Remainder: A Portable Method
余项：一个可携带的方法
This paper takes the remainder out of the normative architecture of SAE and makes it into a judging tool that can be carried away on its own. Every purposeful drawing of a line sorts content to two sides; what this paper judges is whether a given item still lies outside the operation of that particular line. The definition is one sentence: relevant, and outside this line's operation. The test is two questions, one for each conjunct of the definition. Would a change in this item's value make a difference to what the line, by its purpose, ought to do? And is the item, under this line's own criterion, actually placed into a channel that can make a difference under the relevant conditions? The first question is asked against the purpose and not against the criterion — two items sorted to the same side by one criterion are not thereby shown to call for the same treatment under the purpose. Errors divide into two levels: the genus level judges whether the item is a remainder, the type level judges which type it falls under. The two levels may not be mixed, and failure to fall under any type may never be turned around into a finding that the item is not a remainder. The four types are downstream sorting instruments, not a decomposition of the definition; this paper does not claim they are exhaustive, nor does it require them to be mutually exclusive across the board.
The conclusion this paper establishes is conditional: wherever the content bearing on the purpose of a given drawing of a line is not exhausted by that line's own criterion and actual handling, there exists an uncovered portion. This paper does not claim that every purposeful line must leave something over. It does not claim that the routes for eliminating a remainder are closed. It does not rule on how a remainder drives the next drawing of a line. And it passes no evaluation on any remainder. When an item really does enter the handling, this method should be able to say so plainly: relative to the present line, it is a remainder no longer. That is not a weakening of the method. It is part of what makes the method's capacity to be wrong worth anything.
The main deliverable is not four type names. It is a remainder register: which item, relative to which drawing of a line, on what evidence, with what conclusion. A reader who enters their own line into that register is using the method.
Main proposition: a remainder is always the remainder of one particular drawing of a line. Judging whether an item is one takes two questions — would a different value for this item make a difference to what the line, by its purpose, ought to do; and has it been placed into a channel on this line that can make a difference.
Keywords: remainder; drawing of a line; criterion; effective channel; rules for being wrong; line-relativity; portable method
Statement of Layers
What this paper depends on, and what it does not
This paper belongs to the SAE Methodology Series, numbered M-00E. The E stands for export: it sits at the same level as M-00 Via Rho, not as an application of it, but as an output version aimed at a different set of readers.
This paper does not depend on the normative architecture of SAE. The dimensional sequence, the chisel-construct cycle, the whole apparatus that lands on the human being as an end — none of it appears in the body. The reader need not accept any of it, and need not know it exists.
This paper depends on two methodology papers: Negativa (Methodology Paper 0) and Via Rho (Methodology Paper 00). What it depends on is provenance, not prior reading. Every step used below is walked again inside this paper; the reader may go and check those two, or not. Both are listed in the references, by concept DOI, resolving to the latest version.
The entry fee is two sentences
One governs admission, one governs range.
No line drawn, no remainder of that line.
Only when the content bearing on the purpose is not exhausted by that line's criterion and actual handling does the conclusion a remainder must be left follow.
The first sentence says where this method's subject matter begins: with no drawing of a line, there is no remainder relative to it to speak of. The second says under what condition this paper's conclusion holds. The two must not be merged into one. Merged, a reader will take it that any line whatever must leave something over, and this paper does not claim that.
One drawing of a line has three components
The purpose gives relevance. What counts as relevant is settled by what the line is drawn for, not by what the line happens to record.
The object domain gives range. What the line is drawn across.
The criterion gives the way of dividing. How content is sorted to the two sides.
Missing any one of the three, the line is not finished. Section 6 gives a worked example, and there it can be seen that a regulation which supplies only a criterion leaves the reader unable to settle even what to ask.
Three words are used in a fixed way here and do not shift. One drawing of a line and one line refer to the same thing: the thing made of those three components. Operation refers to the stretch in which the line actually places content into an effective channel; it lies inside the line, and is not the whole of it. So the phrase outside this line's operation, in the definition, says that the item has not entered that stretch — not that it lies beyond the line's reach. An item can fall inside the object domain, be gathered onto the admitted side by the criterion, and still be outside the operation.
An example involving no organization whatever. Take a recording rule whose purpose is to tell which of two fixed positions is hotter, whose object domain is the local thermal state at those two positions, and whose criterion keeps only the arithmetic mean of the two readings, with no other handling channel on this line. Swap 10 and 30 between the positions and the record is still 20, while what the purpose calls for differs between the two cases. That is a drawing of a line, all three components present, with no organization, no role, and no decision-maker anywhere in it. It is also the explicit setup of §3.2, and this paper uses it to check whether the test has leaned on institutional furniture it does not need.
Reading a thermometer is not drawing this line; that is using it. The method therefore works where there is no organization and no decision-maker: a definition, an accounting rule, a formal system — each of these draws a line.
Any of the three differs, check separately; a purpose may span lines
One drawing of a line is made of purpose, object domain and criterion, and if any one of the three differs, the two must be checked separately. A different criterion, checked separately; a different purpose, likewise; a different object domain, likewise.
This paper supplies no rule for merging them, and sets up no theory of operational identity for the purpose. What is wanted here is a single working instruction: when must two cases be judged apart. The previous subsection said that with any of the three missing the line is unfinished; this one is its other face — replace any one of the three and what is being checked is a different line, which must be judged afresh, and the two registrations may not be read as one. Judging afresh does not guarantee a different result: the new line may leave the same remainder, or none at all, and that is settled item by item or not at all.
A purpose, on the other hand, may span lines. Two lines with different criteria can serve one purpose, and that shared purpose is exactly what makes the two judgments comparable. Without a shared purpose, setting two lines' results side by side is mere juxtaposition and shows nothing.
This paper does not treat dynamics
This paper judges whether an item still lies outside the operation of the present line. It does not make that item's driving of a further line a precondition of the judgment, and it does not explain when or on what ground a further line occurs. Finding a remainder does not by itself require the line to be rebuilt.
A remainder constitutes a standing imbalance. That is a condition for a further operation, and that condition is not enough to yield the claim that a further operation actually takes place.
Upstream there is a stronger formulation. The abstract of Via Rho says that "the remainder supplies the dynamic condition of Negativa's continued operation (kinetic direction)"; §6.2 of the same paper says that "the conservation of remainder … guarantees the chisel-construct cycle never terminates." This paper does not invoke either sentence as a premise, and does not rule on their modality or their scope.[^1]
This step is not a new move in the series. Value Theory P4 lays down two propositions. One says that its explanations take one form only — "which positions were open at the time, and which path was accessible" — and that what this form refuses is "an engine built into the source" (Value Theory P4, V45). The other says that "conditions are for reading how something occurs, not for saying that it ought to occur. The remainder gives a shape and gives no compulsion; no stretch of remainder forces the next incision" (Value Theory P4, V59). On that basis Meaning Theory P2 states outright that the series takes up the first half of Via Rho's finding and not the second (Meaning Theory P2, §2). This paper is the third place to hold the same line.
But the reason differs from P2's, and that needs saying. P2's reason is that it lies "not in that paper, but in this series' immediate upstream" (Meaning Theory P2, §2) — its upstream is Value Theory, and Value Theory says otherwise. This paper's immediate upstream is Via Rho itself, so that move is unavailable. This paper has one reason only: it has no use for it. The genus-level judgment does not take the item's driving of a further line as a premise, and therefore needs neither half of the dynamic claim, and rules on neither.
[^1]: The dynamic direction is not confined to the abstract and §6.2. It is written into the core formula of §4.5, which joins Negativa, each operation, and the remainder that operation leaves into a closed round, with the remainder figuring there not merely as what the operation produces but as what drives the next one. What this paper sets aside is therefore not a stray remark. That section also states that its subscripts are not markers of physical time but ordinals of logical derivation. On the quotation from §6.2: in the original, the word conservation is followed by a qualification given in symbols, which this paper omits under the discipline against citing the series' internal notation; the omission is marked with an ellipsis and does not alter the sense of what is quoted. The modal relation between "dynamic condition" and "guarantees … never terminates" can be read together only after a sentence-by-sentence check; this paper does not perform that check and argues from neither side. Note also a shared word with two senses: operation in Via Rho means the operation of the chisel-construct cycle; operation in this paper means only the stretch in which a line places content into an effective channel, which is the narrower use.
The discipline of the body
The body observes the following. It uses none of the series' internal notation and borrows none of its analogies; the name SAE appears only in the title, the abstract, this section, the footnotes and the references, and nowhere in the argument. Neither the test nor the type names use institutional vocabulary, and whether the test has taken on institutional furniture is checked against lines involving no organization at all — the arithmetic-mean recording rule of §3.2 and the formal system of §3.3: if a sentence of the test fails to hold on those two lines, it has leaned on organizations, roles or decision-makers it did not need. First-level terms carry a gloss on first appearance. Identifying a remainder is not an evaluation: that an item is a remainder of this line yields nothing about whether it is good or bad; that is judged on other grounds, outside this method, and this method does not forbid evaluating a remainder — it only declines to supply the evaluation. Wherever another author is taken as a point of reference without the original having been checked, the passage is written so that it can be refuted, and answers only for the material this paper actually relied on.
1. What a remainder is
1.1 The definition is one sentence
A remainder is always the remainder of one particular drawing of a line. A content item is a remainder of this line if and only if it is relevant to this line's purpose and lies outside this line's operation.
Three properties follow from that sentence.
First, remainders are plural. Each drawing of a line has its own batch. There is no remainder in general.
Second, they are judged item by item. Given a candidate, a judgment can be offered; judgments are made one candidate at a time, go undetermined where the evidence runs out, and can be corrected when the evidence changes. This is not to say that any candidate whatever is guaranteed a correct two-valued answer. This paper does not promise a procedure that computes a result for every candidate. It promises that each judgment can state what it rests on and how far it goes.
