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\title{\textbf{LTL: Lean Transparency Log}\\[2pt]
\large Where Deductive and Cryptographic Proof Vouch for Each Other}
\author{Olaf Horvath\\ \small \texttt{Olaf.Horvath@zkdefi.org} --- \url{ltl.zkdefi.org} --- \href{https://orcid.org/0009-0004-8008-5805}{ORCID 0009-0004-8008-5805}}
\date{July 2026}

\begin{document}
\maketitle

\begin{abstract}
Interactive theorem provers can certify the functional correctness of
deployed cryptographic code, but the resulting assurance is expensive to
\emph{consume}: re-checking a realistic proof corpus takes a proof
toolchain and hours of kernel time, which excludes almost every
downstream user, including autonomous agents that must decide which
cryptographic library to trust. We describe the \emph{Lean Transparency
Log} (LTL)\footnote{Not to be confused with linear temporal
logic~\cite{pnueli77}; the collision is acknowledged and, in a paper
about verification, difficult to resist.}, a deployed RFC~9162-style
transparency log whose leaves are \emph{replay attestations}: signed
statements that the Lean~4 proofs of a specific Rust repository, at a
specific git commit, re-check with exactly their documented axiom sets.
Consumers verify one signature and a logarithmic inclusion proof in
milliseconds; the hours of kernel time are paid once, by the log
operator. Three design decisions distinguish the LTL from prior
attestation transparency systems: (i)~consumers re-derive verification
verdicts locally from the \emph{observed axiom cones} carried in each
attestation, so the operator is trusted for observations, never for
verdicts; (ii)~axiom cones are matched against per-theorem
\emph{documented boundaries} exactly, in both directions; and (iii)~the
log's tree heads are signed by a binary built from the very Ed25519
implementation whose correctness certificates are leaves of the log, and
each signature embeds the operator's own Merkle self-check of that leaf.
We report a small production deployment covering four verified
production Ed25519 implementations, describe what the accumulated
evidence does and does not establish, and outline the natural next step:
verifying the log's own proof-checking algorithms in Lean and entering
those certificates into the log they defend.
\end{abstract}

\section{Introduction}
\label{sec:intro}

Formal verification of deployed cryptographic code has matured from
research prototypes to substantial artifacts: verified-by-construction
libraries such as HACL*~\cite{hacl} and Fiat-Crypto~\cite{fiatcrypto}
ship in mainstream software, and post-hoc verification pipelines such as
Aeneas~\cite{aeneas} make it possible to state and prove theorems about
\emph{existing} production Rust code. The corpus underlying this paper
is of the latter kind: four production Ed25519
implementations---upstream \code{curve25519-dalek}/\code{ed25519-dalek}
and three deployed forks (Solana, RISC~Zero, Betrusted)---each carry
Lean~4~\cite{lean4} certificates, proven against that fork's own
extracted model, covering field arithmetic over $\mathbb{F}_{2^{255}-19}$,
the complete twisted Edwards laws~\cite{bernsteinlange,twisted}, scalar
arithmetic mod $\ell$, encoding/decoding with constructive point
decompression, and a four-tier characterization of signature
verification~\cite{ed25519,rfc8032} whose strongest tier states: the
extracted verifier accepts iff the signature's $R$ component decompresses
to a valid curve point equal to $[k](-A)+[s]B$.

