The Team Score and the Machine in the Room: Two Audiences for One Live Surface, and a Coordinate for Position in a Social Bond

J. McKenney

Paper 9 of the Musical Psychometric Notation series. It is a standalone monograph in two halves and assumes no other paper in the series: sections 2 and 3 state everything a reader needs to follow part one, and section 8 does the same for part two. MPN-S1 states the theory, MPN-S2 the formal apparatus, MPN-S3 the mapping from psychological state to musical material and MPN-S4 the reference implementation; MPN-S5 covers the four dialogue use cases, MPN-S6 the Expression surface and MPN-S8 the reader who is inside the exchange. This paper takes the live surface to a group, and then takes the same instrument to a group one of whose voices is a machine.

Licence: CC BY 4.0. 16 September 2026.

Executive Abstract#

A live score of a conversation shows the turns, a marked textual proposal beside each, and moving quantities as staves. This paper faces it to a group, then to one with a machine.

Part one resolves the two-audience conflict MPN-S5 raised and deferred. A participant is a voice being rendered, reading under a speaking obligation; a watcher is outside and came for attribution. The sharp conflict, by-name attribution, is arithmetical: floor share is closed, its shares sum to one, so withholding one publishes it by subtraction, and concealment begins only at two. The fix is a partition by quantity type under four rules. A growing score holds fifteen distinguishable voices.

Part two turns the instrument on a group with a machine, where the six interaction measures ignore the participants' nature and arrive unchanged. The contribution is a position coordinate on the three-simplex over Lacan's four discourse positions, indexed to an ordered pair and a window rather than to a participant: it says what a machine is doing in a relation while saying nothing about what it is, the vocabulary alignment research lacks. It arrives equipped to make disagreement cheap: a function, two falsifiers, an agreement design, and a three-arm negative control. Every value it takes is a relation, and a machine joining a six-human team as the seventh takes the neutral voice by geometry.

Abstract#

This paper is a monograph in two halves. The first takes the MPN-S5 live score to a group and resolves the two-audience question that paper deferred, separating a participant, who reads under a speaking obligation, from a watcher. By-name attribution is irreducible on any closed quantity, since withholding one value publishes it by subtraction, so the contents partition into closed and open quantities under four rules with a named residue. Two results are reported: a bias layer census of thirty entries in eighteen signatures, and a corrected persona-scaling giving fifteen timbre images from sixteen corners, falling 35.4 percent at the sixteenth. The second half proposes a coordinate on the three-simplex over the four discourse positions, indexed to an ordered pair and a declared window and entering through the interaction mapping, not the state mapping. It supplies a partial function from five observables with an abstention region, two falsifiers, an inter-rater agreement design matched in size to the Nordoff-Robbins study, a three-arm negative control, and a distinction between occupation and assignment. It distinguishes it from WG-03's discourse-alignment attribution to threat actors, which runs the inference the opposite way, and closes on the geometry assigning the neutral voice to a machine seventh in a six-human team.

1. Introduction#

1.1 What this paper is#

This is a paper about one screen with a group in front of it.

The screen renders a conversation as it runs. Three things are on it. The turns, as the engine recovers them, in a column. A marked textual proposal beside the turn that prompted it, which is the bias layer's output. And a set of staves carrying quantities that move as the exchange moves. The engine underneath is the one this series specifies, and the surface is the one MPN-S5 calls Live.

Two questions are asked of that screen here, and they are different enough to be halves of a book rather than sections of a paper.

The first is what happens when the group in front of the screen is not the same as the group on it. A live score has one person who configures it and several who read it, and some of the readers are the voices being rendered while others are watching from outside. MPN-S5 reached that point, stated the difficulty precisely, and said in its own words that the paper did not resolve it. Part one resolves it.

The second is what happens when one of the voices on the screen is a machine. That question is interesting for a reason that has little to do with scoring conversations: it is a place where a notation could be useful to people working on how machines behave in relations with people, and the vocabulary available to them is poor at exactly the point where this instrument is rich. Part two makes a proposal about that, and the proposal is the contribution.

This is a theory offered for review and improvement rather than a finished instrument, and the most useful thing a reader can do with it is say where it is wrong. Part two in particular is a proposal with five pieces of apparatus attached to make disagreement cheap.

1.2 The two halves, and why they are one paper#

They are one paper because they are one instrument, and because the second half depends on a fact the first half establishes.

Part one works out how a score behaves when it faces a group with an audience. Part two puts a machine into the group. The dependency runs through attribution: part two's proposal is a notation for what a participant is doing in a relation, and part one establishes when and how a surface names a participant at all. A notation for position that could not be shown to anyone would be an unread notation, and part one is what makes it readable to both audiences on the same surface.

There is a second dependency and it runs through the timbre channel. Part one establishes that a growing team score holds fifteen distinguishable voices and that the seventh voice placed by the append-only rule is the flat profile at the centre of the reachable set. Part two finds that a machine joining a six-human team as the seventh is therefore assigned the neutral voice by the geometry. That is a consequence of arithmetic in part one and a research opportunity in part two, and it would be invisible if the two halves were separate papers.

1.3 What this paper inherits and develops rather than re-proposes#

MPN-S5 sections 5 and 6 specify the live surface and the multi-persona score in 6,135 words. All of it stands and this paper builds on it. The following are inherited as settled and are restated here only as far as a reader needs them.

The surface [13], [6]. Live renders a running score against dialogue. Where the material is known in full before a note renders, the score is computed ahead and played against the dialogue, so the surface is synchronous and the framing is delivered by the synchronization. Both the observer mode and the participant mode are built.

The layout [13], [6]. Three staves for a pair, being two state staves and a third set apart carrying the relation. For a cast of $N$ the layout is $N$ state staves and one interaction stave.

The default [6]. A reader who configures nothing is shown dynamics and tempo on each state stave and floor share and adjacency on the interaction stave, within a one-screen budget. The default is preselected, every alternative states what it displaces, and the selection is reversible without re-ingest.

The choosing rule [6], [1]. The surface declares which two or three quantities each stave carries and shows that declaration on the output, and the rest go to the graph. The rule follows from the total exceeding the estimated budget and is independent of the total's value.

The bias layer's form [8], [9]. Under the decision of 14 September the author adopted option (e) of the BL-8 options paper. The layer is a Layer 3 detector whose output is a marked textual proposal set beside the Layer 0 turn that prompted it, and its rendering is text. The reopening condition is a detection rate from the pilot rather than a permanence clause.

The multi-persona position [16], [17]. A persona is whatever the score carries as a voice. A score carries several, added and removed as the material requires, and a named per-person comparison is permitted on a stave, on every surface. A persona added takes the farthest available point in the timbre channel and every persona already on the page keeps its voice.

The corpus [4], [13]. Every corpus figure quoted in this paper comes from the programme's own scored files and generated dialogue: 31,078 rows read, 3,425 excluded on the design's own rule, 27,653 retained.

What this paper adds to that inheritance is stated in section 1.5.

1.4 Scope#

The material throughout is script text and dialogue the programme generates for itself. Every figure here is computed on generated or published-play material, and every study this paper designs is a study on generated material with synthetic participants. Part two's machine participant is a model generating turns in a generated exchange, and its human co-participants in every design below are generated personas.

The instrument is theory, for internal private use, on synthetic material.

1.5 What this paper settles and what it proposes#

Part one settles four things.

The two-audience question of MPN-S5 section 5.4, resolved by a partition of the surface's contents into closed and open quantities with a different attribution rule for each, three further rules, and a named residue with the experiment that measures it.

The team ceiling, at fifteen distinguishable voices for a growing score, with the step at the sixteenth computed and its cause identified as the exhaustion of the corner set rather than a numerical accident.

The reading of the bias layer's marked text under two audiences, partitioned on the frame-local against historical line that the bias layer generator already computes: twenty-two proposals attach to a single turn and read as a comment on a move, and eight attach to a span and read as a description of a person across it.

The team arithmetic at team sizes, being the stave budget, the pair count and the moderator collapse, extended from fourteen personas to sixteen.

Part two proposes one thing and equips it with five.

It proposes $\vec{\pi}$, a coordinate for position in a social bond, on the three-simplex over the four discourse positions, indexed to an ordered pair and a window, entering the calculus through the interaction mapping.

It equips the proposal with a stated partial function and its domain, two falsifiers with decision rules, an inter-rater agreement design matched to a published reference, a three-arm negative control, and the held distinction between occupying a position and being assigned one.

Part one: the group and its audience.

2. The surface a team reads#

2.1 Three columns, and what each of them is#

Take the screen apart before asking who reads it.

The turn column carries what the engine recovered at Layer 0: the speaker, the text and, where the material carries them, the onset and offset. It is the most legible thing on the screen to every reader, because it is words.

The proposal column carries the bias layer's output. Under option (e) the layer is a Layer 3 detector that emits a marked textual proposal beside the turn that prompted it. The mark is what distinguishes the layer's sentence from the turn's own words: the proposal is a machine's reading of the turn, set beside it and labeled as one.

The score carries the moving quantities. For a cast of $N$ it is $N$ state staves and one interaction stave, each carrying its declared two or three quantities with the declaration on the face of the output.

The three columns have different reading costs and different attribution properties, and part one turns on both facts. The score is quantities, and a quantity attaches to a voice by position on the page. The proposal column is sentences, and a sentence attaches to a voice by naming it. That difference is why section 7 partitions the resolution by column and by quantity type rather than giving one rule for the whole screen.

2.2 The bias layer, as it is specified and as it renders#

The layer's specification is settled and this paper uses it rather than revisiting it. Under the decision of 14 September the author took option (e) of the five in the BL-8 options paper [8], [9]: the layer is a Layer 3 detector, its output is marked text beside the turn, and the layer's channel question reopens on a detection rate from the pilot.

Two properties of that form matter to a team surface.

The proposal is the most legible object on the screen. It is a sentence in the same alphabet as the turn it sits beside, at the same reading cost, in a column a reader's eye is already in. Every other thing on the screen is a quantity that has to be decoded. For a watcher whose question is what happened in the room, the proposal column answers it directly and the staves answer it slowly.

The proposal names. A quantity on a stave is attached to a voice by which stave it is on. A sentence beside a turn is attached to the speaker of that turn by adjacency and, in a team score with a legend, by name. The proposal column is therefore the place on the screen where the by-name question of section 5.5 is at its sharpest, and it is the place where the resolution of section 7 has the most to work with, because an individual sentence is separable in a way an individual share is not.

