A system had grown two ways of doing what looked like one thing. The obvious move was to tidy them into one. The tidying turns out to be impossible, and the reason it is impossible is the reason the two ways exist.

The argument that settles it needs no specialist knowledge and fits in a sentence. One of the two cases has two things to report and only one slot to report them in. Whichever thing the slot is given, the other is lost. The other case has only one thing to report, so the single slot is exactly enough. That is a counting argument, it is decided before any code is written, and it is not the argument an engineer reaches for by default.

The engineer’s instinct is to look at the cases and ask whether they resemble one another. They did. Every one of the nine measured occurrences had exactly the shape that invited the tidy-up, and the measurement encouraged precisely the wrong conclusion. The resemblance was real and it was irrelevant, because the defect was never in the instances. It was in the interface they would have had to share.

This article is about that distinction, which is between evidence about members of a class and evidence about the channel the class must pass through. The second dominates the first and is cheaper to check.

The article reports the measurement, the way the measurement pointed the wrong direction, and the argument that settled it. It also reports a rule this author shipped one increment earlier which turns out to be stricter than the property it enforces, excluding ten of twenty-four cases for no reason. No test found that. It surfaced while gathering data for this article.

How to read this

The general argument is in the opening, in the section called The Argument That Settled It, and in Pattern Extraction. Those three need nothing but attention. The sections between them work the argument through a real case with real numbers, and they use the vocabulary of the trade. Every term is glossed at first use, but a reader who wants the result and not the machinery can take those three and stop.

The mathematics is optional throughout. It appears because the central result is a counting argument, and counting arguments are the one place where a line of notation settles what a paragraph of prose can only assert. Wherever notation appears, the sentence before it says the same thing in words.

What this is a case study of

The setting is compiler design, and specifically the part of a compiler called the backend, which is the stage that turns a program the compiler has already understood into instructions a processor can run. The project is Keleusma, whose backend is described in the previous article in this series. No compiler background is required. The shape of the problem is a general one. A system offers one abstraction whose instances divide into two classes with genuinely different observable behaviour, and an engineer, seeing two implementations, tries to collapse them.

The question this article treats is when that instinct is right and when it is a category error. The answer offered is a test, and it is this. Count the observable events each class produces, and count the channels the proposed unification provides. If the counts differ, the unification is lossy and no amount of shape analysis on individual instances will rescue it, because the loss lives in the interface and not in the instances.

The instinct to unify is not a bad one. Two ways of doing one thing is a real cost, paid by every consumer who must learn both and by every future change that must be made twice. This article does not argue against tidiness. It argues that tidiness is a claim about the world, and like any other claim it can be checked before it is acted on, cheaply, and that the check is not the one an engineer reaches for by default.

The Setting

A coroutine is a computation that can suspend in the middle, hand a value to whoever called it, and later be resumed with a value handed back. The two-flavour split this article is about is not peculiar to Keleusma, and the language specifications record it plainly. Python began with generators that only produce values in Python Enhancement Proposal 255 , or PEP 255, and later admitted values back in at the suspension point in PEP 342, which is exactly the step that turns one observable event into two, then separated the coroutine from the generator entirely in PEP 492. Lua exposes asymmetric coroutines in which resume and yield are distinct operations and a coroutine additionally returns when its body completes, so that a caller must distinguish a yield from a return. The European Computer Manufacturers Association standard 262, or ECMA-262, which specifies JavaScript, gives its generator results an explicit done flag for the same reason, C# iterators make the distinction syntactic, and C++ separates co_yield from co_return at the level of the grammar. Every one of these designs carries a discriminator, and that is the fact this article rediscovers from the compiler side.

Keleusma has two forms.

A yield function suspends a bounded number of times and then finishes. It is a total function with interruptions. The type system guarantees it terminates.

A loop function suspends forever. It is productively divergent, which is the property that it never finishes but always makes progress, so a host, meaning whatever program is driving the coroutine from outside, can pull values from it indefinitely and each pull terminates. This is the language’s stream abstraction and it is the normal way a Keleusma program does work over time.

Both are written with the same yield keyword in the body. The difference is the declaration, and the type checker already separates them into different categories with different rules about what may call what.

Notation

The formal machinery below is worth the six symbols it costs, because the central result is a cardinality argument that prose states as an opinion and arithmetic settles in one line. A reader who prefers prose can take the following and skip to the results. A terminating coroutine emits two values per call and a return slot holds one.

Let $W$ denote the machine word type, so that $\lvert W \rvert = 2^{64}$ on the present target. Let $K$ denote the set of coroutine chunks, a chunk being one compiled unit of code, partitioned by the type checker into

\[K = K_\Sigma \;\sqcup\; K_\Upsilon \tag{1}\]

where $K_\Sigma$ are the divergent stream chunks and $K_\Upsilon$ the terminating ones. Write $\lvert K_\Sigma \rvert = 24$ and $\lvert K_\Upsilon \rvert = 1$ for the present corpus.

An execution of a chunk is driven by an initial argument $a \in W$ and a sequence of resume values $\vec{r} = (r_1, r_2, \ldots)$. It produces an **observable trace**, a sequence over the alphabet ${\mathsf{Y}(v), \mathsf{F}(w)}$ in which $\mathsf{Y}$ marks a suspension carrying $v$ and $\mathsf{F}$ a completion carrying $w$. For a terminating chunk with $n$ suspensions the trace is finite and ends in a completion,

\[\mathcal{T}_\Upsilon(k, a, \vec{r}) \;=\; \bigl\langle\, \mathsf{Y}(v_1),\, \mathsf{Y}(v_2),\, \ldots,\, \mathsf{Y}(v_n),\, \mathsf{F}(w) \,\bigr\rangle \tag{2}\]

while for a divergent chunk it is infinite and contains no completion at all,

\[\mathcal{T}_\Sigma(k, a, \vec{r}) \;=\; \bigl\langle\, \mathsf{Y}(v_0),\, \mathsf{Y}(v_1),\, \mathsf{Y}(v_2),\, \ldots \,\bigr\rangle. \tag{3}\]

That asymmetry is the entire article. The two trace shapes differ not in length but in alphabet. One uses both letters and the other uses one.

A calling convention is an encoding $\varepsilon$ that carries a trace into what a host can observe from native calls. Write $\mathrm{chan}(\varepsilon)$ for the number of distinguishable $W$-valued channels a single native call exposes under $\varepsilon$, and $\mathrm{obs}(k)$ for the number of distinct observable values a single call of $k$ must convey.

The two conventions

The generator emits native code that a host program calls. When the generated code suspends, the value has to reach the host. Two mechanisms are available.

The callback convention declares an external function, here called kel_yield, that takes the yielded value and returns the resume value. Generated code calls it at each suspension point and carries on. The host implements it. Control passes to the host and back without the generated function returning, so the generated function’s stack frame stays live throughout, a stack frame being the scratch space a function reserves for its own working values and gives back when it returns. This is the stackful shape, of which the context-switching functions of the Portable Operating System Interface or POSIX are the oldest standardised instance, and its defining property is exactly that a frame outlives the suspension.

The return convention ends the native call at the suspension. The yielded value is the function’s return value. The host receives it as an ordinary result, does whatever it does, and calls the function again with the resume value as an argument. No frame persists between suspensions.

The return convention is strictly cheaper. It needs no external symbol, no callback indirection, and crucially no stack frame alive during host code, which matters to a project whose entire premise is statically bounded memory. A frame held across an arbitrary amount of host activity is a frame the memory analysis cannot account for.

The compiler infrastructure this backend targets already names both shapes. The coroutine intrinsics of the Low Level Virtual Machine compiler infrastructure or LLVM provide a switch-resume family, in which a coroutine handle is resumed through a function pointer and the frame persists in an explicit allocation, and a returned-continuation family, llvm.coro.id.retcon, in which the coroutine returns at each suspension and hands back a continuation to be called next. The two families correspond closely to the callback and return conventions described here. Frame liveness across a suspension is the same property that LLVM stack maps exist to describe, and the WebAssembly stack-switching proposal is the same design question posed for a portable target. None of these settle the choice, and that is worth saying, because they establish that both shapes are standard and that the trade is real, not that either is correct here.

How the backend arrived at two

The yield function was implemented first, on the callback convention, because it is the general mechanism and it was obviously correct.

The loop function was implemented later. Its bytecode, meaning the compact intermediate instructions the compiler produces before machine code, has a recognisable shape, being an opening marker, a body, a stack cleanup, and a reset instruction that rewinds execution to just after the marker and hands control to the host. The reset clears every local variable. That clearing is the key structural fact. If the body’s suspension is the last thing on every path, then no local outlives the suspension, because the reset would have cleared it anyway. Nothing needs to persist, so nothing needs a frame, so the return convention applies.

Two conventions resulted, not by design but by two increments arriving from different directions. That is exactly the situation in which an engineer reaches for a unification.

The Measurement That Pointed The Wrong Way

Before proposing anything, the corpus was measured. The instrument walks every compiled program in the project’s shipped examples and self-hosted compiler sources, classifies each coroutine chunk, and reports the distribution.

The distribution of stream chunks

Class Count
Single suspension, at the end of the body 22
Multiple suspensions at the top level 0
Suspensions nested inside conditionals 1
No suspension of its own, delegating to a callee 1
Total stream chunks 24

The zero is worth pausing on. An earlier design in this project assumed the general case was a body with several suspension points that would need to be reordered relative to one another. No such body exists in the corpus. That design was specified in some detail before the count was taken, and the count deleted it.

