AND and OR gates have unbounded fan-in; NOT gates and Boolean constants are permitted. Its size is polynomial and its depth has a constant bound independent of trajectory length. Approximate families are nonuniform in both $n$ and their real error tolerance $\epsilon$: existence does not require efficiently computing circuit descriptions or numerical tables from a supplied transition matrix.

Formalization targets

The main goal is the conjunction of all results E.1–E.6, including the linear seed guarantee for E.5. This organizational theorem is not an additional numbered result attributed to the paper. All six milestones must be proved for the goal to hold. The exact Markov sampling criterion in Theorem E.5 is one component. A nonnegative real number is dyadic if it equals $a/2^k$ for natural numbers $a,k$, including zero. A chain is path-dyadic when every finite trajectory, including a trajectory of length zero, has a dyadic probability. The target is

$$\mathrm{Path}_n(M)\text{ has an exact randomized }\mathrm{AC}^0\text{ sampler family} \quad\Longleftrightarrow\quad \forall n\geq0\;\forall\gamma\in Q^{n+1},\ p_M(\gamma)\text{ is dyadic}.$$

On the left, the family has polynomial size in $n+1$ and depth bounded independently of $n$. A supporting target records the linear fair-bit budget $O_M(n+1)$ when path-dyadicity holds. Lemma E.4 supplies another target: every path-dyadic chain is the coordinatewise projection of a finite chain whose initial and transition probabilities are all dyadic. Path-dyadicity must not be replaced by the stronger requirement that the original transition matrix itself be dyadic (Appendix E.3).

Theorem E.1 concerns exact sampling of a uniformly random word together with its acceptance indicator:

$$(X,\mathbf 1[X\in L]),\qquad X\sim\operatorname{Unif}(\{0,1\}^n).$$

The sampler uses exactly $n$ fair bits, polynomial size, and depth depending only on the fixed automaton. This is a sampling assertion and does not assert that membership can be computed in constant depth on an arbitrary externally supplied word.

Theorem E.2 concerns the uniform law on nonempty $L_n$. For every $0<\epsilon\leq1/2$, the target is

$$d_{\mathrm{TV}}(Q_{n,\epsilon},\operatorname{Unif}(L_n))\leq\epsilon, \qquad r_{n,\epsilon}\leq n+C_L\log_2(1/\epsilon).$$

Every possible output is accepted. Size is polynomial in $n+1$ and $1/\epsilon$, and depth depends only on the fixed language or its automaton. In particular, the leading coefficient of the $n$ term in the random-bit bound is one (Appendix E.2).

Theorem E.6 is the corresponding approximate target for trajectories conditioned on $X_n\in F$. It requires total variation error at most $\epsilon$, only positive-probability outputs ending in $F$, $O_M(n+\log(1/\epsilon))$ fair bits, polynomial size in $n+1$ and $1/\epsilon$, and depth depending only on the chain. There is no lower bound on the conditioning probability beyond strict positivity (Appendix E.4).

Lemma E.3 supplies the exact representation used in the Markov construction. A transition-dyadic chain is represented by one binary DFA and a fixed state projection. On a uniform input of length $v+ns$, with positive constants $v,s$, the states observed after $v,v+s,\ldots,v+ns$ bits project to exactly the complete trajectory law of the chain. The same DFA and projection work for every length, including zero.

Significance of the targets

These results separate sampling an input together with its finite-state behavior from computing that behavior on a prescribed input. They also identify a precise obstruction to exact sampling with a finite fair-bit seed: each output probability must be dyadic. Approximate sampling allows arbitrary real transition probabilities while retaining the valid trajectory support. The complete trajectory requirement preserves correlations among all time coordinates.

The formalization makes the circuit model, seed accounting, support guarantees and asymptotic quantifiers explicit and reusable. The theorem statements and their separately submitted proofs have passed the platform verifier. The goal proof imports all six milestones; its E.5 milestone in turn imports both the equivalence and the linear seed theorem. Thus the platform dependency graph covers the entire appendix. Platform verification establishes the formal statements; semantic correspondence is documented in the statement descriptions and source references.

Difficulty

A literal step-by-step implementation of a run makes circuit depth grow with $n$, so it does not meet the constant-depth target. Recognition results alone do not furnish the required joint sampler. For approximate word sampling, an error estimate alone is insufficient: the output must remain in the accepted support, and the coefficient-one seed bound must also hold. For exact Markov sampling, unreachable states and non-dyadic individual transitions prevent identifying the desired criterion with entrywise dyadicity. Conditioning on a rare terminal event must not introduce an unstated dependence on its probability.

Formalization scope

States are indexed by finite types $\mathrm{Fin}(q)$. Binary words have type $\mathrm{Fin}(n)\to\mathrm{Bool}$; trajectories have $n+1$ coordinates. Markov vectors and matrices are real-valued and carry normalization and nonnegativity proofs. Sampling probabilities are defined by counting all seeds that give a particular encoded outcome and dividing by $2^r$. Total variation is half the finite sum of absolute probability differences.

The circuit representation is an explicit topologically ordered gate list with checked wire references. Size counts input bits, gates, gate-input wires and output wires; this yields the usual polynomial-size class. State trajectories use a fixed one-hot encoding at each time coordinate. The invalid-output predicate is checked over every seed, so neither arbitrary decoders nor unrepresented outcomes can satisfy an exact or support-preserving claim. The encodings used by the targets are injective. The circuit family is existentially quantified after the fixed finite-state model; the depth and polynomial constants precede $n$ and $\epsilon$.

Zero-length words and zero-transition Markov paths are included. The word target excludes empty accepted slices. The conditioned target excludes zero-probability conditioning events. Definitions of division by zero still exist as total Lean functions, but the hypotheses exclude those cases from approximate claims. Extensions of these results, reusable finite-state probability lemmas, and further circuit constructions are welcome. No sampler-existence axiom, vacuous model or final-state-only replacement is part of the interface.

Selected references

• Jiarui Zhang, Chengwei Liang, Haozhe Jiang, Binhang Yuan and Jingzhao Zhang. Revision Provably Reduces Sequential Computation in Diffusion Language Models. Zenodo preprint, 2026. DOI 10.5281/zenodo.23105365, Appendix E, Theorems E.1, E.2, E.5, E.6 and Lemmas E.3, E.4, printed pages 33–36.

Motivation

Per-occurrence excess and aggregate excess have different financial triggers. One aggregate deductible applies once to the sum of claims, while a per-risk retention is charged against each loss. Combining many small claims may exhaust an aggregate deductible without producing a payment under any separate per-claim policy. HMRC explains how non-proportional reinsurance differs across individual risks and catastrophe or aggregate protections (https://www.gov.uk/hmrc-internal-manuals/general-insurance-manual/gim8060). The valuation question is how the ceded amounts and their pure expected costs are ordered when otherwise identical annual experience and deductibles are used.

Setting

The individual claims x(i), their finite number n, and the deductible d are natural-number monetary quantities. Gross aggregate loss S(n) is the sum of the first n claims. Aggregate recovery is positive excess of S(n) above d; separate per-claim recovery is the sum of each claim's positive excess over d. Finite scenario weights w(s) are real. A scenario maps to an entire loss sequence, and its total claim portfolio is evaluated consistently under both treaty designs. Weighted claims are aggregated for indices from zero to a finite bound B, allowing the scenario count itself to be zero or the probability weights to lack normalisation.

Formalization targets

The formal targets progress from structural boundary identities to financial sensitivity and the full capstone. Supporting theorems remain genuine Open declarations until accepted Lean proofs are submitted; no previously accepted proof is silently assumed.

Boundary identity

The exact theorem name is ActuarialValuation.aggregateLossTotal_zero. An empty collection of claims has zero gross claims and therefore defines the base case for deductible comparisons.

$$S_0=0$$

Structural intermediate target

The exact theorem name is ActuarialValuation.aggregateDeductibleCeded_zero_attachment. With attachment zero, the aggregate excess treaty covers every unit of gross annual claims.

$$C_{\mathrm{agg}}(0)=S_n$$

Main capstone

The exact theorem name is ActuarialValuation.aggregateVsPerClaim_fundamental. The capstone joins the nontrivial pointwise reinsurance coverage inequality to the same ordering of expected pure premiums, without conflating a whole-year treaty with a per-claim deductible.

$$C_{\mathrm{individual}}\le C_{\mathrm{agg}},\quad0\le\Pi_{\mathrm{agg}},\quad\Pi_{\mathrm{individual}}\le\Pi_{\mathrm{agg}}$$

The graph also contains separately stated supporting inequalities and exact definitions, with the intended original derived mathematical identity written out in every milestone.

Significance

The comparison can diagnose material pricing errors, since an aggregate deductible and a per-risk deductible with the same quoted monetary value should not normally have the same technical premium. HMRC's treatment of non-proportional reinsurance (https://www.gov.uk/hmrc-internal-manuals/general-insurance-manual/gim8060) makes this distinction a contractual reality. The formal theorem is useful for ceded reserve reconciliation, claims-data audit, treaty design, and future models including several occurrence covers within a separate annual limit. In each case exact mathematical relationships are more valuable than approximate simulation agreement.

Difficulty

The algebra of positive-part natural subtraction differs from ordinary subtraction over integers or real values. The aggregate deductible is taken once, whereas the per-event premium subtracts the same amount separately from each positive claim. The theorem is a weak inequality and must remain valid even when every event is below the separate retention but the aggregate exceeds the annual attachment. The model explicitly includes zero claims, zero deductible, empty portfolios and nonnegative scenario weights. Signed weights would make the expectation order invalid and are not silently permitted in that result.

Formalization scope

Lean treats all claim amounts, event counts, attachment values and scenario bounds as natural numbers. Real numbers occur only in the finite expected ceded amounts. Each individual-claim sum uses Finset.range and the pure premium is a further finite sum over scenarios. This is an exact deterministic relationship between two reinsurance structures rather than an assumption of independence or identical event distributions. No catastrophe frequency model, claims adjustment expenses or risk load is included. The established contractual idea motivates original derived finite relations, and the formal statements remain open until proved.

The pinned Mathlib revision and the precise Lean type of each definition or proposition are authoritative. The imported source must use targeted Mathlib modules; one large aggregate compiler check never substitutes for testing individual definition imports and obsolete namespace openings. The source-specific intermediate statements separate units and hypotheses. The finite index always counts claims, scenarios, or observations according to the declaration, not an implicit probability measure. In an insurance pricing setting the difference between a gross loss, a ceded payment, a net premium and a credibility risk estimate is financially important. A positive coefficient does not by itself imply the coefficients sum to one. Likewise a deterministic algebraic identity does not grant stochastic independence or a forecast model. These restrictions preserve the distinction between a purely algebraic result and a probabilistic interpretation. The formal theorem declarations have to maintain exactly the same quantifiers, arithmetic operations and endpoint conventions as the descriptions, including meaningful boundary cases. Future extensions can introduce richer distributions and dependence without altering the precise finite identities established here.

Selected references

• S. David Promislow, Fundamentals of Actuarial Mathematics, third edition, Wiley, 2015, printed page 389. https://www.oreilly.com/library/view/fundamentals-of-actuarial/9781118782521/

• HM Revenue & Customs, General Insurance Manual: Non-proportional reinsurance, GOV.UK, 2023. https://www.gov.uk/hmrc-internal-manuals/general-insurance-manual/gim8060

Motivation

Hattendorff's theorem allocates the variance of an insurer's life-contingent loss to policy years. In a fully discrete life insurance model, each policy year's contribution depends on the discounted net amount at risk, the probability of dying in that year, and the conditional probability of surviving the year. Shiu and Xiong (1) derive the classical loss-at-issue formula and the corresponding discrete Hattendorff variance through reserve identities and a summation-by-parts argument. Gerber [2, §6.7] discusses the actuarial allocation of total loss to individual policy years.

The related Missions XVI and XIX address two complementary parts of that argument. XVI specifies death/survival masses for a finite lifetime distribution and proposes proofs of centred annual mortality innovations, cross-year orthogonality and variance. XIX describes a pathwise, year-by-year accounting identity expressing insurer loss, including the terminal reserve, as the opening reserve plus discounted mortality innovations. The present mission is the mathematical bridge combining these structures.

It targets the full finite-horizon reserve-loss expectation and variance theorem under consistent conditional mortality probabilities and the annual prospective reserve recurrence. Unlike a purely algebraic zero-covariance lemma, the mortality model supplies the orthogonality. Unlike a standalone cashflow telescoping lemma, the result calculates the variance of the insurer's actual policy loss.

Setting

Fix a finite scenario space $\Omega$, with nonnegative probabilities $w_\omega$ summing to one, and a curtate future lifetime $K:\Omega\to\mathbb N$. An insured with $K(\omega)=t$ dies during policy year $t$.

Define the in-force and death-year probability masses

$$S_t=\sum_{\omega\in\Omega}w_\omega\mathbf1_{\{K(\omega)\ge t\}}, \qquad D_t=\sum_{\omega\in\Omega}w_\omega\mathbf1_{\{K(\omega)=t\}}.$$

The disjoint survival/death partition gives $S_t=D_t+S_{t+1}$. For each year covered by the policy valuation horizon, assume positive in-force mass, $S_t>0$, so that the conditional death and survival rates are well-defined:

$$q_t=\frac{D_t}{S_t}, \qquad p_t=\frac{S_{t+1}}{S_t}, \qquad q_t+p_t=1.$$

These rates are derived from one joint lifetime distribution. The annual death indicators are therefore linked across years and are not independent. The zero-covariance result must be derived from the nested survival-event structure rather than assumed.

Let $n$ denote the finite number of years covered, $v$ a deterministic annual discount factor, $\pi_t$ the beginning-of-year premium, $b_{t+1}$ the death benefit payable at time $t+1$, and $V_t$ the prospective reserve at time $t$ for a policy still in force.