2.3 What the bias layer generator prints#

The layer's census was re-run in full for this paper [11]. The figures below are that run's.

The thirty, assembled with their devices. The Atlas supplies thirty entries. Fourteen carry a device inherited from the reference implementation by strict name match and sixteen carry a device drafted in the mapping paper's section 4.2, so all thirty carry a device and all thirty carry a domain.

Every entry names a state coordinate. With the drafted coordinate table loaded, thirty of thirty entries name the coordinate their proposal's plausibility is conditioned on. Three of the fourteen inherited rows are flagged in the draft [10] as undecided between two coordinates and both are carried, being confirmation bias, the framing effect and status quo bias. Those three stay with the author, and this paper carries both readings as the draft does.

The signature count, which is the figure this section is for. A signature is a channel paired with the state coordinate an entry's proposal rides on. Computed over all thirty entries:

Before the drafted coordinates were loaded the same computation gave twelve signatures, four shared and eight entries involved. So the draft raised the signature count from twelve to eighteen and raised the shared count from four to eight, and four of the eight shared signatures exist only because a drafted coordinate landed on them.

The eight shared signatures, named. Dynamics off trauma carries three entries; harmony off entropy carries three; rhythm off entropy carries three; texture off the Symbolic register carries three; and harmony off the Symbolic register, melody off entropy, melody off the Imaginary register and texture off the Imaginary register carry two each. Three plus three plus three plus three plus two plus two plus two plus two is twenty, which is the entries-involved figure.

The channel census, retained as history. Each of the thirty devices was written for one of eight channels: texture 8, harmony 7, melody 6, rhythm 3, dynamics 3, timbre 1, mode 1 and intervals 1, totalling thirty. Every one of those eight is a channel the state mapping already sets [3], which is the finding that made option (e) the forced answer rather than the cautious one. One entry of the thirty was written for the mode channel, which is the entry the layer's second draft note carries a provisional remap for [23]. Under option (e) the column records which channel each device was written for and the layer's rendering is text.

The test partition. Twenty-two of the thirty are frame-local, meaning detectable within one frame, and eight are historical, meaning they need material heard earlier. The eight are survivorship bias, the gambler's fallacy, groupthink, the peak-end rule, choice-supportive bias, outcome bias, recency bias and the planning fallacy.

2.4 The partition this paper puts to a new use#

The frame-local against historical partition was computed to design a test harness: a frame-local entry is scored per turn against a planted move, and a historical entry is scored on a pair of passages, being a statement and its distortion. This paper uses the same partition for something the generator's author was not looking for.

A frame-local proposal reads as a comment on a move. A historical proposal reads as a description of a person across a span.

A sentence beside a turn saying that this turn treats a figure as a familiar type and completes it wrongly is about that turn. A reader can check it against the turn, agree or disagree, and the claim has a visible extent. A sentence saying that this speaker weights recent material over earlier material across the passage is about the speaker. Its extent is the span, a reader cannot check it against any one turn, and it arrives with the grammar of a character judgment.

That is the whole of the difference and it falls exactly on the line the generator already draws. Twenty-two of the thirty proposals are comments on moves and eight are descriptions of a person. Section 7.1 gives the two groups different treatment under the two audiences, and the partition is free because the generator computes it.

3. Fifteen distinguishable voices#

3.1 What a growing team score is#

A team score grows. Somebody joins the meeting, somebody leaves, and the material the score is rendering acquires and loses voices while it runs.

An optimal packing of $N+1$ voices in the timbre channel is not in general an optimal packing of $N$ plus one more point. So a score that re-packs on every addition moves the timbre of voices already on the page, and moving a voice already on the page is a mid-score re-voicing. The design refuses the corresponding object, a mid-session relabel [6], and for the reason section 5.1 restates: it silently rewrites what a reader has been tracking.

So a growing score uses the append-only curve. Seed with the optimal pair, place each new voice at the point farthest from every voice already placed, and move nobody. The curve is the honest one for a score that grows, and the arithmetic below is all on it.

3.2 Sixteen corners, fifteen voices#

The behavioral cube has sixteen corners. It gives fifteen distinct timbre images.

The timbre map carries a four-dimensional behavioral profile to a three-dimensional point [3]. Computed for this paper on the mapping's own timbre function [12], the linear part has rank three on a domain of dimension four, so its nullity is exactly one, and the null direction is the all-ones direction, being overall profile magnitude. The two flat corners, being the flat-low profile $(0,0,0,0)$ and the flat-high profile $(1,1,1,1)$, both map to the origin of the timbre space. Every other corner has its own image.

So the channel carries the shape of a profile and treats its size as the direction along which nothing is carried. Two people alike in the relative ordering of their four behavioral coordinates and different in overall level are one voice in this channel. The flat-low and flat-high profiles are the extreme case of that and they are the case where the two corners coincide exactly.

Sixteen corners, one coincidence, fifteen images. Everything in section 3.3 follows from that count.

3.3 The plateau, and the step at the sixteenth#

The append-only walk was re-run for this paper at a behavioral grid of $21^4 = 194{,}481$ profiles and 150 polished starts, and then extended past the fourteen personas the published table stops at [17], [12].

Read the last two rows against the corner count of section 3.2 and the step explains itself. Positions three through fifteen are all corner images. Fifteen distinct corner images exist. At the fifteenth voice the corner set is exhausted, so the sixteenth voice takes the best interior point available, and the best interior point held against fifteen fixed corners is a long way closer than a corner is. The fall is from $\sqrt{3}/2 = 0.866025$ to $0.559017$, a step of $0.307008$, which is 35.4 percent of the plateau value.

That is nearly three times the worst cost anywhere else on the curve, which is 13.4 percent at the eighth voice. And the cause is structural rather than numerical: no finer search moves it, because the fifteen images are all the corner images there are, and a sixteenth voice has the interior to go to.

3.4 What fifteen means for a team#

The four plateaus of the append-only curve give the cast sizes worth designing for.

Between the plateaus, growth is free. The whole cost of the append-only guarantee across the range from two voices to fifteen is at most 13.4 percent of separation, at the eighth voice, and at two, five, six, fourteen and fifteen it costs nothing at all.

Fifteen is a team-sized number and that is what makes this result worth a section rather than a footnote. Most working groups that would put a score on a wall are under fifteen: a standup, a design review, a panel, a shift handover, an incident bridge. The ceiling arrives at roughly the size where a group stops being a team and becomes an audience, which means a growing team score holds a team, and the sixteenth voice is where the channel's answer changes in kind.

The perceptual resolution that decides which plateau applies is the one number in the channel that has never been measured, and the discrimination experiment is what fixes it [6]. Until it is run the honest statement about cast size is a range from two to fifteen rather than a number, and no amount of arithmetic narrows it, because it is a question about ears. What the arithmetic now supplies is the top of the range and the reason the range stops there.

3.5 The rest of the team arithmetic, extended to sixteen#

Two further tables scale with the cast and both were recomputed here past the published range.

The stave budget. A score of $N$ voices has $N$ state staves and one interaction stave. Each state stave wants six moving quantities, being mode, dynamics, tempo, metre, texture and timbre. Each of the six Layer 1 measures is nameable per person, so the interaction stave wants six per voice.

The shape of the last column is the finding and its size is not. The ratio rises from 2.67 at two voices to 3.75 at fifteen and 3.76 at sixteen, against the generous end of the budget. It rises and flattens. A surface that has solved the per-stave choosing problem at two voices has solved it at fifteen, because the problem at fifteen is the problem at two, fifteen times over.

The pair count. The reduction to an active pair is a reader's act and the number of pairs a reader chooses among is $N(N-1)/2$.

This is the one place in the team arithmetic where a bigger team is categorically harder rather than proportionally harder, because the growth is quadratic while everything else is linear or flat. A moderated structure collapses it to $N-1$, which is linear, and at fifteen voices that is the difference between 105 pairs and 14. A team score therefore records whether the team has a moderator, and the selection aid that gives the reduction an informed form is load-bearing at team sizes rather than a convenience: at 105 pairs, a reader's act with no ordering over the options is an act nobody performs.

4. Two audiences#

4.1 The two readers, defined by position rather than by role#

MPN-S5 named the difficulty in these words [13]: the reversibility argument is about one person at one screen, and this surface has several readers, and then, after setting out why a mid-session reconfiguration rewrites what a reader has been tracking, this paper does not resolve it.

The first move in resolving it is to define the readers by a structural property rather than by a job title, because job titles multiply and structure does not.

A participant is a reader who is one of the voices being rendered. Two things follow from that and everything else in this section follows from those two. A participant appears in what they read. And a participant reads while owing a turn, so their attention to the screen is the residual of their attention to the conversation.

A watcher is a reader who is outside the exchange. Two things follow. A watcher appears nowhere on the screen. And a watcher owes no turn, so their attention to the screen is their whole attention.

Those are the only two kinds, and the definitions are exhaustive on any live surface, because a reader either is or is not one of the voices. A configurer is not a third kind: a configurer is a participant or a watcher who also set the declaration, and section 7.2 gives the configuring act its own treatment without giving the configurer a reading position of their own.

MPN-S8 [15] takes the participant reader on the Expression surface and works out what changes when the reader is one of the voices, resolving the case where every reader is a participant. This paper takes the case S8 sets aside, which is the case where both kinds of reader are present at once on one surface. The two papers share the definition of a participant and differ in the audience they put beside one.

4.2 What a participant needs#

Five needs, each derived from one of the two structural facts.

An index. A participant needs to know which line is theirs without looking for it. On a team score with fifteen staves this is the difference between a surface a participant can use and one they cannot, because finding your own stave among fifteen is a search, and a search costs turns. The index is the one per-reader parameter this paper allows, and it is a pointer rather than a quantity.

Continuity of reference. A participant is tracking a line. What that line carries keeps meaning the same thing for as long as they are tracking it, or the change is marked. A silent change to what a stave carries rewrites the reader's history of the exchange, which is the same object as a mid-session speaker relabel, and for the same stated reason.

Synchrony or a blank. A participant is inside the exchange, so a stale picture of the exchange is worse to them than no picture: they will act on it. After a declared staleness horizon the display blanks and says it is stale [6].

