\chapter{Honesty, Axioms, and the Art of Trusting Proofs}
\label{ch:honesty}

\section{``Formally verified'' is a claim, not a spell}

The phrase \emph{formally verified} has marketing gravity, and marketing
gravity attracts abuse. This chapter arms you with the auditor's toolkit:
what a Lean certificate actually rests on, how to interrogate it in one
command, and the specific ways a verification claim can be hollow while
every file compiles. Nothing here is hypothetical --- each failure mode
appears in the wild, and the discipline below is the one the companion
projects hold themselves to.

\begin{pitfall}
This chapter teaches you to interrogate a \emph{certificate}. It does not teach
you to interrogate the \emph{script that interrogates certificates for you} ---
and that script is software you wrote, verified by nothing.
Chapter~\ref{ch:attestation} is about that second problem, which turned out to
be far harder than everything in this chapter, and is where every failure in the
companion projects has occurred.
\end{pitfall}

\section{The trust ledger}

When the kernel accepts \lean{fieldImplementation}, what exactly are you
being asked to believe? Lean can tell you --- precisely:

\begin{lstlisting}[language=Lean]
#print axioms fieldImplementation
-- 'fieldImplementation' depends on axioms:
-- [propext, Classical.choice, Quot.sound]
\end{lstlisting}

\lean{\#print axioms} walks the \emph{entire} dependency tree of a theorem
--- every lemma, every lemma's lemmas, down to bedrock --- and reports every
assumption found there. The three names above are Lean's standard trio
(propositional extensionality, classical choice, quotient soundness):
ordinary classical mathematics, accepted by working mathematicians,
scrutinized by logicians for a century. A certificate reporting exactly
these three is called \textbf{axiom-clean}. Anything \emph{else} in that
list is a custom assumption someone added --- and the whole audit consists
of reading that list and asking whether you believe each entry.

\begin{bigidea}
A machine-checked theorem is a receipt with a complete list of its own
assumptions --- something prose mathematics has never had. But the receipt
only protects people who read it. \textbf{The one-command audit:} run
\lean{\#print axioms} on the headline theorem; anything beyond
\lean{[propext, Classical.choice, Quot.sound]} is where the bodies are
buried. Make this reflex, and no verification claim can hide from you.
\end{bigidea}

\begin{worked}{reading three audit transcripts --- a training set}
Here are three \lean{\#print axioms} outputs. Before reading the
verdicts, write your own one-line assessment of each; the answers
follow.

\begin{lstlisting}
-- Transcript A
'fieldImplementation' depends on axioms:
[propext, Classical.choice, Quot.sound]

-- Transcript B
'mulCorrect' depends on axioms:
[propext, Classical.choice, Quot.sound, sorryAx]

-- Transcript C
'groupLawComplete' depends on axioms:
[propext, Classical.choice, Quot.sound, edDenomNonzero]
\end{lstlisting}

\emph{Transcript A}: the standard trio and nothing else --- axiom-clean.
Remaining questions are the human ones (is the \emph{statement}
non-trivial? is the \emph{model} the shipping code?), but the logical
ledger is closed.

\emph{Transcript B}: \lean{sorryAx} is the placeholder axiom behind
\lean{sorry} --- somewhere in this theorem's dependency tree an
obligation was skipped, and the tree remembers even though the file that
contains it may be ten imports away. This certificate is decorative,
\emph{regardless of anything else about it}. (Note how cheap the
detection was: the tool walked ten thousand dependencies so you didn't
have to.)

\emph{Transcript C}: the interesting one. A custom axiom,
\lean{edDenomNonzero} --- from the name, someone \emph{assumed} the
Edwards denominators are nonzero rather than proving completeness. Two
possible worlds: it is declared in the trusted-base ledger with a
justification (defensible --- but then why does
Chapter~\ref{ch:pyramid} prove it from the non-squareness of $d$ with
modest effort? suspicious laziness at best), or it is undocumented ---
in which case the headline ``group law verified, complete addition!''
is quietly assuming the very fact that makes completeness interesting.
Your move as auditor: \lean{grep} for the axiom's declaration, check
the ledger, and ask the vendor Chapter question (3). The general
skill: \textbf{a custom axiom's \emph{name} tells you which claim to
stop believing until shown the ledger entry.}
\end{worked}

\section{A field guide to hollow certificates}

Compiling proofs can still be worthless. The four classic ways, in
increasing order of subtlety:

\textbf{1. The \lean{sorry}.} Lean's placeholder accepts any goal with a
warning. Fine for work in progress; fatal in a certificate, because
downstream theorems inherit the hole silently. Detection: the compiler
warns, and \lean{\#print axioms} shows \lean{sorryAx}. Trivial to catch ---
if you look.