Assume that, for each year $0\le t<n$, the prospective reserve satisfies

$$\boxed{ V_t+\pi_t=v\left(p_tV_{t+1}+q_tb_{t+1}\right). }$$

The realised insurer loss, valued at issue, is

$$\begin{aligned} L_n(\omega) ={}&\sum_{0\le t<n}\left[ v^{t+1}b_{t+1}\mathbf1_{\{K(\omega)=t\}} -v^t\pi_t\mathbf1_{\{K(\omega)\ge t\}} \right]\\ &+v^nV_n\mathbf1_{\{K(\omega)\ge n\}}. \end{aligned}$$

The terminal reserve is explicitly included if the insured survives beyond the chosen horizon. Thus $L_0=V_0$, a life may remain insured after year $n$, and neither $V_0$ nor $V_n$ is assumed zero. The original Shiu–Xiong complete-life issue-loss formula is recovered only under appropriate terminal/residual liability conditions; the present finite-horizon formula is the reserve-inclusive extension needed for a finite theorem.

Formalization targets

Mortality innovation and pointwise reserve loss

The year-$t$ mortality innovation is

$$I_t(\omega)= \mathbf1_{\{K(\omega)=t\}} -q_t\mathbf1_{\{K(\omega)\ge t\}}.$$

The discounted net amount at risk is

$$\rho_t=v^{t+1}(b_{t+1}-V_{t+1}).$$

By the annual reserve recurrence and the in-force event partition, the change in the insurer's loss across a policy year equals $\rho_tI_t$. Summing over years gives the exact pathwise representation

$$\boxed{ L_n(\omega)=V_0+\sum_{0\le t<n}\rho_tI_t(\omega). }$$

Mission XIX describes this pointwise reserve-loss identity. The present formalisation must bridge XIX's realised-death-year indicator and reserve-derived innovation to XVI's scenario-based mortality innovation. They represent the same object when the death probabilities are calculated from the same $w,K$ model.

From the finite scenario lifetime law and $S_t>0$, the annual innovations satisfy

$$\mathbb E_w[I_t]=0,\qquad \mathbb E_w[I_iI_j]=0\quad(i\ne j),\qquad \mathbb E_w[I_t^2]=D_tp_t.$$

The final factor $p_t$ is the conditional one-year survival probability and is essential to the fully discrete variance formula. It must not be replaced by an assumed independence condition or omitted by importing a continuous-time formula.

Full expectation and variance capstone

Taking expectations of the pathwise identity, using $\sum_\omega{} w_\omega=1$, yields

$$\boxed{\mathbb E_w[L_n]=V_0.}$$

The variance of an insurer loss is unaffected by adding a deterministic opening reserve when the probability weights have unit total mass. Consequently,

$$\begin{aligned} \operatorname{Var}_w(L_n) &=\operatorname{Var}_w\left(\sum_{0\le t<n}\rho_tI_t\right)\\ &=\sum_{0\le t<n}\rho_t^2\mathbb E_w[I_t^2]\\ &=\boxed{\sum_{0\le t<n}v^{2(t+1)} (b_{t+1}-V_{t+1})^2D_tp_t}. \end{aligned}$$

The capstone combines these two results, with no unproved independence assumption. Its mathematical hypotheses explicitly include nonnegative, normalised scenario weights, positive in-force probability for every year in the range, and the one-year reserve recurrence with conditional rates computed from the given lifetime distribution.

The theorem remains valid at $n=0$: the insurer's finite-horizon loss is deterministically $V_0$, and the variance sum is empty and equals zero.

The exact final theorem is ActuarialValuation.finiteHattendorffReserveLoss_fundamental. This result is nontrivial at positive horizons. The formal model does not assert independence, fix the opening reserve to zero, or substitute a deterministic insurer loss.

Significance

The annual decomposition separates valuation assumptions from stochastic fluctuations in mortality. Shiu and Xiong's classical fully discrete treatment (https://doi.org/10.1007/s13385-020-00256-9) emphasises that a survival factor remains in each annual variance contribution. Preserving this factor makes the result useful for actuarial risk allocation and for reconciling the valuation reserve with actual finite-period insurer loss.

The finite-scenario definitions are reusable in downstream insurance valuation missions, for example multiple-decrement reserves and explicit conditional mortality shock models. The result also supplies finite boundary tests for nonzero opening reserves, death exactly at the horizon, and survival beyond the final premium year.

Difficulty

Annual death events arise from the same curtate lifetime, so their indicator variables have nested dependence. Their centred innovations have vanishing cross-year covariance only under the correct conditional in-force rates and probability-weighting conventions. Incorrect denominators or an unproved independence assumption would change the result.

The pathwise identity also accounts for different timing of premiums and benefits, and includes a terminal reserve for survivors. At the last covered year a policy death and survival are disjoint, so the last benefit and terminal reserve must not both be counted. The exact Lean imports must retain the definitions and parameter order of the previously published mortality and loss modules.

Formalization scope

The Lean 4 model uses a finite scenario type with Fintype, real-valued weights, a natural-number curtate lifetime, and the published definition modules from related actuarial missions. Four new definitions and 13 Open theorem obligations form the formal graph, including 12 supporting targets before the conjunction capstone. The accepted proof status of another theorem is never inferred merely from its publication.

The result is finite horizon, finite probability space and fully discrete. The source's complete-life identity would additionally require justified limiting arguments, the disappearance (or valuation) of terminal reserves, integrability and interchange of expectation, variance and limit. None of those conditions is silently assumed here. There is no continuous-time martingale or Lévy-process result in this mission.

The classical full-life result is known mathematically, while the specific finite-horizon Lean declarations remain open proof targets. A compiler-accepted declaration with a sorry body does not establish a proof.

Selected references

• [1] Shiu, Elias S. W. and Xiong, Xiaoyi (2021), “An elementary derivation of Hattendorff's theorem”, European Actuarial Journal, 11, pp. 319–323, equations (1)–(5). Source for the discrete variance survival factor and the reserve-based derivation.
• [2] Gerber, Hans U. (1997), Life Insurance Mathematics, third edition, Springer, Chapter 6, §6.7, Allocation of the Overall Loss to Policy Years.
• [3] Gerber, Hans U., Leung, Bartholomew P. K. and Shiu, Elias S. W. (2003), “Indicator Function and Hattendorff Theorem”, North American Actuarial Journal, 7(1), pp. 38–47. Relates mortality indicator functions, actuarial loss variance and orthogonal yearly increments.

Motivation

A bounded linear operator $u : E \to F$ between Banach spaces is Fredholm if its kernel is finite-dimensional and its range is closed and of finite codimension. Its index

$$\operatorname{ind} u = \dim \operatorname{Ker} u - \dim \operatorname{Coker} u, \qquad \operatorname{Coker} u = F/\operatorname{Im} u,$$

is an integer that survives deformation: it does not change under small perturbations or under compact perturbations, and it is additive under composition. These properties make Fredholm operators the setting for index theorems. Elliptic differential operators on compact manifolds, Toeplitz operators, linearized Cauchy–Riemann operators in symplectic topology, and the linearizations of nonlinear Fredholm maps in transversality arguments (Sard–Smale) all rest on them. Atkinson's theorem characterizes Fredholm operators as those invertible modulo compact operators. For endomorphisms this says that the image in the Calkin algebra $\mathcal{L}(E)/\mathcal{K}(E)$ is invertible.

This mission formalizes the abstract theory on Banach spaces over $\mathbb{R}$ or $\mathbb{C}$, following N. Bourbaki, Théories spectrales, Chapitre III, §§ 3–5 (Springer, 2023). It builds on Mathlib's ContinuousLinearMap.IsFredholm, which already provides the definition and its equivalent descriptions on topological vector spaces. It adds the index and the results listed below.

Setting

$\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$. $E$, $F$, $G$ are Banach spaces over $\mathbb{K}$, and $\mathcal{L}(E;F)$ is the Banach space of bounded linear maps with the operator norm.

• Fredholm operator. $u \in \mathcal{L}(E;F)$ is Fredholm if $\operatorname{Ker} u$ is finite-dimensional and topologically complemented, $\operatorname{Im} u$ is closed and of finite codimension, and $u$ is a strict map (it induces a homeomorphism $E/\operatorname{Ker} u \to \operatorname{Im} u$). On Banach spaces this agrees with Bourbaki's Définition 1 (TS III.41): $u$ has a quasi-inverse $v \in \mathcal{L}(F;E)$, i.e. $1_E - v\circ u$ and $1_F - u\circ v$ have finite rank.
• Index. $\operatorname{ind} u = \dim \operatorname{Ker} u - \dim (F/\operatorname{Im} u) \in \mathbb{Z}$. Bourbaki (TS III.43, formula (1)) uses the opposite sign; this mission uses the sign above, which is the usual one in analysis and geometry.
• Compact operator. $h \in \mathcal{L}(E;F)$ is compact if it maps some neighbourhood of $0$ into a compact set (Mathlib's IsCompactOperator).

Formalization targets

Goal: Atkinson's theorem

$u \in \mathcal{L}(E;F)$ is Fredholm if and only if there is $v \in \mathcal{L}(F;E)$ such that $1_E - v\circ u$ and $1_F - u\circ v$ are compact (Bourbaki III, §5, Corollaire 1, TS III.74).

Milestones

1. Finite codimension gives closed range. If $F/\operatorname{Im} u$ is finite-dimensional, then $\operatorname{Im} u$ is closed and $u$ is strict (§3, Lemme 6, TS III.52).
2. Fredholm criterion. $u$ is Fredholm if and only if $\operatorname{Ker} u$ and $F/\operatorname{Im} u$ are finite-dimensional (§3, Proposition 11, TS III.52).
3. Composition. If $u : E \to F$ and $v : F \to G$ are Fredholm, so is $v\circ u$ (§3, Remarque 2, TS III.41).
4. Additivity of the index. $\operatorname{ind}(v\circ u) = \operatorname{ind} v + \operatorname{ind} u$ (§3, formula (2), TS III.43).
5. Restriction to subspaces of finite codimension. For closed subspaces $E_1 \subset E$, $F_1 \subset F$ of finite codimension with $u(E_1) \subset F_1$, $u$ is Fredholm if and only if its restriction $u_1 : E_1 \to F_1$ is, and then $\operatorname{ind} u - \operatorname{ind} u_1 = \operatorname{codim} E_1 - \operatorname{codim} F_1$ (§3, Proposition 3, TS III.44).
6. Index zero. A Fredholm operator of index $0$ that is injective or surjective is an isomorphism (§3, n° 3, TS III.44).
7. Openness and local constancy. The Fredholm operators form an open subset of $\mathcal{L}(E;F)$, and the index is locally constant on it (§4, Théorème 1, TS III.58).
8. Compact perturbation, kernel side. If $\operatorname{Ker} u$ is finite-dimensional, $\operatorname{Im} u$ is closed and $h$ is compact, then $\operatorname{Ker}(u+h)$ is finite-dimensional and $\operatorname{Im}(u+h)$ is closed (§5, Théorème 1, TS III.72).
9. Compact perturbation, cokernel side. If $\operatorname{Im} u$ has finite codimension and $h$ is compact, then $\operatorname{Im}(u+h)$ is closed and of finite codimension (§5, Théorème 2, TS III.73).
10. Compact perturbation of Fredholm operators. If $u$ is Fredholm and $h$ is compact, then $u+h$ is Fredholm and $\operatorname{ind}(u+h) = \operatorname{ind} u$ (§5, Théorème 3, TS III.73).
11. Index-zero characterization. $u$ is Fredholm of index $0$ if and only if $u$ is the sum of an isomorphism $E \to F$ and a finite-rank operator, if and only if it is the sum of an isomorphism and a compact operator (§5, Corollaire 2, TS III.74).

Significance

The result. Atkinson's theorem identifies the Fredholm operators with the units modulo the compact ideal. Together with the stability of the index under compact and small perturbations, it is the abstract input of every Fredholm index computation. Examples are the Fredholm alternative for integral equations, the index of Toeplitz operators, and the index of elliptic and Cauchy–Riemann operators on compact or punctured surfaces.

Formalizing it. Mathlib defines Fredholm operators on topological vector spaces (ContinuousLinearMap.IsFredholm) and proves that four descriptions agree (isFredholm_tfae). It has no index, no statement about compositions, and no perturbation theory. Its documentation lists the Banach-space criterion (milestone 2) and the restriction statement (milestone 5) as not yet formalized. The library built here is intended for reuse by any later index computation.

Difficulty

The statements combine linear algebra and topology. The index is a difference of dimensions of a subspace and a quotient space, and every statement about it carries a Fredholm hypothesis, since Module.finrank is $0$ on infinite-dimensional spaces. Openness and local constancy are statements about the operator-norm topology on $\mathcal{L}(E;F)$, and the perturbation statements involve operators that are compact but not necessarily of finite rank. The completeness hypotheses are part of the statements: over incomplete normed spaces, a range of finite codimension need not be closed.

Formalization scope

• Namespace FredholmBanach. The definition module FredholmBanach_Setting defines only index. Fredholm operators are Mathlib's ContinuousLinearMap.IsFredholm and compact operators are Mathlib's IsCompactOperator.
• Field and spaces. [RCLike 𝕜], [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E], and likewise for F and G. Bourbaki states several results for locally convex or Fréchet spaces; this mission specializes all of them to Banach spaces.
• Index. index u = finrank 𝕜 u.ker - finrank 𝕜 (F ⧸ u.range) in ℤ. Module.finrank is $0$ on infinite-dimensional spaces, so index is meaningful only for Fredholm u. Every statement about it assumes u.IsFredholm.
• Subspaces and quotients. The codimension of a subspace $E_1$ is finrank 𝕜 (E ⧸ E₁), and "finite codimension" is FiniteDimensional 𝕜 (E ⧸ E₁). The restriction in milestone 5 is ContinuousLinearMap.restrict.
• Isomorphisms are E ≃L[𝕜] F. "Finite rank" is FiniteDimensional 𝕜 (u - e).range.