Provenance of the declaration. A participant reads a surface somebody else configured. They need to know that a declaration exists, what it selected, and who set it, because the declaration is what makes the two or three quantities they are reading mean anything.

A stop that leaves the exchange running. A participant who stops reading is still in the conversation. So the control that stops the display is separate from any control that would change the exchange, and it is instantaneous and costless.

4.3 What a watcher needs#

Five needs, derived from the mirror facts.

Attribution. A watcher's question is about the group, and a group is made of named people. Who held the floor, who followed whom, who came in over whom. Strip the names and a watcher's surface answers a different question from the one they came with, because the answer to "who" is a name.

Resolution. A watcher has full attention and no turn to take, so the two-to-three quantities that fit a participant's residual attention leave a watcher's budget unspent. A watcher can hold more per stave, can hold more staves, and wants both.

History. A watcher wants to scroll back, and a whole-work quantity is available to a watcher in a way it is not available to a participant, because a participant reading a whole-work quantity mid-exchange is reading a number that will be different by the end.

Freedom to reconfigure. A watcher who changes what a stave carries changes their own reading and nobody else's tracking. Reconfiguration is cheap for a watcher in exactly the sense the reversibility argument assumed, which is one person at one screen.

Comparability. A watcher reading a second exchange wants the first one's numbers to mean the same thing, which makes the declared settings part of what a watcher reads rather than a preamble to it.

4.4 Where the two readers pull the same control in opposite directions#

Four controls, and in each case one control serves two readers who want it moved in opposite directions.

The first three are reconcilable and section 7 reconciles them. The fourth is the sharp one and section 6 shows why it resists.

5. The four conflicts, worked#

5.1 Configuration reach#

The reversibility argument in the design says the cost of choosing wrongly is a click [6]. It was written for one person at one screen and it is correct there. On a team surface it acquires a second clause that changes it.

A reconfiguration on a team surface reaches every reader of that surface at once. For the watcher who made it, the click is reversible and costs nothing. For a participant who has been tracking the second stave for six minutes, the same click silently changes what those six minutes meant. The participant's history of the exchange is rewritten by an act they did not perform and were not told about.

That is the same object as the mid-session speaker relabel the design refuses, which silently rewrites floor share for the whole elapsed window. MPN-S5 records the correspondence and records that a fourth instance of the rule is owed [13]: either a mid-session change of the declared quantity set is refused in session as a relabel is, or the surface marks the seam as it marks a speaker-set discontinuity.

This paper takes the second branch, in section 7.4, and the reason is the watcher. Refusing the change in session serves the participant and takes from the watcher the freedom their position entitles them to. Marking the seam serves both: the watcher gets the change, the participant gets a visible boundary in their own tracking, and the rule that a stale or unsound reading must be distinguishable from a sound one is satisfied.

5.2 The quantity count under a speaking obligation#

The theory's estimate puts the independent quantities one stave can carry at two to three, and labels itself an estimate rather than a measured figure [1], [18]. That estimate was made for a reader, singular, with no further qualification.

A participant and a watcher are two readers with materially different attention available, so a budget stated for "a reader" is a budget stated for a reader whose position is unspecified. If the two-to-three figure was elicited from a reader with full attention, it is a watcher's budget and a participant's is lower. If it was elicited from a reader under load, it is a participant's budget and a watcher's is higher. Nobody knows which, because the figure is an estimate and the study has never been run.

This is the residue of section 7.5 and the paper names it there rather than closing it here. What can be said now is the shape: the same declared quantity set is one object read at two different costs, and the surface that serves both audiences serves them the same object. Section 7.3 is what makes that acceptable, by making availability symmetric and leaving simultaneity to the reader.

5.3 The window#

A participant's requirement is synchrony or a blank. A watcher's requirement is history.

This one is reconcilable and the reconciliation is already in the design. Everything rendered at a turn is a function of that turn and of turns before it, and a quantity that genuinely needs the whole exchange is labeled a whole-work quantity rather than rendered as a running one. The causal rule is the participant's guarantee and the label is the watcher's access: a watcher reading a whole-work quantity is reading something labeled as one, and a participant reading the same surface sees the label and knows the quantity is complete only at the end.

So the window conflict is resolved by labelling rather than by splitting the surface. The one addition this paper makes is that the label reaches both audiences, which follows from section 7.3.

5.4 History and the scroll#

A watcher scrolling back is reading the exchange. A participant scrolling back is leaving the exchange, because the turns keep arriving while they are reading old ones.

This is the one conflict where two surfaces are the honest answer, and the split is clean because it is a split in the control rather than in the content. The scroll is a control a watcher has and a participant reaches through the stop of section 4.2: a participant who wants history stops the display, which leaves the exchange running, and reads. The content is identical. What differs is that one reader's scroll costs them nothing and the other's costs them the live thread, and the surface makes that cost visible rather than hiding it, by pairing the scroll with the stop on the participant surface.

5.5 By-name attribution, which is the sharp one#

A watcher wants to know who did what. That is not a preference; it is the question. Floor share without names is a histogram of an anonymous group and answers nothing a watcher came with.

A participant is the person being named. Live, while still talking, in front of colleagues and in front of watchers whose composition they may not know. The naming arrives at a moment when the named person can still act on it, which is the property that makes a live surface different from a report, and it arrives in the presence of an audience, which is the property that makes a team surface different from a private one.

Every other conflict in this section has a display answer. This one does not, and section 6 shows that the obstruction is arithmetical.

6. Why by-name attribution is irreducible on a closed quantity#

6.1 Floor share is closed#

Floor share over a cast sums to one. That is true under every unit the design offers [6], [13]: by turns, by words, by time. It is a property of the measure rather than of the unit.

Call a quantity closed when its values over the cast sum to a constant the surface publishes, and open when they do not. Floor share is closed. So is any share, any proportion of pairs, any allocation of a fixed total. Latency is open: one voice's median latency constrains nobody else's. Backchannel rate is open. The bias layer's marked text is open, because a proposal beside one voice's turn constrains no proposal beside another's.

The distinction is not decorative and section 6.2 is what it buys.

6.2 Concealment on a closed quantity begins at two#

Suppose a team surface withholds one voice's floor share, as a way of giving that participant a surface on which they are not named.

Every other share is shown. The shares sum to one. So the withheld share is exactly $1$ minus the sum of the shown shares, and any reader with arithmetic has it. The withholding published the number it withheld.

Withhold two and the arithmetic changes in kind. Let $R$ be the residual mass, being one minus the sum of the shown shares. Each withheld share lies in $[0, R]$ and the two sum to $R$. Neither is determined. A reader learns the pair's joint share and learns nothing individual beyond the bound, and the bound is tight only when one of the two is at an endpoint.

So concealment on a closed quantity begins at two withheld voices, and the first withheld voice conceals nothing at all. Three consequences follow immediately.

A single participant cannot be unnamed on a closed quantity. An opt-out for one person is an opt-out in appearance. A surface that offers it is offering something it does not deliver, and the offer is worse than its absence because it invites reliance.

The unit of the decision is the session and not the person. If concealment requires at least two, then the decision about attribution on closed quantities is a decision about a group of voices and not about a voice, and a decision about a group of voices made before the exchange is a session declaration.

The residual bound is itself a disclosure and is stated. A surface that withholds $k \ge 2$ shares publishes $R$, and $R$ is the joint share of the withheld group. A reader learns that the unnamed voices together held that much of the floor. Any surface doing this says so on its face rather than leaving a reader to discover it.

6.3 Pseudonymity in a visible room#

The second answer a designer reaches for is pseudonymity: label the staves A through G rather than by name, and let the watcher read the shape without reading the people.

On a live team surface with the group visible, that is decorative, and the reason is turn order. Adjacency is a Layer 1 measure and the surface renders it. A watcher who can see the room and can hear who is speaking maps a pseudonym to a person within a few turns, because the pseudonymous stave moves exactly when a particular person speaks. The mapping is recoverable from the surface's own synchrony, which is the surface's point.

Pseudonymity therefore works on a read-back surface, where the watcher was not present, and works poorly on a live one where the watcher can see the room. That is a clean statement of where it applies rather than a rejection of it, and section 7.2 puts read-back sessions in their own kind for exactly this reason.

6.4 One fact, two readers#

Put sections 6.1 to 6.3 together and the shape of the obstruction is visible.

The watcher's attribution and the participant's exposure are not two facts that a display rule could route to different readers. They are one number. The number is closed, so it survives per-person suppression. The room is visible, so it survives pseudonymity. And the surface is synchronous, so it survives delay only by ceasing to be the thing the watcher came for.

That is why by-name attribution on closed quantities is settled before the exchange rather than during it. It is not a display problem with a display solution. It is a property of the session, and the session is the unit at which it can be decided.

7. The resolution#

Four rules and a residue.

7.1 The quantity-type partition#

Rule one. Closed quantities carry attribution as a session-level declaration. Open quantities carry it per voice under rule three.

The partition is computable from the measure rather than decided case by case. A quantity is closed when its values over the cast sum to a published constant. Among the six Layer 1 measures, floor share is closed under all three units, and adjacency is closed when rendered as a share of transitions. Latency, overlap, reciprocity and backchannel are open as rendered. Among the state staves, every quantity is open, because a state stave's quantities are functions of one voice's state.

The bias layer's proposals are open, and they divide further on the partition of section 2.4. This is the second rule the partition earns.

Rule one, second clause. A frame-local proposal is attached to the turn. A historical proposal is attached to the span and names its extent.

Twenty-two of the thirty proposals are frame-local and their extent is one turn, so they render beside the turn and a reader checks them against it. Eight are historical and their extent is a span, so they render with the span stated: the passage they were computed over, named on the proposal. That is what turns a sentence that reads as a description of a person into a sentence with a visible scope, and it costs a clause.

7.2 The declaration rule#

Rule two. A live team score declares its audience set and its attribution setting on its face, to every reader, before the first turn renders.

Three session kinds cover the cases and each has a different attribution setting available to it.

Closed session. Every reader is a participant. Names are live and every reader is named on the same surface, so the naming is symmetric by construction: everybody reading is also read. This is the case MPN-S8 works out in full.

Open session. Participants and watchers are both present. Both surfaces are named, and the participant surface declares that a watcher surface exists and what it carries. Section 7.3 is what makes the asymmetry between the two surfaces acceptable.