Third, they are line-relative. The status holds relative to that one drawing of a line. Change the line and the judgment must be made again.
1.2 A remainder is relative to the whole line, not to the excluded side
Drawing a line generates two sides. A remainder is relative to the whole line, not to the batch sorted out of it.
This has to be settled at the outset, because the word reads naturally as what spilled over the edge. The definition says outside this line's operation, not on the side that was cut away. An item may sit squarely on the side that was gathered in and still, relative to this line, never have entered the operation.
The cleanest case is recorded but inert, in section 3: the line has recorded it, it is there on the books, it is on the side gathered in — and the line's operation does not run one bit differently for it. It is a remainder, and it is not outside at all.
1.3 Candidates are items of content, not people
The test asks whether a change in the item's value would make a difference to what the line ought to do; what it asks about is an item of content or a property. People are only the bearers of content.
Without that restriction the method collapses on the spot wherever a line sorts people: everyone who has been sorted has thereby been handled. A person admitted through the door is not thereby an item of his content handled; a person turned away is not thereby content that never entered the handling. The same admission can come of weighing his need or of never looking at it; and an item that was weighed can perfectly well end in refusal.
The paralysed philosopher in section 6 is the clearest demonstration. He is a philosopher, the criterion admits, and the person went to that side; while whether he can actually reach the books is an item this line never recorded and never handled. Where the person ends up and whether the content was handled are two different things.
This is used repeatedly in section 6, because what the line there sorts is precisely people.
1.4 Definition on top, test in the middle, four types downstream
Three layers, and the order cannot be inverted.
The genus-level definition (§1.1) stands on its own and mentions no type. The genus-level test (§4.2) is the procedure for judging by it, its two questions answering to the two conjuncts of the definition. The four types (section 3) are downstream instruments: add them, drop them, revise them — the trunk does not move.
The four types are not a decomposition of the definition, and the four do not add up to it. Hitting a type licenses a direct finding because that type's full criteria entail the genus-level condition — not because a type has judging power of its own.
That entailment has to be established type by type and may not be assumed. The short names of barred by rule, recorded but inert and bypass do not visibly contain relevance: an item irrelevant to the purpose can equally well be forbidden a record, and can equally well be recorded and do nothing. So every type's criteria are prefaced by three questions — is it relevant (by the first question of §4.2), is it outside this line's operation (by the second question of §4.2), is the evidence sufficient (checked per §4.3). Fail those three and no shortcut is granted.
A wrong type does not overturn the definition, but neither is it harmless. A new type whose criteria do not in fact entail the genus-level condition will generate genus-level false positives, and judgments that used the shortcut must be checked again when the type is withdrawn.
1.5 This is a finding about a line, not a claim about the world or about knowledge
The remainder of this method is a finding about what a line does. It claims nothing about how anything exists, and turns on nobody's knowing about it.
It is independent of cognition: whether an item is a remainder of this drawing of a line does not turn on whether anyone has spotted it. Nor is it an epistemological claim: this paper does not hold that the world is inexhaustible, and does not hold that reason is bounded. Neither of those is a premise here, and neither is a conclusion.
Several boundaries are written out here.
A line that undertakes only to complete a bounded count or a bounded determination of some property belongs to the class where section 2's antecedent does not hold. Such a line can still be judged item by item per section 4; the conclusion will usually be that there is no remainder. Range of application and range of conclusion are two things: this paper does not put such lines outside the method, it only declines to assert of them that a remainder must be left. And the name of the task guarantees nothing by itself — it must be confirmed that the purpose really does stop there and that the task really was carried out in full.
Where the criterion contains language of discretion, whether a change in value makes the operation run differently is not settled by the written text. Formal evidence is then insufficient and evidence about actual running must be sought; only when both fall short is the exit genus undetermined.
Relevance is relative to the purpose, membership is relative to the object domain, and the two may not stand in for one another. An item can be relevant while its bearer lies outside the object domain, and the reverse.
This method does not treat the back-action on the people sorted — the distortion of trajectory that comes of being badly fitted by a classification. That range is given up deliberately.
1.6 Summary
A remainder is defined in one sentence: relevant, and outside this line's operation. Remainders are plural, judged item by item, and relative to one drawing of a line. They are not confined to the excluded side; their candidates are items of content and not people; and a finding about one is a finding about a line, not a claim about the world or about knowledge. Below the definition sits the test, and only below the test sit the four types.
2. A conditional conclusion, and rejudging on a new line
2.1 The antecedent has one sense only
The purpose is not exhausted by itself has one sense here: the content bearing on the purpose is not covered by that line's criterion and actual handling. What bearing on amounts to is settled by the first question of §4.2.
Coverage and use are taken together — never recorded counts, and recorded but never entering the actual handling counts too. Attainment is excluded: whether the purpose was achieved is an assessment of outcome and falls outside this method.
Once the antecedent is fixed, no drawing of a line may be used to derive therefore necessarily unexhausted. That would reinstall the unconditional universal the paper has already withdrawn.
2.2 The conclusion reaches non-emptiness and no further
Wherever the content bearing on the purpose of a given drawing of a line is not exhausted by that line's own criterion and actual handling, an uncovered portion exists.
The conclusion reaches non-emptiness, not the existence of some one item that can be pointed at. Diffuse non-coverage need not yield any particular item, whereas the three properties require judgment item by item. Pointing is left to the test of section 4 and the trail of §3.7.
The argument is a conditional derivation. A line is drawn for something; wherever the content bearing on that purpose is not covered by the criterion and the actual handling, that content is relevant and outside the operation, and by §1.1 it is a remainder. The argument is in the main an unfolding of the definition this paper works with, not a strong theorem about the whole world, about reason as such, or about all measurement.
That does not make it useless. Its use is to fix the test, and thereby to stop three things from passing themselves off as effective handling: having been mentioned, having been put in a table, and having been renamed.
2.3 A new line must be judged afresh; so must an unchanged criterion whose channel has been connected
There is more than one route by which an item can stop being a remainder of this line, and each of them changes what is being asked about.
Changing the criterion is one. By the Statement of Layers, a different criterion must be checked separately. The line under the new criterion is judged afresh, and the earlier finding is not inherited. Nor does the new index inherit the old antecedent: to say that the new line also has a remainder, it must be shown that it still meets the condition of §2.1; the bare fact that it is a different line does not supply that.
The criterion unchanged while the actual handling channel changes is the other. Two rule texts word for word identical, one of which really has connected some item and one of which only keeps the rule on file; once the latter connects it, that item is, relative to the present line, a remainder no longer. The wording of the criterion did not change. The actual handling did.
From which one sentence follows.
After the criterion changes, the line under the new criterion must be judged afresh; with the criterion unchanged and the actual handling channel changed, it must be judged afresh as well. A later taking-in of some item does not retroactively rewrite the fact that the earlier line did not take it in. That the old line had a remainder does not guarantee the new line has the same one; that the new line has handled some item does not, by itself, show the new line has no others.
2.4 This paper does not claim the routes of elimination are closed
Changing the line only changes what is being asked about — that sentence holds only in the sense of individuation. It says the judgment has to start over. It does not say that nothing in the world can bring an item genuinely into the handling.
The converse matters more. When real handling does occur, this method should be able to say so plainly: relative to the present line, this item is a remainder no longer. That does not weaken the method. A tool that can only say there is more says nothing at all; whether a capacity to be wrong is worth anything is shown precisely by whether the tool dares return a no.
2.5 Summary
The conclusion of this section is conditional: the antecedent is that the purpose is not exhausted by the criterion and the actual handling, and the conclusion reaches non-emptiness. This section has shown two routes by which an item stops being a remainder — changing the criterion is one, connecting the channel is the other — and both require a fresh judgment rather than inheritance; the section shows these two and does not claim there is no third. This paper does not claim either route is closed.
3. Four types
3.1 The four types are sufficient conditions, not necessary ones
Meeting a type's criteria settles the finding: the item is a remainder. Meeting none of them means the typing route has returned no conclusion, and the question goes back to the genus level.
It may never be inferred, from falling under no type, that the item is not a remainder. Nor may a candidate already confirmed at the genus level be pushed back to genus undetermined because it fits no type — those are two different levels of undetermined, and §4.5 keeps them apart.
This paper does not claim the four types cover everything, and does not require them to be mutually exclusive across the board. What needs saying here is that non-exhaustiveness does not by itself yield non-exclusivity: a set of categories can perfectly well fail to exhaust and still exclude one another. This paper need not prove that exclusivity was impossible to begin with; it need only say that exclusivity is not required here. The four types are not treated as a partition. Such local exclusions as have been established stand on their own criteria, and however many types are hit, each one must independently satisfy its full criteria.
The output is a hit set, which may be empty, single or multiple, each entry registered where there is evidence for it. The genus-level finding is made once; no number of type labels adds anything to it. That an item hits both unheld by form and bypass is information. Forcing an item into unheld by form because the means are not to hand yet is a false positive. A long list is not an accurate one.
The four types are not the same kind of thing. Unheld by form, barred by rule and bypass each have a second thing that can be inspected — the recording form, the rule text, another line that acts — and starting from there the material comes in faster than running the genus-level test from scratch. But what is saved is labour, not questions. By §1.4, every type's full criteria take the two questions and the sufficiency of evidence as preconditions; material gathered for the full criteria can serve the genus-level condition at the same time, checks already done need not be repeated, and checks not done are not counted as done because an entry point was hit. Seeing another acting line, a prohibiting clause or a mismatched field is no ground for skipping the two questions. Recorded but inert does not even have that entry point: unfolded, its criteria are already on the books plus no effective channel, and the second half is the genus-level test itself. It stays among the four not because it is a shortcut but because it is the clearest place to display that test.