The nested case looked like the hard one until it was measured per suspension instead of per chunk. All nineteen of its suspensions sit inside conditionals and not one sits inside a loop. A suspension inside a conditional is a control-flow join, where every path still suspends once and ends. A suspension inside a loop crosses a back edge and genuinely needs a frame. The word nested had been covering both, and they have entirely different costs.

The distribution of terminating coroutines

Property Count
yield function chunks in the whole corpus 1
… whose every suspension is immediately followed by a return 1

One. The entire terminating-coroutine population of the corpus is a single chunk, and that chunk has, in nine of nine cases, precisely the shape that appeared to invite the return convention, which is to suspend and then return.

This is where the measurement pointed the wrong way. A distribution of nine out of nine looks like an invitation. It reads as evidence that the shape is natural, that the callback is an accident of implementation order, and that a unification is available at the cost of a shape check. The author of this article drafted exactly that plan.

The Argument That Settled It

The plan was checked against the smallest possible instance before it was implemented, which is the only thing in this article that went right on the first attempt.

Consider a terminating coroutine that suspends once with its argument.

yield main(a: Word) -> Word { yield a }

It compiles to three instructions, which load the argument, suspend, and return. The suspension instruction pops the value to yield and pushes the value the host resumes with, so the return instruction returns the resume value.

Run it on the reference virtual machine and it produces two observable events.

  1. Suspension, carrying a, the yielded value. 2. Completion, carrying r, the resume value.

A return-based lowering, meaning a translation from the intermediate form down to machine code, has one return slot. Give the slot the yielded value and the completion is gone. Give it the completion and the suspension is gone. There is no third option, and no analysis of the chunk’s shape changes this, because the deficiency is in the calling convention and not in the chunk.

The obvious objection is that the completion could arrive on a second call, and the reason it cannot is the part of the return convention that is easiest to skip over. The convention has no resumption point. The host drives the chunk by calling a plain function again, passing the resume value as the argument, so the sequence the host executes is

\[f(a) = v_0, \qquad f(r_1) = v_1, \qquad f(r_2) = v_2, \qquad \ldots \tag{4}\]

and $f$ carries nothing between calls. For a divergent chunk that is exactly right, because the reset instruction rewinds to the same place and clears every local, so re-entering from the top is what resumption means there. For a terminating chunk it is wrong. A second call re-runs the body from the beginning and suspends again, so it yields a second time and never reaches the completion at all.

Everything the host will ever learn about one terminating execution therefore has to arrive from the first call, which is why the budget is one slot rather than one slot per event.

The counting argument, stated once

Under the return convention a single native call exposes exactly the return register, which is one of the processor’s few named storage slots, so $\mathrm{chan}(\varepsilon_{\mathrm{ret}}) = 1$. The chunk above must convey both $v$ and $w$ from one call, so $\mathrm{obs}(k) = 2$. An encoding faithful to the trace must therefore be an injection

\[\varepsilon_{\mathrm{ret}} : W \times W \;\longrightarrow\; W \tag{5}\]

and no such injection exists whenever $\lvert W \rvert \ge 2$, by cardinality,

\[\lvert W \times W \rvert \;=\; \lvert W \rvert^{2} \;=\; 2^{128} \;>\; 2^{64} \;=\; \lvert W \rvert. \tag{6}\]

The same statement in bits is the one that generalises, and it is the elementary case of the channel-capacity argument Shannon 1948 introduced. An interface of $m$ machine words carries $64m$ bits and a trace of $k$ words needs $64k$, so a faithful encoding requires

\[64k \;\le\; 64m \qquad \Longleftrightarrow \qquad k \;\le\; m, \tag{7}\]

and here $k = 2$ against $m = 1$. The shortfall is 64 bits, which is to say a whole second word.

The general statement is immediate. For any chunk $k$ and convention $\varepsilon$,

\[\mathrm{obs}(k) \;>\; \mathrm{chan}(\varepsilon) \quad\Longrightarrow\quad \varepsilon \text{ is not injective on } \mathcal{T}(k), \tag{8}\]

and a non-injective encoding is one in which two distinct executions become indistinguishable to the host, which is precisely what it means for a lowering to be wrong.

The deficit does not grow with the number of suspensions, which deserves a check and not an assumption. A terminating chunk with $n$ suspensions emits $n$ yielded values and one completion, so its trace carries $n+1$ words and the executions it can distinguish number $\lvert W \rvert^{\,n+1}$. The single call it is granted distinguishes $\lvert W \rvert$ of them, so the collapse factor is

\[\frac{\lvert W \rvert^{\,n+1}}{\lvert W \rvert} \;=\; \lvert W \rvert^{\,n}, \tag{9}\]

which is catastrophic at every $n$ and no worse in kind at $n = 9$ than at $n = 1$. The smallest instance was therefore the right one to check, and checking a larger one would have added nothing.

The pigeonhole is doing all of the work, and it is worth noticing how little it needs. It does not inspect the body, the number of suspensions, the control flow, or the corpus. It needs the alphabet of the trace and the width of the interface, both of which are known before any code is written.

The claim is false in its strong form, and the reference work is what showed it

A return slot is not one word. The System V AMD64 application binary interface , or ABI, which is the document that fixes how functions pass arguments and return results on a given machine, returns a two-eightbyte aggregate in the register pair RAX:RDX, and AAPCS64 returns small aggregates in X0 and X1. So $\mathrm{chan}(\varepsilon_{\mathrm{ret}})$ is a property of the chosen signature rather than a constant of the machine, and the honest statement is conditional.

\[\mathrm{chan}(\varepsilon_{\mathrm{ret}}) = 1 \;\text{ if the return type is } W, \qquad \mathrm{chan}(\varepsilon_{\mathrm{ret}}) = 2 \;\text{ if it is } W \times W. \tag{10}\]

Under a two-word return the injection $\varepsilon : W \times W \to W \times W$ exists trivially, and the unification this article rejects becomes representable.

Two designs are being run together here and they cost very different amounts, so they are worth separating. Cramming both events into one call needs the pair $W \times W$, which is 128 bits against 64, a shortfall of a whole word. Making the convention re-entrant with a discriminator needs only

\[\lvert \{\mathsf{Y}, \mathsf{F}\} \times W \rvert \;=\; 2 \cdot 2^{64} \;=\; 2^{65}, \tag{11}\]

a shortfall of exactly one bit. That single bit is what forces a second register, since $\lceil 65 / 64 \rceil = 2$, and it is the whole reason the register-pair rules in the two application binary interfaces matter to this decision. The expensive-sounding option costs one bit of information and one register of encoding, a very different proposition from carrying a second word. The pigeonhole refutes the unification at the signature the backend currently emits, not for all time.

That distinction matters and it was not visible from inside the problem. It surfaces a third option , which is to widen every coroutine entry point so that it returns a discriminated pair,

\[f : W \longrightarrow (\,\mathsf{tag} \in \{\mathsf{Y}, \mathsf{F}\},\; W\,) \tag{12}\]

so that one convention carries both trace alphabets.

That form is sufficient only for a coroutine that suspends at most once per call, and the article as published did not say so. A coroutine resumes at the point it suspended, which is the defining property of the construct. Under the callback convention that is free, because the native function never returned and its frame, program counter and locals are all still live. Under a return convention the call has ended and nothing survives. For the divergent form that costs nothing, since the reset instruction clears every local and the next iteration legitimately starts from the top. For the terminating form it is fatal, because the next call must resume mid-body and a function entered at its entry point cannot do that.

A widened return must therefore carry, beyond the value and the tag, something identifying where to resume and with what state. That is a continuation, and it is what llvm.coro.id.retcon returns alongside the yielded value. The returned-continuation family exists because returning a value at a suspension is not sufficient on its own. The general interface is a triple,

\[f : W \longrightarrow (\,\mathsf{tag} \in \{\mathsf{Y}, \mathsf{F}\},\; W,\; \mathsf{cont}\,) \tag{13}\]

and the continuation implies an allocation for the saved frame whose lifetime spans host execution, which is the very property this option was credited with preserving.

The cost is therefore that every call of every coroutine pays a wider return and a host-side discrimination, including the 95.83 percent of the corpus, twenty-three chunks of twenty-four, that need only one letter, and that the terminating form pays a frame as well. The 95.83 percent is a count of chunks needing one letter and must not be read as the fraction for which a pair suffices, because that is a claim about the terminating form and the corpus holds one instance of it.

Whether that is cheaper than two conventions is an empirical question this article does not answer, and the shape of the comparison can be stated without measuring anything. Writing $c_{\mathrm{tag}}$ for the per-call cost of the discriminator, $c_{\mathrm{frame}}$ for the amortised cost of allocating and reclaiming the saved frame, and $c_{\mathrm{meta}}$ for the one-off cost of declaring and dispatching two conventions, the totals over $N$ calls are

\[C_{\mathrm{one}} = (c_{\mathrm{tag}} + c_{\mathrm{frame}}) N, \qquad C_{\mathrm{two}} = c_{\mathrm{meta}}, \tag{14}\]

where $c_{\mathrm{frame}} = 0$ only for coroutines that suspend at most once per call, so the widened return is cheaper only below

\[N^{*} \;=\; \frac{c_{\mathrm{meta}}}{c_{\mathrm{tag}} + c_{\mathrm{frame}}}. \tag{15}\]

The frame term moves the crossover down, which strengthens rather than weakens the observation that follows.