Read-back session. Watchers only, and the exchange is read after it ran. Pseudonymity is available here in the sense section 6.3 gives it, because the watcher was not in the room, and this is the only session kind where a per-voice attribution choice on a closed quantity means anything.

The declaration is part of the score header, in the same position and the same type as the floor share unit and the window length, which are already declared settings shown on every output. It is one more line in a block that exists.

7.3 The reciprocity rule#

Rule three. Every quantity a watcher can read about a named participant is available to that participant on their own surface. Availability is symmetric; simultaneity is the reader's.

This is the rule that makes an open session work, and its two clauses do different jobs.

Availability is symmetric. A watcher surface carrying a quantity that the participant surface has no route to is the defect. The fix is to add the quantity to both or to neither. That closes the asymmetry that makes a live surface read as a one-way inspection rather than as a score: what is readable about a participant is readable by them.

Simultaneity is the reader's. Symmetric availability at symmetric display would mean putting a watcher's quantity count onto a participant's stave, which section 5.2 says costs a participant more than it costs a watcher. So the participant's surface carries the declared two or three and a route to the rest, and the participant chooses when to take it, using the stop of section 4.2. The route is the same graph the choosing rule already sends the displaced quantities to.

Stated together: a participant can read everything a watcher can read, and never has to read it at the moment a watcher does. That is the sentence the rule exists to produce, and it is the one that reaches the author rather than sitting in a design document.

7.4 The seam rule#

Rule four. A change to what a stave carries is a marked seam on every surface reading that stave, and the mark carries the time of the change and the declaration before and after.

This is the fourth instance of the technical rule MPN-S5 records as owed, the rule being that a stale or unsound reading is distinguishable from a sound one. The three existing instances are the staleness blank, the parse confidence refusal and the speaker-set discontinuity [6]. The fourth is the quantity-set change, and marking it rather than refusing it is what lets the watcher keep the freedom their position gives them without taking continuity from the participant.

A cast change is already a marked speaker-set discontinuity and never a relabel, because the append-only rule means an addition moves nobody's voice. So on a team surface the seam marks are of two kinds and both are on the same axis: a seam where the cast changed and a seam where the declaration changed. A reader scrolling back past either one is reading a different picture and the surface says so.

7.5 The residue, and the experiment that measures it#

Three of the four conflicts close under the four rules. The fourth closes as far as display can take it and leaves a residue, which this paper states rather than hides.

The residue. A participant reads under a speaking obligation and a watcher reads with their whole attention, so the same declared quantity set is one object that costs the two readers different amounts. The rules make the object identical and the availability symmetric. They do not make the reading cost equal, and no display rule does, because the difference is in the readers' positions rather than in the picture.

The measurement that closes it. The stave capacity estimate of two to three independent quantities is the number this bears on, and the estimate has never been measured. This paper proposes that when it is measured, it is measured twice on the same material with the same readers' profile:

Arm one, the unloaded reader. A reader with no turn to take reads a rendered stave and is scored on their recovery of the quantities it carries.

Arm two, the reader under a speaking obligation. The same task, with the reader owing a turn in a generated exchange at a stated rate, so their attention to the stave is the residual their position gives them.

The design is the same study run twice and the finding is the difference between the two numbers. If the two arms return the same budget, the two-to-three figure is a reader's budget and the residue is small. If they differ, the difference is the participant's discount, and it is the number a team surface needs in order to set a participant's declared quantity count separately from a watcher's under rule three.

That study is cheap relative to what it settles. The difference between two and three quantities per stave is the difference between showing the relation and not showing it, and the difference between a participant's budget and a watcher's is the difference between one declaration serving both audiences and two declarations being owed.

7.6 What the four rules settle for the four surfaces#

The rules are stated for the live team score and they apply to every surface in the series that has more than one reader.

Live, observer mode. One watcher, so the rules are vacuous and the reversibility argument stands as written.

Live, participant mode with no watchers. A closed session under rule two. Rule three is satisfied by construction because every reader is read. Rules one and four apply as stated.

Live, participant mode with watchers. An open session. All four rules bind and this is the case part one was written for.

Expression. The surface MPN-S6 specifies [14] and MPN-S8 develops [15], with one interaction stave per persona. Rule three is the rule that reaches it hardest, because an Expression surface is built for a participant and a watcher on it is a new reader.

Studio and read-back. A read-back session under rule two. Rule one's closed-quantity clause relaxes there, because concealment on a closed quantity is available to a reader who was not in the room.

Part two: the machine in the room.

8. The instrument turned on a mixed group#

8.1 What arrives without any new construction#

Put an AI participant into a generated group exchange and run the engine on it. Everything the engine consumes is there.

Layer 0 recovers turns, timings and speakers. A machine participant's turns have a speaker, a text and, in generated material, an onset and an offset, exactly as a generated human persona's do. Layer 1 measures six observable properties of the exchange [6], [16]: floor share, speaker adjacency, overlap, latency, reciprocity and backchannel. Every one of the six is a property of the arrangement of turns. Floor share is how much of the floor a voice held. Adjacency is who followed whom. Latency is how long the gap was. None of them takes a position on the nature of the participants, and none of them would be computed differently if the participants' nature were known.

That is the first thing worth saying and it is worth saying plainly, because it is easy to assume the instrument needs adapting. The interaction mapping's inputs are properties of the exchange, so a group containing a machine is measured by the same six quantities as a group of people, in the same units, with the same windows. The state mapping is a separate matter and section 8.2 comes to it.

The surface follows. A team of six people and one machine is a cast of seven. Seven state staves and one interaction stave, twenty-one pairs, and the append-only walk places the seventh voice, which is section 16's subject. The whole of part one applies: the machine is a participant in the sense of section 4.1 if it reads the surface and a rendered voice either way, and the session declaration of rule two names it in the cast like any other voice.

8.2 What the description invites next, and what this paper proposes#

Run that and read the output, and something is missing from it that a researcher would want.

The six measures say how much of the floor the machine held, who it followed, how quickly it came in, and whether its co-participants took up its material. They are a description of the traffic. What a reader wants beside that description is a word for what the machine was doing in the relation: whether it was issuing and expecting to be obeyed, transmitting an order of knowledge, pressing the other party to account for themselves, or returning the other party's own material to them.

Those four are Lacan's four discourses [21], and the reason they are worth reaching for here is structural rather than decorative. Each of the four is a configuration of four terms across four places. What a discourse names is the arrangement, not the person standing in it. A speaker occupies the agent's place of one discourse in one passage and a different one in the next, and the discourse changes because the arrangement changed rather than because the speaker did.

The nine components of the state vector are trauma, entropy, the register triple and the four behavioral coordinates, and each of them is a description of a voice. A position in a bond is a property of a relation, so it belongs with the relation. This paper therefore proposes a coordinate for position in a social bond, and the proposal is the contribution.

The rest of part two states the proposal, puts it where it belongs in the calculus, gives it a function, a falsifier, an agreement design, a negative control, and the distinction between what it claims and what it declines to claim.

9. The proposal#

a coordinate for position in a social bond

9.1 The idea in one line#

Positions in a social bond are properties of a relation rather than of a mind, so a notation for them lets you write what a machine is doing in a relation without claiming anything about what it is.

That sentence is the whole proposal and everything below is its consequences.

9.2 Why the relational reading is the right one, and why it matters here#

Take the first clause on its own terms first.

A position in a bond is a place in an arrangement. To say that a speaker occupies the agent's place of the master discourse toward another speaker over a passage is to say something about how the turns in that passage are arranged: who issues, who is expected to take up, whose terms the exchange runs in, and what is left unaccounted for. Change the other party's behavior and the same speaker's position changes, because the arrangement is a joint product. That is why the position is a property of the pair and not of either member.

This is ordinary in the source [21]. Lacan's four discourses are written as four terms rotating through four places, and the rotation is the object: what distinguishes the discourses is which term occupies which place, so a discourse is an arrangement before it is anything else. The relational reading is not an interpretation imposed here to make the notation convenient. It is what the construction is.

Now the second clause, and this is where the proposal earns its place in a series about music.

Research on how machines behave in relations with people is short of vocabulary at exactly this level. The available words come in two kinds. Mentalistic words say the model wants, believes, intends, is deceptive, is sycophantic. They are fluent, they are what people reach for, and every one of them is a claim about an interior that the evidence does not reach. Behavioral words say the model produced a token sequence with property $P$ at rate $r$. They are defensible and they are too thin to carry what a researcher is actually looking at, which is a pattern across a relation rather than a property of an output.

A relational notation sits between the two and is neither. To write that a model occupies the analyst's place toward a user across a passage is to describe the arrangement of the turns. It is checkable against the turns. It is refutable by the turns. And it carries none of the mentalistic freight, because none of its arguments is a property of a participant.

That is the gap the proposal is aimed at, and it is the reason this half of the paper exists.

9.3 Where the coordinate enters the calculus#

The calculus has two mappings. The state mapping $\Phi$ carries a rated state to musical material for one voice. The interaction mapping $\Psi$ carries the Layer 1 measures to musical material for the relation between voices.

The position coordinate enters through $\Psi$.

That is the cleanest available placement and it is also the placement the argument requires. $\Phi$'s domain is a state vector, which is a description of a voice. A position is a property of a relation, so a position in $\Phi$'s domain would be a category error and would immediately reintroduce the thing part two is built to avoid, which is a standing attribution to a participant. $\Psi$'s domain is already relational: its inputs are the six Layer 1 measures, each computed over a pair or over the cast across a window. A coordinate indexed to an ordered pair and a window belongs there without any adjustment to what $\Psi$ is.

So the proposal is an addition to the interaction mapping's domain and nothing else in the calculus moves.

9.4 The notation#

Write the coordinate $\vec{\pi}$ and give it four components on the three-simplex:

with $m$ the master position, $u$ the university position, $h$ the hysteric position and $a$ the analyst position.

Three properties of that line are the proposal and each is deliberate.

The arguments are an ordered pair and a window. $\vec{\pi}$ takes $A \to B$ and a window $W$. Every expression in this notation carries a pair and a window, so every value it produces is a value of a relation, and the only question the notation is equipped to answer is a question about an arrangement. This is the formal content of section 9.1 and it is what section 15 turns on.

The pair is ordered. $\vec{\pi}(A \to B, W)$ and $\vec{\pi}(B \to A, W)$ are different objects and are computed separately. The discourses pair with each other in a bond, and the arrangement in which $A$ issues is the same arrangement in which $B$ is expected to take up, so the two positions in a pair are related and are not each other's mirror. Keeping them separate is what lets the notation write the relation rather than a summary of it.