3.2 Unheld by form
The recording form this line provides for the item cannot preserve the difference being asked about.
The ground is formal, not an estimate, and it is a positive check: take two cases that differ in the respect being asked about and see whether this line's recording rule gives them the same record. Same record, and the record has already erased the difference; no way of reading it will recover it.
Take a recording rule whose field is a counter holding the total number of events, where what is asked is whether A came before B. AB and BA are both recorded as 2. The ground is an explicit rule plus two different objects yielding the same record — not nobody has tried, so presumably not; and raising the precision of the count will not turn a count into an order.
Three stipulations must hold, and the finding fails without any of them: order really is relevant to that purpose; what is being judged is order and not count; and this line has no other channel that actually handles it. If only the main field fails to hold it while another effective branch is handling it, a mismatch in one field does not settle that the line as a whole has not handled it.
No further restriction is added here to the effect that the recording form must feed an effective channel. Whether the record preserves the difference asked about, and whether later handling uses that record, are two questions: the first belongs to this type, the second to recorded but inert. If a free-text field really can record the order, then what is blocking it is not the form, and the finding belongs somewhere else.
Infeasibility, excessive cost and not measured yet do not by themselves put a case under this type; it must still be checked against the other types' criteria. Where the genus is confirmed and no type is hit, the entry is confirmed remainder, untyped; only where relevance or the actual operation has yet to be established is the entry genus undetermined.
An example involving no organization. Two positions are fixed, their local temperatures are 10 and 30, and the recording rule keeps only the arithmetic mean. Swap them and the record is still 20. If the line's purpose also bears on which side is hotter, and this line has no other channel, that spatial difference is unheld by form. If the purpose is only to produce one mean reading, no remainder is asserted on this ground.
The type is not named unmeasurable, and not unrepresentable. Either name reads as a limit on knowing, and collides with §1.5, which says outright that this paper does not hold that reason is bounded. What the type says is that this line's form does not hold it, not that no form could.
3.3 Barred by rule
This line's form could record it, and a rule says not to.
Could record is relative to whether this line's form can hold it. It must be written down whose rule it is, and whether the rule is actually enforced: where a rule exists but the item is still recorded and still used, the case does not fall under this type automatically.
The local exclusion between this type and unheld by form comes out of the antecedents themselves: one says cannot, the other says can, and for one candidate, one form and one range they cannot both hold.
This exclusion does not rest on the thought that forbidding what cannot be done would be a redundant rule. Redundant and non-existent are different: in the world a formal mismatch and a prohibition can perfectly well coexist. It is only that under the definitions this paper adopts, one candidate does not hit both types.
An accounting example. A historical-cost schedule has no field for market price, so the form cannot preserve the difference between current market price and historical cost — that is unheld by form; and the standard also states in writing that measurement is at historical cost. The second looks like barred by rule, but this type's antecedent requires that the form could record it, and here it could not, so barred by rule does not hold. This paper finds unheld by form and notes in the register that a prohibition also exists. This is not both types holding with one preferred — the other type's antecedent was simply never met.
An example involving no organization: a formal system whose alphabet and formation rules admit a certain class of formulas — the form could record it — and which carries a further explicit restriction forbidding the introduction of that class in derivations — a rule says not to. If the system's purpose bears on the propositions that class can express, and the system has no other channel that handles them, those propositions are remainders of this drawing of a line, of type barred by rule. Three antecedents — the form could record it, an explicit prohibition, relevance with no channel — and the finding fails without any of them; saying only that some string is not accepted settles nothing about this type.
3.4 Recorded but inert
This line has recorded it, and in this line's actual operation it does nothing.
It is not written as takes no part in the operation on paper. That is the formal reading, and it would shrink the type to the box where a zero weight has been written in. On the factual reading taken in §4.4, the type keeps its real extension, including the large class where a weight is written down and nobody looks at it.
The exact form of the paradigm case: recorded only; within the specified scope there is neither an effective handling weight nor any veto, review or other actual handling channel.
This type demonstrates one thing: a record is not a handling. It does not demonstrate anything else. In particular, a remainder has not been handled on this line and this imbalance has not been made good are two different findings. The same issue written into the report year after year and never entering an actual decision satisfies this type; and over that period there may perfectly well have been repeated discussion, adjustments elsewhere, movement on another line. Gave rise to nothing afterwards is not something this type can exhibit, and this paper does not undertake to exhibit it.
An example involving no organization: an accounting rule requires a field to be registered, and no subsequent calculation refers to it.
Institutional scholarship has neighbouring work. Bromley and Powell (2012) distinguish policy–practice decoupling from means–ends decoupling.[^2] This type is close to the first in shape and is not of the same kind — that one says a rule is not carried out, this one says an item is recorded and does nothing; see section 5. What makes the type stand is not novelty but that it is the clearest place to display the test.
[^2]: The two definitions are quoted from the full text: on policy–practice, "decoupling occurs when rules are unimplemented or routinely violated"; on means–ends, "Decoupling also occurs in the relationship between means and ends, when policies are implemented but the link between formal policies and the intended outcome is opaque." Taken from the author-hosted full-text PDF (patriciabromley.com); this paper has not checked the journal pagination of these two sentences against the journal edition, and so gives no page numbers here.
3.5 Bypass
First, a term. The table is what this line's written criterion has registered as fillable, assignable a value, or citable. Put another way, the table is the set of slots the criterion sets up. On the temperature line that keeps only the arithmetic mean there is one slot on the table, the mean; the table of a formal system is its well-formed formulas.
Bypass: the item is not on this line's table, and its acting occurs by way of another line.
Typing under this head requires separately showing that another line is present, and that the other line has a purpose, an object domain and a criterion of its own. Typing under the first three heads does not require another acting line, but neither do those types claim there is none. An item can fall under recorded but inert — on this line's table, with no effective handling channel — while still acting on another line; the other line does not thereby disappear.
The other line is called the channel through which the item acts, not the mechanism that excluded it. That it makes the item act does not yield that it caused this line's omission; that has to be established from the facts. A chain of natural causation is not, without further argument, a purposeful drawing of a line either: a fault that never entered the quality criterion reduces output — that shows that what was sorted out still has causal purchase, and to find bypass it must be said what the other line actually is.
An example involving no organization: two lines over one interval, each with its own three components, and no organization anywhere in it. The first has the purpose of establishing whether any point in the interval exceeds a threshold, while its criterion keeps only the arithmetic mean, and it has no other channel. The second has the purpose of preventing local overheating, its criterion is that any point above the threshold raises an alarm, and the alarm does not feed back into the first line.
Take two states with the same mean of twenty, one with every point below the threshold and one with a single point far above it. That difference makes the answer the first line ought to give, by its purpose, a different one — the first question passes. It leaves no trace in the mean and the line has no other channel — the second question fails. It is not on the first line's table, whose only slot is that one mean. And it does act, by way of the second line. All four in place, and the finding relative to the first line is bypass.
This example has to be read in that order. If the first line's purpose is only to produce a mean reading, the local excess is irrelevant to it, the first question does not pass, and there is no remainder to speak of, let alone a bypass. The candidate has to be cut precisely as well: that only the mean survives does not mean the whole local-reading property never entered the handling — every point's reading went into taking that mean. What is cut out is the difference the mean cannot tell apart, not the local-reading property itself.
3.6 The table separates recorded but inert from bypass
The exclusion the table condition gives is this: recorded but inert requires that the item already be on this line's table, bypass requires that it not be. So long as the table means the same state in both places, the two cannot both hold. That is an exclusion given by the conditions, not a type picked first so that the output comes out single.
The exclusion governs that pair only and does not extend to the other two. Unheld by form asks whether the record preserved the difference asked about; barred by rule asks whether the rule permits recording; neither is the same question as being on the table, and this paper does not derive their exclusion from recorded but inert out of the table condition.
Note that the table is defined formally and recorded but inert is defined factually, so there is a gap between them — and the gap is a good thing. Suppose the slot is set up on paper and nobody has ever filled it. Not unheld by form: the form holds it. Not barred by rule: nothing forbids it. Not recorded but inert: it was never actually recorded. Not a bypass: it is on the table. None of the four is hit, while by the genus-level test it is relevant and has not entered an effective channel — so the entry is confirmed remainder, untyped. That is the ready-made display of §3.1's sufficient-not-necessary clause.
Being on or off the table is not the same as being known to whoever is doing the checking. An item can lie outside this line and have been common knowledge for years and still satisfy bypass; an item already on the books may be discovered, in this very check and for the first time, never to have entered any effective handling. How an item stands to a line is one question; how the checker came by the candidate is another.
3.7 Bypass supplies a trail
The first three types can be judged starting from this line's form, rules and records; judging bypass additionally requires checking another acting line. That structure therefore supplies a trail: raise a candidate from acting that shows up elsewhere, then come back to this line and check its relevance and whether it went unhandled.
This paper develops only this one route of active search, and does not claim it is the only possible route to finding remainders. The genus-level test does not require that a candidate already act along another line. Raise a candidate by comparing the purpose against the existing criterion, then check relevance and non-handling, and §4.5 settles it just as well, with no second line needed anywhere.
One point about a vague criterion needs separating out, because it is easily read as a ready-made regularity. Vagueness does not by itself generate another line. What it removes is the written criterion's power to settle the case, and what turns up in that position may be another line, or may be nothing but discretion with no line to go on. The dividing point is whether that unwritten practice has a purpose, an object domain and a criterion of its own: if it does — if it is stable, repeatable, and lets one say in advance how a case will come out — then it is a second line. But establishing that a second line is present satisfies one of bypass's conditions only. The candidate must still pass the two genus-level questions, and its status on this line's table and the fact that its acting really does occur by way of that line must still be checked; all four in place, and only then is bypass registered. That a second line stably drives some practice, and that the item is relevant to this line, are two different things: if what this line ought to do by its purpose does not change with the item's value, the first question does not pass, and all that has been shown is that another standard is at work — the item does not thereby acquire remainder status on this line. Conversely, if the other line's result has fed back into this line's effective channel, §4.4 says the bypass label cannot be kept either.