\paragraph{What the certificates state.} Concretely, the corpus is a
stack of theorems about extracted code, each stated through a
\emph{denotation} from machine representation to mathematics. Field
elements are five 51-bit limbs denoting
$\den{(a_0,\dots,a_4)}=\sum_i a_i2^{51i}\bmod p$, and every
operation carries a two-clause specification---the value is right
\emph{and} the representation invariant is preserved, e.g.
\begin{align*}
\footnotesize
\forall a\,b.\ &\mathrm{bnd}\,a \Rightarrow \mathrm{bnd}\,b \Rightarrow{}\\[-2pt]
&\exists c.\ \code{mul}\,a\,b = \code{ok}\,c \,\wedge\, \mathrm{bnd}\,c
\,\wedge\, \den{c} = \den{a}\cdot\den{b}.
\end{align*}
Point operations are proven to implement the complete twisted Edwards
addition law on
$E : -x^2+y^2 = 1+d\,x^2y^2$ over $\mathbb{F}_p$,
\begin{equation*}
\small
(x_1,y_1)+(x_2,y_2)=
\Bigl(\tfrac{x_1y_2+x_2y_1}{1+d\,x_1x_2y_1y_2},\;
\tfrac{y_1y_2+x_1x_2}{1-d\,x_1x_2y_1y_2}\Bigr),
\end{equation*}
including the completeness fact that makes it branch-free ($a=-1$ is a
square and $d$ a non-square in $\mathbb{F}_p$, so the denominators never
vanish~\cite{bernsteinlange}).
At the apex, writing $\code{accept}(A,m,R,s)$ for ``the extracted
verifier returns \code{ok}'', with $k$ the scalar produced by the hash
oracle $H(R,A,m)$ and no properties assumed of $H$, the byte-level tier
states
\begin{equation*}
\footnotesize
\code{accept}(A,m,R,s) \;\Leftrightarrow\; \code{compress}([s]B-[k]A) = R,
\end{equation*}
and the strongest tier lifts byte equality to the group:
\begin{equation*}
\footnotesize
\code{accept}(A,m,R,s) \;\Leftrightarrow\; \code{decompress}(R) = [k](-A)+[s]B,
\end{equation*}
with decompression itself proven (exact byte parsing, the
$(p{+}3)/8$-power square root, sign-bit root selection). Each theorem's
axiom cone is pinned exactly---the standard three below the apex, the
enumerated oracle boundary at the apex tiers.

The economics of \emph{consuming} such evidence are poor: re-checking
one fork's certificates takes $\approx$30 minutes of Lean kernel time
and a pinned toolchain. A wallet, a package manager, or an autonomous
agent choosing a cryptographic backend cannot pay this per
decision---and need not: a deterministic re-check yields a fact that can
be attested once and distributed.

This is the classic transparency-log trade---Certificate
Transparency~\cite{rfc6962,rfc9162} for certificate issuance, Sigstore's
Rekor~\cite{sigstore} for signing events and supply-chain
attestations~\cite{intoto}, key transparency~\cite{coniks}, checksum
databases---applied to a payload with different trust semantics:
\emph{evidence of machine-checked mathematical truth, together with its
exact assumption set}. We claim no novelty for any component (the hash
structure and proof algorithms are RFC~9162 verbatim); the contribution
is the composition, its trust model, and a deployed, reproducible
instance. The LTL does not mechanize cryptographic security
proofs---that inter-disciplinary bridge is already being built by
EasyCrypt and its relatives~\cite{easycrypt}. It bridges the adjacent,
mostly empty gap: type-theory-certified artifacts have no distribution
infrastructure, and cryptographic transparency infrastructure has never
carried kernel-checked mathematics---each discipline solving the
other's problem.

\S\ref{sec:log} describes the log and its trust model,
\S\ref{sec:dogfood} the self-referential signing loop,
\S\ref{sec:deploy} the deployment and what its evidence establishes,
and \S\ref{sec:next} the verification agenda for the log itself.