Contributions welcome: reusable general-purpose lemmas about bounded and compact operators between Banach spaces.

Selected references

• N. Bourbaki, Théories spectrales, Chapitres III à V, Springer (2023), Chapitre III, §§ 3–5. https://doi.org/10.1007/978-3-031-19505-1
• F. V. Atkinson, The normal solubility of linear equations in normed spaces, Mat. Sb. (N.S.) 28(70):1 (1951), 3–14 (in Russian), MR0042059. https://mathnet.ru/eng/sm5589
• T. Kato, Perturbation Theory for Linear Operators, Springer (1995 reprint of the 2nd ed.). https://doi.org/10.1007/978-3-642-66282-9
• D. McDuff, D. Salamon, J-holomorphic Curves and Symplectic Topology, 2nd ed., AMS Colloquium Publications 52 (2012). https://doi.org/10.1090/coll/052

Motivation

Let $S : S^1 \to \mathbb{R}^{2n\times 2n}$ be a loop of symmetric matrices. The asymptotic operator

$$A_S = -J_0\,\partial_t - S(t)$$

acts on loops $S^1 \to \mathbb{R}^{2n}$. It appears wherever a pseudoholomorphic curve approaches a periodic orbit. Near a puncture, a finite-energy curve in a symplectization converges to a Reeb orbit, and in a trivialization along that orbit the linearized Reeb flow is $\Psi' = J_0 S \Psi$. The operator $A_S$ is the Hessian of the action functional at that orbit. The rate of approach and the winding of the curve around the orbit come from its eigenfunctions (Hofer–Wysocki–Zehnder 1995). In Floer theory the operator $-J_0\partial_t - S$ is the Hessian of the symplectic action at a $1$-periodic orbit, and the gradient-flow operator $\partial_s + A_{S(s)}$ is the linearized Floer operator (Robbin–Salamon 1995).

The operator links functional analysis to the linear algebra of stage I. $A_S$ is nondegenerate, i.e. $0$ is not an eigenvalue, exactly when the time-one map $\Psi(1)$ of the linearized flow has no eigenvalue $1$. Then $\Psi|_{[0,1]}$ lies in $\mathrm{SP}(n)$ and has a Conley–Zehnder index. The Fredholm index formula for Cauchy–Riemann operators on punctured surfaces is written in terms of the Conley–Zehnder indices of the asymptotic operators at the punctures (Wendl, Lectures on SFT, Lecture 3).

This is stage II of a series that formalizes the Fredholm theory of linearized Cauchy–Riemann operators. Stage I (Conley–Zehnder Index I: Axiomatic Characterization on Paths of Symplectic Matrices) characterizes the Conley–Zehnder index on $\mathrm{SP}(n)$. This stage sets up the spectral theory of asymptotic operators: the operator itself, self-adjointness, discrete spectrum, compact resolvent, and the dictionary between the kernel of $A_S$ and the time-one map. Later stages are the Fredholm theory of $\partial_s + A_{S(s)}$ in Banach spaces, elliptic regularity for $\bar\partial$, and the index formula.

Setting

$S^1 = \mathbb{R}/\mathbb{Z}$ with its Haar probability measure. $\mathbb{R}^{2n}$ has coordinates $(q,p)$ and the Euclidean inner product, and

$$J_0 = \begin{pmatrix} 0 & -\mathrm{Id}\\ \mathrm{Id} & 0\end{pmatrix}.$$

$S : S^1 \to \mathbb{R}^{2n\times 2n}$ is continuous with $S(t)^{\mathsf T} = S(t)$ for every $t$.

• Fourier coefficients. For $f \in L^2(S^1,\mathbb{R}^{2n})$ and $k \in \mathbb{Z}$, $\hat f_k = \int_{S^1} e^{-2\pi i k t} f(t)\,dt \in \mathbb{C}^{2n}$.
• Domain. $W^{1,2}(S^1,\mathbb{R}^{2n})$ is the set of $f \in L^2$ with $\sum_{k\in\mathbb{Z}} (1+k^2)\,|\hat f_k|^2 < \infty$.
• The operator. For $f \in W^{1,2}$, $A_S f = g$ means $\hat g_k = -2\pi i k\,J_0 \hat f_k - (Sf)^\wedge_k$ for every $k$, i.e. $g = -J_0 f' - S f$ with the derivative taken in $L^2$. This is Wendl's (3.4) with period $1$.
• Fundamental solution. $\Psi : \mathbb{R} \to \mathbb{R}^{2n\times2n}$ with $\Psi(0) = \mathrm{Id}$ and $\Psi'(t) = J_0 S(t)\Psi(t)$. These are the solutions of $(-J_0\partial_t - S)\Psi = 0$. For constant $S$, $\Psi(t) = \exp(tJ_0S)$ is the path in the signature property of stage I.

Formalization targets

Goal: spectral theory of the asymptotic operator

For every continuous loop $S$ of symmetric matrices:

1. $A_S$ with domain $W^{1,2}$ is self-adjoint on $L^2(S^1,\mathbb{R}^{2n})$;
2. $L^2(S^1,\mathbb{R}^{2n})$ has an orthonormal basis of eigenvectors of $A_S$ lying in $W^{1,2}$; the eigenvalues are real, each has finite multiplicity, and they accumulate only at $\pm\infty$;
3. $\ker A_S \cong \ker(\Psi(1) - \mathrm{Id})$, where $\Psi$ is the fundamental solution.

Milestones

1. Well-defined operator. Exactly one partially defined linear map on $L^2$ has graph $\{(f, A_S f)\}$.
2. Domain. The domain of $A_S$ is $W^{1,2}$, and it is dense in $L^2$.
3. Symmetry. $\langle A_S f, g\rangle = \langle f, A_S g\rangle$ for $f, g \in W^{1,2}$.
4. Self-adjointness.
5. Discrete spectrum. An orthonormal eigenbasis exists, with eigenvalues of finite multiplicity tending to $\pm\infty$.
6. Compact resolvent. If $\mu \in \mathbb{R}$ is not an eigenvalue, $A_S - \mu : W^{1,2} \to L^2$ is bijective and its inverse is a compact operator on $L^2$.
7. Unbounded spectrum. For $n \geq 1$ there are eigenvalues of both signs with arbitrarily large absolute value.
8. The unperturbed operator. The eigenvalues of $-J_0\partial_t$ are exactly $2\pi k$, $k\in\mathbb{Z}$, and each eigenspace has dimension $2n$.
9. Fundamental solution. $\Psi$ exists and is unique for every continuous loop $S$.
10. Symplecticity. If $S$ is symmetric, $\Psi(t) \in \mathrm{Sp}(2n)$ for every $t$.
11. Kernel. $\ker A_S \cong \ker(\Psi(1) - \mathrm{Id})$.
12. Nondegeneracy. $0$ is not an eigenvalue of $A_S$ if and only if $\Psi|_{[0,1]} \in \mathrm{SP}(n)$.

Significance

The result. These are the analytic facts that every use of asymptotic operators relies on. Self-adjointness and the discrete spectrum define the eigenvalue branches and the spectral flow. The compact resolvent makes $A_S - \mu$ Fredholm of index $0$ for every real $\mu$, and the weighted Sobolev spaces of the later Fredholm theory are chosen with weights avoiding the spectrum. The kernel dictionary identifies nondegenerate asymptotic operators with the paths of stage I. It is the bridge through which Conley–Zehnder indices enter the index formula.

Formalizing it. All results here are classical. Mathlib has partially defined linear maps and their adjoints, the Fourier basis of $L^2$ on the circle, and compact operators. It has no Sobolev spaces on the circle, no unbounded operator with discrete spectrum beyond trivial cases, and no first-order differential operator treated as an unbounded operator. This stage builds the first such example. The same pattern applies to other first-order elliptic operators on $S^1$, and the library is reusable for spectral flow and for Floer-type operators.

Difficulty

$A_S$ is unbounded and not defined on all of $L^2$, so self-adjointness is a statement about the domain of the adjoint, not a symmetry identity. The spectral statements concern the resolvent of a first-order differential operator with non-constant coefficients, as an operator on $L^2$. The kernel dictionary relates $L^2$ classes, defined through Fourier coefficients, to classical solutions of a linear ODE. The statements about $\Psi$ concern linear ODEs whose coefficients are only continuous.

Formalization scope

• Namespace AsymptoticOperator; all definitions are in AsymptoticOperator_Setting, which imports stage I's ConleyZehnder_Setting (for Mat n, J₀, IsSymplectic and SP).
• The circle is UnitAddCircle ($\mathbb{R}/\mathbb{Z}$) with AddCircle.haarAddCircle. $L^2(S^1,\mathbb{R}^{2n})$ is Lp (EuclideanSpace ℝ (Fin n ⊕ Fin n)) 2 haarAddCircle, a real Hilbert space.
• Fourier coefficients are Mathlib's fourierCoeff, applied to the componentwise complexification $\mathbb{R}^{2n} \to \mathbb{C}^{2n}$. $J_0$ and $S(t)$ act on $\mathbb{C}^{2n}$ through the real-to-complex coercion.
• The operator is a LinearPMap, defined by choice as the map whose graph is the set of pairs $(f, g)$ satisfying the coefficient identity. The fallback is $0$, and milestone 1 makes it unreachable. Self-adjointness is Mathlib's IsSelfAdjoint for LinearPMap, and symmetry is IsFormalAdjoint.
• Eigenspaces are kernels of $A_S - \mu$ on the domain. The discrete spectrum is stated as a HilbertBasis of eigenvectors whose eigenvalues tend to $\infty$ in absolute value along the cofinite filter. This covers $n = 0$, where $L^2 = 0$.
• Regularity of $S$. Wendl assumes $S$ smooth; Hofer–Wysocki–Zehnder (§3, (35)) take $S$ continuous, as here. HWZ state self-adjointness, compact resolvent and discreteness of the spectrum for $n = 1$; this mission states them for all $n$.
• Sign conventions follow Wendl §3.2: $A_S = -J_0\partial_t - S$ and $\Psi' = J_0S\Psi$.

Contributions welcome: reusable lemmas on Sobolev spaces of vector-valued functions on the circle (Fourier characterization, continuity of $W^{1,2}$ functions), on unbounded self-adjoint operators with compact resolvent, and on linear ODEs with continuous periodic coefficients.

Selected references

• C. Wendl, Lectures on Symplectic Field Theory, arXiv:1612.01009, Lecture 3 (§§3.2–3.4). https://arxiv.org/abs/1612.01009
• H. Hofer, K. Wysocki, E. Zehnder, Properties of pseudoholomorphic curves in symplectisations II: Embedding controls and algebraic invariants, Geom. Funct. Anal. 5 (1995), 270–328. https://doi.org/10.1007/BF01895669
• J. Robbin, D. Salamon, The spectral flow and the Maslov index, Bull. London Math. Soc. 27 (1995), 1–33. https://doi.org/10.1112/blms/27.1.1
• D. Salamon, Lectures on Floer homology, in: Symplectic Geometry and Topology, IAS/Park City Math. Ser. 7, AMS (1999), 143–229. https://doi.org/10.1090/pcms/007/05
• J. Gutt, Generalized Conley–Zehnder index, Ann. Fac. Sci. Toulouse Math. (6) 23 (2014), 907–932. https://doi.org/10.5802/afst.1430

Motivation

A small insurance fund or participant with limited resources can gain or lose capital repeatedly and be forced to stop at one of two barriers. This is the familiar gambler's ruin problem, useful as a discrete illustration of capital adequacy, finite-state absorption and the distinction between ruin and eventual success. Promislow's treatment derives fair and biased cases using finite capital and a bounded opponent, and Grinstead and Snell present related random-walk absorption probabilities (https://stats.libretexts.org/Bookshelves/Probability_Theory/Introductory_Probability_%28Grinstead_and_Snell%29/12%3A_Random_Walks/12.02%3A_Gambler%27s_Ruin). The practical distinction is whether a probability refers to absorption at some eventual time or only to absorption within a fixed number of additional bets.

Setting

A unit-stake capital process occupies a natural-number capital state i between zero and a positive upper barrier N. With probability p the next bet raises capital by one unit; with complementary probability one minus p it lowers capital by one unit. Capital zero and capital N are absorbing. The fair case has p equal to one-half and a linear harmonic success function. In the biased case, a down-to-up odds ratio r is defined by (1−p)/p. An additional finite-horizon model records separately the chance of hitting the upper barrier and the chance of hitting zero before a given step count, leaving unresolved paths in the interior.

The fair finite-capital eventual upper-absorption harmonic candidate is named actuarial_gamblerFairSuccess in the Lean development. A fair unit-stake random walk on capital states zero through positive target N has eventual upper-barrier success probability i/N when starting from integer capital i in the bounded interval. This source defines the harmonic candidate, not the underlying stochastic stopping-time construction.

The downward-to-upward transition probability ratio is named actuarial_gamblerOddsRatio in the Lean development. The fixed probability of an up step is p; the probability of a down step is 1−p. Their ratio (1−p)/p controls the biased harmonic solution and requires p>0 to be interpreted as transition odds.

The biased finite-capital upper-absorption harmonic candidate is named actuarial_gamblerBiasedSuccess in the Lean development. For a biased unit-stake walk with r=(1−p)/p different from one and a positive upper barrier N, the formula gives the eventual probability of hitting N before zero: (1−r^i)/(1−r^N). The denominator must be nonzero. The fair p=1/2 case requires the separate linear formula.

The probability of hitting upper capital target within finitely many steps is named actuarial_gamblerFiniteSuccess in the Lean development. The bounded random walk makes unit upward jumps with probability p and unit downward jumps with probability 1−p. It is absorbed at capital zero or at or above target N. This finite-horizon recursion records only the probability of reaching the success barrier by time n; ongoing paths at the horizon remain unresolved.