But this dividing point governs the bypass label only; it does not settle the genus-level exit. If no acting line with a purpose, object domain and criterion of its own can be established, all that follows is that the bypass label cannot be had on that ground; the candidate's genus-level status is still judged per §4.5. Taking formal and actual evidence together, the entry is genus undetermined only where relevance or the state of the effective channel still cannot be established. That no stable second criterion can be generalized out of a pattern of discretion, and whether this occasion's material can show that the item actually entered this line's effective channel, are two different things: establish the latter and the finding is not a remainder; establish relevance together with the definite absence of any effective channel and the finding is remainder. Neither end requires finding a second line first.
3.8 What a new type has to satisfy
The four types are instruments and can be added to. But addability with no threshold amounts to immunization: take every counterexample into a fifth type and the apparatus can never again be wrong.
A new type must satisfy five conditions. Criteria distinguishing it from each of the existing four; not being a sub-case of an existing type; carrying its own way of being wrong; pointing out a new route into the operation at the genus level, or pointing out where the genus-level test shared by the four types falls short; and a stated limit of application. The fourth is this paper's own additional requirement on typing, not a necessity.
And it must be established that the type's criteria entail the genus-level condition; otherwise the type does not acquire the shortcut licence to confirm a remainder directly.
A case that fails these is not uniformly ruled a boundary problem of an old type. It is registered, according to the facts, as one of three: a boundary problem of an old type, a confirmed remainder that is untyped, or evidence pending.
3.9 Summary
The four types are four downstream sorting instruments, not a partition of the remainder. Satisfying a type's full criteria settles the finding — and those criteria themselves take the two genus-level questions as preconditions; failing to satisfy them leaves the typing undecided while the genus still has to be judged separately. What the table separates is recorded but inert from bypass, and bypass additionally requires an acting line to be present, presence being one of its conditions only. The four types can be added to, and adding has a threshold, whose work is to prevent immunization.
4. The test, and being wrong
4.1 Two levels of error, not to be mixed
Whether it is a remainder is judged at the genus level by the test in §4.2; which type it falls under is judged at the type level by each type's full criteria.
What the genus level carries is the correctability of genus-level judgments in the individual case — that is, the procedure can reverse a finding. It does not carry the same kind of falsifiability on behalf of every proposition in the paper, and the conditions under which those others hold are all written down here: the two local exclusions among the types are derived by unpacking their respective antecedents (§3.3 and §3.6), so the only way they can fail is a misstated antecedent, and they carry no empirical risk; the conditional derivation of section 2 unfolds the definition this paper works with, and §2.2 has already declared it is not a strong theorem; and the conditions of application of the trail in §3.7 are given in section 7. All that is said here is that they do not share the same correctability as genus-level judgments in the individual case — not that three items have been left without a sequel.
One inference is expressly forbidden: finding that an item is not a remainder because it fits no type.
There are two conditions under which an item fails to be a remainder: content irrelevant to this line; and content that has, relative to this line, entered its operation. The genus always carries the index to a line — a remainder of one line does not stop being that line's remainder by being absorbed into another. Cross-line phenomena may be described separately, but they do not enter the genus's exit conditions.
4.2 The test
The test is two questions, answering to the two conjuncts of the definition.
First question: relevance. An item is relevant to this line if a change in its value makes a difference to what this line, by its purpose, ought to do.
Two things must be nailed down here, or the question fails in either of two directions.
First, what is asked is a dependence, not an attainment. Ought to do means what this line ought to do under this purpose; it does not mean whether the line brought it off. Whether the purpose was achieved is, per §2.1, an assessment of outcome, and is already excluded. So the first question can be asked even where the purpose is unachieved, or will never be achieved: what the variation is held against is the purpose itself, not its realization. Every finding of irrelevance that runs on the thing gets done either way, the result is no different has slipped out at this point.
Second, the question is asked against the purpose, not against the criterion. Two items sorted to the same side by this criterion are thereby shown only that the criterion did not tell them apart, not that what ought to be done under the purpose is the same for both. Suppose a criterion admits anyone whose attainment reaches five, while the competence the purpose bears on requires eight: six and nine carry the same record under the criterion, and fall on opposite sides of what the purpose calls for. Every finding of irrelevance inferred from both were gathered onto the same side by the criterion is an error made at this point.
Second question: the effective channel. Whether an item has entered this drawing of a line's operation is settled by whether, under this line's own criterion, it has actually been placed into a channel that can make a difference under the relevant conditions.
Every qualifier in that sentence is doing work. Under this line's own criterion is the index to a line. Actually been placed is the factual reading. Can make a difference under the relevant conditions rules out channels that carry a formal position and change nothing whatever the value.
The two questions do not stand in for one another. The second's result may not be run backwards into the first — that an item has not been handled by this line does not show it is relevant to this line; and that it has been handled does not make it relevant either. A field can genuinely drive the calculation while what the line ought to do under its purpose never looks at it.
The two questions have four exits, and they are not two separate verdicts later combined: the question is which sufficient condition is established first.
Establish a no to the first question and the item is excluded on relevance, registered as not a remainder (relevance no). Establish a yes to the second and remainder status is ruled out; register not a remainder, and relevance may stand unestablished and be recorded as such — a remainder is the conjunction of relevant and outside the operation, the second conjunct has been denied, and the state of the first no longer changes the result. Establish a yes to the first and a no to the second, and the finding is remainder. Only where none of these three is established and the necessary material is still short is the entry genus undetermined.
So it is not the case that insufficient material on either question yields undetermined. Any one sufficient condition, once established on adequate material, fixes the genus-level status.
4.3 Evidence comes in three places: purpose, candidate, channel
Each question has its own material, and they may not be mixed. The first question requires that the purpose be established and that the cut of the candidate be fixed; the second requires that the channel be established.
The purpose first. The purpose is not assigned by the analyst on a whim; it must be established and fixed at registration. Five rules.
First, both the purpose declared in writing and the purpose shown by actual running count as evidence.
Second, where the two disagree, record both and check them separately. The same batch of content can yield different conclusions against the declared purpose and against the actual one; that difference is itself informative and should not be flattened into one.
Third, where a line has multiple purposes, or purposes in conflict, check them separately and do not merge them into one overall purpose.
Fourth, where both kinds of evidence fall short and the analyst supplies the purpose, the register must note that it was supplied, and the conclusion holds only relative to that purpose.
Fifth, the purpose is fixed at registration and may not afterwards be extended in order to make some item relevant. To widen the purpose is to have another line, registered afresh per the Statement of Layers, with no automatic inheritance from the old one.
The fifth rule is the crux of this section. If a stronger purpose may be supplied midway, any item at all can be called relevant and the first question loses its grip: see something that has not been handled, then go back and write the purpose wide enough to just take it in — a remainder got that way was manufactured, not found.
Beyond the five there is one more guard, against the same trick entering by another door. Since the purpose shown by actual running counts as evidence, it can be read straight off the criterion: what this line is really doing is just what its criterion prescribes. A purpose read off that way coincides with the criterion. Coincidence does not by itself make the first question return no for every candidate — content inside the criterion can still be relevant: if the purpose is to give the event count accurately, then a change in the count changes what the line ought to record, and the first question is of course yes. The risk lies elsewhere. Inferring the purpose from the criterion alone narrows, in advance, a relevance that was supposed to be checked independently, confining it to the portion the existing criterion already accommodates, and so conceals the omissions that show up only against an independent purpose. And even where purpose and criterion do coincide, §1.5 still requires confirming that the task really was carried out in full; the name of the task guarantees no absence of omission. So the actual purpose must be corroborated by material from outside the criterion — the executor's own account, how this line fits with other arrangements, a higher written rule — and may not be read back off the criterion alone. Where the criterion is all there is, what gets registered is a criterion, not a purpose; the purpose cell stands empty, and by §6.2 the line is not finished.
The candidate next. With the purpose nailed down, the same door is still open on the candidate's side. For one underlying property, a coarse cut and a fine cut can give opposite results. Take the admission line of section 6, and stipulate separately that attainment of zero fails the competence the purpose requires while six and nine both meet it, other conditions held fixed: take the full range of attainment as the range, and zero and six call for different treatment under the purpose, so the first question passes; take the difference between six and nine inside the admitted group as the range, and the first question does not pass. Both of those are judged against the purpose, not against who got admitted — the latter is a result of the criterion, and by §4.2's second point cannot be used to answer the first question. The cut itself takes part in the finding.
Hence the sixth rule. The candidate's range of variation must be written down and fixed at registration along with the candidate, and may not afterwards be narrowed to make it irrelevant or widened to make it relevant. One property can yield different conclusions over different ranges; register the two separately and do not read them as one.
The logical content of that rule is this: conclusions on different ranges may not be inherited from one another unconditionally. Among conclusions already established, the guaranteed inheritance is one pair. With the purpose the same, other conditions fixed, and the ranges nested, relevance established on the smaller range is inherited upward to the larger — the pair of values that established it is still inside the larger range; and irrelevance established on the larger range is inherited downward to a sub-range — any two values in the sub-range are already covered by that established universal. Those two are the two faces of one dependence.
What is not inherited is the state of lacking evidence. Having found no evidence of relevance is not the same as having established irrelevance, and so it is not inherited downward; from having found no relevance on the fine cut it does not follow that there is none on the coarse cut, and from relevance on the coarse cut it does not follow which stretch within it is relevant. Without this paragraph the sixth rule is only a registration requirement; with it, the rule blocks cutting finer until irrelevance is reached.