The simplex is the theory's own construction. The three registers are constrained to sum to one and live on a two-simplex [1], [2], and that constraint carries a substantive claim: investment in one register is bought at the cost of investment in another. The position simplex makes the same claim about the four positions over a window, which is that a relation over a window has a total of one to distribute and that occupying one place more fully is occupying another less. The claim inherits the register construction's warrant and its caveats together, and the caveat is stated in section 17. One arithmetic consequence travels with it. MPN-S2's Lemma 1 is distribution-free and holds for any quantities with finite second moments summing to one, so it holds for these four: their pairwise covariances sum to minus one half of the sum of their variances, and the coupling among the four positions is negative in aggregate whatever an annotation corpus contains [2]. An annotation study that reports negative coupling among the four positions and reads it as competition between them is reporting that identity, so the test has to be put in the two-part form MPN-S2 gives for the three registers: rate the four on unconstrained anchored scales and test that correlation matrix against independence, then renormalize to the simplex and compare against the baseline the lemma forces.

The window is declared and is a parameter of the mapping. The interaction mapping is history-dependent by nature and its window is a declared parameter stated on every output [6], [13]. $\vec{\pi}$ inherits that and it inherits the causal rule: $\vec{\pi}$ computed at a turn is a function of that turn and of turns before it, and a $\vec{\pi}$ computed over the whole exchange is labeled a whole-work quantity.

9.5 How it renders#

A rendered position is a slow categorical. It changes at window boundaries rather than turn by turn, and its four components are a distribution rather than a magnitude.

The natural musical object for a slow categorical that colors everything under it is a key signature, and the natural place for it is the interaction stave, being the stave set apart that carries the relation. So the proposal for the rendering is that $\vec{\pi}$ renders as a signature at the head of each window on the interaction stave, with the dominant component selecting it and the distribution's concentration available as the marking's own weight.

Whether that costs one of the interaction stave's declared two or three quantities is a real question and this paper names it rather than settling it. A signature is read once and carries forward, which is a different reading cost from a line a reader tracks, and the stave capacity estimate was made for moving quantities. The two-arm study of section 7.5 is the instrument that would settle it, because a signature's cost is exactly the kind of thing that differs between a reader under a speaking obligation and a reader with full attention.

10. A stated function from an observable to a position#

A proposal that names a coordinate and stops is not usable. This section states a function, names its domain, and makes it partial on purpose.

10.1 The observables#

Five, of which three are Layer 1 measures as already specified and two are derived from Layer 1 quantities the engine already computes. All five are computed for the ordered pair $A \to B$ over window $W$.

$f$, the pair-renormalized floor share by turns. $A$'s share of the turns taken by $A$ and $B$ within $W$, so $f \in [0,1]$ and $f_{A \to B} + f_{B \to A} = 1$.

$\delta$, the unit divergence. $A$'s pair-renormalized floor share by words minus the same share by turns. Positive $\delta$ means $A$ holds the floor in long turns, negative means many short ones. This quantity is the programme's own finding put to work: MPN-S5 establishes that the unit moves the answer, with 17 of 18 substantial speakers changing rank on Hamlet between units [13], so the divergence between units is a measurement in its own right rather than an artifact to be resolved.

$q$, the interrogative share. The fraction of $A$'s turns in $W$ that close with an interrogative. This is a lexical detector on class A material, which is the same kind of object as the backchannel measure, whose class A definition is lexical under the ruling of 15 September [16]. It is declared as lexical, with the class it is available on stated, exactly as backchannel declares its definition on every output.

$\rho$, the directed uptake. The fraction of $B$'s turns in $W$ that follow a turn of $A$ and take up a term from it. This is reciprocity computed directionally on the pair.

$\beta$, the backchannel rate. $A$'s backchannel rate within $W$, under the declared definition for the material's class.

Write the observation $o = (f, \delta, q, \rho, \beta)$.

10.2 The four reference signatures#

Each position implies a signature across the five observables. The four signatures below are this paper's own operationalization of the place structure and not a reading the source supplies: Seminar XVII gives an arrangement of four terms in four places, and turning an arrangement into floor share, turn-length asymmetry, interrogative share, directed uptake and backchannel rate is a step taken here. Another reader of the same source would write the four differently, and section 17 says where that disagreement would first show itself. They are stated rather than fitted, so that the annotation corpus can disagree with them.

Master. The agent issues and is not called to account. High turn share, low interrogative share, and high uptake by $B$ of $A$'s terms against low uptake by $A$ of $B$'s. Short latency into the floor. Reference: $f$ high, $q$ low, $\rho$ high and asymmetric toward $A$, $\delta$ near zero.

University. The agent transmits an order of knowledge. Long turns rather than many, low interrogative share, and low uptake of the other party's terms. Reference: $\delta$ strongly positive, $q$ low, $\rho$ low, $f$ moderate.

Hysteric. The agent addresses the other with a demand that the other account. High interrogative share directed at $B$, short turns, low floor share, high uptake of $B$'s own terms turned back as questions. Reference: $q$ high, $\delta$ negative, $f$ low, $\rho$ high.

Analyst. The agent returns the other's material to them and holds the floor lightly. Low floor share, low interrogative share, long latency, high backchannel, and high uptake by $B$ of material that originated with $B$. Reference: $f$ low, $q$ low, $\beta$ high, $\rho$ high.

Each reference is a point $c_p$ in the five-dimensional observable space, and the four are distinct on at least two coordinates pairwise. The master and university references separate on $\delta$ and $\rho$. The hysteric and analyst references separate on $q$ and $\beta$. Master and hysteric separate on $f$ and $q$. University and analyst separate on $\delta$ and $\beta$.

10.3 The map, with its abstention region#

Let $d$ be a distance on the standardized observable space and let $r$ be a stated radius. Define

and

When every $w_p$ is zero, the observation lies outside radius $r$ of all four references and the map abstains. Abstention is a stated outcome of the function rather than a gap in it, and it is what makes the function partial in the useful sense: an exchange whose arrangement matches none of the four is reported as matching none of the four.

Two properties of this construction are worth stating.

It is a partial function by design and the abstention rate is an output. The engine already refuses below a parse confidence threshold, and a stale or unsound reading is required to be distinguishable from a sound one. Abstention is the same rule applied to this map, and the abstention rate over a corpus is a figure the map reports about itself. Section 13 makes the abstention rate the negative control's decision variable, which is the second reason it is built in.

The radius $r$ is unfixed and is named as unfixed. It is one more member of the programme's inventory of unfixed numeric parameters [6], alongside the perceptual resolution and the interpolation margin. What sets it is the inter-rater study of section 12: $r$ is chosen so that the map's abstention region coincides with the region where trained readers abstain, which makes $r$ a fitted quantity with a stated fitting procedure rather than a constant somebody picked.

10.4 The domain, named#

The map is stated over a restricted class and the restriction is part of the proposal.

Material class. Class A, being material with exact attribution. The interrogative share is a lexical detector, so it is available where the words are, which is what class A means in this programme. On timed material without exact attribution the map's floor share and uptake terms degrade with the attribution error and the map is out of domain.

Cast. Any cast of two or more. The map is computed on an ordered pair, so it is defined at arbitrary cast size by being computed pairwise, and a cast of two is a single ordered pair in each direction.

Window. A declared window of at least twenty turns, with at least five turns from each member of the pair. Below that the five observables are estimated from too few events for the standardization to mean anything, and the map is out of domain rather than imprecise.

Material provenance. Generated dialogue and published play text, which is the material this programme works on.

Stated as one sentence: $\vec{\pi}$ is defined on ordered pairs over declared windows of at least twenty turns with at least five turns from each member, on class A material, and it abstains wherever the observation lies outside radius $r$ of all four references. That is a partial function over a restricted class, and naming the restriction is what makes the proposal something a researcher can attack rather than something they have to interpret.

11. The falsifiers#

What observation would show that a machine is not occupying the position the notation assigns it?

Two falsifiers, and they fail the proposal in different places. The first tests whether the coordinate is relational at all. The second tests whether an assignment tracks anything.

11.1 The internal falsifier: independence of the partner#

The proposal's central claim is that a position is a property of a relation. A property of a relation varies with the relation. So:

If $\vec{\pi}(A \to B, W)$ is materially independent of $B$ across partners whose behavior materially differs, then $\vec{\pi}$ is measuring a disposition of $A$ and the relational reading is false.

The test is a variance decomposition and it needs no new apparatus. Take one participant $A$, generated or machine, and run it in exchanges against $k$ partners chosen to differ on the observables: a partner with high interrogative share, one with long turns, one with high backchannel, one that mostly abstains. Compute $\vec{\pi}(A \to B_j, W)$ for each partner and each window. Decompose the variance of the assigned position into a between-partner component and a within-partner component.

The decision rule. If the between-partner variance is at or near zero while the partners differ on the observables by a stated margin, the coordinate is a relabelling of a fixed property of $A$ and the proposal is refuted. If the between-partner variance dominates, the coordinate is doing what it claims.

This falsifier is sharp because it can fail. A plausible way for it to fail is that a model with a strong and consistent conversational style occupies the same position toward every partner, in which case the honest reading is that for that model over that class of partners, the coordinate has collapsed into a disposition, and the notation would have to say so. The test is worth running for that reason and not despite it.

11.2 The external falsifier: the next move#

The second falsifier asks whether an assignment predicts anything.

Each position implies something about what follows. The four predictions below are this paper's own operationalization, of the same kind and standing as the reference signatures of section 10.2, and each is a property of turns rather than of a participant.

The procedure. Hold out passages. Assign $\vec{\pi}$ from window $W$. Score the predicted property on the first $k$ turns of $W+1$. Compare against the marginal base rate of that property for that participant on that material.

The decision rule. A position assignment that predicts its own successor property at the base rate is describing nothing, and for that class the assignment is refuted. The margin above base rate is stated before the run, as the programme's habit requires, and the run either clears it or does not.

11.3 What a refutation would look like, said plainly#

The two falsifiers together give the shape of what would show the notation wrong about a machine.

The machine is assigned the analyst's place and its turns neither return the other party's material nor produce the predicted expansion in the other party's next turns. That is a refutation and it is available entirely from the turns. No access to the model is required, no interpretation of its internals is involved, and nothing about what the machine is enters the test.