\textbf{2. The smuggled axiom.} Stuck on a lemma? \lean{axiom} makes it
true. Sometimes legitimate (see the trusted-base section below); rotten when
undisclosed --- a proof ``of'' the group law that axiomatizes the hard
half of the group law is theater. Detection: \lean{\#print axioms}, always.

\textbf{3. The trivial specification.} The most instructive one. Consider:

\begin{lstlisting}[language=Lean]
theorem mul_correct : ∀ a b, ∃ c, mul a b = c := by
  intro a b; exact ⟨_, rfl⟩   -- checks! and says NOTHING
\end{lstlisting}

Kernel-approved, axiom-clean, and utterly empty: ``mul returns whatever it
returns.'' No tool catches this, because nothing is wrong \emph{formally}
--- the defect is that the statement doesn't say what the reader assumes it
says. The only detector is a human reading the \emph{statement} (never mind
the proof) and asking: \emph{if the code were wrong, would this theorem
fail?} For the trivial spec, the answer is no --- a buggy \code{mul}
satisfies it identically.

\begin{worked}{the substitution test --- executing ``would a wrong program pass?''}
The italicized question deserves a mechanical procedure, because you will
ask it of every certificate you ever audit. The procedure: \emph{invent
the worst program of the right type, substitute it into the statement,
and see whether the statement notices.} Run it on three candidate specs
for field multiplication, with the adversarial implementation
$\code{mul}_0 :=$ ``ignore the inputs, return the zero array.''

\emph{Spec A (the trivial one):}
$\forall a\, b,\ \exists c,\ \code{mul}\ a\ b = c$. Substitute
$\code{mul}_0$: is there a $c$ with $\code{mul}_0\,a\,b = c$? Yes ---
$c = (0,0,0,0,0)$, for every input. \textbf{Spec A holds for the
adversary.} Verdict: empty.

\emph{Spec B (existence-of-ok):}
$\forall a\, b,\ \mathrm{Bnd}(a) \to \mathrm{Bnd}(b) \to
\exists c,\ \code{mul}\ a\ b = \mathtt{.ok}\ c \wedge \mathrm{Bnd}(c)$.
Substitute: $\code{mul}_0$ always returns \lean{.ok} of the zero array,
and the zero array is certainly bounded. \textbf{Spec B holds for the
adversary too.} Verdict: overflow-safety only --- honest as far as it
goes, but a reader who took it for correctness was reading the title,
not the theorem.

\emph{Spec C (the commuting square):} add the clause
$\denote{c} = \denote{a} \cdot \denote{b}$. Substitute and pick the
one-line counterexample: $a = b = (1,0,0,0,0)$, so
$\denote{a} \cdot \denote{b} = 1 \cdot 1 = 1$, while
$\denote{\code{mul}_0\,a\,b} = \denote{(0,\dots,0)} = 0 \neq 1$ in
$\Fp$. \textbf{Spec C refutes the adversary.} Verdict: this one has
teeth.

The test took four lines of substitution per spec and required no Lean at
all --- it is a \emph{reading} skill. Note also what it is \emph{not}: it
cannot certify a spec as strong (a cleverer adversary might slip through
where $\code{mul}_0$ was caught); it is a smoke detector, not a proof.
But every hollow certificate in the field-guide above fails it within a
minute, which is a remarkable return on four lines of pen and paper.
\end{worked}

\textbf{4. The wrong model.} The proof is real, the spec is strong --- but
the thing verified isn't the thing that ships: a hand-transcription
(Chapter~\ref{ch:rust}), a stale extraction, a simplified semantics.
Detection: trace the chain from artifact to source --- extraction scripts,
pinned tool versions, regeneration instructions. If the chain can't be
replayed, the claim is about an orphan.

\begin{pitfall}
Rank these by danger and notice the inversion: the crude failures
(\lean{sorry}, smuggled axioms) are machine-detectable in seconds, while
the subtle ones (trivial specs, wrong models) defeat every automated check
and yield only to a thoughtful reader. Verification does not eliminate the
need for human judgment; it \emph{concentrates} all of it into two small,
well-lit places --- the statement and the model. That concentration is the
gift; squandering it by not reading the statement is the sin.
\end{pitfall}

\section{Honest boundaries: the trusted base}

Real projects meet real limits: SHA-512's compression function, SIMD
backends no translator models, foreign function calls. The honest move is
not to pretend, but to \emph{declare}: state the unverified piece as an
explicit assumption, document it in a ledger (the companion repos call
theirs \code{TRUSTED-BASE.md}), and let \lean{\#print axioms} carry the
disclosure to every downstream theorem automatically.