The fifth and sixth rules guard the two directions of one manufacture: one manufactures a remainder, the other manufactures its absence.
The channel last. Written text stands differently in different lines, and the evidence must be sorted by what is being asked about.
Where the operation is wholly settled by the written text — a definition, a formal system, an algorithm's output relation — the text is the fact and formal evidence is complete. Where the line is executed by people or institutions, the text is preliminary evidence only and can be overturned by execution records. Where the criterion contains language of discretion, the text cannot answer at all.
A written rule can settle whether the form has reserved an effective acting channel for the item; whether the channel is actually connected and running has still to be checked separately. Where the text has been confirmed to describe accurately the mechanism actually executed on this occasion, the possibility of acting may be judged from the formal relation. One thing slips past easily here: not an empirical prediction and no need to confirm actual execution are not the same sentence. An algorithm's output relation can be settled formally, and whether the algorithm was actually run is not thereby shown, however formally it was written.
4.4 A formal position is not a handling
A formal position alone does not amount to effective handling having taken place. It is when relevant content has, relative to this line, no effective acting channel within the scope checked that the finding still a remainder follows.
The reason belongs in the open. Acknowledging a thing's standing and letting it act on the operation are two things, and the first does not entail the second. An item occupies a position in the written text, carries a written weight, and in the actual operation no change in its value leads to any difference whatever — find that handled, and the method has gone silent exactly where it most needed to speak, and §1.2's sentence, that an item gathered onto the admitted side may still never have entered the operation, fails along with it.
But a zero weight carries a scope. Suppose one line has two channels, one outputting 0 times x and one outputting x; the zero in the first channel may not settle, on behalf of the whole line, that x does not act. This is the same discipline as §3.2's question of whether another effective channel exists.
Three corrections follow.
First, a difference in output is not required every time. An input can genuinely take part in the handling and the final result still not change; a same result may not be run backwards into a finding that the input went unused. What is checked is whether it entered a channel that can make a difference under the relevant conditions.
Second, recording lines are another matter. Where the task of the line under examination just is to produce a record, actually producing that record can be that line's handling. A record does not guarantee the effective handling asked about is not a record never counts as handling on any line. Sort out which line is asking, per the Statement of Layers.
Third, a cross-line fact does not vouch for itself. Acting on another line is not by itself enough to show that this line has handled the item; if that acting's result has actually entered this line's effective channel, it is checked afresh by this line's test. The names of two lines cannot testify to whether the two are actually joined.
The word change does not enter the definition. It is only a means of observation: set two versions of the criterion and of actual practice side by side and see whether the item has begun to act.
In organizational and institutional material, having been placed into an effective channel usually shows up as bearing on how resources are allotted, how cases are settled, or who answers for what. That is a common way it shows up; it is not the test.
4.5 The walk, item by item
Given one item of content or one property, walk the two questions of §4.2; there are four exits. It is not two verdicts later combined; it is a matter of which sufficient condition is established first.
Establish a no to the first — irrelevant to this line — and the item is excluded on relevance, registered as not a remainder (relevance no), and goes no further. Establish a yes to the second and remainder status is ruled out: register not a remainder, and record separately where relevance stands. Establish a yes to the first and a no to the second, and the item is locked as a remainder; then match it against the four types and output a hit set, which may be empty. Where none of the three is established and the necessary material is still short, the entry is genus undetermined.
The second question is one gate, not two gates for text and practice. The incompleteness of a written text is not in itself enough to push back into undetermined an actual operation that adequate material has established; §1.5's discretion exit says that the text alone may be insufficient, not that other material can never make it good.
Genus undetermined and confirmed remainder, untyped are two different things and may not be merged. The first is a shortage of evidence, the second is evidence enough with no shortcut catching. An undetermined state of knowledge is not to be rewritten as the object's having no status.
4.6 Misjudgment in application, and checking rules back
What correctability guarantees is that the procedure can reverse a finding: a genus-level judgment can be found wrong on later material and put right. This paper does not separately claim that the method as a whole has acquired some independent condition of overall falsification. The genus-level definition and the decision procedure use the same set of conditions, so the shapes below say, in the first instance, that something has gone wrong in the use of evidence, in the implementation of a rule, or in internal consistency — not that an empirical hypothesis about the world has been refuted.
Misjudgments sort by question, two to each. On the first question: content relevant to the purpose is found irrelevant and excluded; or content irrelevant to the purpose is treated as relevant and then found to be a remainder. On the second: relevant content that has not been placed into an effective channel is found not to be a remainder; or content that has been placed into one is found to be a remainder. All four can be caught on re-checked material and reversed, and that is the capacity this method has to keep — a tool that can only say there is more says nothing at all.
Where the error comes from a decision rule rather than from short material, the rule must be revised and every judgment that depended on it checked again.
Adding a type label cannot repair an error in a genus-level judgment. The genus-level finding does not pass through the types; adding a fifth type changes the hit set and not the genus-level conclusion. A type whose criteria do not in fact entail the genus-level condition will simply go on generating genus-level false positives until the type is withdrawn and the judgments that used it are checked again.
4.7 The remainder register
The main instrument this paper delivers is this register, not four names.
The three rows below are a filled-in demonstration on explicitly stipulated setups, not a survey of real execution material. What the evidence column cites is where in this paper the setup is given; the facts about running and the channel ranges in them are all stipulated. In actual use, that column must instead carry re-checkable documents, clauses, record numbers or points in time.
| Candidate and its range | This line's three components | Scope checked | Relevance | Placed into an effective channel? | Genus-level finding | Hit set | Evidence and location |
|---|---|---|---|---|---|---|---|
| Which of two events came first (restricted to the orders AB and BA) | Purpose: a distinction must be drawn by order; object domain: the events this record covers; criterion: a single counter holding the total | All channels of the current version of this recording rule | Yes: what the purpose calls for differs with the order | No: AB and BA are both recorded as 2 | Remainder | Unheld by form | Setup at §3.2. The purpose is stipulated by that setup and not read off the criterion; the absence of any other branch on this line is a stipulated condition of it |
| The value of input x (restricted to the range over which the two channels' outputs differ) | Purpose: the result is to vary with x; object domain: this line's inputs; criterion: two parallel channels, one outputting 0 times x, the other outputting x | The two channels of this line | Yes: the result the purpose calls for varies with x | Yes: the second channel outputs x, and different values give different outputs | Not a remainder | Empty | Setup at §4.4. A zero-weight channel cannot settle for the whole line that nothing acts, so both channels had to be checked |
| Whether the predicate philosopher still holds after its bearer's death | Purpose: to let in those who can make good use of the books; object domain: those seeking entry; criterion: philosopher or not, two-valued | All identity checks during the period the regulation was in force | Not established: what is missing is that the purpose says nothing about whether the dead remain among those who can make good use | Not established: what is missing is that the case never arose, so there are no execution records | Undetermined | Not applicable, genus not locked | Setup at §6.4. The two questions are short of different material and each must be written out; that the wording of the criterion is unsettled does not by itself make the item's remainder status unsettled |
Every entry must state the version, the point or period in time, and the actual range of channels checked for the line in question; the evidence column must give both where the material is and what judgment it supports, and not merely the kind of evidence or the act of checking; and the source of the purpose, or the fact that it was stipulated here, must be stated too.
The columns are filled as follows. The candidate column takes one item of content or one property — not a person, and not the name of a phenomenon — and, per the sixth rule of §4.3, carries its range of variation with it: cutting specifically makes the judgment easier, while the cut itself takes part in the finding of relevance, so how specifically it is cut must be registered with the candidate and not adjusted afterwards. All three component cells must be filled; with the purpose empty the relevance column has nothing to stand on, and the purpose may not be read back off the criterion. The scope-checked column nails down what range this check actually covered, and heads off a very specific failure: two lines with identical rule texts and different actual connections leaving near-identical entries in the register. Relevance is judged by §4.2's first question, against the registered purpose and not against the criterion. Placement into an effective channel is judged by §4.2's second question. The genus-level finding is still one of three, and the four procedural exits coexist with the three statuses: an item excluded on a no to the first question is registered as not a remainder (relevance no), and that negative judgment must leave a re-checkable record like any other; an item settled by a yes to the second is registered as not a remainder, with the relevance column carrying whatever its state then is. The hit set may be empty, and empty is not not a remainder; where the genus is undetermined, this column reads not applicable rather than empty. Evidence and location are filled per the paragraph above, with a note on whether they suffice.
A reader who enters their own line into this register is using the method. The four types govern one column only, the hit set.
4.8 A precedent, and a gap
Star and Bowker (2007) define residual categories as categories that cannot be formally represented within a given classification system. This supports the step of system-relativity, and it also gives one local rule for being wrong: whatever can be formally represented within that classification system is not that system's residual category.[^3]
[^3]: The definition is from their 2007 paper; it has not been checked sentence by sentence against Sorting Things Out (1999). This paper relies on it only to support the step of system-relativity, and derives from it no normative mechanism for barred by rule.
The gap is that every such rule this paper has seen holds only relative to its own generating operation. This paper does not possess a rule for being wrong that is general across traditions; §4.2 is this paper's candidate, and its range is bounded by this paper's definitions.
4.9 Summary
The test is two questions: relevance and the effective channel. The first is asked against the purpose and not against the criterion, and what it asks is a dependence, not an attainment. Evidence comes in three places: the purpose, the cut of the candidate, and the channel. The four exits are settled by which sufficient condition is established first, and not by whether either question is short of material. A formal position is not a handling. Errors divide into the genus level and the type level, each with its own undetermined exit; where the error lies in a decision rule rather than in the material, the rule must be revised and the judgments depending on it checked again, and adding a type label repairs no genus-level error. What is delivered is a register.