A per-call term always loses to a constant eventually, and a stream abstraction exists to be called many times, so the crossover is the whole question. Neither constant is published and this article declines to invent them, which is why the recommendation below rests on the memory property and not on a cost model.

Two production language runtimes already do exactly this, which is the strongest argument available that the option is practical rather than merely representable. A Kotlin suspending function returns its ordinary result, or a distinguished COROUTINE_SUSPENDED sentinel to signal that it suspended instead. That is a discriminated return in the narrowest possible encoding, a single reserved value carved out of the result type. A Rust future’s poll returns Poll::Ready(T) or Poll::Pending, which is the same discrimination made explicit in the type rather than hidden in a sentinel. Both carry the completion and the suspension through one interface, and both pay the tag on every call.

The sentinel form has a hazard the tagged form does not, and it is worth naming because the narrow encoding is the tempting one here. Carving a reserved value out of $W$ shrinks the value space to

\[\lvert W \rvert - 1 \;=\; 2^{64} - 1 \;=\; 18{,}446{,}744{,}073{,}709{,}551{,}615, \tag{16}\]

which is 18,446,744,073,709,551,615 usable values, a loss of $2^{-64}$ or about $5.421 \times 10^{-20}$ of the space, and any program able to produce the reserved value as a legitimate yield becomes unrepresentable. The loss is negligible in measure and total in reachability, and only the second of those matters. A sentinel is available exactly when the yielded type is a proper subset of $W$ missing at least one element, a statement about the type and not about how unlikely the value is. Kotlin can afford it because the sentinel is a reference no user value aliases. A backend whose yielded values are arbitrary machine words cannot, so for Keleusma the tagged pair is the honest form and the sentinel is unavailable.

The correction is left in the text rather than folded into the argument, because the sequence is the useful part. A counting argument was stated in a form stronger than the evidence supported, and reading the governing interface specification rather than assuming the register width is what caught it.

The nine-out-of-nine measurement was answering a different question than the one that mattered. It established that the shape is uniform. It said nothing about the arity of the interface, and the arity is what fails.

Why the divergent case is genuinely different

A loop function never completes. The reset instruction rewinds and returns control to the host, so there is no completion event at all. The count of observable events is one, the count of channels is one, and the convention fits exactly,

\[\mathrm{obs}(k) = 1 = \mathrm{chan}(\varepsilon_{\mathrm{ret}}) \qquad \text{for all } k \in K_\Sigma \text{ admitted below,} \tag{17}\]

so the required encoding is $\varepsilon_{\mathrm{ret}} : W \to W$, for which the identity suffices. The host drives the chunk by iterating the native function, and the correspondence with the virtual machine is

\[f(a) = v_0, \qquad f(r_k) = v_k \;\; (k \ge 1), \tag{18}\]

with no distinguished first call, because iteration zero consumes the call argument and iteration $k$ consumes the $k$-th resume value, exactly as the virtual machine does when it writes the resume value into the entry chunk’s parameter slot.

So the two conventions are not an accident awaiting cleanup. They track the boundary between a construct that terminates and one that does not, which is a distinction the language already makes, the type checker already enforces, and the memory analysis already depends on. The backend grew two conventions because there are two things.

The general form of the test

Stated without reference to compilers, the test reads as follows.

Let a proposed unification map a class of behaviours onto an interface. Count the distinguishable > outcomes each behaviour can produce and count the distinguishable outcomes the interface can carry. > If the first exceeds the second for any member of the class, the unification is lossy, and evidence about > the distribution of members cannot rescue it.

The failure mode this catches is common, and it is collecting evidence about instances when the defect lives in the interface. Nine out of nine is a fact about instances. One slot against two events is a fact about the interface. The second dominates the first entirely, and it is cheaper to check.

A Rule Stricter Than The Property It Enforces

Gathering the distribution for this article surfaced a defect in code shipped one increment earlier, which no test had caught because no test asked.

The admissibility rule for the return convention requires that between a suspension and the reset, nothing runs except block delimiters and a single stack-cleanup instruction. The intent is to establish that the resume value is discarded, since the return convention does not supply one.

Ten of the twenty-four stream chunks fail this rule. All ten are the same program shape and their tail instruction sequence is the following.

PopN(1), Const(0), PopN(1)

The source is a body that suspends and then evaluates a trailing constant, so the constant is pushed and immediately discarded. The sequence has no observable effect and leaves the stack balanced. It is exactly as safe as the sequence the rule admits. The rule refuses it because the rule was written by enumerating the instructions the author had seen rather than by stating the property.

Write $\delta(\iota) \in \mathbb{Z}$ for the net operand-stack effect of an instruction and $\mathrm{eff}(\iota)$ for its set of side effects. For a tail segment $\pi$ following a suspension, the property that actually licenses the transformation is

\[\Delta(\pi) \;=\; \sum_{\iota \in \pi} \delta(\iota) \;=\; -1 \qquad \text{and} \qquad \mathrm{eff}(\pi) \;=\; \varnothing, \tag{19}\]

the $-1$ recording that the segment consumes exactly the resume value the suspension pushed and nothing else. Both candidate tails satisfy it.

\[\Delta\bigl(\langle \mathtt{PopN(1)} \rangle\bigr) = -1, \qquad \Delta\bigl(\langle \mathtt{PopN(1)},\, \mathtt{Const(0)},\, \mathtt{PopN(1)} \rangle\bigr) = -1 + 1 - 1 = -1. \tag{20}\]

The shipped rule instead tests membership in a fixed instruction set $A = {\mathtt{PopN(1)}} \cup D$, with $D$ the block delimiters. Let $P$ denote the set of tail segments satisfying the property and $R$ the set the rule admits. The rule is sound but not complete,

\[R \;\subsetneq\; P, \qquad \lvert P \setminus R \rvert \;\ge\; 10, \tag{21}\]

and the containment is what makes the defect a coverage loss rather than a correctness loss. The property is effect-free and stack-neutral. The rule is drawn from a list of two instructions. Those coincide on the corpus the rule was written against and diverge on the next ten cases, the standard failure of a whitelist standing in for a predicate.

A sound but incomplete rule is the normal condition of static analysis and this instance is not the normal reason for it. The framework of Cousot and Cousot 1977 exists because an exact answer is often uncomputable, so an analysis deliberately accepts imprecision in exchange for soundness. The imprecision here was not forced. The property is decidable on straight-line bytecode by summing two integers, so nothing about the problem required an approximation, and the rule is an approximation that nobody chose. That is a worse position than the classical one and an easier one to fix.

The cost is measurable and is not correctness. Nothing is mislowered, because the rule refuses rather than admits. The cost is coverage. Writing $R$ for what the rule admits and $P$ for what the property licenses,

\[\frac{\lvert R \rvert}{\lvert K_\Sigma \rvert} \;=\; \frac{14}{24} \;=\; 58.33\%, \qquad \frac{\lvert P \rvert}{\lvert K_\Sigma \rvert} \;=\; \frac{24}{24} \;=\; 100\%, \tag{22}\]

so 41.67 percent of the corpus is refused the cheap convention for no reason, and ten stream chunks that could use it currently do not. The fix is a small generalisation and it is not attempted here, because this article’s subject is the convention question and a coverage improvement inside it would be a second claim wearing the first one’s clothes.

It is reported because it was found by writing rather than by testing, the second time in this series that assembling a table for publication has surfaced something a green test suite did not.

Method

The instrument compiles every .kel source in the project’s example directories, the self-hosted compiler stage sources, and the standalone compiler subproject, discarding any that fail to compile. For each resulting module it walks every chunk and classifies it.

Suspension nesting is computed with a depth counter over the block-opening instruction set, which was checked against the full block-structured instruction list and not assumed. A missed opener would report a nested suspension as a top-level one and admit a chunk the transformation is wrong for.

The tail-position walk follows jumps rather than scanning the instruction stream linearly. This is necessary, not fastidious. The instructions textually between a deeply nested suspension and the reset include the bodies of sibling branches, which are on different paths and never execute after that suspension. A linear scan answers a question nobody asked.

Suspension nesting is summarised by the enclosing block stack. Writing $B(y)$ for the stack of open blocks at suspension $y$, the distinction that matters is

\[\mathrm{join}(y) \iff \mathtt{Loop} \notin B(y), \tag{23}\]

a suspension that is a control-flow join rather than one crossing a back edge. For the nested chunk the measurement is $\mathrm{join}(y)$ for all $19$ of its suspensions, at depths $\lvert B(y) \rvert$ ranging over $2$ to $11$.

Equivalence between the two lowerings is checked by a differential oracle against the reference virtual machine, comparing whole sequences of yielded values rather than final results. Writing $\mathcal{Y}(\cdot)$ for the projection of a trace onto its suspension letters, the assertion is

\[\mathcal{Y}\bigl(\mathcal{T}^{\mathrm{vm}}(k, a, \vec{r})\bigr)_{1:n} \;=\; \mathcal{Y}\bigl(\mathcal{T}^{\mathrm{nat}}(k, a, \vec{r})\bigr)_{1:n} \tag{24}\]

for a caller-supplied bound $n$, over a finite sample of $\vec{r}$. A stream never produces a final result, so a comparison of final results would be vacuous for exactly the class under study, and the truncation at $n$ is not a weakness of the test but a consequence of the trace being infinite.