The probability of reaching zero capital within finitely many steps is named actuarial_gamblerFiniteRuin in the Lean development. The second finite-horizon recursion tracks absorption at zero capital, the ruin boundary. Successful absorption at the upper target contributes zero ruin probability. At an interior state it mixes successor ruin probabilities, and paths still active at horizon n contribute to neither finite success nor finite ruin.

Formalization targets

The formal targets are ordered from elementary boundary and consistency claims to their combined capstone. The exact Lean identifiers below are the authoritative local targets; none of the listed theorem bodies is already accepted as a machine-checked proof. The mathematical results themselves are established actuarial or probability identities.

Baseline identities

The declaration ActuarialValuation.gamblerFairSuccess_zero states the following relation. The fair candidate's numerator is zero at the lower absorbing boundary, so the proposed upper-barrier success probability is zero even if the target denominator is algebraically zero.

$$H_{\rm fair}(0)=0$$

Stronger structural properties

The declaration ActuarialValuation.gamblerFiniteRuin_nonneg states the following relation. Ruin starts at the boundary indicators zero or one and evolves through nonnegative weighted averages at every interior capital state. Therefore finite-horizon absorption-at-zero probability cannot be negative under valid win probabilities.

$$0\le p\le1\Rightarrow R_n(i)\ge0$$

Capstone valuation statement

The declaration ActuarialValuation.gamblerBoundedRuin_fundamental states the following relation. The capstone formalises the correct first-step harmonic equations for the fair and biased finite-capital random walk and nonnegative bounded finite-horizon absorption-event probabilities. Unlike an infinite-horizon gambler's ruin theorem, it does not assert that every path has already been absorbed by a finite horizon or identify finite success directly with the harmonic candidate.

$$H_{1/2}(i)=\tfrac12(H_{1/2}(i+1)+H_{1/2}(i-1)),\quad H_p(i)=pH_p(i+1)+(1-p)H_p(i-1)$$

Significance

The exact harmonic functions identify candidate upper-absorption probabilities for fair and biased walks, while the finite-horizon recursions quantify how much absorption can have occurred after a specified number of steps. The latter distinction matters for capital planning because some solvent paths may still be active when an assessment period ends. Standard treatments show the importance of the bias ratio and the two absorbing barriers (https://stats.libretexts.org/Bookshelves/Probability_Theory/Introductory_Probability_%28Grinstead_and_Snell%29/12%3A_Random_Walks/12.02%3A_Gambler%27s_Ruin). The mathematical identities themselves are known; a Lean representation offers checkable conventions around capital zero, positive target, finite horizon and the singular fair limit. Its state-recursion definitions can be reused in finite Markov reward and insurance capital models.

Difficulty

The fair success expression cannot simply be obtained by substituting p=one-half into the biased quotient: both its numerator and denominator become zero. The separate linear expression needs its own fair-case convention. A second easy mistake is to regard finite-horizon success and ruin as exhaustive complementary events. They are disjoint, but an interior path may reach neither barrier during the available number of steps. At extreme probabilities zero or one, the odds ratio may be undefined or degenerate, and an eventual absorption assertion requires hypotheses not present in a finite first-step equality. These boundary distinctions are central to the stated target.

Formalization scope

Lean uses natural numbers for capital, positive upper barrier and finite horizon, with real-valued probabilities and harmonic candidate functions. Both boundary states are absorbing; ruin is capital equal to zero, unlike the strictly negative surplus convention in another insurance reserve model. The biased quotient is used only when its normalising denominator is nonzero, and the fair case is represented separately. The source does not assert almost-sure absorption or equality of finite-horizon success with an eventual harmonic function, and it does not equate unresolved paths with failure. Mathlib finite recursion and rational function arithmetic support the scope. Reusable contributions include independent success/ruin state operators, boundary-state lemmas and nonnegative weighted continuation bounds.

The five local definitions form the available mathematical vocabulary: actuarial_gamblerFairSuccess, actuarial_gamblerOddsRatio, actuarial_gamblerBiasedSuccess, actuarial_gamblerFiniteSuccess, actuarial_gamblerFiniteRuin. Their support conditions, nonzero denominators, and variable types are fixed by the Lean declarations rather than inferred from examples. The nontrivial mathematical content must not be replaced by a vacuous premise or by a special case that bypasses the model boundaries. The existing paper mathematics is known, while these particular theorem declarations remain formalisation targets with no accepted proofs. Contributions that strengthen reusable finite identities or expose missing boundary hypotheses are appropriate, provided they preserve the exact authoritative theorem types.

Selected references

• S. David Promislow, Fundamentals of Actuarial Mathematics, third edition, Wiley, 2015. https://www.oreilly.com/library/view/fundamentals-of-actuarial/9781118782521/

• Charles M. Grinstead and J. Laurie Snell, Introduction to Probability, American Mathematical Society, 1997. https://stats.libretexts.org/Bookshelves/Probability_Theory/Introductory_Probability_%28Grinstead_and_Snell%29/12%3A_Random_Walks/12.02%3A_Gambler%27s_Ruin

Motivation

Non-proportional reinsurance allows an insurer to retain ordinary claims while transferring part of the cost of larger claims to a reinsurer. Such contracts specify both an attachment point and a maximum amount the reinsurer will pay. Confusing the retention with the limit changes ceded payments, claims reporting and the net expected treaty cost. HMRC describes common non-proportional arrangements and their layers in its general-insurance technical guidance (https://www.gov.uk/hmrc-internal-manuals/general-insurance-manual/gim8080). Pricing systems must distinguish an unlimited excess claim from the finite reimbursement available under one layer, including claims exactly at the attachment or at the upper layer boundary.

Setting

A gross claim x, attachment d and layer width L are nonnegative natural-number amounts in the same currency unit. An unlimited excess contract pays only the positive part of x minus d. A capped ceded layer pays the smaller of that gross excess and L. Amounts beyond the top of the layer, d plus L, form a separate higher excess. The expected treaty cost is evaluated under a bounded severity distribution with nonnegative mass w(x) at each integer claim amount from zero through the finite bound B. An unlimited stop-loss expected cost uses the same distribution without a layer cap.

The single-claim positive excess over an attachment is named actuarial_layerExcessLoss in the Lean development. The claim amount remaining above the insurer's retained attachment is natural-number positive-part subtraction. Below attachment it is zero, matching the gross excess payment of an unlimited stop-loss treaty.

The capped single-layer reinsurance payment is named actuarial_layerCededPayment in the Lean development. For a claim amount x, a reinsurance layer of width or limit L attaching at d reimburses the lesser of excess x−d positive part and the layer limit. It pays zero below attachment and cannot pay more than L.

The remaining gross excess above the top of the layer is named actuarial_layerHigherExcess in the Lean development. Any part of the claim above the layer's upper attachment d+L is left for higher reinsurance layers or the ultimate insurer. It is the positive excess loss above the sum of the lower attachment and the stated layer width.

The finite aggregate expected ceded loss cost of a layer is named actuarial_layerExpectedCeded in the Lean development. The finite bounded claim-severity mass model computes the net expected excess-of-loss reinsurance premium by weighting each loss payment of the capped layer by its claim probability. Expected ceded cost excludes risk loading and expenses.

The finite unlimited stop-loss expected payment is named actuarial_layerExpectedExcess in the Lean development. The net stop-loss premium above an attachment d is the finite weighted expected excess claim amount without an upper layer cap. The difference of this value at d and at d+L will equal the capped layer's expected ceded cost.

Formalization targets

The formal targets are ordered from elementary boundary and consistency claims to their combined capstone. The exact Lean identifiers below are the authoritative local targets; none of the listed theorem bodies is already accepted as a machine-checked proof. The mathematical results themselves are established actuarial or probability identities.

Baseline identities

The declaration ActuarialValuation.layerExcessLoss_below states the following relation. When the claim amount does not exceed the attachment, no monetary loss lies above the retention. Saturating natural-number subtraction gives zero excess exactly.

$$x\le d\Rightarrow E_d(x)=0$$

Stronger structural properties

The declaration ActuarialValuation.layerExpectedCeded_stoploss_difference states the following relation. The expected capped ceded payment equals the expected unlimited excess above attachment d, less the excess above the upper layer attachment d+L. The identity follows termwise from the exact pointwise loss partition and finite linearity, with no distribution-normalisation assumption.

$$\mathrm{EL}_{d,L}=\mathrm{SL}(d)-\mathrm{SL}(d+L)$$

Capstone valuation statement

The declaration ActuarialValuation.layerCededValuation_fundamental states the following relation. The capstone unifies a bounded single-claim excess-of-loss treaty's payment decomposition with the exact net layer premium as difference of two unlimited stop-loss premiums, and establishes its nonnegative bound against gross excess. It does not assert an expense-loaded reinsurance price or infinite-support integration.

$$E_d(x)=C_{d,L}(x)+H_{d,L}(x),\quad \mathrm{EL}_{d,L}=\mathrm{SL}(d)-\mathrm{SL}(d+L)$$

Significance

The difference between net unlimited excess payments at two attachments gives an immediately auditable valuation of a finite reinsurance layer. A treaty administrator can reconcile the covered amount, the higher excess and the original unlimited amount without making assumptions about claim frequencies or total policyholder losses. This separation is useful in exposure rating and in modelling multiple reinsurance towers, including arrangements described in the HMRC guidance at https://www.gov.uk/hmrc-internal-manuals/general-insurance-manual/gim8080. The identity is already a standard insurance-payment relation; formalising its finite version would produce reusable payment functions and expected-cost inequalities rather than announcing a new result about treaty markets. It would also distinguish claim indemnities from premiums loaded for operating costs and capital.

Difficulty

An intuitive subtraction of the reinsurance limit from the gross excess can become negative and cannot describe the remaining claim burden below the top of a layer. Likewise, a formula that gives every loss above attachment an unlimited reimbursement silently overstates the provider's maximum liability. Natural-number subtraction in Lean saturates at zero, so any stated decomposition must respect positive-part semantics at both the lower and upper treaty thresholds. There are also two independent directions of monotonicity: increased layer width raises coverage, while a larger retention generally reduces the covered part. The capstone fixes only the former and the precise difference of stop-loss expected amounts.

Formalization scope

The source uses natural numbers for money, including the deductible and treaty limit, and real numbers for bounded claim coefficients and expected ceded payment. There is no independent hidden frequency distribution, and the finite claim grid can include zero severity and zero layer width. Nonnegative weights are explicitly required for order and positivity conclusions, while pointwise payment identities hold without probability normalisation. The formalisation does not collapse a capped layer into a trivial unlimited excess contract, nor assume an expense-loaded market premium is equal to an actuarial expectation. Reusable contributions include positive-part payments, min-limited cover, finite layer aggregation and higher-attachment functions. Later models may combine several layers or add claims-inflation adjustments.

The five local definitions form the available mathematical vocabulary: actuarial_layerExcessLoss, actuarial_layerCededPayment, actuarial_layerHigherExcess, actuarial_layerExpectedCeded, actuarial_layerExpectedExcess. Their support conditions, nonzero denominators, and variable types are fixed by the Lean declarations rather than inferred from examples. The nontrivial mathematical content must not be replaced by a vacuous premise or by a special case that bypasses the model boundaries. The existing paper mathematics is known, while these particular theorem declarations remain formalisation targets with no accepted proofs. Contributions that strengthen reusable finite identities or expose missing boundary hypotheses are appropriate, provided they preserve the exact authoritative theorem types.

Selected references

• S. David Promislow, Fundamentals of Actuarial Mathematics, third edition, Wiley, 2015. https://www.oreilly.com/library/view/fundamentals-of-actuarial/9781118782521/

• HM Revenue & Customs, General Insurance Manual: Non-proportional reinsurance, UK government technical guidance, accessed 2026. https://www.gov.uk/hmrc-internal-manuals/general-insurance-manual/gim8080

Motivation

An insurer can expect premium income to exceed claims and nevertheless become insolvent because the timing and size of claim payments are uncertain. Ruin theory asks about the chance of crossing an insolvency boundary before a specified date or eventually. The classic actuarial approach uses the distribution of yearly net cashflows and, when it exists, an exponential adjustment condition to bound risk as starting capital increases. Finite-state random walks provide a related illustration of first-passage probability and absorption (https://stats.libretexts.org/Bookshelves/Probability_Theory/Introductory_Probability_%28Grinstead_and_Snell%29/12%3A_Random_Walks/12.02%3A_Gambler%27s_Ruin). The mission isolates the finite-horizon inequality that can be stated without imposing the measure-theoretic hypotheses of an infinite stopping-time theorem.

Setting

An annual aggregate claim X is a nonnegative integer taking values zero through a fixed maximum B, with annual probabilities w(k). The insurer starts at signed integer surplus u, receives nonnegative integer premium c each year and pays an independent claim from the same finite distribution. The next signed surplus is current surplus plus c minus that claim. Strict ruin occurs as soon as surplus becomes negative, including at time zero if the initial reserve is negative. A finite-horizon ruin probability assigns one to an already-ruined state, zero to a solvent state at zero horizon, and a probability-weighted continuation at other states.

The one-period integer insurance surplus update is named actuarial_ruinNextSurplus in the Lean development. The insurer begins a period with an integer surplus, receives a fixed nonnegative integer premium, then pays the period's aggregate integer claim. The next surplus can be negative, so it is represented as an integer rather than a natural number with saturating subtraction.

The total annual claim probability on a bounded integer grid is named actuarial_ruinClaimMass in the Lean development. The finite annual aggregate-claim model has possible integer loss amounts 0 through B. Its total probability is the sum of those coefficients, required to equal one in the genuine probability interpretation of the recursion.

The exponential moment of one period's net insurance loss is named actuarial_ruinAdjustmentMoment in the Lean development. The adjustment coefficient R compares the annual aggregate claim X with the premium c via the exponential moment E[exp(R(X−c))]. When this quantity is no greater than one and R≥0, the exponential capital bound becomes a supermartingale-style one-step estimate.