5. Comparisons
5.1 How this section is written
This section does not sort these bodies of work into types. The reason is internal to this paper: the four types are shortcuts for judging remainders relative to one drawing of a line, and what they sort are candidates on some line, not a thinker's position as a whole. To push a thinker's work entire into one of the four is to use a line-indexed instrument with no line specified.
Nor does this section claim that any of them is an instance of this method. Mutual illumination is the direction, and it does not have to be redeemed in both directions — a theory that helps this paper see one problem clearly has earned its place in the body, and no what this paper adds column need be manufactured in return.
The test of pertinence is that the two sides are handling the same structure — that is, that a local structural correspondence can be specified — and no proof of isomorphism is required. Where reading them together makes at least one side clearer, that is mutual illumination; where a body of work does not handle the same structure but marks this method's outer edge, that is a boundary marker.
Where the original has not been checked, this section writes only what this paper actually relied on, and puts the claim in a form that can be refuted. That is not modesty; it is drawing the revisable range in advance: if it is wrong, what changes is a qualifying clause, not a position. But limiting the range cannot substitute for provenance: wherever this paper says what it relied on, it must be able to point to which work that was, and that work is listed in the references; a formulation whose source cannot be pointed to is not used as a basis for comparison.
The table below records only where each stands on five axes. It passes no judgment.
| Generated relative to a line | Requires a subject | Judged item by item | Has a rule for being wrong | The leftover still acts | |
|---|---|---|---|---|---|
| This paper | Yes | No | Yes; undetermined where evidence is short | Yes | Not required at the genus level; bypass additionally requires showing action by way of another line |
| Callon | Yes | Not required | Not developed | Not checked here; no judgment offered | Yes |
| Law | Yes | Not required | Not developed | See §5.4 | Yes |
| Star and Bowker | Yes, system-relative | No | In part | Yes, formal representability | Yes |
| Gödel | Yes, relative to a system | No | Yes | Not applicable | Open |
| The Goodhart line | Yes | Some mechanisms require one | In part | Yes, mechanism conditions | Yes |
| Messick and Borsboom | Yes | No | Yes | Yes, validity conditions | Yes |
| Bromley and Powell | Yes | Not required | Not developed | Open | Yes |
| Barad | Yes, one layer down | No | Open | Open | Open |
| Schelling through Žižek | No | Not applicable | No | No | Yes |
5.2 Callon: closest in shape
Callon's (1998) framing and overflowing are the closest in shape to this paper, and the first thing an outside reader thinks of, so this passage answers it head on.
His framing is the bounding act that makes a transaction calculable; overflowing is the effects and connections that escape that framing. Overflowing is relative to a particular framing, and the practice of framing itself produces externalities.[^4] All three run with this paper, and none of them needs a psychoanalytic subject.
[^4]: This paper has not checked the whole chapter; what it relied on here is that chapter's account of framing and overflowing. The chapter is listed in the references; the same text also has a journal citation, with pagination identical to the book's, given in that entry. If what is said here does not match the original, this section should be revised accordingly.
What this paper delivers here is a two-question test applied item by item, rules for being wrong, and sorting instruments. It has not checked the whole chapter, and therefore offers no judgment on whether he has elsewhere a procedure doing the same job. That answer is both true and modest enough — whether this paper's instrument works does not turn on whether another has one like it, and putting a problem into a shape that is easier to check item by item does not require first proving that nobody else has done so.
His formulation runs in the grammar of product and consequence: frame, then overflow, then frame again. This paper writes the handling and the sorting-out as registered together at the moment of definition. What is recorded here is only the difference between two formulations; nothing about simultaneity is inferred from the order of exposition, and in what this paper relied on he takes no position on it. How his distinction between cold and hot situations stands to this paper's test is not developed here.
5.3 Star and Bowker: taking in is not handling
This pair illuminates two things.
First, a classification equips itself with a single place for taking things in. The explicit other or not elsewhere specified in a classification system is the box the system prepares for itself. Being taken in is not entering the operation — tick other and nothing follows, which is the commonest institutional form of recorded but inert. This runs with the principle of §4.4: a formal position does exist, and whether anything entered the operation is another matter.
Second, a blind spot does not display itself. A system is not aware of what it is unaware of. That fits the character of this method: the genus-level test can be run only once a candidate is given, and what it judges is whether that candidate is a remainder, not which candidates there are. Where candidates come from — comparison, questioning, acting that shows up elsewhere — the method does not govern. That follows directly from this method's being, overall, an instrument of verification.
They have a further concept, about the distortion of trajectory that a badly fitting classification imposes on those classified. This paper does not treat that layer.
5.4 Law: a different question asked, and a weaker claim made here
Law (2004) distinguishes manifest absence from Otherness: the former already acknowledged and already said, the latter not yet acknowledged. His Otherness holds relative to a particular assemblage of method, and he holds that any assemblage produces Otherness.[^5]
This paper is not asking the same thing. He asks whether something is acknowledged or not yet acknowledged; this paper asks whether something has entered the operation of this drawing of a line. Both questions carry an index, and by this paper's criteria a relevant item that has been stated outright and has also definitely not been handled is still a remainder.
One difference in modality is worth writing out, and it runs against this paper: he holds that any assemblage produces Otherness, while this paper does not hold that any drawing of a line must leave a remainder, and draws its conclusion only where the antecedent holds. This paper's claim is the weaker one.
What this paper delivers here is a two-question test applied item by item, rules for being wrong, and sorting instruments. It has not checked the whole book, and therefore offers no judgment on whether he has elsewhere a procedure doing the same job, and draws from this comparison no conclusion that he lacks an index or lacks criteria for being wrong. Whether his enactment and gathering are separated by finer stages of handling has likewise not been checked; if the answer turns out to be no, the comparison in this section should be revised.
He also holds a background commitment about the relation between method and reality. This paper does not adopt it and does not restate it here.
[^5]: This paper has not checked the whole book. What it relied on here is the statement of that distinction in Law (2004), listed in the references; Law (2003) is a related shorter piece from which this paper draws no independent claim, so this section cites only 2004. If the actual content of the distinction does not match what is said here, this section should be revised accordingly.
5.5 Gödel: entering the expression is not obtaining a result of some handling
In its modern standard form: in a formal system that is consistent, has a recursively enumerable axiom set, and has sufficient arithmetical expressive and proof-theoretic strength, there exists a proposition which the system's language can express and which the system can neither prove nor refute.[^6]
[^6]: Theorem VI of Gödel's 1931 paper takes ω-consistency as its condition; weakening that to plain consistency is Rosser's (1936) improvement. What the body cites is the standard form the latter yields, generally called the Gödel–Rosser theorem. This paper uses only the interface of that result and does not touch its proof.
The word outside has to be written accurately here: it means outside the range of what the system can prove or refute, not outside its language. That is useful to this paper, because what this paper is doing is exactly to hold apart three things: having been expressed, having been settled, and having acquired some standing in another line. Entering the expression is not obtaining the result of some particular handling.
One more thing must not be let past: a proposition's proof-theoretic status does not automatically mean it has no purchase in the world. One may not judge outside by provability in the first half and inside by real-world effect in the second.
This paper adds no new conclusion to incompleteness in return.
5.6 The Goodhart line: the gap between proxy and purpose
Where a proxy quantity fails to cover some difference the purpose it proxies for bears on, a candidate can be raised on that ground; whether the item is a remainder of the line that takes the proxy as its criterion still has to be checked against this paper's genus-level test. Behaviour that gathers around a proxy comes in through gaps of exactly this kind.
In the account this paper relied on, Manheim and Garrabrant (2018) already sort this line's failures into four mechanisms, each with its own conditions.[^7] That runs with this paper's way of working: neither treats all failures as one thing, and both require it to be said which one. This paper does not equate those mechanisms wholesale with bypass, and compares them one by one only where relevance is clear.
Under the discipline this section passes no evaluation. The gap between proxy and purpose is a structure, not a fault; a line that uses a proxy is not thereby to be accused of having done something wrong.
[^7]: What this paper relied on is Manheim and Garrabrant (2018), listed in the references; it has not checked Goodhart's original or the primary sources of the individual mechanisms. If the names and conditions of the mechanisms do not match that paper, this section should be revised accordingly.
5.7 Messick and Borsboom: they evaluate, this paper only registers
In the account this paper relied on, Messick (1989) distinguishes two kinds of mismatch: one where the measurement fails to cover important aspects of what it claims to measure, and one where scores vary because of factors outside the construct measured. The first is very nearly co-referential with the antecedent of section 2.
The difference is in the act. They evaluate — this is a deficiency of validity. This paper only registers — this item is relevant and has not entered the operation, and whether that is good or bad is judged on other grounds. This is the most useful interface this paper offers to readers on the side of scales and assessments, and the place where its neutrality discipline is easiest to see from outside.
There is a stronger way of writing it (Borsboom, Mellenbergh and van Heerden 2004), which ties the validity of a measurement to the existence of the attribute measured and its causal effect on the measurement outcome. This paper does not adopt that, because it claims nothing about how anything exists; but it is a ready comparison, showing that a rule for being wrong can be set stronger than this paper's, at the cost of a heavier ontological commitment.
5.8 Bromley and Powell: decoupling
Bromley and Powell (2012) distinguish two kinds of decoupling: policy–practice decoupling, where rules are unimplemented or routinely violated; and means–ends decoupling, where policies are implemented while the link between formal policies and the intended outcome is opaque.[^2]
The two sit at different places relative to this paper, and have to be treated separately.
Means–ends decoupling is close to the antecedent of section 2 — it has been done as prescribed, and the content the purpose bears on is not covered. It belongs on the same side as Messick's first kind of mismatch in §5.7, not on the same side as recorded but inert.