\section{The Lean Transparency Log}
\label{sec:log}

\begin{figure*}[t]
\centering
\begin{tikzpicture}[every node/.style={font=\small}]
  \tikzset{box/.style={draw, rounded corners=2pt, minimum width=1.6cm, minimum height=0.56cm}}
  % leaves: 0-3 failed run (gray), 4-7 verified (black)
  \foreach \i/\c in {0/gray, 1/gray, 2/gray, 3/gray, 4/black, 5/black, 6/black, 7/black}
    \node[box, draw=\c, text=\c] (l\i) at (\i*2.05,0) {leaf \i};
  \node[box, draw=black, very thick, fill=black!6] at (4*2.05,0) {\textbf{leaf 4}};
  \node[text=gray, font=\footnotesize] at (3*2.05/2 - 1.02,-0.62) {failed-audit run --- retained forever};
  \node[font=\footnotesize] at (5.5*2.05 + 0.35,-0.62) {verified attestations (16/16, boundary-exact)};
  % internal nodes
  \foreach \i in {0,...,3}
    \node[box] (m\i) at (\i*4.1+1.025,1.25) {node};
  \foreach \i in {0,1}
    \node[box] (n\i) at (\i*8.2+3.075,2.5) {node};
  \node[box, thick] (root) at (7.175,3.75) {root};
  \foreach \a/\b in {l0/m0, l1/m0, l2/m1, l3/m1, l4/m2, l5/m2, l6/m3, l7/m3,
                     m0/n0, m1/n0, m2/n1, m3/n1, n0/root, n1/root}
    \draw[gray!70] (\a) -- (\b);
  % signature head
  \node[box, thick, minimum width=5.2cm] (sig) at (7.175,4.95)
    {signed tree head: $\mathrm{Ed25519}(\mathrm{root})$};
  \draw[-{Stealth}] (root) -- (sig);
  % the self-certification loop, routed through clear space on the right
  \draw[-{Stealth}, thick, dashed]
    (sig.east) .. controls (13.2,4.95) and (12.4,1.6) .. (l4.north east);
  \node[align=left, font=\footnotesize, anchor=west] at (12.45,3.1)
    {signed by the binary built\\ from the source attested at\\ leaf 4; the operator verifies\\ that leaf's inclusion first};
\end{tikzpicture}
\caption{The deployed 8-leaf log. Leaves 0--3 record an early replay whose
audit step failed (two tooling defects, since fixed); an append-only ledger
retains them. The tree head is signed by a binary built from the
implementation attested at leaf 4, and each signature embeds the operator's
own inclusion check of that leaf against the tree being signed.}
\label{fig:loop}
\end{figure*}

\paragraph{Roles.} The system has exactly two roles with deliberately
asymmetric costs and capabilities. The \emph{operator} (one per log)
owns a Lean toolchain, replays proof corpora, holds the log's signing
key, and bears append-only obligations. \emph{Consumers} (unbounded)
hold the operator's public key, receive small evidence files, and verify
with $\approx$25 lines of standard-library code. Nothing a consumer does
requires a theorem prover.

\paragraph{Leaves: replay attestations.} A leaf is the canonical JSON
serialization of an attestation recording: the subject repository URL
and \emph{git commit} (which cryptographically pins the entire source
tree); the toolchain versions; the resource-control regime under which
the replay ran; and, per certificate, its name, replay status, and the
\emph{observed axiom cone}---the exact output of Lean's
\code{\#print axioms} for that theorem. For the corpus above each
attestation carries 16 certificates.

\paragraph{Boundary-exact auditing.} Every certificate has a
\emph{documented} allowed axiom set. Foundational certificates must
carry exactly Lean's three standard axioms
(\code{propext}, \code{Classical.choice}, \code{Quot.sound}); the four
signature-tier certificates additionally carry a per-fork, explicitly
enumerated boundary (an opaque SHA-512 oracle and opaque wire-format
types---e.g., eleven axioms in total for the upstream fork). A cone is
\emph{clean} iff it equals its allowed set; deviation in either
direction---an unexpected axiom, or a missing boundary axiom---is
flagged. Each source repository enforces the same discipline in its own
check scripts; the log mirrors those sets and treats the repositories'
scripts as authoritative.

\paragraph{Observation, not verdict.} Attestations include the
operator's pass/fail judgment, but consumers ignore it: the consumer's
tooling re-derives every verdict locally by comparing the
\emph{observed} cone against the consumer's own copy of the allowed
sets. An operator that labels a dirty cone ``clean'' gains nothing; an
attestation that omits observed cones is treated as unverifiable. This
narrows the trust placed in the operator to: ``the reported
\code{\#print axioms} output is what the kernel printed for this commit.''

\paragraph{Tree, heads, receipts.} Leaves are accumulated in an
RFC~9162 Merkle tree ($\mathrm{SHA}$-256 with $0x00/0x01$
leaf/node domain separation). The operator signs tree heads; consumers
receive a \emph{receipt}: leaf index, sibling path, and the signed head.
Consumers additionally maintain a local pin store: same-size heads must
match the pinned root exactly (a mismatch is reported as equivocation
and is unrecoverable); growth requires a verified consistency proof from
the pinned size; shrinkage is rejected. A freshness policy bounds head
age. The full log is also published as a git repository (one file per
leaf, plus every head ever signed), so any cloner can recompute every
prefix root from the public leaves and audit the complete head history
without consistency proofs---a low-infrastructure witness
mechanism~\cite{rfc9162}; a standalone $\approx$150-line
standard-library verifier ships in the mirror.

