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How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Full-history cost and the bias from truncating memory

Example

On a uniform grid with $N$ steps, direct L1 evaluation uses exactly $N (N + 1) / 2$ history-weight terms and stores the $N + 1$ nodal values. If only the most recent $m$ increments are retained, then for $u (t) = t$ and $n > m$ the truncated L1 Caputo value is

${\tilde{δ}}_{h}^{α} u^{n} = \frac{(m h)^{1 - α}}{Γ (2 - α)},$

whereas the full value is $t_{n}^{1 - α} / Γ (2 - α)$ . The lost history therefore contributes the exact bias

$\frac{t_{n}^{1 - α} - (m h)^{1 - α}}{Γ (2 - α)} .$

The truncated implementation uses $\sum_{n = 1}^{N} \min (n, m)$ terms and at most $m + 1$ recent nodal values. No error guarantee for other trajectories is claimed.

Facts & Assumptions

Given: $0 < α < 1$ , $T > 0$ , an integer $N \geq 1$ , $h = T / N$ , and an integer $1 \leq m \leq N$ .

[F1]

Absolute continuity on a compact interval: absolute continuity means that for every $ε > 0$ some $δ > 0$ makes the sum of absolute endpoint increments smaller than $ε$ for any finite disjoint interval family of total length below $δ$ .

[F2]

Fractional integral and Caputo derivative: $D_{C}^{α} u = Γ (1 - α)^{- 1} \int_{0}^{t} (t - s)^{- α} u^{'} (s) d s$ almost everywhere; at points where this integral is finite, its pointwise integral value is the representative used below.

[F3]

Uniform-grid and graded-grid L1 discretization: uniform-grid weights are $a_{j} = (j + 1)^{1 - α} - j^{1 - α}$ and each L1 step sums all retained increments with these weights.

[F4]

The real Gamma functional equation $Γ (s + 1) = s Γ (s)$ : $Γ (2 - α) = (1 - α) Γ (1 - α)$ .

[F5]

Real powers for positive bases, with the zero-base positive-exponent convention and Continuity and derivatives of positive-base real powers: $(t^{1})^{'} = 1$ for $t > 0$ .

[F6]

Integral of the fractional power kernel: $\int_{0}^{t} (t - s)^{- α} d s = t^{1 - α} / (1 - α)$ .

[F7]

Finite sums and finite products, by recursion: the displayed finite sums add their indexed terms.

[F8]

Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule: reindexing a finite sum by a bijection of its finite index set preserves its value.

Verification

technique · direct

1.1givenF1algebra

Fix any $ε > 0$ . For any finite disjoint intervals, the sum of absolute endpoint increments of $u (t) = t$ equals the sum of their lengths. Thus [F1]'s defining condition holds with $δ = ε$ , and $u$ is absolutely continuous.

1.2givenF2F4F5F6algebra

By [F5], $u^{'} = 1$ on $(0, T)$ , so $u^{'} \in L^{1} (0, T)$ . Using the pointwise representative from [F2], the kernel integral [F6] gives $D_{C}^{α} u (t) = \frac{1}{Γ (1 - α)} \int_{0}^{t} (t - s)^{- α} d s = \frac{t^{1 - α}}{(1 - α) Γ (1 - α)} = \frac{t^{1 - α}}{Γ (2 - α)},$ using [F4]. This integral is finite at each $t \in [0, T]$ .

1.3givenF7F8algebra

The direct full-history step $n$ uses one term for each increment $k = 1, \dots, n$ . Let $S = \sum_{n = 1}^{N} n$ . Reindexing by $n \mapsto N + 1 - n$ using [F8] also gives $S = \sum_{n = 1}^{N} (N + 1 - n)$ . Adding these equalities termwise gives $2 S = \sum_{n = 1}^{N} (N + 1) = N (N + 1)$ by [F7], so the full-history term count is $N (N + 1) / 2$ . Retaining the sequence requires $N + 1$ nodal values.

2.1step 1.2F3algebra

For $n > m$ , the truncated version of [F3] retains exactly the $m$ terms with index lag $j = 0, \dots, m - 1$ . Since $u^{k} - u^{k - 1} = h$ , ${\tilde{δ}}_{h}^{α} u^{n} = \frac{h^{1 - α}}{Γ (2 - α)} \sum_{j = 0}^{m - 1} a_{j} = \frac{h^{1 - α} m^{1 - α}}{Γ (2 - α)},$ where the sum telescopes. The full sum through $j = n - 1$ likewise telescopes to $n^{1 - α}$ and equals the exact derivative from step 1.2. This gives the bias stated.

3.1givenstep 2.1F7algebra∎

At step $n$ the truncated evaluation uses $\min (n, m)$ terms, so [F7] gives its total count as $\sum_{n = 1}^{N} \min (n, m)$ . It can retain only the $m$ most recent increments, equivalently $m + 1$ adjacent nodal values. For $n \leq m$ no term is omitted, and at $m = N$ the bias is zero at every step.

Depends on

Used by

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Dependency tree · two levels

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Sources

Yumin Lin and Chuanju Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, Journal of Computational Physics 225 (2007), §3 (standard reference, not scraped)