The lundberg exponential benchmark for an integer initial surplus is named actuarial_ruinExponentialBound in the Lean development. The benchmark at current surplus u is exp(−Ru). For R≥0 this is at least one if u is negative, allowing it to bound an already-ruined state whose ruin indicator equals one; for a nonnegative initial reserve it falls exponentially with additional starting capital.

The first-passage ruin probability over a finite number of annual periods is named actuarial_ruinProbabilityFinite in the Lean development. The probability of ruin by horizon n records whether surplus was ever strictly negative. Already-ruined integer states have probability one at any horizon. Otherwise the next-year claim distribution mixes the ruin probabilities of all possible updated signed surpluses, recursively for n further years. The horizon is finite and annual claims are modelled as conditionally independent copies of w.

Formalization targets

The formal targets are ordered from elementary boundary and consistency claims to their combined capstone. The exact Lean identifiers below are the authoritative local targets; none of the listed theorem bodies is already accepted as a machine-checked proof. The mathematical results themselves are established actuarial or probability identities.

Baseline identities

The declaration ActuarialValuation.ruinNextSurplus_zero_claim states the following relation. If the insurance portfolio generates no claim payment during a period, only the fixed premium c affects the insurer's capital. The signed next-period surplus is therefore the old surplus plus the earned premium.

$$U'=U+c$$

Stronger structural properties

The declaration ActuarialValuation.ruinProbabilityFinite_le_one states the following relation. With an annual claim mass function summing to one, every solvent state averages predecessor-horizon ruin probabilities with nonnegative weights of total one. Since the base and already-ruined values lie at most one, induction establishes that the finite-horizon ruin probability never exceeds one.

$$w\ge0,\ \sum_kw_k=1\Rightarrow\psi_n(u)\le1$$

Capstone valuation statement

The declaration ActuarialValuation.ruinFiniteHorizon_fundamental states the following relation. The capstone formalises a bounded annual integer-claim surplus recursion with an absorbing strict-negative ruin condition. Proper nonnegative annual claim probabilities keep finite-horizon ruin probability within the unit interval. A nonnegative exponential adjustment parameter satisfying E[exp(R(X−c))]≤1 then bounds it by exp(−Ru) for every finite horizon and signed starting capital.

$$0\le\psi_n(u)\le1,\quad M(R)\le1\Rightarrow\psi_n(u)\le e^{-Ru}$$

Significance

A valid exponential upper bound gives a transparent relationship between initial capital and probability of failure over every finite number of years. It can be used to compare capital buffers or to test an insurer's response to a changed premium and annual claims distribution. The bounded recursive setting is particularly relevant when annual severity is itself computed by exact portfolio convolution. Classical first-passage models distinguish eventual absorption from a finite-horizon crossing event (https://stats.libretexts.org/Bookshelves/Probability_Theory/Introductory_Probability_%28Grinstead_and_Snell%29/12%3A_Random_Walks/12.02%3A_Gambler%27s_Ruin). The formal target is the known exponential comparison property, not a new adjustment-root existence result. It yields reusable recursion and exponential-moment definitions for subsequent finite insurance reserve analysis.

Difficulty

A tempting representation of surplus as a natural number is incorrect for insolvency, because saturating subtraction hides the negative state that defines ruin. A second mistake is to present the exponential capital function as an unconditional bound. It can fail when the claim-loss exponential moment exceeds one, even in a deterministic one-year example. The argument also needs the negative-surplus boundary, at which ruin probability is already one. These complications are visible at finite horizons and cannot be repaired merely by assuming that premium income has a positive expected margin.

Formalization scope

The Lean source uses signed integers for surplus and natural numbers for annual claim amounts, premium, severity bound and horizon. Annual probabilities and adjustment exponent are real. The event is strictly negative first-passage ruin, distinct from reaching exactly zero in a gambler's ruin model. A nonnegative adjustment exponent and an explicit upper bound of one on its net-claim exponential moment are mandatory. The source gives the finite recursive quantity algebraically, not an independently constructed probability space, an infinite-horizon supremum or a theorem that a positive adjustment root exists. Reusable developments include finite weighted expectation operators, signed reserve transitions and nonnegative probability bounds; meaningful contributions may extend these to random premiums or general annual claim distributions.

The five local definitions form the available mathematical vocabulary: actuarial_ruinNextSurplus, actuarial_ruinClaimMass, actuarial_ruinAdjustmentMoment, actuarial_ruinExponentialBound, actuarial_ruinProbabilityFinite. Their support conditions, nonzero denominators, and variable types are fixed by the Lean declarations rather than inferred from examples. The nontrivial mathematical content must not be replaced by a vacuous premise or by a special case that bypasses the model boundaries. The existing paper mathematics is known, while these particular theorem declarations remain formalisation targets with no accepted proofs. Contributions that strengthen reusable finite identities or expose missing boundary hypotheses are appropriate, provided they preserve the exact authoritative theorem types.

Selected references

• S. David Promislow, Fundamentals of Actuarial Mathematics, third edition, Wiley, 2015. https://www.oreilly.com/library/view/fundamentals-of-actuarial/9781118782521/

• Charles M. Grinstead and J. Laurie Snell, Introduction to Probability, American Mathematical Society, 1997. https://stats.libretexts.org/Bookshelves/Probability_Theory/Introductory_Probability_%28Grinstead_and_Snell%29/12%3A_Random_Walks/12.02%3A_Gambler%27s_Ruin

Motivation

Insurers model annual claim frequency to price collections of policies and assess how often unusually large numbers of claims may arrive together. A Poisson count model has equal mean and variance, but heterogeneity between policyholders can produce materially more dispersion. The negative-binomial family is an established alternative, including as a marginal count law obtained by mixing the Poisson intensity over a Gamma distribution. The mathematical relationship, with explicit Gamma shape and rate parameters, is set out in the Stanford discussion of mixture models (https://web.stanford.edu/class/bios221/MixtureLecture.html). A reliable formal description must specify which negative-binomial probability parameter is being used, since actuarial and statistical literature adopt incompatible success and failure conventions.

Setting

A negative-binomial count variable N records a nonnegative integer number of claims. The integer shape parameter r is strictly positive, and a real count probability parameter p lies between zero and one, with the upper endpoint excluded when calculating moments. In this convention p is associated with powers of the claim count n, while 1 minus p is associated with powers of the shape r. The Gamma-mixed Poisson model starts from a Gamma-distributed intensity with positive rate beta and integer shape r; at one future exposure unit its negative-binomial count parameter equals the reciprocal of beta plus one.

The negative-binomial claim count coefficient is named actuarial_negBinCountMass in the Lean development. Defines a negative-binomial frequency coefficient for n claims and positive integer shape r, with p the count probability parameter and 1−p the Bernoulli success probability. The finite choose expression is mathematically the conventional formula only for shape r≥1.

The expected claim count under nb size and count parameter is named actuarial_negBinCountMean in the Lean development. For 0≤p<1 and positive integer shape, the negative-binomial mean count equals r p divided by one minus p. This p denotes count rather than Bernoulli success probability, so the formula differs from sources that reverse the parametrisation.

The negative-binomial count variance is named actuarial_negBinCountVariance in the Lean development. For valid negative-binomial parameters, the variance exceeds the mean because uncertainty in the Poisson event frequency introduces additional dispersion. The exact variance is r p divided by (1−p) squared.

The unit-exposure gamma-poisson predictive count parameter is named actuarial_negBinGammaProbability in the Lean development. A Gamma mixing distribution with positive rate beta and positive integer shape yields a negative-binomial mixed-Poisson count law at one exposure unit with the count parameter p=1/(beta+1). This is the reciprocal rate-plus-one, not beta/(beta+1), which is the complementary success parameter.

The integer-shape gamma-poisson predictive count coefficient is named actuarial_negBinGammaPredictive in the Lean development. The model uses a shape/rate Gamma prior for a Poisson rate at unit exposure. The predictive count mass is the integer-shape negative-binomial coefficient evaluated at the rate-derived count parameter. The declaration encodes the known mixture result algebraically without asserting an integral identity.

Formalization targets

The formal targets are ordered from elementary boundary and consistency claims to their combined capstone. The exact Lean identifiers below are the authoritative local targets; none of the listed theorem bodies is already accepted as a machine-checked proof. The mathematical results themselves are established actuarial or probability identities.

Baseline identities

The declaration ActuarialValuation.negBinGammaProbability_pos states the following relation. Prior Gamma rate b positive implies denominator b+1 strictly positive, so its reciprocal is a valid strictly positive negative-binomial count parameter.

$$\beta>0\Rightarrow p>0$$

Stronger structural properties

The declaration ActuarialValuation.negBinGammaMean_unit_exposure states the following relation. The frequency mean for the negative-binomial predictive law derived from a Gamma shape/rate prior at one future exposure is shape divided by rate. This agrees with the prior Gamma mean Poisson intensity, not shape times rate.

$$\mu=r/\beta$$

Capstone valuation statement

The declaration ActuarialValuation.negBinClaimFrequency_fundamental states the following relation. The capstone combines the exact integer-shape negative-binomial count-mass recursion and the unit-exposure Gamma-Poisson mixture parameterisation. It verifies both predictive moments and coefficient nonnegativity, without claiming an infinite PMF normalisation proof or a Gamma mixture integral in Lean.

$$(n+1)g(n+1)=(n+r)pg(n),\quad\mu=r/\beta,\quad\sigma^2=\mu+r/\beta^2$$

Significance

The ability to compute neighbouring frequency coefficients from one another supports actuarial claim-table construction without repeatedly evaluating large factorial ratios. The difference between count variance and mean provides a direct overdispersion diagnostic when comparing heterogeneous risks with a Poisson benchmark. Under a Gamma prior, the mixed frequency mean and variance recover the Gamma shape/rate moments, linking pricing assumptions to uncertainty in latent risk intensity. Panjer's work on recursive compound distribution evaluation shows why stable frequency recurrences matter for downstream aggregate insurance claims (https://doi.org/10.1017/S0515036100006796). The results are known mathematics; an exact formalisation would make the parameter conventions and real-valued boundary assumptions machine checkable and provide coefficient lemmas reusable by collective-risk computations.

Difficulty

The main hazard is a parameterisation mismatch. Some publications place p on the Bernoulli success count and others put p on the recorded total of failures or claims; the mean and Gamma mapping change accordingly. The integer-binomial coefficient must also be interpreted with positive shape: allowing r=0 while retaining the expression choose(n+r−1,n) introduces natural subtraction at a degenerate boundary. Moment formulas include one-minus-p in a denominator and therefore cannot be stated as ordinary probability facts at p=1. Finally, the algebraic coefficient sequence alone does not supply a probability-space proof that Gamma integration produces the count law.

Formalization scope

The Lean representation uses natural numbers for integer shape and count, and real numbers for the probability parameter, Gamma prior rate and moment expressions. The combinatorial coefficient is Nat.choose, and all arithmetic is within Mathlib. The zero-shape index is excluded from the conventional distribution claim, although some sign statements remain algebraically valid there. This mission does not assert an infinite normalisation sum, an unrestricted non-integer shape model or a vacuous Gamma-mixture identity proved by defining the mixture as the desired result. Reusable contributions include count-ratio recurrences, nonnegative binomial coefficient calculations and parameter conversion lemmas. Future work could connect those components to a genuine Gamma integral or a separately formalised Panjer aggregate recursion.

The five local definitions form the available mathematical vocabulary: actuarial_negBinCountMass, actuarial_negBinCountMean, actuarial_negBinCountVariance, actuarial_negBinGammaProbability, actuarial_negBinGammaPredictive. Their support conditions, nonzero denominators, and variable types are fixed by the Lean declarations rather than inferred from examples. The nontrivial mathematical content must not be replaced by a vacuous premise or by a special case that bypasses the model boundaries. The existing paper mathematics is known, while these particular theorem declarations remain formalisation targets with no accepted proofs. Contributions that strengthen reusable finite identities or expose missing boundary hypotheses are appropriate, provided they preserve the exact authoritative theorem types.

Selected references

• S. David Promislow, Fundamentals of Actuarial Mathematics, third edition, Wiley, 2015. https://www.oreilly.com/library/view/fundamentals-of-actuarial/9781118782521/

• Harry H. Panjer, Recursive evaluation of a family of compound distributions, ASTIN Bulletin, 1981. https://doi.org/10.1017/S0515036100006796

Motivation

A pair of insurance portfolios can have the same expected aggregate claim while requiring different ceded premiums or capital to withstand severe outcomes. Expected loss alone cannot identify the difference: the distribution of loss across small and large events also matters. The stop-loss order compares expected loss above every retention level and provides a concrete way to rank riskiness for excess-of-loss contracts. Müller studied wider relationships among such risk orders and their stop-loss transforms (https://doi.org/10.1016/0167-6687(96)90002-5). The present object is a local, explicit change in an integer loss distribution that preserves its first moment but shifts probability from a middle amount towards two neighbouring outcomes.

Setting

A bounded aggregate-loss distribution has nonnegative coefficients w(s) for integer losses s between zero and a finite maximum B, with total coefficient mass usually equal to one. Its gross expected claim is the weighted sum of loss amounts and its stop-loss premium at integer deductible d is the expected positive excess above d. A proposed new coefficient array moves an amount delta out of one interior integer claim centre c, placing equal halves at the adjacent claim sizes c minus one and c plus one. Strictly positive centre and a top neighbour no higher than B keep the change within the declared support.

The bounded discrete expected excess above an integer attachment is named actuarial_orderStopLossPremium in the Lean development. The actuarial stop-loss transform multiplies each aggregate loss's excess over the attachment point by its real coefficient and sums over the finite covered loss grid. A nonnegative probability mass law gives its net excess-of-loss reinsurance premium.

The bounded aggregate loss expectation is named actuarial_orderAggregateMean in the Lean development. The finite aggregate claims mean uses the claim amount at each integer index as its payment and multiplies it by its probability mass. Comparing mean claims is necessary when interpreting stop-loss order as mean-preserving convex order.