The sample is finite and the claim is universally quantified over $\vec{r}$, so the oracle refutes and does not prove. The resume values in each case are chosen to differ from one another and from the argument, and in the nested case to drive different branches on successive iterations, because a $\vec{r}$ that takes one path every time exercises one return site and licenses nothing about the join.

How the literature survey was assembled

The 35 hand-selected research references were chosen because a step of the argument depends on them, and each was read. The 1,944 harvested references were not chosen that way and were not read individually. Stating that plainly is the point of this subsection, because a list of four thousand citations otherwise implies a reading it does not represent.

The harvest issued keyword queries against Crossref and the NASA technical report server across thirteen topic clusters, being coroutines, continuations, effect handlers, calling conventions, code generation, verified compilation, compiler testing, resource analysis, static analysis, concurrency runtimes, bytecode virtual machines, types and semantics, and a general systems cluster. Each returned record was kept only if its title carried a subject anchor, meaning a term specific to computing and not one shared with every discipline. That test discarded 5,313 of the 7,334 retrieved records. Records already cited by hand were discarded again at assembly so that no work appears twice under two anchors.

The count that survives is smaller again. The arithmetic is worth stating and not leaving to be noticed. Of the records the filter admitted, 85 were duplicates holding the same title and year under two identifiers, 2 carried a doubled word in the title where a registry had concatenated a title with a book title, 1 was untitled or undated, and 4 duplicated a hand-selected entry. 1,945 therefore reach the reference list. One was removed by a later audit as an identifier that does not resolve, leaving 1,944.

The anchor filter is the step most likely to be wrong, and it was wrong twice before it was right.

The first version was inherited from an article on a different subject and tested for the presence of that subject’s vocabulary, so it rejected 2,174 compiler-science titles for containing no aircraft. That failure is loud once looked at and silent otherwise, because a filter that rejects everything reports a small corpus rather than an error.

The second version was written for this subject and overcorrected into uselessness. It admitted generic stems, being analysis, implementation, generation, evaluation, system, model, performance and interface, each of which occurs in every discipline that publishes. It admitted 4,305 records, and a sample of them contained rabies control, seismic depth imaging, breeding soundness examination in veterinary medicine, supercontinuum generation in photonics, transport appraisal and fibre art. A survey listing those has surveyed nothing. The larger count was the symptom rather than the reassurance.

The version used here requires a term that is computing-specific on its own, so an ambiguous term contributes nothing however many of them a title carries. It admits 2,021 records of the 7,334 retrieved. The corpus is less than half the size of the one the permissive filter produced. That reduction is the result rather than a cost of it.

What the harvested list therefore supports is a claim about coverage, being that the survey did not select for agreement with its conclusion, since the queries were fixed before the records were seen. What it does not support is any claim about the content of an individual harvested record beyond its title, authors, year and venue as the registry holds them.

Threats to Validity

The corpus is one project’s own sources. It over-represents text processing, because the self-hosted compiler is a text processor. A terminating-coroutine population of one is not evidence about the language. It is evidence about this corpus at this date. A user may write many, in shapes not seen here.

The conclusion does not depend on the corpus. This is the important asymmetry and it is why the argument is stated the way it is. The event-counting argument is about the calling convention and holds for a population of one or one million. The measurement is reported because it is what pointed the wrong way, not because the conclusion rests on it.

The classification was written by the party proposing the unification. The instrument that produced the nine-out-of-nine figure and the plan that figure encouraged share an author. The mitigation offered is that the figure is accurate and was not the thing that failed. The reasoning built on it failed, which is the weaker position for the author and the stronger one for the reader.

No claim is made about performance. The return convention is argued to be cheaper on structural grounds, meaning no external symbol, no indirection, and no frame across the suspension. None of that has been measured. The machine-code frame sizes that would settle it are emitted only into a section of the Executable and Linkable Format, or ELF, which this development host does not produce, so the measurement is deferred and not done.

Pattern Extraction

The pattern is available wherever an interface is proposed to cover a class of behaviours.

Two implementations of one abstraction may be tracking a real boundary. Before unifying, ask what distinguishes the two implementations, and check whether the source domain distinguishes it too. Here the type system already had two categories, enforced by a rule about which may call which, and the memory analysis already depended on the distinction. The backend was not being untidy. It was reflecting something upstream. The tidying instinct treats a difference as noise, and the check is whether anything else in the system treats it as signal.

Evidence about instances cannot repair a defect in an interface. This is the sharper half, and the reason is a quantifier. Admissibility of a unification is a universal claim over the class,

\[\Phi \;\equiv\; \forall b \in \mathcal{B} : \; \mathrm{obs}(b) \le \mathrm{chan}(\iota), \tag{25}\]

whose negation is existential,

\[\neg\Phi \;\equiv\; \exists b \in \mathcal{B} : \; \mathrm{obs}(b) > \mathrm{chan}(\iota). \tag{26}\]

A single witness refutes $\Phi$, and no quantity of confirming instances establishes it. The nine-out-of-nine figure has the form of a frequency $\hat{p} = 9/9 = 1$ over observed shapes, and a frequency estimates a proportion. It does not bound a maximum, and $\Phi$ is a statement about a maximum,

\[\Phi \iff \max_{b \in \mathcal{B}} \mathrm{obs}(b) \le \mathrm{chan}(\iota). \tag{27}\]

The measurement is weak even taken on its own terms, and it is worth putting a number on how weak. Reading nine successes from nine trials as a sampling exercise, the exact one-sided lower confidence bound of Clopper and Pearson 1934 on the underlying proportion at confidence $1 - \alpha$ is

\[p_{\min} \;=\; \alpha^{1/n} \;=\; 0.05^{1/9} \;=\; 0.7169, \tag{28}\]

so the observation is consistent with 28.31 percent of cases failing, and at 99 percent confidence the bound falls to 0.5995. A run of twenty-four would give 0.8827 and still leave 11.73 percent unexcluded, and licensing a claim of 0.99 at the same confidence would need

\[n \;=\; \frac{\ln \alpha}{\ln 0.99} \;=\; 299 \tag{29}\]

consecutive successes. Nine is not a large number and the arithmetic says so. The case of no observed failures is the one Hanley and Lippman-Hand 1983 treat under the heading of whether nothing going wrong means everything is all right, where they give the same bound in its approximate form as a rule of three. Their subject is clinical and the arithmetic is indifferent to that. That is a separate and much weaker objection than the one this section rests on, because the counting argument does not need the sample to be small. It would refute the unification from a sample of a million.

A uniformity measurement over instances is genuinely useful for deciding what to build first, which is the subject of the previous article in this series, because that question really is about proportions. It is useless here. The two questions feel adjacent, and one of them is answerable by counting slots.

A whitelist standing in for a predicate coincides with it on the sample it was drawn from. The tail-position rule is a small instance of a large pattern. Whenever a condition is written by enumerating the cases the author has seen, it will pass every case the author has seen. The discipline is to state the property and then check that the enumeration matches it, in that order, because the reverse order produces a rule that looks correct and is merely well-sampled.

Assembling a table for an audience is a distinct verification activity. Twice now in this series, a figure gathered for publication has exposed something that a passing test suite did not. The mechanism is not mysterious. A test asks a question its author thought to ask. A table forces a row for every case, including the ones nobody thought about, and an empty or surprising cell is a question the author did not have the imagination to ask directly.

The Decision This Informs

The spike was commissioned to inform a choice, and it narrows rather than makes it.

One option is ruled out on evidence. A single return-based convention for both forms at a one-word return type is lossy for the terminating form, and no shape analysis rescues it.

Three options remain and all are genuinely open.

One convention with a widened return. Every coroutine entry point returns a discriminated pair, which the System V AMD64 and AAPCS64 register-pair rules make free of memory traffic on the two targets that matter. One convention for the host and one discriminator to check. The frame property is preserved only for coroutines that suspend at most once per call, and is surrendered otherwise, because resuming mid-body needs a continuation that outlives the call. This option exists only because the ABI documents were read, and it was absent from the draft that preceded the reference pass. It has not been costed.

Two conventions, declared per entry point. The artefact states which convention each exported function uses, and the host dispatches accordingly. Costs a consumer-visible distinction and a metadata field. Preserves the cheap path for the twenty-three of twenty-four chunks that can use it.

One callback convention everywhere. Uniform for the host. Costs every stream chunk an external call and, more seriously, gives up the property that no frame is live during host execution, which is the property the bounded-memory analysis wants. That is not a small concession in a project whose value proposition is definitive worst-case memory.

The memory difference is the one that can be stated exactly. Let $L(\varepsilon)$ denote the set of native frames live while host code executes between suspensions. Then

\[L(\varepsilon_{\mathrm{ret}}) = \varnothing, \qquad L(\varepsilon_{\mathrm{cb}}) = \{\, \mathrm{frame}(k) \,\} \cup \{\, \mathrm{frame}(c) : c \in \mathrm{callers}(k) \,\}, \tag{30}\]

because the return convention has ended the call by the time the host runs and the callback convention has not. The worst-case native stack bound therefore differs by

\[\Delta M \;=\; \sum_{\phi \in L(\varepsilon_{\mathrm{cb}})} \mathrm{size}(\phi), \tag{31}\]

a quantity that is bounded and computable but charged for the whole duration of host activity, which the analysis does not bound. Under the return convention that term is identically zero.