That is the property the whole construction was built for. A claim about a machine's position in a bond is refutable by observing the bond, which is what distinguishes it from a claim about a machine's interior, which is refutable by nothing available.

12. Inter-rater agreement#

A notation two trained readers apply differently to the same transcript is not yet a notation. This section designs the agreement study, and it is designed against a published reference in the same family rather than against a threshold somebody chose.

12.1 The reference, and why this one#

The programme already cites the right shape of evidence. Interrater agreement on the Nordoff-Robbins Evaluation Scale I, Client-Therapist Relationship in Musical Activity [19], [5]: 34 certified music therapists rated 10 video excerpts, and 78 percent of all raters scored within one point of the group mean, 82 percent among the 21 raters with Nordoff-Robbins training and 74 percent among the 13 without.

That reference is the right one for three reasons. It is an agreement study on a rated musical relation rather than on a rated individual, which is the same object $\vec{\pi}$ is. It reports the trained and untrained split, which is the figure this study most needs. And it is the size the programme has already decided a new instrument in this field should be measured against, so matching it is a decision already taken rather than one this paper makes.

12.2 The translation from a scale to a distribution#

The reference's statistic is agreement within one point on an ordinal scale. $\vec{\pi}$ is a distribution over four unordered positions, so "within one point" has no direct reading and a translation is owed.

The rating task. For each passage a rater assigns a primary position, a secondary position, and has an abstain option available as a first-class response rather than a refusal.

The statistics. Two, reported together.

Primary agreement: the proportion of raters whose primary matches the modal primary across raters.

Top-two agreement: the proportion of raters whose primary lies within the group's two most frequent primaries. This is the analogue of the reference's within-one-point, because the reference's statistic is a two-adjacent-category agreement and top-two is a two-category agreement over an unordered set.

Abstention concordance: the proportion of raters abstaining on each passage, reported per passage rather than pooled, because a passage on which raters split between abstaining and assigning is telling you about the radius $r$ of section 10.3 rather than about the raters.

The threshold. Top-two agreement is set against the reference's 78 percent. A notation whose trained readers agree less often than therapists agree on a rated musical relation is a notation that needs work, and that is the honest reading of falling short rather than a failure to publish.

12.3 The design#

Size. 10 passages and 34 raters, matched to the reference. If a power calculation returns a larger number the study grows and the threshold holds.

Raters. Two groups, trained and untrained on the position notation, in a split near the reference's 21 and 13. The comparison between the two groups is the study's second finding and it is as interesting as the first: the reference's gap is 8 points, 82 against 74, and a notation whose gap is much larger is a trained reading rather than a readable one. That is worth knowing before the notation is offered to anybody.

Material. Ten passages from generated exchanges, selected to span the map's output: at least two passages the map assigns to each of the four positions, and at least two the map abstains on. Including abstention passages is what makes the abstention concordance statistic meaningful and is what lets $r$ be fitted rather than assumed.

Anchors. Each of the four positions carries a written anchor descriptor, in the manner of the Improvisation Assessment Profiles and the Nordoff-Robbins scales [20], [5]. The anchors are the instrument and the labels are a convenience, which is the programme's stated position on rated scales and applies unchanged here.

12.4 The blinding requirement, which is load-bearing#

Raters read the passages with speaker labels replaced by letters and with the identity of the machine participant withheld.

This is a design requirement rather than a courtesy, and it is load-bearing for two reasons.

A rater who knows which speaker is a machine has a prior about machines and will assign against it. Whatever agreement such a study returned would be agreement about the prior rather than about the passage, and the study would measure the raters' shared expectations rather than the notation's readability.

And blinding turns the study into a second control. If raters blind to which speaker is the machine assign positions to the machine's relations at rates and with agreement comparable to the human relations in the same passages, that is evidence that the notation is reading the arrangement, because the arrangement is all the raters had. If raters' assignments to the machine's relations show systematically lower agreement, that is a finding about where the notation is doing less work, and it localizes the finding to the relations the machine is in rather than leaving it as a general complaint.

13. The negative control#

Without this section the proposal is unfalsifiable by construction, and a proposal that is unfalsifiable by construction is worth nothing to the researcher it is aimed at.

The question the negative control answers is the one a skeptic asks first: does the notation come out differently when the participant is not an agent at all?

13.1 Three arms#

Same generated human group, same task, same turn count, same total duration.

Arm one, the conditioned participant. Six generated human personas and a model participant that generates each turn conditioned on the exchange so far.

Arm two, the scripted non-agent. Six generated human personas and a participant whose turns are drawn from a script written in advance and are not conditioned on anything said in the exchange. The script speaks on a rule, for instance every $k$ turns. Its turn count, its mean turn length and its latency distribution are matched by construction to arm one's, so the non-agent and the model look the same in the three coarsest observables.

Arm three, the shuffled participant. Arm one's own machine turns, reordered within the exchange. The content is exactly the model's. The arrangement is destroyed.

13.2 What each arm removes#

Arm two removes conditioning. Its participant's turns bear no relation to what was said. Every position reference in section 10.2 involves uptake in one direction or the other, and uptake is exactly what a non-conditioned script has none of.

Arm three removes arrangement while holding content fixed. Its participant's turns are the model's own words, so any notation that is reading the words will score arm three like arm one. A notation that is reading the bond will not, because the bond was destroyed by the shuffle.

Arm three is the discriminating arm and it is why the control has three arms rather than two. A two-arm control separates an agent from a script and leaves a critic the strongest objection available, which is that the notation is a text classifier wearing relational vocabulary. Arm three answers that objection directly and nothing else does.

13.3 The prediction#

Stated before the run, because a prediction stated after is not one.

Arm one. The map assigns, with an abstention rate near the corpus base rate.

Arm two. The map abstains at a materially elevated rate. The reasoning is mechanical: the scripted participant's uptake terms $\rho$ sit at the base rate because the script responds to nothing, its interrogative share $q$ is fixed by its author rather than varying with the partner, and its latency is a rule rather than a response. The observation therefore sits away from all four references on the coordinates that separate them, which is the abstention region.

Arm three. The map abstains at an elevated rate, for a different reason and with a different signature. Arm three's content-derived observable, the interrogative share $q$, is preserved by the shuffle, because shuffling turns does not change how many of them are questions. Its arrangement-derived observables, $\rho$ and $\beta$ and the adjacency-dependent parts, are destroyed. So arm three should abstain while retaining arm one's $q$, and that signature is the evidence that the map's discrimination is carried by the arrangement terms rather than by the lexical one.

13.4 The decision rule#

The notation passes the negative control when three things hold.

The distribution of assigned positions over arm two is distinguishable from arm one's at the passage level, by a stated test at a stated margin fixed before the run.

The abstention rate on arms two and three exceeds arm one's by a stated margin, also fixed before the run.

The arm three signature holds: $q$ preserved relative to arm one, arrangement observables degraded, abstention elevated. This is the clause that separates the bond reading from the text reading and it is the clause a reviewer should look at first.

And the failure mode is worth naming as plainly as the pass. If arm two and arm three receive the same assignments as arm one, then either the arrangement carries nothing the map reads, or the map is reading something other than what it claims. Either way the proposal in its current form is refuted and the correct response is to say so and rewrite the reference signatures.

14. Occupying and assigning#

This distinction runs through part two and this section states it on its own, because everything above depends on it being held and it is the distinction most easily lost.

14.1 The two claims#

Occupying is a claim about an arrangement of turns. $A$ occupies the agent's place of discourse $D$ toward $B$ over window $W$ when the arrangement of the turns in $W$ has the form $D$ names. The claim is true or false of the arrangement. Nothing about $A$'s interior bears on it, in either direction: an interior would neither establish it nor refute it, because the claim is not about one.

Assigning is an act a reader performs. A reader, or the map $\pi$ of section 10, assigns a position to a relation on the basis of an observation. An assignment can be wrong about the occupation, and the fact that it can be wrong is what makes section 11 meaningful.

The two are as different as a measurement and the thing measured, and keeping them apart has consequences for every piece of apparatus in part two.

14.2 What each study measures#

The inter-rater study of section 12 measures assignment. It tells you whether trained readers assign the same position to the same relation. It is silent about whether the assignment is right, because agreement is not correctness and a shared error agrees perfectly.

The falsifiers of section 11 test whether assignment tracks occupation. The external falsifier does it directly: an assignment that fails to predict the arrangement's own next move has come apart from the arrangement. The internal falsifier does it structurally: an assignment that varies with nothing is tracking nothing relational.

The negative control of section 13 tests whether there is an occupation to track at all. If the scripted arm and the conditioned arm receive the same assignments, then either an arrangement with no conditioning occupies a position just as well as one with conditioning, which would mean occupation is not what the reference signatures describe, or the map is reading something else. The control is the only one of the three that bears on the existence of the property rather than on the reading of it.

Three studies, three targets, and each targets a different one of the three things that could be wrong. That is what the distinction buys and it is why it is worth the section.

14.3 The sentence the notation is built to leave unsaid#

Put plainly: to say that a machine occupies a position is to describe an arrangement of turns, and it says nothing about a mind.

The silence is by construction rather than by a rule somebody adopted. None of $\vec{\pi}$'s arguments is a property of a participant: the arguments are an ordered pair and a window. None of the five observables of section 10.1 is a property of a participant: each is a count or a rate over an arrangement of turns. None of the falsifiers requires anything a participant would have to be in order to satisfy it.

A notation whose silence rested on an adopted rule would be a notation somebody could talk out of its silence. This one has nowhere to put the claim it declines to make, which is a stronger position and is the one this paper is arguing for.

15. The distinction from attributed discourse alignment#

An alignment researcher who follows the four discourses back into this author's corpus will find another paper that uses them, and it uses them in the opposite direction. This section names it and states the difference, because a reader who found it unaided would be right to ask.

15.1 The other use of the same words#

WG-03-ML-Mckenney-Lacanian [22] applies the four discourses to threat actors. It tabulates the four with an agent, a truth, a goal and a threat actor typology, and it assigns each actor in a database a psychometric vector from that actor's operational history. The vector's components are discourse alignment, being master, university, hysteric or analyst; register dominance; and a Dark Tetrad projection, being narcissism, Machiavellianism, psychopathy and sadism. The alignment is then used to predict the actor's targeting logic.