Policy–practice decoupling looks close to recorded but inert in shape and is not of the same kind, because what the two speak of does not sit at the same level. They are speaking of a rule not being carried out; this paper's recorded but inert is speaking of an item recorded by this line while the line's operation runs no differently for it. A fully implemented rule can carry a recorded but inert item quite happily — the field is filled in entry by entry as prescribed, and no subsequent calculation refers to it once; conversely, a rule that is not carried out does not by itself supply a recorded but inert candidate, since it must still be said which item of content, and then the two questions of §4.2 must be run.
The grounds differ accordingly. This paper's ground is the effective channel, item by item, relative to one drawing of a line; decoupling is a description of an organizational state, and what they give is which kind of decoupling is more likely under which conditions, not a decision procedure for individual cases. This paper does not declare the two the same thing because the shapes are close, and does not declare that it adds a layer because the grounds differ.
5.9 Two boundary markers
Barad (2007) marks the outer edge below this method's operations. In the account this paper relied on, her cut settles what there is — relata do not exist prior to relations, and boundaries and properties acquire determinate meaning only within a particular apparatus. This paper does not go to that layer: it presupposes that something is there, and only judges whether, relative to this drawing of a line, that something is inside the operation. Her cut comes before this paper, not inside it.
The line running from Schelling to Žižek (1996) marks the outer edge before this method's operations. In the account this paper relied on, the leftover in that line is there prior to any operation, and is singular and ontological. This paper does not deny that such a leftover exists; it simply does not take it as this method's object — this method's object begins with the drawing of a line. That is a choice of range, not a judgment about the world.
One further body of work was once in this section and is now removed. An impossibility result says that no aggregation rule can satisfy all the conditions at once, which is not the same kind of proposition as an item's not having entered the operation of some line; by the test of §5.1 this paper currently cannot specify a local structural correspondence, and so does not keep it by force.
5.10 Summary
The closest is Callon; what this paper delivers at that point is a decision procedure, and it offers no judgment on whether he has one doing the same job. Star and Bowker give this paper two things: taking in is not handling, and a blind spot does not display itself. Law asks a different question, and this paper's claim is the weaker. Gödel supplies the distinction between entering the expression and obtaining a result of some handling. The Goodhart line, validity theory, and the means–ends kind of decoupling each approach one structure from one side — through the gap between proxy and purpose, the gap between measurement and construct, and the gap between policy and intended outcome; the policy–practice kind of decoupling is not on that side, and is not of the same kind as recorded but inert either. The two boundary markers mark the edges below and before this method.
6. One drawing of a line, walked through
This section uses no empirical case. A thought experiment with everything stipulated is enough to show how a judgment is made and when it cannot be; demonstration on real institutions is left to later interface or reading papers.
6.1 The setup
A small Greek city states: to enter the library, one must be a philosopher.
That is a drawing of a line. The stipulations are as follows, and without any of them the inferences below fail. At the identity check in this example, being a philosopher is the sole standard; if yes, entry is registered, if no, it is not. Whether the books can actually be reached is checked elsewhere.
So three things are separate from the start: establishing the identity, registering the entry, and actual reachability.
6.2 A criterion given, no purpose given, the line unfinished
The regulation gives a criterion only. The purpose is not stated, and the object domain is stated only as far as one, not which ones — so by the Statement of Layers this line is unfinished and nothing can be judged yet. The object-domain layer is left to §6.3.
Supply the purpose and there is a line to judge. And one criterion with three purposes does not give three readings of one line: by the Statement of Layers a different purpose must be checked separately, so those are three different lines, each raising an entirely different question of relevance.
The first line: the purpose is to let philosophers in. Criterion and purpose coincide, which puts it in the class of §1.5, and the conclusion is usually that there is no remainder. But with a restriction: only where the purpose really does stop at completing this already-specified identity determination and entry registration, and where that really was carried out in full, is the conclusion had within that range. The name a task of determining a property guarantees by itself neither the absence of omission nor the absence of shortfall in execution.
The second line: the purpose is to let in those who can make good use of the books. Remainders pour out at once: the comprehension that medical or geometrical training brings, the learning needs of those still studying, the reading capacity of someone who does not present under the name of philosopher — all of these are content relevant to the purpose and unrecordable by this two-valued criterion. By §1.3 the candidates are those capacities and needs, not the physician, the geometer and the young person who carry them; people are bearers and do not occupy the candidate column.
The third line: the purpose is to keep the scrolls from being damaged. A different batch of remainders altogether: whether one is careful with things, whether one can read, whether one's hands are clean. Being a philosopher is nearly irrelevant under this purpose.
The three lines are alike in the criterion only. Section 2's antecedent is a cell the reader fills in by hand, once for each line.
6.3 The layer the object domain left unwritten
Does one include those from outside the city? The regulation does not say.
The written text cannot answer, so what one has at first is a batch of genus undetermined. But by §4.3, insufficient text does not mean nothing can be judged — whether the gatekeeper in fact lets foreigners through is something that can be looked into. Undetermined formally, determinable in fact.
Here §1.3 has to be used: candidates are items of content, not people. A foreigner let through or turned away is where a person ended up, not the handling status of an item of content. Three questions have to be asked apart: whether foreigners fall within the range actually checked; whether their status becomes an object of the classification; and whether their need for the books, or their capacity to make good use of them, entered an effective channel as relevant content.
The decision rule follows and is nailed down. Where the material shows only admission or refusal and says nothing about how the item of content took part in the handling, no genus-level judgment about that content may be made on it. Where the material suffices to show the content entered an effective channel, the finding is not a remainder. Only where relevance is established and there is definitely no effective channel is the finding remainder.
This is not asking the method to first take a view on who ought to be let into the library. Quite the reverse: only when handled-or-not is kept from being settled by proxy through outcomes favourable or unfavourable to a person does identifying a remainder is not an evaluation survive.
Incidentally, whether one is a member of the city is itself an item that can be run as a candidate. Depending on the stipulations it can fall under bypass — not on this line's table, acting by way of another line; it can fall under barred by rule — a separate written general provision on foreigners; and it may be excluded as irrelevant to the purpose. One and the same thing falls into different boxes as the stipulations change, and the reader is forced back to the test rather than to memorizing four names.
6.4 Four cases with the purpose fixed
Below, the purpose is fixed as letting in those who can make good use of the books. Four cases, each plugging one hole.
The foreigner case plugs the unwritten object domain; the finding is above — genus undetermined, determinable from the actual operation.
The just-deceased philosopher case plugs the point in time. The criterion does not say when the determination is made, and does not say whether the predicate lapses at death. The written text cannot answer, and actual practice cannot be looked up either, because the case never arose. This is a genus undetermined of evidential underdetermination — not a failure to fit a box, but no way to look into the material for either question; and the two questions are short of different things. The first is short because the purpose says nothing about whether the dead remain among those who can make good use; the second is short because there are no execution records to look into at all. The register has to write those two out separately and may not compress them into one line reading insufficient evidence.
The paralysed philosopher case plugs the admitted side. Three things are stipulated here, the second of them about the purpose and stipulated independently of the identity criterion: two philosophers carry the same identity record while differing in whether they can actually reach the books; under the other conditions held fixed in this example, one who can reach the books meets the competence the purpose requires and one who cannot does not; and this line has no other effective channel that handles the difference.
The second must be stipulated and cannot be derived. From being able to reach is a necessary condition of making good use one gets only that whoever cannot reach fails to meet it; one does not get that whoever can reach thereby meets it — the two may perfectly well both still lack some other condition of making good use, in which case what the purpose calls for is the same for both and the difference turns out irrelevant. This example stipulates that reachability is here the deciding condition, and only then does relevance stand. Nor may the gap be filled by the fact that both are philosophers: fill it that way and the identity criterion is once again answering on the purpose's behalf.
So, the first question: a different value for this item makes a difference to what the line, by its purpose, ought to do — one ought to be let through and one ought not. Nothing here asked whether anyone actually was let through, and nothing asked whether the line brought anything off; what was asked is only what ought to be done under the purpose.
The second question: whether the books can actually be reached is an item this line never recorded from beginning to end; the only slot on its table is the identity bit, and the line has no other channel. The second question fails. The finding is remainder, of type unheld by form.
It must be said precisely which difference is not held. Whether the books can be reached could perfectly well be recorded in a two-valued field; the problem is not that a two-valued form cannot hold it in general, but that the only two-valued form on this line records being a philosopher and does not record this separate difference. Nor does the word paralysed by itself establish the absence of a channel; that rests on the third stipulation.
This case is the pivot of the section. Admitted on paper, unable to get in in fact — what it displays is precisely §1.3: where the person ends up and whether the content was handled are two different things.
The brilliant student case plugs the fact that the predicate is itself a matter of degree.
Take the identity layer first. Does he count as a philosopher? — that question applies the criterion to a person, which by §1.3 is not a candidate, so no genus-level finding can be made here at all. To make one, the candidate has to be named first: whether his teacher's standing is distinguished.
For that item the written evidence is short: the criterion is two-valued while philosopher is a predicate of degree, there is no line between student and master, and the text does not say whether a teacher's standing counts. But by §1.5, insufficient text means going to look for evidence about actual practice, not exiting straight to undetermined. What the gatekeeper actually does can be looked into. If that unwritten practice is stable enough to have a purpose, an object domain and a criterion of its own, then after checking the full conditions of §3.7 this item may be registered as a bypass relative to the written line — the second line's presence is one condition among them, and the two questions and the table status still have to be walked separately. If no such line can be generalized out, all that follows is that the bypass label cannot be had on that ground; the genus-level status is still judged per §4.5. If this occasion's material suffices to show the item entered this line's effective channel, the finding is not a remainder; if it suffices to show the item is relevant with no channel, the finding is remainder. Only where both kinds of evidence are short is the entry genus undetermined. This is §3.7's dividing point on site.