\begin{lstlisting}[language=Lean]
-- Declared, documented, and visible in every audit forever:
axiom sha512_spec : ∀ msg, Sha512.hash msg = SHA512_ideal msg
\end{lstlisting}

A signature-layer certificate honestly reads: \emph{EdDSA verification is
correct, GIVEN the declared oracles} --- with every \emph{given}
machine-visible. The companion repos push the posture one step further
than this example: their SHA-512 is opaque with \emph{no properties
assumed at all} --- not even ``behaves ideally''; the apex theorems hold
for whatever bytes it produces --- and the backend question was
\emph{eliminated} rather than assumed (extraction pins the serial path,
which is real translated code, proven like everything below it). Compare the two
postures: ``everything verified!'' (and hope nobody checks) versus ``these
two assumptions, this ledger, audit me'' --- the second is both humbler and
\emph{stronger}, because its claim survives the audit.

The companion projects add one more layer of candor worth copying: a
\emph{failure map}. Their control repository documents the dead ends ---
tactic patterns that exhaust memory, extraction scopes that drag in the
world, proof styles that don't scale --- each with the tell that identifies
it early. Knowledge of where the cliffs are is part of the method, and
pretending the cliffs don't exist is how the next person walks off one.

\begin{aha}
Notice the running theme: at every level, the methodology converts
\emph{invisible} trust into \emph{visible} trust. Extraction made the
code-to-model step visible; two-clause specs made operating envelopes
visible; \lean{\#print axioms} makes logical debts visible; the trusted-base
ledger makes engineering limits visible. Formal verification's deepest
product is not certainty --- it is \textbf{legibility of exactly what
remains uncertain}.
\end{aha}

\begin{tryit}
Audit the real thing. In \code{dalek-ed25519-verified}, run
\lean{\#print axioms} on \code{fieldImplementation} and
\code{edwardsImplementation} --- confirm the clean trio. Then read
\code{TRUSTED-BASE.md} and match each entry to where it would surface in an
audit. Finally, write a deliberately trivial spec for \code{add}, prove it
in one line, and observe that every automated check passes. Keep that file
open for one full minute. That minute is the chapter.
\end{tryit}

\section*{Exercises}

\exercise{For each hollow-certificate species, name its detector: (a)
\lean{sorry}; (b) smuggled axiom; (c) trivial spec; (d) wrong model. Which
two can a CI pipeline catch mechanically, and what CI check would you write
for each?}

\exercise{Strengthen this spec until a buggy implementation would fail it:
\lean{theorem sub_ok : ∀ a b, ∃ c, sub a b = .ok c}. (List what's missing:
bounds hypotheses? bounds propagation? the value equation? Compare with
Chapter~\ref{ch:denotation}'s two-clause shape.)}

\exercise{A vendor's whitepaper says: ``Our signature library is formally
verified in Lean.'' Draft the five questions you would send them, in
priority order, and the answer you would require for each before relying on
the claim. (You now know all five.)}

\exercise{(Discussion) The trusted-base \lean{axiom} for SHA-512 and the
smuggled \lean{axiom} for a hard lemma are the \emph{same language feature}.
Articulate the difference in one sentence --- it is not technical.}

\exercise{(Paper --- the substitution test, on the toy system) For the
mini-arithmetic of Chapter~\ref{ch:denotation}'s exercises, a vendor
ships: $\forall a\, b,\ \mathrm{Bnd}(a) \to \mathrm{Bnd}(b) \to
\denote{\code{add}\,a\,b} + \denote{(0,0)} = \denote{\code{add}\,a\,b}$.
It compiles; it is axiom-clean. Run the substitution test: does the
buggy \lean{add'} of \code{exercises/Ch12.lean} pass this spec? What is
the spec actually about?}

\section*{Solutions and pathways}
\solutionsintro

\solhead{11.1}
\pathway For each species, ask \emph{where the defect physically lives}
--- in a warning, in a dependency list, in the meaning of a statement, in
the relation between artifact and world --- and match it to the tool that
inspects that place.
\answer (a) \lean{sorry}: detector is the compiler warning and
\lean{sorryAx} in \lean{\#print axioms}; CI check: grep build output for
\lean{uses 'sorry'} and fail (the companion repos' \code{check.sh} does
exactly this --- warnings, unlike comments, cannot be spoofed by prose).
(b) Smuggled axiom: detector is \lean{\#print axioms} on every shipped
certificate; CI check: assert the output is exactly
\lean{[propext, Classical.choice, Quot.sound]}, plus a source-level gate
refusing \lean{axiom} declarations outside the sanctioned model
directory. (c) Trivial spec: \emph{no mechanical detector} --- the
substitution test, executed by a human reading the statement. (d) Wrong
model: no mechanical detector either --- the replay test: can a stranger
re-extract from pinned sources and reproduce the artifact? CI can
\emph{support} it (pin versions, re-run extraction, diff), but the
judgment ``this model is the shipping code'' remains human. The pattern
of the answers: mechanical failures get mechanical detectors; semantic
failures get disciplined humans. Budget your skepticism accordingly.