The aggregate probability within a finite loss bound is named actuarial_orderAggregateMass in the Lean development. The sum of all coefficients on the finite aggregate-loss support is the bounded probability mass, whose value is one for a normalised distribution. An explicit mass operator makes the mean-preserving spread's total-mass conservation separately checkable.

The mean-preserving symmetric transfer from one claim amount to adjacent values is named actuarial_orderNeighbourSpread in the Lean development. A fixed amount δ of probability mass is moved away from an interior integer claim level centre: half is allocated one unit below, and half one unit above. Under 1≤centre and centre+1≤B these three loss levels lie within the support and the transfer preserves the aggregate mean.

The finite-grid stop-loss ordering of aggregate risks is named actuarial_orderStopLossDominates in the Lean development. The first aggregate loss distribution is no riskier in stop-loss order than the second when its expected excess payment never exceeds the other's for any integer attachment d. Equal means and unit total masses can be imposed separately to obtain a mean-preserving spread interpretation.

Formalization targets

The formal targets are ordered from elementary boundary and consistency claims to their combined capstone. The exact Lean identifiers below are the authoritative local targets; none of the listed theorem bodies is already accepted as a machine-checked proof. The mathematical results themselves are established actuarial or probability identities.

Baseline identities

The declaration ActuarialValuation.orderStopLossPremium_zero states the following relation. Since every nonnegative integer loss exceeds or equals attachment zero, the stop-loss payment equals the gross loss in each scenario. Thus the expected excess premium at attachment zero coincides exactly with the original aggregate expected claim.

$$\Pi_w(0)=\mu_w$$

Stronger structural properties

The declaration ActuarialValuation.orderNeighbourSpread_stoploss states the following relation. The payment function (s−d) positive-part is convex in the claim amount s. Transferring nonnegative mass symmetrically from centre to its adjacent lower and higher losses can only increase its expected value, with a difference equal to δ/2 times the nonnegative discrete second difference at c.

$$\Pi_{w'}(d)-\Pi_w(d)=\frac{\delta}{2}[(c-1-d)_++(c+1-d)_+-2(c-d)_+]\ge0$$

Capstone valuation statement

The declaration ActuarialValuation.orderNeighbourSpread_fundamental states the following relation. The capstone shows that a feasible symmetric probability transfer to adjacent losses produces a valid nonnegative bounded coefficient distribution, preserves the aggregate total probability and mean, yet weakly raises every stop-loss expected excess premium. This is an explicit constructive mean-preserving spread and a finite-lattice convex-order result.

$$w'\ge0,\ M_{w'}=M_w,\ \mu_{w'}=\mu_w,\ w\preceq_{\rm sl}w'$$

Significance

A mean-preserving spread separates the effect of greater claim variability from a change in the underlying expected loss. By conserving total mass and gross mean, the comparison can be attributed to the redistribution of outcomes rather than an increase in total claim cost. Its downstream consequence is a weak rise in the expected ceded payout for every positive-part reinsurance retention. This connects aggregate frequency/severity valuation to ordered risks and tail measures, consistent with wider stop-loss ordering results studied in https://doi.org/10.1016/0167-6687(96)90002-5. Formalising the finite construction provides a reusable bounded-grid example of stochastic-order consequences without requiring an axiomatic treatment of all convex functions.

Difficulty

The obvious claim that any symmetric redistribution increases expected excess is false if the transfer is allowed to cross the boundary of the support or remove more probability than the central cell contains. At centre zero the intended lower neighbouring loss is negative and no longer exists in a natural-number grid. If delta exceeds the centre's initial coefficient, the resulting array is not a probability distribution. Equality of expected claim amounts by itself also does not imply stop-loss order. The formal statement must account for the actual positive-part payment at each possible deductible, including deductions outside the claim support.

Formalization scope

Lean represents loss indices, attachment points and the spread centre as naturals, and probability coefficients and expected claims as reals. The claim grid is finite, and the full probability-normalisation equation is separate from local mass conservation. The capstone explicitly requires nonnegative starting coefficients and a feasible interior transfer. It does not substitute a trivial comparison between identical distributions, nor claim general continuous convex order or a complete converse representation of risk ordering. Mathlib finite sums and arithmetic over nonnegative parts are enough for the declared scope. Reusable infrastructure includes finite aggregate means, bounded mass totals and stop-loss order on arrays, suitable for later reinsurance layers and capital-risk measures.

The five local definitions form the available mathematical vocabulary: actuarial_orderStopLossPremium, actuarial_orderAggregateMean, actuarial_orderAggregateMass, actuarial_orderNeighbourSpread, actuarial_orderStopLossDominates. Their support conditions, nonzero denominators, and variable types are fixed by the Lean declarations rather than inferred from examples. The nontrivial mathematical content must not be replaced by a vacuous premise or by a special case that bypasses the model boundaries. The existing paper mathematics is known, while these particular theorem declarations remain formalisation targets with no accepted proofs. Contributions that strengthen reusable finite identities or expose missing boundary hypotheses are appropriate, provided they preserve the exact authoritative theorem types.

Selected references

• S. David Promislow, Fundamentals of Actuarial Mathematics, third edition, Wiley, 2015. https://www.oreilly.com/library/view/fundamentals-of-actuarial/9781118782521/

• Alfred Müller, Orderings of risks: A comparative study via stop-loss transforms, Insurance: Mathematics and Economics, 1996. https://doi.org/10.1016/0167-6687(96)90002-5

Motivation

Insurers routinely estimate next-period frequency from policyholders who have accumulated unequal historical exposure. A raw observed rate can swing markedly when the exposure base is small, while a collective rate may ignore evidence of persistent individual risk differences. The Gamma–Poisson model gives a precisely defined compromise: a posterior expected claim intensity that responds to observed counts and earned exposure. The literature on credibility analysis documents the distinction between process variation and variation across risks, including the role of Bayesian conjugacy (https://openacttexts.github.io/LDAVer2/ChapCredibility.html). The relevant question is how the posterior parameters and credibility premium behave under sequential reporting, zero-data boundaries and different exposure amounts.

Setting

The latent claim frequency is measured in expected events per insured exposure unit. A Gamma prior has shape alpha greater than zero and rate beta greater than zero, so beta is the reciprocal of the Gamma scale. Conditional on this frequency, the observed number of claims C is Poisson with mean equal to exposure E times the intensity. Distinct observed blocks contribute counts C and D and exposures E and F. The formal declarations use natural numbers for counts and real numbers for rates, exposures and expected frequencies. The posterior shape/rate update and credibility-weight equations are the objects of the stated algebraic model.

The gamma posterior shape after observing claim count is named actuarial_gammaPoissonPosteriorShape in the Lean development. The Gamma prior uses the shape-and-rate parameterisation and starts with prior shape α. Observing a nonnegative integer aggregate count C increases the posterior shape to α+C. This is an algebraic conjugacy update rather than a complete formalisation of the Gamma probability density.

The gamma posterior rate after exposure volume is named actuarial_gammaPoissonPosteriorRate in the Lean development. The Gamma distribution's second parameter is a rate, not a scale. The posterior rate is prior rate β plus observed insured exposure E; this explicit convention prevents the common reciprocal-scale error in Poisson-Gamma credibility.

The posterior expected claim frequency per exposure unit is named actuarial_gammaPoissonPosteriorMean in the Lean development. For a Poisson count model and shape/rate Gamma prior, the posterior expected frequency per unit exposure is updated shape divided by updated rate. Positive prior rate and nonnegative exposure ensure an ordinary positive denominator.

The exposure-dependent exact bayesian credibility coefficient is named actuarial_gammaPoissonExperienceWeight in the Lean development. The exact credibility weight in the Poisson-Gamma conjugate model is E divided by β+E. When β>0 and E≥0 this lies between zero and one, approaching greater reliance on claim experience as accumulated exposure increases.

The weighted experience and collective frequency is named actuarial_gammaPoissonCredibilityBlend in the Lean development. The credibility premium combines observed aggregate claims per unit exposure C/E and the Gamma prior population frequency α/β, with respective weights E/(β+E) and β/(β+E). The usual representation requires E>0 and β>0; at zero exposure the claim rate C/E is not a valid observed experience statistic.

Formalization targets

The formal targets are ordered from elementary boundary and consistency claims to their combined capstone. The exact Lean identifiers below are the authoritative local targets; none of the listed theorem bodies is already accepted as a machine-checked proof. The mathematical results themselves are established actuarial or probability identities.

Baseline identities

The declaration ActuarialValuation.gammaPoissonPosteriorShape_zero states the following relation. When no claims are observed, the sufficient statistic added to the shape is zero. The updated Gamma posterior shape therefore equals its prior shape, even before imposing positivity required by the probabilistic model.

$$\alpha+0=\alpha$$

Stronger structural properties

The declaration ActuarialValuation.gammaPoissonCredibilityBlend_exact states the following relation. With positive prior rate and positive observed exposure, the observed claim frequency and collective mean are both well-defined. Substituting weight E/(β+E) and simplifying proves the exact Bayesian credibility decomposition of posterior mean (α+C)/(β+E).

$$Z\,C/E+(1-Z)\alpha/\beta=(\alpha+C)/(\beta+E)$$

Capstone valuation statement

The declaration ActuarialValuation.gammaPoissonCredibility_fundamental states the following relation. The capstone combines exact Gamma-Poisson experience credibility at positive exposure with associative sequential count/exposure updates and admissibility of the Bayesian experience weight. It formalises the conjugate-model parameter and premium algebra without claiming a full measure-theoretic Gamma likelihood integral.

$$\hat\lambda=ZC/E+(1-Z)\alpha/\beta,\ \alpha'=\alpha+C,\ \beta'=\beta+E,\ 0\le Z\le1$$

Significance

Exact agreement between the posterior mean and the collective/individual premium blend allows underwriting calculations to be reconciled across reporting intervals. Exposure aggregation must not alter the end-of-period estimate when the underlying sufficient statistics are unchanged. The additional Gamma predictive parameter variance explains why a future claim count can exhibit dispersion beyond that of a Poisson distribution with deterministic intensity. These are established properties of conjugate credibility models, rather than new claims about the existence of a posterior distribution; the mathematical background is described in https://openacttexts.github.io/LDAVer2/ChapCredibility.html. The formal output would certify the numeric parameter identities and boundary assumptions in Lean, allowing subsequent developments to reuse the same frequency and credibility conventions.

Difficulty

The common failure is to confuse a Gamma rate with a Gamma scale: this reverses the update and the ratio used as the experience credibility weight. Another false shortcut is to take the product of a zero exposure weight and an undefined observed count divided by zero as a meaningful zero experience contribution. Those algebraic terms are distinguishable in Lean's totalised division and in actuarial modelling. The exact mixture identity needs positive observed exposure, whereas the no-data statement requires both zero exposure and zero observed claims. Strict positivity is also essential when the total rate appears in a denominator.

Formalization scope

The Lean development uses Mathlib real arithmetic and natural claim counts. The source represents the posterior parameter update directly and does not formalise an underlying Gamma density, conditional-expectation kernel or integration proof of conjugacy. In particular, no vacuous statistical assumption is allowed to stand in for a parameter identity, and the nonzero-exposure requirement is retained where observed frequency occurs. Reusable contributions include real-valued exposure-ratio identities, independent positivity facts, sequential sufficient-statistic updates and sharply stated data boundaries. Further work could construct the probability-space interpretation or the negative-binomial predictive count law without changing the current capstone.

The five local definitions form the available mathematical vocabulary: actuarial_gammaPoissonPosteriorShape, actuarial_gammaPoissonPosteriorRate, actuarial_gammaPoissonPosteriorMean, actuarial_gammaPoissonExperienceWeight, actuarial_gammaPoissonCredibilityBlend. Their support conditions, nonzero denominators, and variable types are fixed by the Lean declarations rather than inferred from examples. The nontrivial mathematical content must not be replaced by a vacuous premise or by a special case that bypasses the model boundaries. The existing paper mathematics is known, while these particular theorem declarations remain formalisation targets with no accepted proofs. Contributions that strengthen reusable finite identities or expose missing boundary hypotheses are appropriate, provided they preserve the exact authoritative theorem types.

Selected references

• S. David Promislow, Fundamentals of Actuarial Mathematics, third edition, Wiley, 2015. https://www.oreilly.com/library/view/fundamentals-of-actuarial/9781118782521/

• OpenActTexts contributors, Loss Data Analytics: Experience Rating Using Credibility Theory, online actuarial textbook, accessed 2026. https://openacttexts.github.io/LDAVer2/ChapCredibility.html

Actuarial Mathematics XXIII: Countable-Lifetime Hattendorff Limit

Mathematical source and original extension

Elias S. W. Shiu and Xiaoyi Xiong (2021), An elementary derivation of Hattendorff's theorem, European Actuarial Journal 11, 319–323, https://doi.org/10.1007/s13385-020-00256-9. In that paper equation (1) defines the fully discrete insurer's complete-life loss; equation (2) gives the Hattendorff variance formula, and equations (3)–(5) connect annual reserve recursion to the net amount at risk by summation by parts.

The present mission proposes an original countable-outcome and L²-convergence formalisation of that discrete formula. It is not asserted that the authors stated this exact Lean theorem, and it does not inherit proofs from the earlier finite-scenario missions.

Unbounded lifetime and conditional mortality

Let $K\in\mathbb N$ be the realised curtate lifetime, the number of full years remaining until the year of death. Its possible values are not uniformly bounded. Set $w_k=\Pr(K=k)$, where $w_k\ge0$ and $\sum_{k=0}^{\infty}w_k=1$. Assume the survival tail is positive in every year:

$$S_t=\sum_{k\ge t}w_k>0,\qquad q_t=\frac{w_t}{S_t},\qquad p_t=\frac{S_{t+1}}{S_t}=1-q_t.$$

These are unconditional death masses $w_t$ and conditional mortality and survival rates $q_t,p_t$. Some individual $w_t$ may be zero. The outcome space is genuinely countable; a finite Fintype of realised death years would force a bounded support and undermine the claimed infinite-horizon extension.