Now the content layer. One step here has to be taken correctly, and once taken correctly the conclusion runs against intuition.
First cut the candidate out and fix its range of variation, per the sixth rule of §4.3: what is being judged is the difference between six and nine inside the admitted group, not the full range of the attainment property. Change the range and the conclusion changes — stipulate separately that attainment of zero fails the competence the purpose requires (stipulated against the purpose, not against who got admitted), and then with the full range of attainment as the range, zero and six call for different treatment under the purpose and the first question passes at once — so the range must be registered with the candidate and may not be adjusted afterwards.
Then judge the first question, and judge it against the purpose, not against the criterion. Stipulate two things independently of the criterion: six and nine both meet the competence the purpose specifies, and the purpose does not call for further ranking among those who meet it. On that, as this item's value varies between six and nine, what the line ought to do under its purpose does not change — both ought to be let through. The first question does not pass, the item is excluded here, and no remainder is found.
The order may not be reversed. One may not say that both are recorded as one by the criterion and both were admitted, therefore there is no difference — that is letting the criterion's sorting result answer on the purpose's behalf, exactly the move nailed down at §4.2's second point. Suppose the criterion records anyone at five or above as one while the competence the purpose bears on requires eight: six and nine still carry the same record under the criterion, and fall on opposite sides under the purpose. What this example's finding of irrelevance rests on is the two independent stipulations of the previous paragraph, not on both being recorded as one.
To make it a remainder it would have to be made relevant, and that means changing the purpose: say, that the order of access to the books is set by level of attainment. But by the fifth rule of §4.3 the purpose is fixed at registration and may not afterwards be extended to make some item relevant. Add the ordering and that is another line, with its three components registered afresh. On that line the difference between six and nine makes the order that ought to be set a different one, and the first question passes; while the criterion is only that two-valued record, six and nine are recorded alike as one, the record cannot recover the precision being asked about, and the line has no other channel — the second question fails. So on the ordering line it is a remainder, of type unheld by form.
The two lines must be registered separately and may not be read as one. This is the thing in the section most worth keeping: one and the same difference is irrelevant on the admission line and a remainder on the ordering line. What settles its status is the purpose and the cut, not the item itself.
This runs with the AB-and-BA setup of §3.2: whether what is asked about is the count, the order, or some finer difference has to be fixed first — and in this example one more thing has to be fixed besides, namely what the line ought to do under its purpose.
What this case delivers is the character of the method. It does not answer whether he counts as a philosopher after all, and it does not announce that differences inside the admitted group are remainders; it only takes the problem apart far enough to be answered item by item. The method does not make your judgments for you. It lets you see where your criterion cannot make a judgment, and which judgments were never this line's business at all.
Among the four cases there is no positive instance of barred by rule or recorded but inert. That is as it should be: this section walks the method through, it does not fill the boxes. By §3.1 the hit set may be empty; by §4.7 the value of the register lies in every column having grounds, not in some column being filled fullest.
6.5 Adding a class of researchers
The city later changes the criterion to admit philosophers or researchers.
By the Statement of Layers this is another line, not the original one amended. The two lines share one purpose, so the two judgments are comparable — the shared purpose is exactly what makes them so.
The comparison has to be made item by item, and changing the line does not by itself guarantee any conclusion. Which item of content from the old line has entered effective handling on the new one has to be named specifically, along with the channel through which it entered; one more category by itself does not show that any item of content was handled. Whether the new line leaves unhandled content the old one did not has to be checked item by item as well: what counts as research shows that the new criterion's sense is yet to be settled, and a rule whose sense is unsettled and an item confirmed at the genus level as a remainder are two different states — to find it a remainder, relevance and the absence of an effective channel still have to be checked. And the final result of philosophers or researchers can still be two-valued, so one more category is no ground for saying the new problem did not exist under the old two-valued criterion.
Section 2 has already laid down that a new index does not inherit the old antecedent. This section may not, in order to demonstrate that a remainder remains, turn that around into something that happens every time a line is changed.
6.6 Three stipulated setups
The three below are not cases; they are part of the argument.
AB and BA both recorded as 2 with the question being which came first supports the formal argument for unheld by form. Two channels in one line, one outputting zero times x and one outputting x, supports the rule that a zero weight carries a scope. Two rule texts word for word identical, one really connected and one only kept on file, supports the point that written text cannot vouch for actual practice, and also supports §2.3's rejudging — the criterion can stay unchanged while the actual handling changes.
6.7 Summary
A regulation that gives a criterion only judges nothing. Supply the purpose and remainders appear; a different purpose is a different line, to be checked separately, and the batch it yields can be entirely different. Where the object domain is unwritten a batch of genus undetermined appears, part of which the actual operation can make good. The admitted side has remainders too. Where the predicate is a matter of degree, the relevance question has to be passed first, and passed against the purpose and not against the criterion: that a record failed to preserve some difference does not make that difference relevant to the line; and that two people were sorted to the same side by the criterion does not make what the purpose calls for the same for both. To make the difference relevant, one must change the line. And once the line is changed, everything is judged afresh.
7. What this method can discover, and on what condition
The trail this paper develops — working back from acting on another line to an omission on this one — needs the corresponding lead to enter the search. Where the lead has not yet appeared or has not been recognized, the trail cannot by itself guarantee that every relevant candidate is raised.
That shows the discovery capacity is conditional. It does not show that what has not been raised does not exist, and it does not prove that discovery must wait on this one way of showing up.
Nothing unknown is used to certify that objects of some class already satisfy the genus-level condition. Not having been seen does not automatically make something a remainder by this method — it must still be relevant to the line in question and lie outside that line's operation. Where there is no concrete candidate at hand, the limit of the method's capacity can be stated; remainder status cannot be issued in advance to an object out of nothing.
If a case is offered that is confirmed at the genus level and untyped, two situations must be kept apart. In the first, the candidate has actually been checked one by one against the four types' full criteria and the hit set came back empty — an empty set is an output this sorting sub-procedure allows, not a failure to have handled the candidate. To call did not obtain a positive label did not enter the handling is precisely the step §4.4 forbids, and this paper does not permit it when speaking of itself either. In the second, the material is insufficient to complete the type-level check, which is not checked, registered according to the facts as evidence pending. Neither amounts to a remainder of the method as a whole. Being checked and being caught by some type are two different kinds of taking in.
This is not the no method is perfect sort of modesty. It is a capacity limit with an explicit input condition: the condition can be written down, and can therefore be tested.
8. What the method is for
Identifying a remainder is not an evaluation, and it yields neither must be eliminated nor must be preserved. A reader who has come this far will ask: then what can it do?
What can be answered without passing an evaluation is this. It splits off, from is this measure any good, the prior step that can be stated item by item — which item of content, relative to which drawing of a line, on what material, with what conclusion; and where typing is possible, with a type attached. How good or bad is judged with another rule.
Where it is most useful is not in proving that however things are revised a remainder will always be there; this paper does not claim that. It is useful because it stops three things passing themselves off as one another: already classified, already recorded, and already actually handled.
And when real handling does occur, this method should be able to say so plainly: relative to the present line, this item is a remainder no longer.
References
I · Works in this series called on directly
Qin, H. (2026). Negativa: On negation prior to being (SAE Methodology Paper 0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.19544619
Qin, H. (2026). Via Rho: The way of the remainder (SAE Methodology Paper 00) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.19657439
Qin, H. (2026). On the dynamics of value (SAE Value Theory Paper 4) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.22050204
Qin, H. (2026). The finite bearer's carrying-forward (SAE Meaning Theory Paper 2) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.22713651
All four are cited by concept DOI, resolving to the latest version. Calls in the body always carry the paper number and the proposition number.
A concept DOI guarantees only navigation to the latest version and cannot by itself fix a quotation, so the version DOIs this paper relied on are recorded separately: Negativa, 10.5281/zenodo.19544620; Via Rho, 10.5281/zenodo.19657440; Value Theory P4, 10.5281/zenodo.22050205; Meaning Theory P2, 10.5281/zenodo.22713652. Those version numbers are taken from the version records obtained by resolving each concept DOI on Zenodo on 18 September 2026; each of the four has exactly one version, so the version text and the published text are the same. The three passages quoted from Via Rho (one sentence from the abstract, one from §6.2, and the core formula of §4.5), V45 and V59 of Value Theory P4, and the sentence from §2 of Meaning Theory P2 were checked sentence by sentence in September 2026, and section numbers and wording agree.
II · External works
The entries below follow standard citation. Where this paper has not checked the original, the body has been written, under the discipline of the Statement of Layers, to answer only for what this paper relied on; where citation details differ from the originals, the originals govern.
Barad, K. (2007). Meeting the universe halfway: Quantum physics and the entanglement of matter and meaning. Duke University Press.
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Bromley, P., & Powell, W. W. (2012). From smoke and mirrors to walking the talk: Decoupling in the contemporary world. The Academy of Management Annals, 6(1), 483–530. https://doi.org/10.1080/19416520.2012.684462
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Manheim, D., & Garrabrant, S. (2018). Categorizing variants of Goodhart's law. arXiv:1803.04585. https://arxiv.org/abs/1803.04585
Messick, S. (1989). Validity. In R. L. Linn (Ed.), Educational measurement (3rd ed., pp. 13–103). American Council on Education / Macmillan.
Rosser, B. (1936). Extensions of some theorems of Gödel and Church. The Journal of Symbolic Logic, 1(3), 87–91.
Star, S. L., & Bowker, G. C. (2007). Enacting silence: Residual categories as a challenge for ethics, information systems, and communication. Ethics and Information Technology, 9(4), 273–280. https://doi.org/10.1007/s10676-007-9141-7
Žižek, S. (1996). The indivisible remainder: An essay on Schelling and related matters. Verso.