\solhead{11.2}
\pathway Apply the substitution test to your own strengthening: after
each added clause, ask which adversarial implementation still passes.
Stop when the only survivor is a correct one.
\answer Stepwise. Start:
$\exists c,\ \code{sub}\ a\ b = \mathtt{.ok}\ c$ --- passes for
``return zero always.'' Add bounds hypotheses and propagation
($\mathrm{Bnd54}$ on inputs, $\mathrm{Bnd56}(c)$): still passes for
return-zero. Add the value clause
$\denote{c} = \denote{a} - \denote{b}$: return-zero dies
($a = (1,0,\dots), b = 0$ gives $1 \neq 0$), and so does every other
wrong-value implementation, by definition. Final spec --- the full
two-clause square:
\[
\mathrm{Bnd54}(a) \to \mathrm{Bnd54}(b) \to \exists c,\;
\code{sub}\,a\,b = \mathtt{.ok}\,c \,\wedge\, \mathrm{Bnd56}(c)
\,\wedge\, \denote{c} = \denote{a} - \denote{b}.
\]
Each clause pulls real weight: \lean{.ok} excludes panics/overflow, the
bound feeds the next operation, the value equation carries correctness.
Delete any one and re-run the substitution test to see what sneaks back
in --- that exercise-within-the-exercise is the fastest way to internalize
why all three clauses recur through the entire companion codebase.

\solhead{11.3}
\pathway Order the questions by how cheaply they kill a hollow claim:
axioms first (one command), then sorries, then statement, then model,
then reproducibility.
\answer The five, in priority order, each with its required answer:
(1) \emph{``What does \lean{\#print axioms} report on the headline
theorems?''} --- exactly the standard trio, or each extra axiom
documented in a trusted-base ledger. (2) \emph{``Does any shipped file
contain \lean{sorry}, and does CI fail on the warning?''} --- no, and
yes. (3) \emph{``Show me the exact statement of the main theorem''} ---
it must contain a universally quantified value equation against an
independent mathematical definition (substitution test on the spot).
(4) \emph{``What artifact was verified, and how does it relate to the
code you ship?''} --- a model extracted mechanically from the shipping
sources, tool versions pinned. (5) \emph{``Can I reproduce the check?''}
--- a one-command script re-extracts, re-compiles, re-audits. A vendor
who answers all five without flinching is, empirically, not hiding
anything --- and a vendor who stumbles on (3) has told you which chapter
of this book to reread on the flight home.

\solhead{11.4}
\pathway Strip away everything technical --- both are \lean{axiom} ---
and what remains is a relationship between author and reader.
\answer One sentence: \emph{the trusted-base axiom is disclosed as a
debt --- named, documented, and priced for the reader's own judgment ---
while the smuggled axiom is disclosure's opposite: a debt hidden so the
certificate can impersonate a stronger claim.} Same instruction to the
kernel; opposite speech act to the human. (Which is why the audit
command matters so much: it makes the speech act verifiable.)

\solhead{11.5}
\pathway Substitution test, by the book: does the statement mention the
program's \emph{relationship to anything external}? Look closely at what
appears on both sides of the equation.
\answer \lean{add'} passes --- and so does every function of the correct
type, including ``return $(3,1)$ always.'' The spec has the form
$X + \denote{(0,0)} = X$ where $X := \denote{\code{add}\,a\,b}$ appears
identically on both sides; since $\denote{(0,0)} = 0$, it reduces to
$X + 0 = X$ --- a true fact \emph{about the field}, in which \code{add}
occurs only as an inert label. The spec is about $\Zmod{15}$'s additive
identity, dressed in the vendor's function name. This species ---
\emph{true theorem, wrong subject} --- is the subtlest hollow
certificate: it even survives a careless substitution test if you only
check that the statement mentions \code{add}. The refined test: does the
implementation appear on \emph{one side only} of an equation whose other
side is independent of it? If it cancels, it was never being tested.

\begin{checkpoint}
You should now be able to: run and interpret the one-command audit; name
the standard three axioms and greet anything else with suspicion; explain
why trivial specs and wrong models defeat automation and what defeats
\emph{them}; and argue --- with conviction --- why a declared trusted base
is stronger, not weaker, than a claim of totality.
\end{checkpoint}