Mortality surprises and reserve losses

The annual mortality martingale difference is defined pathwise by

$$I_t(k)=\mathbf1_{\{k=t\}}-q_t\mathbf1_{\{k\ge t\}}.$$

For $t<k$, $I_t(k)=-q_t$; at death $t=k$, $I_t(k)=1-q_t$; and after death $t>k$, it is zero. Let the discounted net amount at risk for the payment at year-end $t+1$ be $\rho_t=v^{t+1}(b_{t+1}-V_{t+1})$. This matches Shiu–Xiong's year-end death-benefit timing. The truncated centred reserve loss and the pathwise complete-life centred loss are

$$Z_n(k)=\sum_{t=0}^{n-1}\rho_tI_t(k),\qquad Z_\infty(k)=Z_{k+1}(k).$$

The complete-life variable is well-defined on each realised finite $k$ and $Z_n(k)=Z_\infty(k)$ whenever $n>k$. These definitions only model the centred innovation component. A separate cashflow reserve-recursion bridge, as in earlier missions, is needed to identify a given insurer contract loss $L$ with $V_0+Z_\infty$; this mission does not silently assert that for arbitrary premium sequences.

Supporting results and convergence condition

The discrete expectations are weighted countable sums. Show, under summability and positive in-force mass,

$$\mathbb E[I_t]=0,\quad \mathbb E[I_i I_j]=0\;(i\ne j),\quad \mathbb E[I_t^2]=w_tp_t.$$

Using finite sums and cross-year orthogonality first gives

$$\mathbb E[Z_n^2]=\sum_{t=0}^{n-1}\rho_t^2w_tp_t.$$

The capstone introduces the mathematically substantive L² risk condition

$$\sum_{t=0}^{\infty}\rho_t^2w_tp_t<\infty.$$

This implies that the sequence of truncated centred losses is Cauchy in the probability-weighted L² space; its limit equals the pathwise stabilised $Z_\infty$. Hence

$$\boxed{\mathbb E[Z_\infty]=0,\qquad \operatorname{Var}(Z_\infty)=\mathbb E[Z_\infty^2] =\sum_{t=0}^{\infty}\rho_t^2w_tp_t.}$$

The capstone explicitly assumes nonnegative weights, unit total mass, positive survival tails, countable weight summability and risk-variance summability. The formal statement must not be weakened to merely a pointwise convergence or an unjustified interchange of infinite sums.

Exact-rational validation and boundaries

A geometric law with $q=\frac14$, $p=\frac34$, $w_k=qp^k$ and $\rho_t=2$ gives

$$Z_\infty(k)=2(p-qk),\qquad \mathbb E[Z_\infty]=0,\quad\operatorname{Var}(Z_\infty)=3.$$

Its finite-year variance is $3(1-p^n)$. The repository includes exact-rational tests for 13 horizons. The model allows an individually zero death mass but requires positive tails. Every scenario terminates in some finite year; adding a $K=\infty$ outcome with positive weight would require a separate transversality condition and is not part of this mission.

Prove2Me proof structure

The mission supplies four mathematical definitions, eleven supporting milestone theorems and one combined capstone theorem. Prove the indicator identities, the countable weighted first and second moments, finite orthogonality and finite allocation, and finally the L² passage to the unbounded lifetime. The target Lean namespace is `ActuarialValuation` in the pinned Mathlib environment. Finite-year results from earlier actuarial missions are background, not automatic countable-measure proof imports. None of the present open theorem statements is an accepted proof.

Motivation: from actuarial reserves to Hattendorff's theorem

A prospective insurance reserve is the conditional expected amount needed to fund the remaining liabilities of an in-force policy. Annual realised claims differ from their conditional expectations, and the variation in total discounted insurer loss can be allocated to these unexpected annual outcomes. Hattendorff's theorem expresses that allocation as a sum of year-specific variance contributions. The fully discrete case contains an important factor: the conditional probability of surviving the policy year.

Shiu and Xiong's article An elementary derivation of Hattendorff's theorem [1] gives a direct discrete derivation. With curtate lifetime $K$, annual death benefits $b_{j+1}$ paid at time $j+1$, premiums $\pi_j$ paid at time $j$, discount factor $v$ and prospective reserve $V_{j+1}$, their formulas include

$$L=v^{K+1}b_{K+1}-\sum_{j=0}^{K}v^j\pi_j \qquad\text{and}\qquad \rho_j=v^{j+1}(b_{j+1}-V_{j+1}).$$

In particular, the variance identity they derive has a mortality-year survival factor,

$$\operatorname{Var}(L)=\mathbb E\!\left[\rho_K^2p_{x+K}\right].$$

This mission establishes the finite-scenario mortality-innovation component of that result. It derives the orthogonality of annual death surprises rather than assuming it. It does not yet identify their weighted sum with the full insurance loss $L-V_0$; that requires the separate time-dependent reserve recursion and summation-by-parts identities in [1, equation (3)]. Those insurance cashflow bridge obligations are explicitly reserved for the next mathematical stage.

Finite lifetime model

Let $\Omega$ be a finite scenario space, $w_\omega\ge0$ its weights and $K:\Omega\to\mathbb N$ a curtate future lifetime, where $K=j$ means death during year $j$, at the end of which the death benefit is payable.

For policy duration $t$, define the in-force probability mass and the death-year mass:

$$S_t=\sum_{\omega\in\Omega}w_\omega\mathbf1_{\{K(\omega)\ge t\}}, \qquad D_t=\sum_{\omega\in\Omega}w_\omega\mathbf1_{\{K(\omega)=t\}}.$$

For probabilistic interpretation assume in addition $\sum_\omega w_\omega=1$. The exact algebraic statements do not generally require weight normalisation, though they explicitly require nonnegative weights for probability inequalities.

The disjoint identity

$$\{K\ge t\}=\{K=t\}\mathbin{\dot\cup}\{K\ge t+1\}$$

gives $S_t=D_t+S_{t+1}$. If $S_t>0$, the conditional one-year death and survival probabilities are

$$q_t=\frac{D_t}{S_t},\qquad p_t=\frac{S_{t+1}}{S_t},\qquad q_t+p_t=1.$$

Division by $S_t=0$ is totalised as a Lean real expression but is not interpreted as conditioning on an impossible event. All substantive annual-innovation results impose positive in-force mass for the years used.

A centred innovation for every policy year

For each outcome and policy year define the mortality innovation

$$I_t(\omega)= \mathbf1_{\{K(\omega)=t\}} -q_t\mathbf1_{\{K(\omega)\ge t\}}.$$

It equals $1-q_t$ for death in year $t$, equals $-q_t$ for an individual surviving that year, and equals zero after death. Under $S_t>0$,

$$\mathbb E_w[I_t]=D_t-q_tS_t=0.$$

The orthogonality is more than a generic assumption. For $i<j$, if $I_j\ne0$ then the policy is still in force at $j$, so the preceding year $i$ must have been survived and $I_i=-q_i$. If $I_j=0$, the product is also zero. Thus, pointwise,

$$I_iI_j=-q_iI_j,\qquad i<j.$$

Taking the weighted expectation and using the centring of $I_j$ gives

$$\mathbb E_w[I_iI_j]=0,\qquad i<j,\quad S_j>0.$$

These innovations can be uncorrelated despite being highly dependent events generated by the same curtate lifetime. They must not be claimed independent.

The one-year discrete second moment

By conditioning on the death-year and survival outcomes, the squared mortality surprise has expected value

$$\begin{aligned} \mathbb E_w[I_t^2] &=D_t(1-q_t)^2+S_{t+1}q_t^2\\ &=D_t\,p_t,\qquad S_t>0. \end{aligned}$$

The final conditional survival factor $p_t$ is characteristic of the fully discrete actuarial Hattendorff identity. It is not an arbitrary variance weight, and it cannot be dropped by borrowing a continuous-time formula.

For a finite horizon $n$, let $\rho_t\in\mathbb R$ be any deterministic year-specific net-amount-at-risk coefficient. Define the discounted innovation portfolio

$$Z_n(\omega)=\sum_{0\le t<n}\rho_t I_t(\omega).$$

When $S_t>0$ for each $t<n$, the annual zero-mean and cross-year orthogonality identities yield

$$\boxed{ \mathbb E_w[Z_n]=0,\qquad \operatorname{Var}_w(Z_n)= \sum_{0\le t<n}\rho_t^2D_tp_t. }$$

For nonnegative unit-sum weights this is the usual finite-scenario variance. The capstone couples the derived pairwise cross-year zero moments with the resulting finite-horizon variance formula. No reserve balance, actuarial premium equation or conditional L² filtration theorem is silently assumed.

Formal proof structure and scope

The formal graph contains seven definitions: survival and death masses, conditional death and survival rates, the mortality innovation, its finite discounted portfolio and a finite weighted variance. Supporting theorems establish event partitions, nonnegativity, $q_t+p_t=1$, behaviour after death, innovation centring, a pointwise earlier/later product identity, orthogonality, the discrete second-moment formula, zero horizon and portfolio moments. The capstone packages zero covariance and the mortality-weighted variance allocation.

The variance theorem is valid for arbitrary deterministic coefficient sequence $\rho$ and any finite number of years with positive survival mass; it does not presume every life dies before that horizon. $n=0$ is covered by the empty sum. For standard actuarial application choose $\rho_t=v^{t+1}(b_{t+1}-V_{t+1})$, as in [1]. A future bridge must show that this particular innovation portfolio equals the centred insurer loss under the correct reserve recursion and endpoint conventions.

The root theorem is therefore a genuine probabilistic predecessor, not an assertion that the entire traditional Hattendorff theorem has already been proved. The mission is proposed for formal proof: private remote statement elaboration and finite rational tests, if accepted, do not constitute public Prove2Me proof acceptance.

References

[1] Shiu, Elias S. W. and Xiong, Xiaoyi (2021), “An elementary derivation of Hattendorff's theorem”, European Actuarial Journal, 11, pp. 319–323, equations (1)–(3). This is the direct source for the fully discrete variance factor, net amount at risk and finite summation-by-parts strategy.

[2] Gerber, Hans U. (1997), Life Insurance Mathematics, third edition, Springer, Chapter 6, §6.7, Allocation of the Overall Loss to Policy Years. The reserve-change and annual-loss decomposition provides broader actuarial context.

[3] Gerber, Hans U., Leung, Bartholomew P. K. and Shiu, Elias S. W. (2003), “Indicator Function and Hattendorff Theorem”, North American Actuarial Journal, 7(1), pp. 38–47. Explains the central role of indicator functions and the relationship to uncorrelated martingale increments.

Motivation: from one-step reinsurance decisions to an optimal policy

An insurer may adjust its reinsurance retention or other risk-management decision repeatedly, rather than optimising a single future year's payments. Current action choices affect both the immediate net reward and the state of the business next year, which changes the value of all later decisions. A finite-horizon Markov decision process makes this multi-year optimisation precise.

Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming, Chapter 4 [1], develops the finite-horizon Bellman equations and existence of optimal deterministic Markov policies for finite decision problems. Mission XV introduces a finite-action Bellman operator and excess-of-loss claim splitting, but is still under moderator review. This mission formalises the full finite-horizon backward recursion and optimal deterministic policy existence, without importing an unreviewed declaration.

The reinsurance motivation comes from Brachetta and Ceci's optimal excess-of-loss study [2]. Their original problem is a continuous-time stochastic-factor and HJB verification problem. The present mission is strictly a finite-state, finite-action, discrete-time precursor. Neither the stochastic-factor diffusion nor the jump-process verification theorem is asserted here.

Finite states, actions and one-year valuation

Let $S$ be a finite state type, and let $A$ be a finite nonempty set of admissible control actions. A state can encode a simplified surplus, rating, claims regime or any finite risk-class abstraction. At the beginning of a policy year the insurer is in state $s$ and selects an action $a$.

Let $P(s,a,t)$ be the transition weight for next state $t$, and $r(s,a)$ the immediate reward. The usual Markov model requires $P(s,a,t)\ge0$ and $\sum_tP(s,a,t)=1$ for every $s,a$. The general algebraic definitions accept real matrices, while the economic probabilistic interpretation requires valid row-stochasticity.

If $U(t)$ is the continuation value in state $t$, the return from action $a$ is

$$Q(s,a;U)=r(s,a)+v\sum_{t\in S}P(s,a,t)U(t),$$

where $v$ is a deterministic annual discount factor. The discount is applied to following-year continuation, not to the current reward. For policy interpretation $0\le v\le1$ is conventional, but finite-horizon algebra makes sense at other fixed real $v$ as well.

The one-year Bellman optimisation is

$$(TU)(s)=\max_{a\in A}Q(s,a;U).$$

The maximum is attained because $A$ is finite and nonempty. Both this attainment and the inequality $Q(s,a;U)\le(TU)(s)$ are explicit supporting theorem obligations.

Finite-horizon policy evaluation

Let $h(s)$ denote a terminal benefit or terminal value. A deterministic nonstationary Markov policy is a mapping

$$\pi:\mathbb N\times S\longrightarrow A.$$

Here $\pi_n(s)$ is the action chosen when $n+1$ decisions remain, so the index counts backwards towards the terminal time. This makes the recursion valid for a homogeneous one-step transition model without reversing calendar-time transitions.

For a fixed policy, recursively define

$$\begin{aligned} V^\pi_0(s)&=h(s),\\ V^\pi_{n+1}(s) &=r(s,\pi_n(s)) +v\sum_{t\in S}P(s,\pi_n(s),t)V^\pi_n(t). \end{aligned}$$

The policy is nonstationary because its choice at the same state may differ according to remaining time $n$. It is Markov because the action depends only on current state and remaining horizon, not on the entire historical path. The mission does not introduce a path-space filtration or general history-dependent randomised policy, so the precise capstone compares deterministic nonstationary Markov policies.