\section{The self-certifying signature}
\label{sec:dogfood}

Tree heads are Ed25519 signatures, and this creates an opportunity for
coherence: the log \emph{contains} correctness certificates for an
Ed25519 implementation. The LTL's heads are therefore signed by a binary
built from the pinned source tree of exactly the implementation attested
in the log (serial backend pinned, matching the verified extraction),
and---before signing---the operator runs the same Merkle inclusion
verification a consumer runs, on the newest leaf attesting the signing
implementation, against the tree about to be signed. The verdict is
embedded in the signature block:

{\small
\begin{verbatim}
signing_backend: verified-dalek-serial
signing_library_source_commit: aa0f6ab...
signing_library_leaf_index: 4
signing_library_certificates_proven: 16/16
self_inclusion: verified
\end{verbatim}}

\noindent The signature vouches for the tree; the tree vouches for the
code that produced the signature; and the two vouchings are different
proof modalities (cryptographic and deductive), so the loop is
self-referential without being circular
(Fig.~\ref{fig:loop}). We state the honest extent of this claim
precisely: the Lean certificates cover the \emph{verification} path of
the library (the theorems' subject is the extraction image of that
path); the \emph{signing} path is not covered by any certificate and is
declared trusted base---the design merely ensures the trusted signing
code is the attested artifact rather than an unrelated third
implementation, and that consumers can check as much. Signature
\emph{verification} on consumer machines can optionally run through the
same certified-source binary, with the backend that actually ran
recorded in every result and a fail-closed policy flag available.
First-append bootstrapping is handled honestly: heads signed before the
signing library's attestation enters the log record
\code{self\_inclusion:\ library\_not\_in\_log}.



\section{Deployment and evidence}
\label{sec:deploy}

The LTL is deployed\footnote{Service:
\url{https://ltl.zkdefi.org} (read-only HTTP API and
documentation). Mirror: \url{https://github.com/saymrwulf/lean-transparency-log}.
Operator/consumer tooling and a twelve-lecture course:
\url{https://github.com/saymrwulf/proof-aware-crypto-tooling-agent}. The
underlying proof corpora are in the \code{saymrwulf/*-ed25519-verified}
repositories; every claim in this paper is re-checkable from these
artifacts.} with eight leaves: one attestation per fork from each of two
full replay runs (58--64 Lean files and $\approx$1{,}800\,s per
fork, under hard memory caps and core pinning). In the second run all
four forks reported 16/16 certificates proven with boundary-exact cones,
pinned to exact commits. The first run is deliberately still in the log: its audit step
failed due to two defects in the \emph{operator tooling} (a path issue
and a parser that mishandled Lean's line-wrapped axiom lists for the
eleven-axiom cones), and the operator signed attestations recording the
failure rather than suppressing the run. We consider the resulting
ledger---four failure leaves permanently beside four success
leaves---a feature of the trust model, and note that both defects were
fail-closed: valid proofs were rejected; invalid ones were never
accepted.

\paragraph{What a verified receipt establishes.} Under the assumptions
enumerated below, a consumer who verifies a receipt knows: \emph{the
operator whose key I pinned attests that the Lean certificates of
repository $X$ at commit $Y$ re-check, with per-certificate observed
axiom cones as included---and this statement is part of the log
presented to every other consumer}. Combined with local verdict
re-derivation, this yields source-level assurance for the pinned
commit. It deliberately does \emph{not} establish: correctness of any
binary (consumers build from the pinned source; compilers are trusted
base), correctness of SHA-512 (an opaque oracle in the theorems),
correctness of the wire-format parsers (their outcomes are hypotheses of
the signature tiers), signing-side correctness, or side-channel
properties. The assumption set, in full: the Lean kernel and its three
axioms plus mathlib; faithfulness of the Charon/Aeneas
extraction~\cite{aeneas}; each fork's documented oracle boundary;
operator key custody and the trust-on-first-use key distribution
(mitigated by publishing the key in two independent locations);
collision resistance of SHA-256 for the log; unforgeability of Ed25519
for the heads; and the consumer's own $\approx$25-line verifier.