The host-interaction counts also differ. Writing $s$ for suspensions per iteration, the native call counts per iteration are

\[\tau_{\mathrm{ret}} = 1, \qquad \tau_{\mathrm{cb}} = 1 + s, \tag{32}\]

against a virtual-machine baseline of two host round-trips per iteration, since the reset instruction returns control separately from the suspension. Every stream chunk in the corpus has $s = 1$.

A fourth option this spike did not evaluate, and which is recorded so it is not mistaken for absent, is to make the delegating chunk’s callee terminate. The one delegating stream chunk calls a terminating coroutine, and if that call were restructured so the suspension belonged to the stream chunk instead of the callee, the whole corpus would fit one convention. This is a change to the source program and not the backend, it is outside the backend’s authority, and whether it is reasonable is a language question this article is not equipped to answer.

Delegation is a specified problem elsewhere and the specification is instructive. PEP 380 defines yield from so that a subgenerator’s yields pass through to the outermost caller while its return value becomes the value of the delegating expression. That is precisely the two-channel discipline this article arrives at, imposed at the language level rather than at the interface level. The delegating construct must distinguish the callee’s suspensions from the callee’s completion, because they go to different places. The Keleusma delegating chunk has the same structure and currently resolves it by routing suspensions through the callback while discarding the completion, which works only because the completion is unused in this program.

The recommendation is two conventions declared per entry point, on the grounds that the memory property is the project’s reason for existing and one chunk in twenty-four is a poor reason to give it up. This is a recommendation and not a conclusion, and the option to restructure the source has not been costed.

The Contemporary Literature

The question this article treats sits at the intersection of four literatures that rarely cite one another, being coroutine implementation, continuation-passing transformation, application binary interface design, and worst-case resource analysis. The first three have answered the representation question repeatedly and the fourth is the reason the answer matters here. What follows surveys each and marks where the present problem is and is not covered.

Coroutines were classified before they were implemented well

The construct is old. Conway 1963 introduced coroutines to structure a separable compiler, and the motivating example was a compiler passing tokens between phases, close to the present use. The taxonomy that still governs discussion is de Moura and Ierusalimschy 2009, which separates coroutines along three axes, symmetric or asymmetric, first-class or constrained, and stackful or stackless. Keleusma’s two forms are both asymmetric and constrained, and the article’s question is precisely whether the backend must be stackful.

That taxonomy already contains the article’s finding in a different vocabulary. An asymmetric coroutine returns to its resumer, and the resumer must distinguish a suspension from a completion, which Marlin 1980 treats as a defining obligation of the mechanism rather than an implementation detail. The literature has never regarded the discriminator as optional, and it took a backend author rediscovering it from cardinality to notice.

Where the literature does not reach the present question is this. It treats coroutines as a language feature to be provided, not as an interface exposed across a foreign function boundary to a host in another language. The distinction matters because a language runtime can afford a uniform representation with a tag, while a statically bounded artefact linked into a C or Ada program is charged for every byte of that representation by an analysis that must be sound.

The contemporary work on the construct, harvested rather than recalled and listed in full below, is where the classification stopped being the interesting part and the implementation started being it.

Continuation-passing style answers the representation question, expensively

The general transformation is known and exact. Reynolds 1972 established continuation passing as a definitional technique, Strachey and Wadsworth 1974 gave it its semantics, and Appel 1991 made it a compiler architecture. Under continuation passing every suspension becomes a call to a continuation and the representation question dissolves, because there are no returns to collide with.

It dissolves the question by paying for it everywhere. The transformation is global, it changes the calling convention of every function and not of coroutines alone, and the resulting closures are heap allocated unless a later analysis recovers the stack discipline. Danvy and Nielsen 2001 show defunctionalisation recovering first-order code from the higher-order form, the step that makes continuation passing implementable without a garbage collector, and it is the step a bounded-memory project would have to trust.

Delimited control is the sharper tool. Danvy and Filinski 1990 introduced shift and reset, Felleisen 1988 the prompt-based formulation, and Dybvig et al. 2007 a monadic framework with an implementation strategy. Flatt et al. 2007 report adding delimited and composable control to a production system, the closest thing in the literature to a cost report. A delimited continuation is exactly a coroutine frame, and the delimiter is exactly the boundary this article’s reset instruction draws.

The continuation literature harvested for this survey is listed below. It is the thinnest of the modern clusters, at 15 records from 2015 onward against 38 earlier ones, and the imbalance is itself informative. The representation question was answered in the 1970s and 1990s and the modern work has moved to the handler formulation surveyed further down.

The stackless transformation is what production compilers actually ship

The state-machine transformation, in which a coroutine becomes a function over an explicit state record, is the dominant implementation strategy and it is well documented. Syme et al. 2011 describe the F# asynchronous model, Bierman et al. 2012 formalise the C# async transformation, and Elizarov et al. 2021 give the design rationale for Kotlin coroutines including the suspension sentinel discussed above. Prokopec and Liu 2018 extend the treatment to coroutines with snapshots and give a cost model.

Every one of these reports the same discriminator problem and solves it the same way, by returning a tagged or sentinel value. That convergence is the strongest available evidence for the widened-return option this article identifies, and it was invisible from inside the backend.

It is also evidence for the correction recorded in the Epistemic State, and reading it carefully would have prevented the error. Not one of these systems returns a bare tagged value. Each pairs the tag with a state object that survives the call, being the state machine in C# and F# and the continuation in Kotlin, because a coroutine that suspends more than once must resume mid-body. The literature converged on the triple, and this article originally read it as converging on the pair.

The alternative is stackful, which the operating-systems literature has costed thoroughly. von Behren et al. 2003 argue for threads with dynamically sized stacks against event-driven code, and the stack-growth machinery they describe is what a stackful coroutine needs. For a project whose memory bound must be static, that machinery is not available.

The code-generation literature this article’s backend sits inside is the largest of the survey’s topical clusters, at 201 contemporary records, exceeded only by the general systems cluster listed at the end. It is here because the stackless transformation is a code-generation decision before it is a language-design one.

Effect handlers are the modern generalisation, and they carry the same obligation

Algebraic effects subsume coroutines. Plotkin and Pretnar 2009 give the handler formulation, Leijen 2017b a type-directed compilation, Hillerström and Lindley 2016 a row-typed treatment, and Sivaramakrishnan et al. 2021 a production retrofit onto OCaml with performance figures.

The relevance here is narrow and worth stating precisely. An effect handler’s return clause and its operation clauses are syntactically distinct, which is the same discriminator appearing a third time, now in the type system rather than the calling convention. A language that adopted handlers would not face this article’s question, because the distinction would already be in the source. Keleusma does not adopt them, and the V0.4 architecture does not propose to.

Verified compilation constrains what the backend may do

The project’s stance is that the backend refuses what it cannot lower rather than approximating it, and that stance has a literature. Leroy 2009a reports CompCert and its formally verified back end in Leroy 2009b, and Kumar et al. 2014 report CakeML, a verified implementation of ML with a verified compiler. Necula 1997 formalises proof-carrying code, in which the artefact carries its own evidence, and Sewell et al. 2013 report translation validation for a verified operating-system kernel, which is the technique a project unable to verify its compiler can still use.

None of these treats the calling convention as a proof obligation, and that is the gap the present project sits in. CompCert’s correctness theorem is about observable behaviour of whole programs under a fixed calling convention, and it does not tell you which convention to choose when a source construct emits two observable event kinds. The theorem shape assumed here, that a lowering is correct when yielded sequences agree, is a coinductive statement over infinite traces of the kind Leroy and Grall 2009 formalise for big-step semantics of diverging programs. That paper is the right formal setting for the divergent case and this article does not use it as such, which is a real weakness rather than an omission of politeness.

Worst-case resource analysis is why the convention matters at all

If native code did not have to carry a bound, the callback convention would simply win on generality. Wilhelm et al. 2008 survey the worst-case execution-time problem and the tools that address it, and it remains the standard reference for what a sound timing analysis requires. For the memory side, Regehr et al. 2005 eliminate stack overflow by abstract interpretation of machine code, exactly the analysis a native Keleusma artefact must support and exactly what a frame held across arbitrary host execution defeats.

This is the literature that makes the choice non-obvious, and it is the one the coroutine implementation literature does not cite. A runtime that may allocate is free to choose the general mechanism. A bounded artefact is not, and the trade appears only when both literatures are read together.

Compiler testing supplies the oracle and warns about its limits

The differential method used here is standard. Yang et al. 2011 report Csmith and the compiler defects it found, Le et al. 2014 introduce equivalence modulo inputs, and Chen et al. 2020c survey compiler testing as a field.

The warning these carry is directly applicable. A differential oracle refutes and does not prove, and its power is bounded by the inputs supplied. This article’s equivalence assertion is checked over a finite sample of resume sequences chosen by hand, the weakest form of the technique. Equivalence modulo inputs is the natural strengthening and it has not been applied.

The calling convention itself has a literature, and it is mostly about stability

The article treats a calling convention as a design choice, which is the compiler author’s view of it. The larger literature treats it as an interface contract whose principal property is that it must not change, because every separately compiled artefact already linked against it would break. That is the sense in which adding a discriminator to the return convention is expensive, and it is a cost the cardinality argument does not see. The cardinality argument says a wider convention is available. The interface literature says a wider convention is a different convention, and that the cost of adopting it falls on everything already compiled.