Optimal backward induction

Define $V_n(s)$ by the finite Bellman recursion

$$\begin{aligned} V_0(s)&=h(s),\\ V_{n+1}(s)&= \max_{a\in A}\left[ r(s,a)+v\sum_{t\in S}P(s,a,t)V_n(t) \right]. \end{aligned}$$

Under $P(s,a,t)\ge0$ and $v\ge0$, a larger continuation function cannot reduce the value of any action; taking a finite maximum preserves this monotonicity. That simple inequality allows induction on the remaining horizon to show

$$V^\pi_n(s)\le V_n(s) \qquad\text{for every deterministic Markov policy }\pi.$$

The second substantive step is existence of an attaining policy. Since $A$ is finite and nonempty, at each remaining horizon and each state there is at least one maximising action. Choosing such actions and combining them recursively gives a single nonstationary deterministic Markov policy $\pi^\star$ satisfying

$$V^{\pi^\star}_n(s)=V_n(s) \qquad\text{for all states }s.$$

The capstone combines the universal upper bound and existence of an attaining policy. This is genuinely more than a one-step maximum: it is an optimality result over an arbitrary finite number of sequential decisions.

A complete optimality result against all history-dependent randomised policies would need an additional equivalence argument showing no improvement from extra history or randomisation. Such an extension is well-known for suitable finite Markov models but is not silently claimed by this narrower Lean target.

Formal graph and limiting cases

Four proposed definitions encode the one-step action return, the finite Bellman maximum, policy evaluation and recursively optimal horizon value. Eleven supporting theorem targets establish zero-discount behaviour, action domination and attainment, monotonicity of action and Bellman values, zero-horizon and successor recursions, policy domination and existence of an attaining policy. The capstone packages the latter two results.

At $n=0$ there are no decisions and both values are precisely $h$. At $v=0$, any continuation is irrelevant to the one-step value. Nonnegative weights and discount are explicit hypotheses for monotonicity and policy domination, rather than false claims for signed matrix weights. The existence of an attaining maximiser in each finite action set does not itself require stochastic row-sum assumptions.

To obtain a reinsurance model from this graph, one must specify how each action corresponds to an admissible retention, how the retention changes reinsurance premiums and retained claim distributions, and how those assumptions determine the transition matrix $P$ and rewards $r$. That specialised bridge is separate from the finite-horizon mathematical result.

All theorems are proposed Lean proof targets. Preliminary remote elaboration establishes statement well-formedness only if verified by the authoritative compiler. It does not constitute a completed proof or approval of the mission's mathematical design.

References

[1] Puterman, Martin L. (1994), Markov Decision Processes: Discrete Stochastic Dynamic Programming, Chapter 4, Finite-Horizon Markov Decision Processes, Wiley. This is the direct mathematical reference for finite-horizon Bellman recursions, backward induction and existence of optimal deterministic policies under finite-state/action assumptions.

[2] Brachetta, Matteo and Ceci, Claudia (2019), “Optimal Excess-of-Loss Reinsurance for Stochastic Factor Risk Models”, Risks 7(2), article 48. This motivates insurance retention as a controlled decision but uses a continuous-time stochastic-factor/HJB model not established by this mission.

[3] Schäffeler, Maximilian and Abdulaziz, Mohammad (2022), “Formally Verified Solution Methods for Markov Decision Processes”, computer-assisted formalisation of Bellman equations, value iteration and optimal policy existence in Isabelle/HOL. This is methodology context only; no proof code is imported into Lean from that project.

Motivation

A tall $N\times n$ matrix with independent standard normal entries is, with high probability, a near-isometry: all its singular values lie in $[\sqrt N-\sqrt n,\ \sqrt N+\sqrt n]$ up to Gaussian fluctuations. This is Gordon's theorem together with Gaussian concentration (Vershynin 2012, Theorem 5.32 and Corollary 5.35), the non-asymptotic counterpart of the Bai–Yin law and the model result for the whole theory of non-asymptotic random matrices. In randomized numerical linear algebra the same two ingredients give the expected spectral norm of $S\Gamma T$ (Halko–Martinsson–Tropp 2011, Proposition 10.1 (10.2)), its second moment (Tropp–Webber 2023, Lemma B.1), and the lower tail of $\sigma_{\min}$ of a Gaussian sketch, which controls the inverse moments $\mathbb E\,\sigma_{\min}^{-2p}$ that appear in the analysis of sketching algorithms with rank truncation. None of these results is available in Mathlib; this mission proves them.

Setting

A standard Gaussian matrix $A\in\mathbb R^{N\times n}$ has independent $\mathcal N(0,1)$ entries; its law is $\gamma_{N,n}$. Its extreme singular values are

$$s_{\min}(A)=\inf_{\|x\|_2=1}\|Ax\|_2,\qquad s_{\max}(A)=\sup_{\|x\|_2=1}\|Ax\|_2=\|A\|,$$

the infimum and supremum over the unit sphere of $\mathbb R^n$ ($n\ge1$); $\|A\|$ is the spectral norm. For a wide matrix $G\in\mathbb R^{k\times t}$, $k\le t$, the usual $\sigma_{\min}(G)$ is $s_{\min}(G^{\mathsf T})$. The Frobenius norm is $\|A\|_F=(\sum_{i,j}A_{ij}^2)^{1/2}$; a function $h$ on matrices is $L$-Lipschitz if $|h(X)-h(Y)|\le L\|X-Y\|_F$. A chi-square variable with $d$ degrees of freedom is $\chi_d^2=\sum_{i=1}^d g_i^2$ for independent standard normals $g_i$. The letter $e$ denotes Euler's number.

Formalization targets

Goal — deviation of the extreme singular values (Vershynin, Corollary 5.35)

For every $t\ge0$, with probability at least $1-2e^{-t^2/2}$,

$$\sqrt N-\sqrt n-t\ \le\ s_{\min}(A)\qquad\text{and}\qquad s_{\max}(A)\ \le\ \sqrt N+\sqrt n+t .$$

This is the goal because it is the statement users of Gaussian matrices quote: a quantitative near-isometry with explicit sub-Gaussian deviations. It follows from the expectation bounds (Gordon) and concentration, both of which are milestones.

Milestones

• Gaussian concentration for Lipschitz functions of a Gaussian matrix: $\mathbb P\{h(G)\ge\mathbb E h(G)+Lt\}\le e^{-t^2/2}$ (HMT Proposition 10.3; Vershynin Proposition 5.34).
• Gordon–Chevet bound. $\mathbb E\|S\Gamma T\|\le\|S\|\|T\|_F+\|S\|_F\|T\|$ (HMT Proposition 10.1 (10.2), A.2) and its second-moment form $(\mathbb E\|S\Gamma T\|^2)^{1/2}\le\|S\|\|T\|_F+\|S\|_F\|T\|$ (Tropp–Webber Lemma B.1).
• Gordon's theorem. $\sqrt N-\sqrt n\le\mathbb E s_{\min}(A)\le\mathbb E s_{\max}(A)\le\sqrt N+\sqrt n$ (Vershynin Theorem 5.32).
• One-sided tail of $s_{\min}$. $\mathbb P\{s_{\min}(A)\le\sqrt N-\sqrt n-u\}\le e^{-u^2/2}$ for $u\ge0$ (Davidson–Szarek Theorem II.13; one-sided form of Vershynin Corollary 5.35).
• Chi-square lower tail. $\mathbb P\{\chi_d^2\le u\}\le(eu/d)^{d/2}$ for $0\le u\le d$ (Dasgupta–Gupta Lemma 2.2).
• Small-ball bound for $s_{\min}$. $\mathbb P\{s_{\min}(A)\le s\}\le n\bigl(\frac{e\,ns^2}{N-n+1}\bigr)^{(N-n+1)/2}$ for $s\ge0$, $ns^2\le N-n+1$, from the invertibility-via-distance lemma of Rudelson–Vershynin (Lemma 3.5), rotational invariance and the chi-square lower tail.

Significance

The results. Gordon's theorem is sharp: $\mathbb E s_{\min}$ and $\mathbb E s_{\max}$ are pinned to within $O(1)$ of the Bai–Yin limits for every $N,n$. Gaussian concentration upgrades it to tail bounds with the optimal sub-Gaussian rate, and the Chevet-type bound is the same mechanism for $S\Gamma T$, where it yields the expected spectral error of the randomized SVD. The small-ball bound covers the other regime, $s\to0$, with polynomial decay of the exact order $N-n+1$; together the two tails make every negative moment $\mathbb E s_{\min}(A)^{-2p}$, $2p<N-n+1$, finite and explicitly bounded, which is what the early-truncated analysis needs.

Formalizing them. Mathlib has the standard Gaussian measure on finite-dimensional inner product spaces but neither Gaussian concentration for Lipschitz functions, nor Slepian's or Gordon's comparison inequalities, nor any result on singular values of random matrices. These are foundational pieces of high-dimensional probability whose formalization would be reused far beyond this series.

Difficulty

The upper bound in Gordon's theorem is proved by Slepian's comparison lemma for the Gaussian process $(u,v)\mapsto\langle Au,v\rangle$ on $S^{n-1}\times S^{N-1}$, and the lower bound by Gordon's minimax comparison inequality for $\max_v\min_u\langle Au,v\rangle$; both require a theory of Gaussian processes indexed by continuous parameter sets, or at least a discretization argument with careful limits. Gaussian concentration (Tsirelson–Ibragimov–Sudakov) is usually proved through the Gaussian logarithmic Sobolev inequality or the Ornstein–Uhlenbeck semigroup, with a smoothing step to pass from smooth to Lipschitz functions. Both are long developments with no Mathlib support. The elementary alternatives do not reach the stated constants: an $\varepsilon$-net argument bounds $\mathbb E\|A\|$ by $C(\sqrt N+\sqrt n)$ with $C>1$, and a union bound over rows for the small-ball estimate is exactly how the stated bound is obtained, but it needs the fact that the distance from a Gaussian vector to a fixed $(n-1)$-dimensional subspace is a $\chi_{N-n+1}$ variable, which again rests on rotational invariance.

Formalization scope

Matrices are sampled from the product measure $\gamma_{N,n}$ on arrays. $s_{\min}$ is sMin, the real infimum over the unit sphere of $\mathbb R^n$; $n\ge1$ is assumed so the sphere is nonempty, and $N\ge n$ is not assumed for Gordon's theorem (for $N<n$ one has $s_{\min}=0$ and the lower bound is trivial, as in the source). $s_{\max}$ is Mathlib's $\ell_2$ operator norm. Expectation bounds are stated together with integrability of their integrands, so the Lean convention that non-integrable functions integrate to $0$ cannot trivialize them. Probabilities are measures in $[0,\infty]$ compared with $\operatorname{ofReal}$ of the real bound; the goal's lower bound $1-2e^{-t^2/2}$ uses truncated subtraction and is vacuous for $t\le\sqrt{2\log2}$, exactly as in the source. Gaussian concentration is stated with $L>0$ and $t\ge0$, which the sources leave implicit (for $L=0$ or $t<0$ the event has probability one). Results that the sketching literature states for wide $k\times t$ matrices are stated here for tall $N\times n$ matrices via $\sigma_{\min}(G)=s_{\min}(G^{\mathsf T})$, with $(N,n)=(t,k)$.

Welcome contributions: Gaussian concentration for Lipschitz functions on $\mathbb R^d$ (any proof), Slepian's and Gordon's inequalities, the $1$-Lipschitz property of $s_{\min}$ and $s_{\max}$, the chi-square Chernoff bounds, and the law of the distance from a Gaussian vector to a subspace.

Selected references

• N. Halko, P.-G. Martinsson, J. A. Tropp, Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions, SIAM Review 53(2), 217–288, 2011. https://arxiv.org/abs/0909.4061
• J. A. Tropp, R. J. Webber, Randomized algorithms for low-rank matrix approximation: design, analysis, and applications, 2023. https://arxiv.org/abs/2306.12418
• R. Vershynin, Introduction to the non-asymptotic analysis of random matrices, in Compressed Sensing: Theory and Applications, Cambridge University Press, 2012. https://arxiv.org/abs/1011.3027
• L. Isserlis, On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables, Biometrika 12, 134–139, 1918. https://doi.org/10.1093/biomet/12.1-2.134
• S. Dasgupta, A. Gupta, An elementary proof of a theorem of Johnson and Lindenstrauss, Random Structures & Algorithms 22, 60–65, 2003. https://doi.org/10.1002/rsa.10073
• M. Rudelson, R. Vershynin, The Littlewood–Offord problem and invertibility of random matrices, Advances in Mathematics 218, 600–633, 2008. https://arxiv.org/abs/math/0703503
• Y. Chen, J. Dongarra, Condition numbers of Gaussian random matrices, SIAM J. Matrix Anal. Appl. 27(3), 603–620, 2005. https://doi.org/10.1137/040616413
• A. Edelman, Eigenvalues and condition numbers of random matrices, SIAM J. Matrix Anal. Appl. 9(4), 543–560, 1988. https://doi.org/10.1137/0609045
• Y. Gordon, Some inequalities for Gaussian processes and applications, Israel J. Math. 50, 265–289, 1985. https://doi.org/10.1007/BF02759761
• K. R. Davidson, S. J. Szarek, Local operator theory, random matrices and Banach spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, 317–366, 2001. https://doi.org/10.1016/S1874-5849(01)80010-3
• M. Ledoux, The Concentration of Measure Phenomenon, AMS Mathematical Surveys and Monographs 89, 2001. https://doi.org/10.1090/surv/089

Shared definitions: the items of this mission import Definitions.Def_GaussianMatrix_basic, a draft of Mission I; submit that proposal first so the import resolves at compile time.

Series: I. Rotation Invariance and Moment Identities · II. Inverse Moments of the Pseudoinverse · III. Extreme Singular Values and Concentration. All three missions share the definition bundle GaussianMatrix_basic, published with Mission I, and the namespace GaussianMatrix.