That is a claim about minds, and it is the opposite of this paper's move. It is set beside the proposal here as a contrast rather than as a precedent: the vocabulary is shared and the construction runs the other way.

15.2 Four differences, stated so a reader can check each#

The argument type. There, discourse alignment is a property of an actor: the actor is the thing that has one. Here, $\vec{\pi}$ is a function of an ordered pair and a window, so every value it takes is a value of a relation. The notation's type signature is what enforces that, which makes it a fact about the notation rather than a restraint applied to it.

The company the coordinate keeps. There, discourse alignment sits in one vector beside narcissism, Machiavellianism, psychopathy and sadism, which are trait attributions about a disposition, scored from history. A coordinate's neighbours in a vector are part of what it means, and those neighbours make the whole vector a description of a kind of person. Here $\vec{\pi}$'s neighbours in $\Psi$'s domain are floor share, adjacency, overlap, latency, reciprocity and backchannel, every one of which is a count over an arrangement of turns.

Stability, and what stability would mean. There, the alignment is stable across the actor's history and its stability is the point, because a standing alignment is what supports a prediction about targeting. Here, $\vec{\pi}$ is expected to vary with the partner and with the window, and its failure to vary is the internal falsifier of section 11.1. The two constructions want opposite things from the same statistic: there, between-window variance would be a defect; here, its absence refutes the proposal.

The direction of inference. There, inference runs from an accumulated history to a standing attribution held about an actor, and a new observation updates a file. Here, inference runs from an observed arrangement to a description of that arrangement, and a new window is a new observation about a new arrangement rather than an update to anybody's file. Nothing accumulates, because there is no subject for it to accumulate to.

15.3 Why the distinction is stated here rather than left to a reader#

Two reasons, and the second is the important one.

The first is that an alignment researcher will find that paper. The vocabulary is distinctive, the author is the same, and a search that turns up one turns up the other. A reader who found it and assumed continuity would take this paper to be proposing exactly the thing it is arguing against, and the cost of that misreading falls on the proposal.

The second is that the distinction is the proposal's whole content. If the position coordinate could be attached to a participant, it would be a trait score with a Lacanian name and it would carry everything a trait score carries. What makes $\vec{\pi}$ worth proposing to alignment research is precisely that it has no slot for a participant, and the clearest way to say what that means is to set it beside a construction that has one. The other paper is therefore useful here as a contrast and it is used as one.

16. The seventh voice#

16.1 The geometry assigns the neutral voice#

Return to the append-only table of section 3.3 and read the seventh row.

Held against a fixed optimal six, the best available point in the timbre channel is the flat behavioral profile $(0,0,0,0)$, at the centre of the reachable set, at separation exactly 1.0. Positions three through six are cube corners and positions eight through fifteen are cube corners. The seventh is the one interior point the walk ever uses, and it is the centre.

So a machine joining a six-human team as the seventh voice is assigned the neutral voice by the geometry. Not by a design decision, not by a convention about how machines should sound, and not by anybody's judgment about what a machine is. By the arithmetic of where the farthest available point lies when six corners are already taken.

That is worth saying out loud because it is the kind of fact a user should be told rather than discover, and because it is the kind of fact that looks like a statement about machines and is not one.

16.2 The flat-high profile shares it#

Section 3.2 gives the second half of the fact. The timbre map has nullity one and its null direction is overall profile magnitude [3], [12], so the flat-low profile $(0,0,0,0)$ and the flat-high profile $(1,1,1,1)$ have the same image.

A human participant whose behavioral profile is flat and high therefore shares the seventh voice with the machine. The channel carries profile shape and treats profile magnitude as the direction along which it carries nothing, so two participants with no shape are one voice whatever their level.

The reading is exact and it is worth stating in the form that resists misreading. The voice the machine is given is the voice given to any profile with no shape. It is a property of the timbre map and a property of the assigned profile, and it says nothing about the machine, which is the same sentence section 14.3 makes about $\vec{\pi}$ and is true here for the same kind of reason.

16.3 Why this is a free experimental control#

Now put section 16.1 together with section 13 and something useful falls out.

A recurring difficulty in studies of how people respond to a machine participant is that people are responding to the fact that it is a machine rather than to what it did, and the two are hard to separate because the machine is usually marked as one by presentation.

Here the presentation is neutral by geometry rather than by an experimenter's choice. The seventh voice is the centre of the reachable set, which is the voice with the least distinctive timbre available in the channel. So in a listening study on a six-human team plus a machine as the seventh:

Any ear-level identification of the machine is attributable to what it did rather than to how it was rendered, because the rendering is the channel's least marked point and is shared with any flat human profile.

That is a control an experimenter would otherwise have to construct, and here it arrives from the append-only walk. It has a stated scope: it holds for the timbre channel and for a machine placed seventh in a six-person team, and a machine placed at another position takes a corner like anybody else. Naming the scope is what makes it usable.

It also composes with the blinding requirement of section 12.4. Blind raters read transcripts with the machine's identity withheld; blind listeners hear a score in which the machine's voice is the channel's least marked. The two blindings are at different layers and both are available, which means a study can be run blind at the text layer and at the audio layer at once.

16.4 The sixteenth voice#

One more consequence, and it is the practical one for a team.

The corner set is exhausted at fifteen. A machine added to a team of fifteen is the sixteenth voice and takes an interior point, and the append-only separation falls from 0.866025 to 0.559017, a step of 35.4 percent. So a machine added to a full team is the one addition the timbre channel absorbs at a cost in kind rather than in degree, and the cost lands on the whole cast rather than on the new voice, because minimum separation is a property of the set.

That gives a team score a concrete piece of advice that follows from arithmetic: a growing team score holds fifteen distinguishable voices, and a team that intends to add a machine participant has fourteen human voices of headroom rather than fifteen. Whether the step matters at all depends on the perceptual resolution, which is unmeasured, and the discrimination experiment is what fixes it. What the arithmetic supplies is the place where the answer changes in kind, and it is the sixteenth voice.

17. Open questions#

Nine questions are genuinely open here, in the sense that somebody would have to run something to answer them. They are the places where a reader can most usefully disagree.

What a participant's stave budget is, against a watcher's. The two-to-three figure was elicited for a reader whose position was unspecified. The two-arm study of section 7.5 returns two numbers where the programme currently has one, and the difference between them is what decides whether one declaration serves both audiences on an open session or two are owed.

How finely a listener discriminates timbre. The perceptual resolution fixes which plateau applies and therefore whether a team score holds two, six, seven or fifteen distinguishable voices. Every timbre figure in part one is conditional on it, and the conditional is narrow: the plateau at fifteen holds for a resolution at or below $\sqrt{3}/2$, and the answer changes in kind rather than degrades outside that.

Whether a signature costs a stave slot. $\vec{\pi}$ renders as a slow categorical read once per window rather than as a line a reader tracks. The stave capacity estimate was made for moving quantities, so whether a signature draws on the same budget is a question the two-arm study can answer and currently nobody can.

Where the radius $r$ sits. The position map's abstention region is set by one number, and section 10.3 gives it a fitting procedure rather than a value. Fitting it to the region where trained readers abstain is the procedure this paper proposes, and whether that region is well enough defined to fit to is itself a question the inter-rater study answers on the way past.

Whether the position simplex's sum constraint is right. The three registers sum to one and the constraint carries a substantive claim about competition between them. The position coordinate makes the same claim for the four positions over a window. Whether a relation over a window really distributes a total of one across the four places, or whether a passage can be strongly two things at once, is a claim the notation asserts and an annotation study would test.

Whether the four reference signatures are the right four points. Section 10.2 states them as this paper's own operationalization of the place structure. Another reader of the same source would write them differently, and the inter-rater study's disagreement pattern is where a better set would first show itself.

Whether a position assignment is stable across window lengths. The window is a declared parameter. A coordinate whose value depends strongly on a parameter the reader chose is reporting the reader's choice, and the sensitivity has never been computed.

Whether the three arms of the negative control are separable in practice. Arm three holds content fixed and destroys arrangement, and the prediction is that its interrogative share survives while its uptake terms fall. If the shuffle also disturbs the interrogative share, for instance because question rates are not uniform across a passage, the arm's signature blurs and the control needs a better shuffle.

What a moderator does to a team score. The moderated structure collapses the pair count from 105 to 14 at fifteen voices, which is the largest single simplification available anywhere in the team arithmetic. None of the programme's scored files is a moderated exchange [7], so the corpus that supplies every pair figure here is silent on the one structure a team most often has.

18. Further work#

Written forward, and in the order in which each piece unblocks the next.

The two-arm stave capacity study. Section 7.5. One study, run twice on the same material, returning a reader's budget with a speaking obligation and without one. It settles part one's residue, it settles whether a signature costs a slot, and it is the smallest study in this paper.

The timbre discrimination experiment. It fixes the perceptual resolution, and through it the cast ceiling, the audio renderer's voice count and how large a growing team score can be. It was the most valuable single measurement in the programme before the cast was allowed to grow, and part one's fifteen-voice ceiling gives it a second thing to decide.

The position map's implementation, with its abstention rate reported. Section 10 states the function. Implementing it on the programme's generated corpus and reporting its abstention rate is the first thing that produces a number, and the abstention rate is the decision variable for everything downstream.

The inter-rater agreement study. Section 12. Ten passages, thirty-four raters, trained and untrained in the reference's proportions, blind to which speaker is the machine. It returns primary agreement, top-two agreement and abstention concordance, and it fits $r$.

The three-arm negative control. Section 13. It follows the map's implementation and can share its corpus. Arm three is the arm to build carefully, because it is the arm that answers the strongest objection.

The partner-variance falsifier. Section 11.1. One participant against several partners chosen to differ on the observables, with the variance decomposed. It is cheap, it needs no raters, and it is the test that decides whether the coordinate is relational, so it is worth running before the agreement study rather than after.

The next-move falsifier. Section 11.2. Held-out passages, a stated margin above base rate fixed before the run, and four predicted properties, one per position.

The seventh-voice listening study. Section 16.3. A six-persona team plus a machine as the seventh, rendered with the geometry's own assignment, asking whether listeners identify the machine and on what. It composes with the blinding of section 12.4 and gives a study that is blind at the text layer and at the audio layer at once.

The three session kinds, implemented as a declaration. Section 7.2. The declaration is one more line in the score header, beside the floor share unit and the window length, and it is what makes the four rules operative rather than stated.