Keleusma is pre-1.0 and has no external artefacts to break, so the article is right that the cost is not yet being paid. It will be, and the decision made now is the one that will be expensive to revisit.

Static analysis supplies the vocabulary for a rule stricter than its property

The rule this article examines admits 14 of 24 chunks while the property it enforces holds for all 24, which is a soundness-preserving imprecision. The literature that names this condition, measures it and reduces it is large and the article uses only one paper from it.

The article’s position within this literature is unusual and worth being explicit about. The classical justification for an over-strict rule is that the exact property is uncomputable, so an approximation is forced. This article’s property is decidable on straight-line bytecode by summing two integers, so the imprecision is not forced and the rule is simply cruder than it needs to be. That is a better position to be in than the classical one and it is a different problem.

The bytecode layer is where the property is actually decidable

The backend consumes a verified bytecode, and the whole argument about stack effects lives at that level and not at the source or the machine level. The virtual-machine literature is therefore where the article’s decidability claim has to be checked, and it is also the differential oracle’s other side, since the oracle compares the virtual machine against the native code.

Types and semantics decide whether the question arises at all

The observation the article contributes is that two source forms have different trace alphabets, which is a statement about the source language’s type system before it is a statement about the backend. A language whose types separate a stream from a terminating coroutine can select the convention per form. A language whose types do not cannot, and must widen the convention globally or pay for a general mechanism. The type system is therefore the mechanism by which the choice becomes available, and the literature on what type systems can express about control effects is where the boundary of that availability is drawn.

Concurrency runtimes are where the general mechanism is affordable

The callback convention holds a frame across arbitrary host execution, which is what a runtime with a scheduler does routinely and what a statically bounded artefact cannot do. The contrast is the article’s central trade seen from the other side, so the runtime literature is listed here as the case where the general answer is correct.

The historical layer

The clusters above are drawn from work published in 2015 or later, which is the survey’s contemporary window. The older records the harvest returned are listed here in one place instead of distributed through the sections above, because for most of these clusters the pre-2015 work is the foundation the contemporary work cites rather than a competing account. There are 679 of them.

The surrounding systems literature

The harvest returned a large body of work that is adjacent rather than central, being compiler, runtime and systems research that shares the article’s vocabulary without addressing its question. It is listed rather than discarded because the selection that produced it is reported in full, and a survey that presents only the records supporting its thesis has selected twice, once by query and once by judgement, while reporting one selection.

What the survey shows

The representation question is settled in the literature and the selection question is not. Four independent traditions converge on a discriminated return for terminating coroutines, and none of them addresses whether a bounded-memory artefact should pay that discrimination on every call of a construct that never terminates. The present article contributes the observation that the two source forms have different trace alphabets and that the choice can therefore be made per form rather than globally, which the surveyed work does not consider because no surveyed language separates the two forms in its type system.

That separation is Keleusma’s own contribution to the problem, and it is the reason the question is even askable here. It is also why the survey cannot answer it.

The scale of the survey should be read for what it is. It lists 1,979 references, of which 35 were selected because the argument depends on them and the rest were harvested by query across thirteen clusters. The harvested majority establishes coverage and not agreement. A reader looking for the works that carry the argument should read the 35, which are named in the prose above. A reader checking whether the survey was assembled to flatter its conclusion should note that the queries were fixed before any record was seen, that every admitted record is listed including the 680 that are merely adjacent, and that the selection procedure is reported in Method with the count it discarded.

The Source Base

The generator is a detached package inside the Keleusma repository, consuming the reference compiler’s in-memory module representation and emitting LLVM intermediate representation through the inkwell bindings. The bytecode it consumes has already passed a structural verifier, in the tradition the Java Virtual Machine Specification sets out for bytecode verification, so the backend declines what it cannot lower and does not approximate it. The classification instrument is a test in the same package, reporting rather than asserting, with one guard that fails if the corpus walk finds nothing, because a broken path and a real zero look identical in a report.

The differential oracle drives both the virtual machine and the just-in-time compiled native code over identical bytecode and compares the whole sequence of yielded values. The stream drivers are bounded by a caller-supplied count, because a productively divergent program will otherwise run until something kills it, and a hung test reports nothing at all, worse than a failing one.

Epistemic State

Measured, and reproducible from the instrument. The corpus yields 24 stream chunks, of which 22 have a single suspension at the end of the body, 0 have multiple top-level suspensions, 1 has suspensions nested in conditionals, and 1 delegates with no suspension of its own. The nested chunk has 19 suspensions, all inside conditionals and none inside a loop, at nesting depths from 2 to 11. The corpus contains exactly 1 terminating coroutine chunk, in which 9 of 9 suspensions are immediately followed by a return. Ten stream chunks are refused by the current tail-position rule for a tail sequence that is effect-free and stack-neutral.

Derived, and checkable from the definitions. That a terminating coroutine produces two distinguishable observable events while a return-based convention provides one channel, and that the second therefore cannot carry the first, by $\lvert W \times W \rvert = \lvert W \rvert^2 > \lvert W \rvert$ for $\lvert W \rvert \ge 2$. That a divergent coroutine produces one, because its trace alphabet omits the completion letter. That the admissibility rule satisfies $R \subsetneq P$, so its defect is coverage and not soundness. That $\Delta(\pi) = -1$ for both candidate tail segments. That a suspension inside a conditional is a control-flow join while one inside a loop crosses a back edge.

Assumed, and marked as such. That the return convention is cheaper at run time. The structural grounds are stated and the frame-size measurement that would establish it was not performed, because the machine-code section carrying it is not emitted on this host.

Verified, and the verification method itself produced a finding twice. All 35 hand-selected research identifiers were resolved against the registry and compared with the work they are cited as. Thirty-five of thirty-five resolve to that work, an error rate of zero. The 1,944 harvested identifiers were not audited that way and do not need to be, because they were transcribed from the registry that issued them rather than recalled, so the failure mode the audit exists to catch cannot arise. What the audit would still catch, and did not run for, is a harvested record cited for a claim it does not support, and no harvested record is cited for any claim.

Sampled, because checking all of them would be checking the registry against itself. A random sample of 250 harvested identifiers was resolved, and 250 of 250 exist. What the sample does establish is a transcription check on the pipeline in this repository, since a fault there would affect a constant fraction of records and not a rare one, and no such fraction appeared.

Twenty-two of the 250, or 8.8 percent, resolved only through the registry. The identifier resolver did not serve them. Twenty refused the connection outright, of which 14 were Defense Technical Information Center deposits, and 2 returned a not-found response from a publisher that no longer serves the landing page. In every one of the 22 the identifier is registered and the record is correct. The defect is therefore in the resolution path and not in the citation. It is reported because a reader checking a sample of these references by clicking them will meet roughly one failure in eleven, and that failure is not evidence of a bad citation.

Two passes were run with two different instruments and they flagged different records, which is itself the lesson. A title-overlap heuristic flagged four. A later check comparing author surname and year flagged five. The union is eight distinct records and every one of them was vindicated on individual inspection, for four separate reasons.

Artefact Records What the registry does
Title split from subtitle 3 Stores “CakeML”, “Coroutines” or “Capriccio” while the citation carries the full descriptive title
Wrong registration agency 1 Proceedings in the LIPIcs series deposit with DataCite, so a Crossref lookup returns nothing for a valid identifier
Surname particle dropped 1 Stores de Moura as “Moura”, so an anchor built from the full surname fails a substring test
Registry typo 1 Stores “Dyvbig” for Dybvig, so the article is right and the registry is wrong
Identifier resolves to a reprint 2 Reynolds 1972 and Strachey and Wadsworth 1974 carry identifiers for their 1998 and 2000 reprints in Higher-Order and Symbolic Computation, so the registry year is later than the year cited

The last category is the one worth carrying forward, because it is not an instrument error at all. The identifier is correct, the work is correct, and the year genuinely differs, since a foundational paper and its journal reprint are two publications of one text. A checker comparing years will flag every reprinted classic in any bibliography, and a bibliography of foundational work is mostly reprinted classics.

Not one of the eight was a citation defect, and reporting any of them as one would have been a measurement mistake of exactly the kind this series keeps documenting. It is recorded because the previous article reported a 5.5 percent rate by the same method and did not separate these artefacts from genuine mismatches, which means that figure is an upper bound rather than an estimate and the true rate there may be lower.

Corrected during writing, and left visible. The plan this article set out to support was to unify the two conventions, and it survived until it was traced against a three-instruction example. The central claim was also stated in a form stronger than the evidence supported, assuming a one-word return until the governing application binary interface documents were read, and the correction opened an option the draft did not contain. An earlier design in this series specified a reordering transformation for bodies with several suspension points, of which the corpus contains none. A classification that called nineteen suspensions “the general case” was collapsing two structurally different situations, and the per-suspension measurement separated them.

Corrected after publication, and left visible. Two defects reached the published text and both are repaired above rather than silently.

The widened-return option was published under-costed. It was presented as costing a wider return and a host-side discrimination while preserving the frame property. A widened return does not give reentrancy. A terminating coroutine that suspends more than once must resume mid-body, which needs saved state, which is a frame. The honest interface is a triple carrying a continuation and not a pair carrying a tag, the per-call model omitted the frame term entirely, and the claim that the frame property is preserved holds only for a coroutine that suspends at most once per call. The error was in the direction that made that option look more attractive than it is, and the correction therefore strengthens the recommendation rather than disturbing it. The pigeonhole argument, the semantic-boundary finding and every measurement are unaffected.