\paragraph{An observational by-product: proof portability.} Because the
four corpora prove the same theorems against four independent
extractions, the diff between proof files measures how portable proofs
are across real forks. Pure-mathematics files (e.g., a carry-telescope
lemma file) are byte-identical across all four; extraction-facing proof
scripts diverge sharply where the forks' code or the extractor's naming
differs (e.g., 215 changed lines for the byte-parser proofs on the two
forks whose extraction produces a closure-based loader; 121 lines for
the signature-glue proofs on the same-crate fork; 27 lines---all
annotation---between the two structurally closest forks, documenting the
one fork's \code{black\_box} optimization barrier). Per-target
verification, in other words, is doing measurable work exactly where the
targets actually differ.

\section{Related work}
\label{sec:related}

Certificate Transparency~\cite{rfc6962,rfc9162} supplies the data
structure and proof algorithms, used here unchanged. Rekor within
Sigstore~\cite{sigstore} is the closest deployed system: a transparency
log over signing events and supply-chain attestations such as
in-toto~\cite{intoto} link metadata; its payloads attest \emph{process}
(who signed, how an artifact was built), whereas LTL leaves attest
kernel-checked \emph{mathematical statements together with their
assumption sets}, and the consumer re-derives verdicts rather than
trusting labels. Key transparency~\cite{coniks} and checksum databases
share the pattern with different payloads. Proof-carrying
code~\cite{pcc} ships proofs to consumers who check them; the LTL serves
consumers who cannot run any checker, replacing proof transport with
attestation, inclusion, and signature---at the cost of trusting the
operator's kernel run, a cost the design minimizes but does not
eliminate. Cheval, Moreira and Ryan formally verify transparency
protocols themselves~\cite{cheval}; our direction is the complement
(we log the verification), and \S\ref{sec:next} proposes meeting in the
middle. Verified Merkle tree \emph{implementations} exist, notably in
EverCrypt~\cite{evercrypt}; \S\ref{sec:next} builds on that precedent
rather than claiming it.

\section{Limitations and next steps}
\label{sec:next}

\paragraph{Limitations.} The deployment is small (one operator, eight
leaves, four subject repositories) and the operator is a single party;
split-view defense currently rests on consumer-side pinning plus the
public git mirror rather than an independent witness network. Key
distribution is trust-on-first-use. The signing path of the dogfood
binary is unverified (declared, not proven). The corpus itself stops at
source-level assurance---reproducible builds and side-channel evidence
remain open---and ML-DSA slots in the head format are deliberately
recorded as unavailable rather than backed by an unverified
implementation.

\paragraph{Verifying the accumulator itself.} The natural next step
applies the corpus's own discipline to the log's cryptographic half,
which is currently the \emph{unproven} half of the composition. All of
the following are tractable Lean targets: (i)~completeness of the
RFC~9162 inclusion verifier (honest proofs verify---no assumptions);
(ii)~\emph{soundness as an explicit reduction}: an accepting inclusion
proof for a leaf outside the tree yields a SHA-256 collision, making
collision resistance a documented boundary axiom audited exactly like
the SHA-512 oracle in the Ed25519 tiers; (iii)~the analogous statement
for the consistency verifier (acceptance implies prefix, modulo
collisions); (iv)~domain separation as a lemma; and (v)~total
correctness of the consumer's pin-store state machine (equivocation
detection, rollback rejection). Verified Merkle implementations in
F*~\cite{evercrypt} and algorithm verifications in other systems show
these proofs are well within reach; the LTL-specific closure is
\emph{where the certificates go}: into the log they defend, checked by
the certified checker they specify, alongside a consumer policy flag
requiring the certified verifier. At that point both proving traditions
in the composition run on certified code, and the remaining trusted base
is two hash assumptions, a compiler, an extraction pipeline, and one
key.

\paragraph*{Acknowledgments.} The proof corpora, tooling, deployment,
and this paper were produced with substantial assistance from Claude
(Anthropic). All claims are enforced by the referenced check scripts and
are independently re-checkable from the public artifacts.

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\end{document}