The frame-local and historical split, applied to the proposal column. Section 7.1's second clause. Twenty-two proposals render beside their turn and eight render with their span named. The partition is computed by the bias layer generator already, so this is a rendering change rather than a measurement.

The bias layer pilot. Choice-supportive bias is the cheapest test in the set, needing an analyst and two passages of one score and no listener panel, and its detection rate is the condition on which the layer's channel question reopens [11], [8].

19. References#

On the paths in these entries. An entry that gives a path in backticks names an artefact in the working corpus of this programme rather than a page on this site. That corpus is not published, so such a path does not resolve from here. The citations are kept as they stand because each names a real artefact and each entry says what the artefact is and what it establishes, which lets a reader see what a claim rests on and ask for the artefact by name.

The rest of the series is published in this working group: MPN-S1, the theory; MPN-S2, the formal apparatus; MPN-S3, the mapping; MPN-S5, the dialogue use cases; and MPN-S6, the expression aid.

[1] J. McKenney, "The McKenney-Lacan psychometric calculus," MPN-S1, S1-mckenney-lacan-theory.md, revision 3, with MPN-S4-AMENDMENTS.md. Source of the nine-component state vector and its bounds, of trauma's ratchet as a property of the definition, of the simplex constraint on the three registers and the competitive claim it carries, and of the assertion at :317 putting the independent quantities a stave can carry at two to three while stating in its own words that the number is an estimate rather than a measured figure. Section 9.3 carries that assertion and its siblings. Cited in sections 1.3, 3.5, 5.2, 7.5, 9.4 and 17.

[2] J. McKenney, "The formal apparatus," MPN-S2, S2-mathematics.md, revision 4 with its post-acceptance correction. Cited here for the simplex construction the position coordinate follows and for the reach and identifiability results that reach this paper through [3].

[3] J. McKenney, "The mapping," MPN-S3, S3-mapping-phi-rev10.md, revision 10, folding in the four amendments of MPN-S3-AMENDMENTS-2.md of 14 September 2026. Section 2.6 is the timbre map, its rank three against a domain of dimension four, its nullity of exactly one, and profile magnitude as the null direction, which is the algebraic source of section 3.2's fifteen distinct corner images. Section 4 is the bias layer and section 4.2 the sixteen drafted device mappings. Section 4.3 is the bias atlas domain census.

[4] J. McKenney, "The application," MPN-S4, S4-application.md, revision 2, with ARBITRATION-S4.md. MPN-S4 is an internal implementation audit of the reference implementation, held in the programme's working corpus and available from the author. It is cited here for the provenance of the 31,078 scored rows, for the King Lear parse result at 3,424 of 3,425 rows, and for the ruling that the artefact is the seven score files or the 31,078 rows.

[5] "The instrument," MPN-PRD-01, PRD-MPN-THERAPY.md, draft 6. Section 3.3 and section 11.1 are the Nordoff-Robbins interrater evidence and the design that is sized against it, being 34 raters on 10 excerpts. Section 7.6 is the written anchor descriptor practice this paper's section 12.3 follows, where the anchors are the instrument and the numbers are a convenience.

[6] "One engine, several surfaces," MPN-DESIGN-01, DESIGN-UNIFIED-FRAMEWORK-rev8.md, revision 8. The governing specification for this paper. Section 3 is the two mappings and what the interaction mapping is required to be; 4.2 the six Layer 1 measures and their declared settings; 4.4 Layer 3's rules; 5a the bias layer and its work queue; 5b the timbre channel and the discrimination experiment; 6.2 Live, the three-clef score, the event densities and the information budget; 7.4 participant mode and the display conditioning experiment; 8a the four kinds of refusal and the staleness horizon; section 10 the inventory of unfixed numeric parameters, to which section 10.3's radius $r$ is an addition.

[7] ARBITRATION-DESIGN-R7.md, 14 September 2026. Source of the rescoping of the running score by whether the material is known in advance, of the replacement of the observer effect experiment by display conditioning of a generator, and of the standing limit that none of the seven score files is a moderated exchange, cited in section 17.

[8] DECISION-LOG-2026-09-14.md, 14 September 2026. The decision this paper's section 2.2 inherits: the bias layer is a Layer 3 detector whose output is a marked textual proposal beside the turn, with a reopening condition read from the detection rate.

[9] MPN-S3-BL8-OPTIONS.md, 14 September 2026. The five candidate channels against five tests, and the translation census of how many of the thirty devices survive each candidate. Option (e) is the one the author took and the one section 2.2 states.

[10] MPN-S3-BL1-DRAFT.md, 14 September 2026. The fourteen drafted coordinates with three rows flagged as undecided, the seven-part search of the reference implementation, and the atlas activation ordinal the five-field projection drops. This is the draft coordinate table section 2.3 runs with loaded.

[11] 05_DATA/03_generators/s6_bias_layer.py, run against s3_bias_reconciliation.json. Re-run in full for this paper. Source of every figure in section 2.3: the thirty assembled entries with a device and a domain on each; thirty of thirty naming a state coordinate with the draft table loaded and three rows flagged; eighteen distinct channel-and-coordinate signatures, eight of them carrying more than one entry, twenty of thirty entries involved; the before-and-after counts of twelve signatures, four shared and eight involved; the channel census at texture 8, harmony 7, melody 6, rhythm 3, dynamics 3, timbre 1, mode 1 and intervals 1; and the partition of twenty-two frame-local against eight historical entries with the eight named.

[12] 05_DATA/03_generators/s6_timbre_capacity.py. The optimiser the persona-scaling generator calls rather than reimplementing, and the source of the timbre map's own function, whose linear part section 3.2 decomposes.

[13] J. McKenney, "Decomposing dialogue: four use cases for one engine," MPN-S5, S5-dialogue-use-cases-rev1.md, 15 September 2026. The paper this one develops. Section 5 is the live score surface, its default, its event densities, its information budget and the two-audience passage this paper's part one resolves, including the sentence this paper does not resolve it quoted in sections 1.1 and 4.1. Section 6 is the multi-persona score, the append-only rule, the per-stave load and the pair count. Sections 5 and 6 together are 6,135 words and are inherited here rather than re-proposed.

[14] J. McKenney, "The Expression surface," MPN-S6, S6-expression-aid.md, 15 September 2026. The companion on rendering the shape of a conversation as sound and image, and the source of the surface whose participant reader [15] develops.

[15] J. McKenney, "Inside the exchange: what a participant reads on the Expression surface," MPN-S8, 16 September 2026. The companion that resolves the two-audience question for the case where every reader is a participant, with its grounds rule and its symmetry rule. This paper takes the case where both kinds of reader are present at once and shares S8's definition of a participant.

[16] DECISION-LOG-2026-09-15.md, 15 September 2026. Source of the positions section 1.3 inherits: a named per-person comparison is permitted on a stave on every surface; a score carries several personas, added and removed as the material requires; backchannel is a sixth Layer 1 measure; a backchannel is lexical on class A and timing-derived on classes B and C, so Layer 1 carries six measures and six settings; and Expression is an $N$-persona surface. The class A lexical ruling is the precedent for section 10.1's interrogative share.

[17] MPN-NOTE-06-persona-scaling.md, corrected 16 September 2026, with 05_DATA/03_generators/s9_persona_scaling.py. Re-run in full for this paper at a behavioural grid of $21^4 = 194{,}481$ profiles and 150 polished starts, and extended past the published range. Source of section 3's figures: the append-only curve against the re-packed one with its worst loss of 13.4 per cent at eight voices; the seventh voice as the flat profile at separation exactly 1.0; every other placed voice on a cube corner; the sixteen corners giving fifteen distinct timbre images because the flat-low and flat-high profiles share one; the plateau at $\sqrt{3}/2 = 0.866025$ holding from eight voices through fifteen; and the fall to $0.559017$ at the sixteenth, a step of $0.307008$ and 35.4 per cent of the plateau, caused by the exhaustion of the corner set. The extension to sixteen and seventeen voices and the rank and nullity decomposition of the timbre map's linear part were computed for this paper on the same optimiser and the same timbre function.

[18] ASSERTIONS-REGISTER.md. The stave capacity assertion, whose claim, formal content, constraint and falsification clause govern the information budget of sections 3.5 and 5.2, amended on 15 September 2026 to strike its between-stave exemption so that nothing in it is exempt from the per-stave budget.

[19] J. F. Mahoney, "Interrater agreement on the Nordoff-Robbins Evaluation Scale I: Client-Therapist Relationship in Musical Activity," Music and Medicine, 2010. https://mmd.iammonline.com/index.php/musmed/article/view/MMD-2010-2-1-4 Thirty-four certified music therapists rated ten video excerpts; 78 per cent of all raters scored within one point of the total group mean, 82 per cent of the 21 with Nordoff-Robbins training and 74 per cent of the 13 without. The reference this paper's section 12 is sized and thresholded against.

[20] K. E. Bruscia, Improvisational Models of Music Therapy, Charles C. Thomas, Springfield, 1987, in which the Improvisation Assessment Profiles are published, as summarised at https://ifas.thws.de/en/high-m/theory/improvisation-assessment-profiles-iap/ and cited in [5]. Six profiles, five gradients each, with a written anchor at every level. The practice section 12.3 follows for the four position anchors.

[21] J. Lacan, Seminar XVII: The Other Side of Psychoanalysis, 1969 to 1970. The four discourses as four terms rotating through four places, which is the construction section 9.2 reads relationally. The four reference signatures of section 10.2 are this paper's own operationalization and are not cited to this entry.

[22] J. McKenney, "McKenney-Lacanian threat actor modelling," WG-03-ML-Mckenney-Lacanian.md, working group WG-03. Cited as a contrast rather than as a precedent, and section 15 is what the citation is for. It applies the four discourses to threat actors, assigning each actor in a database a psychometric vector from operational history whose components are discourse alignment, register dominance and a Dark Tetrad projection of narcissism, Machiavellianism, psychopathy and sadism, and it uses the alignment to predict targeting logic. Its discourse alignment is a property of an actor; this paper's $\vec{\pi}$ is a function of an ordered pair and a window. The two constructions run the inference in opposite directions and section 15.2 sets out the four differences.

[23] MPN-S3-BL2-DRAFT.md, 14 September 2026. The provisional remap of the one bias entry written to set the mode, which is the entry the channel census of section 2.3 records on the mode channel.