The corpus could not have exposed it, and that is the article’s own lesson turned back on the article. The corpus holds exactly one terminating chunk and it suspends at most once per call, so a tagged pair genuinely does suffice for it, because resuming mid-body and resuming at entry after the sole suspension coincide. The general case never appears. A distribution over instances cannot answer a question about an interface, which this article establishes at length and then failed to apply when costing an option. It was raised before publication and did not reach the text.

The option count contradicted the list beneath it. The published text read that two options remained and then set out three. The sentence was introduced by a later editing pass that replaced a colon-led label with a full sentence and asserted a count the list never had. It now reads three.

What this article does not establish. Which of the remaining options is correct, since that turns on an ABI decision belonging to a workstream this article does not speak for. Whether the widened-return option is cheaper than two conventions, which is an empirical question and is not measured here. Whether the source-restructuring option is reasonable. Whether the corpus resembles the population the generator will serve.

The literature survey has two layers with different standing and only one of them bears on the argument.

The 35 hand-selected works were chosen by the author’s judgement of relevance, were read, and are the ones the prose reasons from. That layer is a narrative review. No protocol was registered and no completeness is claimed for it.

The 1,944 harvested works were retrieved by fixed queries against two registries and admitted by a stated filter, which is reproducible in a way the narrative layer is not. It is still not a systematic review. The query set was written by the same author whose judgement the narrative layer rests on, only two registries were searched, no grey literature was sought, no inclusion criteria beyond the title filter were applied, and no harvested record was read.

A reader should therefore treat the assertion that four traditions converge on a discriminated return as an observation over the works cited rather than over the field. The specific negative claim, that none of the surveyed work addresses per-form selection in a bounded-memory setting, is the one most exposed to an omission. The harvest makes that claim more exposed rather than less, because the harvested titles were not read and a paper answering the question could sit in the list unremarked. It is stated as a gap in the survey rather than a gap in the literature.

The strongest claim the evidence supports is that the return convention cannot serve terminating coroutines, on an argument that does not depend on the corpus at all. The weakest link is the cost comparison between the two remaining options, which rests on an unmeasured performance claim and on a corpus of twenty-four.

Out of Scope

The design of the lowering for any individual instruction. Register allocation, scheduling and peephole optimisation. The worst-case execution time analysis, which is a separate subject with a separate method. The exact frame-size comparison between the two conventions, which requires a target this host does not emit. Whether the delegating program should be restructured, which is a language question. The Keleusma language itself, which is covered in the getting-started article and the self-hosting strategy.

Conclusion

An apparent design wart turned out to be a semantic boundary, and the evidence that made it look like a wart was accurate. The measurement said nine out of nine, and nine out of nine was true. It was a fact about the shape of instances, offered in answer to a question about the arity of an interface, and the two are not the same question.

The check that settled it cost one traced example and produced an argument that holds independently of the corpus, which the measurement it overturned does not. Where a claim can be settled by counting the channels an interface provides against the outcomes a behaviour produces, that count dominates any distribution over instances, and it is available before any code is written.

The article also reports a rule shipped one increment earlier that is stricter than its own stated purpose, refusing ten of twenty-four cases whose tail sequence is provably harmless. It was found by building a table, not by running a test, which is now the second such finding in this series and no longer looks like a coincidence.

Equations

This section collects the 32 display equations of the article in the order they appear, each under its number and reproduced exactly as it stands in the text. A title in bold names an established result, and a title in italics beginning with the word Bespoke names a construction made for this analysis. Of the 32, 9 are established and 23 are bespoke. Cardinalities are counts of distinguishable values of the 64-bit machine word type, so a power of two here counts values while a multiple of sixty-four counts bits, and the two are related by a logarithm rather than by a factor.

Equation 1, Bespoke Partition of Chunks by Termination. The article partitions the compiled coroutine chunks into divergent stream chunks and terminating chunks, the distinction the type checker already enforces.

\[K = K_\Sigma \;\sqcup\; K_\Upsilon \tag{1}\]

Equation 2, Bespoke Trace of a Terminating Chunk. The article models a terminating execution as a finite trace of suspensions that ends in exactly one completion, so it uses both letters of the trace alphabet.

\[\mathcal{T}_\Upsilon(k, a, \vec{r}) \;=\; \bigl\langle\, \mathsf{Y}(v_1),\, \mathsf{Y}(v_2),\, \ldots,\, \mathsf{Y}(v_n),\, \mathsf{F}(w) \,\bigr\rangle \tag{2}\]

Equation 3, Bespoke Trace of a Divergent Chunk. The article models a divergent execution as an infinite trace of suspensions with no completion, so it uses only one letter of the trace alphabet.

\[\mathcal{T}_\Sigma(k, a, \vec{r}) \;=\; \bigl\langle\, \mathsf{Y}(v_0),\, \mathsf{Y}(v_1),\, \mathsf{Y}(v_2),\, \ldots \,\bigr\rangle. \tag{3}\]

Equation 4, Bespoke Host Call Sequence Under the Return Convention. Under the return convention the host calls a stateless native function again with each resume value, so nothing carries over between calls.

\[f(a) = v_0, \qquad f(r_1) = v_1, \qquad f(r_2) = v_2, \qquad \ldots \tag{4}\]

Equation 5, Bespoke Faithful Encoding Required of a One-Word Return. A return convention that is faithful to a terminating trace must map the pair of a yielded value and a completion value injectively into a single machine word.

\[\varepsilon_{\mathrm{ret}} : W \times W \;\longrightarrow\; W \tag{5}\]

Equation 6, Cardinality of a Pair of Machine Words. The set of pairs of 64-bit words has the square of the number of words as its size, which exceeds the size of a single word, so by the pigeonhole principle no injection from pairs into one word exists.

\[\lvert W \times W \rvert \;=\; \lvert W \rvert^{2} \;=\; 2^{128} \;>\; 2^{64} \;=\; \lvert W \rvert. \tag{6}\]

Equation 7, Bit Count Condition for a Faithful Encoding. An interface of a given number of machine words can carry a trace faithfully only when the trace needs no more words than the interface provides, the elementary case of the channel-capacity argument. Shannon 1948

\[64k \;\le\; 64m \qquad \Longleftrightarrow \qquad k \;\le\; m, \tag{7}\]

Equation 8, Bespoke Pigeonhole Test for a Calling Convention. The article states its general test, that a convention exposing fewer distinguishable channels than a chunk has observable values per call cannot be injective on that chunk’s traces.

\[\mathrm{obs}(k) \;>\; \mathrm{chan}(\varepsilon) \quad\Longrightarrow\quad \varepsilon \text{ is not injective on } \mathcal{T}(k), \tag{8}\]

Equation 9, Bespoke Collapse Factor for a Terminating Chunk. A terminating trace with a given number of suspensions carries one more word than suspensions while one call distinguishes a single word, so the article’s collapse factor grows as a power of the word count in the number of suspensions.

\[\frac{\lvert W \rvert^{\,n+1}}{\lvert W \rvert} \;=\; \lvert W \rvert^{\,n}, \tag{9}\]

Equation 10, Return Channels Fixed by the Application Binary Interface. The System V AMD64 application binary interface and the Arm 64-bit procedure call standard return a word in one register and a two-word aggregate in a register pair, so the number of return channels depends on the declared return type. System V AMD64 application binary interface AAPCS64

\[\mathrm{chan}(\varepsilon_{\mathrm{ret}}) = 1 \;\text{ if the return type is } W, \qquad \mathrm{chan}(\varepsilon_{\mathrm{ret}}) = 2 \;\text{ if it is } W \times W. \tag{10}\]

Equation 11, Cardinality of a Tagged Word. Attaching a two-valued tag to a 64-bit word doubles the number of distinguishable values, which needs one bit more than a single word holds.

\[\lvert \{\mathsf{Y}, \mathsf{F}\} \times W \rvert \;=\; 2 \cdot 2^{64} \;=\; 2^{65}, \tag{11}\]

Equation 12, Bespoke Widened Return with a Discriminator. The article’s third option widens every coroutine entry point to return a tag that distinguishes suspension from completion together with a word, which suffices only for a coroutine that suspends at most once per call.

\[f : W \longrightarrow (\,\mathsf{tag} \in \{\mathsf{Y}, \mathsf{F}\},\; W\,) \tag{12}\]

Equation 13, Bespoke Widened Return with a Continuation. A terminating coroutine must resume mid-body, so the article extends the widened return with a continuation in the manner of the returned-continuation lowering of the LLVM compiler infrastructure. the Low Level Virtual Machine compiler infrastructure

\[f : W \longrightarrow (\,\mathsf{tag} \in \{\mathsf{Y}, \mathsf{F}\},\; W,\; \mathsf{cont}\,) \tag{13}\]

Equation 14, Bespoke Total Cost of One Widened Convention Against Two Conventions. The article’s cost model charges the widened return a per-call tag and frame cost over all calls and charges two conventions a single fixed metadata cost.

\[C_{\mathrm{one}} = (c_{\mathrm{tag}} + c_{\mathrm{frame}}) N, \qquad C_{\mathrm{two}} = c_{\mathrm{meta}}, \tag{14}\]

Equation 15, Bespoke Crossover Call Count. Equating the two totals of Equation 14 gives the number of calls below which the widened return is the cheaper design.

\[N^{*} \;=\; \frac{c_{\mathrm{meta}}}{c_{\mathrm{tag}} + c_{\mathrm{frame}}}. \tag{15}\]

Equation 16, Value Space Remaining After Reserving a Sentinel. Reserving one word as a sentinel leaves one fewer than two to the sixty-fourth power usable values, a negligible loss in measure that nonetheless makes the reserved value unrepresentable as a yield.

\[\lvert W \rvert - 1 \;=\; 2^{64} - 1 \;=\; 18{,}446{,}744{,}073{,}709{,}551{,}615, \tag{16}\]

Equation 17, Bespoke Fit of the Return Convention to a Divergent Chunk. A divergent chunk has one observable value per call and the one-word return supplies one channel, so the identity encoding suffices for every admitted stream chunk.

\[\mathrm{obs}(k) = 1 = \mathrm{chan}(\varepsilon_{\mathrm{ret}}) \qquad \text{for all } k \in K_\Sigma \text{ admitted below,} \tag{17}\]

Equation 18, Bespoke Host Correspondence for a Divergent Chunk. The first call consumes the initial argument and each later call consumes the next resume value, exactly as the virtual machine writes the resume value into the entry chunk’s parameter slot.

\[f(a) = v_0, \qquad f(r_k) = v_k \;\; (k \ge 1), \tag{18}\]

Equation 19, Bespoke Admissibility Property for a Tail Segment. The article states the property that licenses the return lowering, that the tail after a suspension has no side effects and a net stack effect of minus one so that it consumes exactly the resume value.

\[\Delta(\pi) \;=\; \sum_{\iota \in \pi} \delta(\iota) \;=\; -1 \qquad \text{and} \qquad \mathrm{eff}(\pi) \;=\; \varnothing, \tag{19}\]

Equation 20, Bespoke Stack Effect of the Two Candidate Tails. The admissibility property of Equation 19 is evaluated for the single discard and for the discard, constant and discard sequence, and both have a net stack effect of minus one.

\[\Delta\bigl(\langle \mathtt{PopN(1)} \rangle\bigr) = -1, \qquad \Delta\bigl(\langle \mathtt{PopN(1)},\, \mathtt{Const(0)},\, \mathtt{PopN(1)} \rangle\bigr) = -1 + 1 - 1 = -1. \tag{20}\]

Equation 21, Bespoke Soundness Without Completeness of the Shipped Rule. The set of tails the shipped whitelist admits is a proper subset of the set the property licenses, and at least ten licensed tails are refused.

\[R \;\subsetneq\; P, \qquad \lvert P \setminus R \rvert \;\ge\; 10, \tag{21}\]

Equation 22, Bespoke Coverage of the Shipped Rule Against the Property. The fractions of stream chunks admitted by the shipped rule and by the property show that ten of twenty-four chunks are refused the cheap convention without cause.

\[\frac{\lvert R \rvert}{\lvert K_\Sigma \rvert} \;=\; \frac{14}{24} \;=\; 58.33\%, \qquad \frac{\lvert P \rvert}{\lvert K_\Sigma \rvert} \;=\; \frac{24}{24} \;=\; 100\%, \tag{22}\]

Equation 23, Bespoke Join Predicate for a Suspension. A suspension is classified as a control-flow join when no loop block is open on the enclosing block stack, so it crosses no back edge.

\[\mathrm{join}(y) \iff \mathtt{Loop} \notin B(y), \tag{23}\]

Equation 24, Bespoke Differential Oracle on Yielded Sequences. The native lowering is checked against the reference virtual machine by comparing the first several yielded values of the two traces for the same argument and resume values.

\[\mathcal{Y}\bigl(\mathcal{T}^{\mathrm{vm}}(k, a, \vec{r})\bigr)_{1:n} \;=\; \mathcal{Y}\bigl(\mathcal{T}^{\mathrm{nat}}(k, a, \vec{r})\bigr)_{1:n} \tag{24}\]

Equation 25, Bespoke Admissibility of a Unification as a Universal Claim. The article states admissibility of a unification as the claim that every behaviour in the class has no more observable values than the interface has channels.

\[\Phi \;\equiv\; \forall b \in \mathcal{B} : \; \mathrm{obs}(b) \le \mathrm{chan}(\iota), \tag{25}\]

Equation 26, Negation of a Universal Claim. The negation of a universal claim is an existential one, so a single behaviour that exceeds the interface refutes the unification.

\[\neg\Phi \;\equiv\; \exists b \in \mathcal{B} : \; \mathrm{obs}(b) > \mathrm{chan}(\iota). \tag{26}\]

Equation 27, Universal Bound as a Bound on the Maximum. A bound holds for every member of a finite class exactly when it holds for the maximum over the class, which a frequency of observed successes does not estimate.

\[\Phi \iff \max_{b \in \mathcal{B}} \mathrm{obs}(b) \le \mathrm{chan}(\iota). \tag{27}\]

Equation 28, Clopper-Pearson Lower Bound After No Failures. After all of a number of trials succeed, the exact one-sided lower confidence bound on the success proportion is the significance level raised to the reciprocal of the number of trials. Clopper and Pearson 1934

\[p_{\min} \;=\; \alpha^{1/n} \;=\; 0.05^{1/9} \;=\; 0.7169, \tag{28}\]

Equation 29, Trials Needed to Bound a Proportion Above a Threshold. Inverting the bound of Equation 28 gives the number of consecutive successes needed before a proportion of ninety-nine percent is supported at the stated confidence. Clopper and Pearson 1934 Hanley and Lippman-Hand 1983

\[n \;=\; \frac{\ln \alpha}{\ln 0.99} \;=\; 299 \tag{29}\]

Equation 30, Bespoke Native Frames Live During Host Execution. The article observes that the return convention leaves no native frame live while the host runs, whereas the callback convention leaves the chunk’s frame and those of its callers live.

\[L(\varepsilon_{\mathrm{ret}}) = \varnothing, \qquad L(\varepsilon_{\mathrm{cb}}) = \{\, \mathrm{frame}(k) \,\} \cup \{\, \mathrm{frame}(c) : c \in \mathrm{callers}(k) \,\}, \tag{30}\]

Equation 31, Bespoke Difference in Worst-Case Native Stack. The worst-case native stack bound of the callback convention exceeds that of the return convention by the total size of the frames live during host execution.

\[\Delta M \;=\; \sum_{\phi \in L(\varepsilon_{\mathrm{cb}})} \mathrm{size}(\phi), \tag{31}\]

Equation 32, Bespoke Native Calls per Iteration. The return convention makes one native call per iteration and the callback convention makes one more for each suspension in the iteration.

\[\tau_{\mathrm{ret}} = 1, \qquad \tau_{\mathrm{cb}} = 1 + s, \tag{32}\]

Notation

  • $W$, the machine word type, with $\lvert W \rvert = 2^{64}$ distinguishable values on the present target
  • $K_\Sigma$, the set of divergent stream chunks, which never complete
  • $K_\Upsilon$, the set of terminating chunks, which complete after finitely many suspensions
  • $\mathsf{Y}$, the trace letter for a suspension carrying a yielded value, with $\mathsf{F}$ the letter for a completion
  • $\mathcal{T}_\Upsilon(k, a, \vec{r})$, the observable trace of chunk $k$ driven by initial argument $a$ and resume values $\vec{r}$, written with $\Sigma$ for a divergent chunk
  • $\varepsilon_{\mathrm{ret}}$, the return-based calling convention, with $\varepsilon_{\mathrm{cb}}$ the callback convention
  • $\mathrm{chan}(\varepsilon)$, the number of distinguishable word-valued channels a single native call exposes under convention $\varepsilon$
  • $\mathrm{obs}(k)$, the number of distinct observable values a single call of chunk $k$ must convey
  • $\mathsf{cont}$, a continuation identifying where and with what state a coroutine resumes
  • $c_{\mathrm{tag}}$, the per-call cost of the discriminator, with $c_{\mathrm{frame}}$ the amortised per-call cost of a saved frame and $c_{\mathrm{meta}}$ the one-off cost of two conventions
  • $N^{*}$, the crossover number of calls below which a single widened convention is cheaper
  • $\delta(\iota)$, the net operand stack effect of instruction $\iota$, summed over a tail segment $\pi$ as $\Delta(\pi)$
  • $\mathrm{eff}(\pi)$, the set of side effects of tail segment $\pi$
  • $R \;\subsetneq\; P$, the set $R$ of tail segments the shipped rule admits and the set $P$ the property licenses
  • $B(y)$, the stack of open blocks at suspension $y$
  • $\mathcal{Y}$, the projection of a trace onto its suspension letters, applied to the virtual machine trace and the native trace
  • $\mathcal{B}$, a class of behaviours $b$ that a proposed unification maps onto an interface, whose channel count is written $\mathrm{chan}(\iota)$ in Equations 25 to 27
  • $\alpha$, the significance level, one minus the confidence
  • $L(\varepsilon_{\mathrm{cb}})$, the set of native frames live while host code executes under the callback convention
  • $\Delta M$, the difference in worst-case native stack between the callback and return conventions
  • $\tau_{\mathrm{ret}}$, the native calls per iteration under the return convention, with $\tau_{\mathrm{cb}}$ the count under the callback convention and $s$ the suspensions per iteration

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