The Mapping: From Psychological State to Musical Material, Parameter by Parameter

J. McKenney

Paper 3 of the Musical Psychometric Notation series. MPN-S1 states the theory and MPN-S2 the formal apparatus; this paper states the mapping from psychological state to musical parameter, and MPN-S5 and MPN-S6 apply it. The audit of the reference implementation against this mapping is MPN-S4, an internal record held in the programme's working corpus and available from the author.

Licence: CC BY 4.0. 15 September 2026.

Executive Abstract#

A composer scoring a character makes decisions that stay in the ear rather than on the page: how loud, how fast, in what mode, in what color, how much theme to state, how thickly to score it. This paper writes each as an explicit function of a described psychological state, so a reader can disagree with a line rather than a general claim.

Six parameters are specified in full. Dynamics rides on trauma, which only accumulates, so a marking rises or holds. Tempo and metre ride on entropy. Mode comes from the dominant Lacanian register, trauma a second-stage switch. Fragmentation and density are orthogonal sums of trauma and entropy; harmonic position walks the twenty-four consonant triads; and timbre is specified here for the first time.

Four findings run against the mapping. Every mode reaching a score is a function of something other than the registers, printed name and notated pitches agreeing on 36.0 percent of states. The timbre channel carries three of DISC's four coordinates, losing profile magnitude. The tie rule is given in closed form, since a sixth of states sit on a tie. Chain position and harmonic distance vary independently, so a larger state change can produce a smaller move.

The modal section sets out four incompatible register-to-mode tables, two of them exact inversions; a blinded listening pack asks whether listeners converge with each other rather than agree with the theory. The paper withdraws a call-graph analyzer that flagged five live functions as dead.

Abstract#

This paper states the transformation from psychological state to musical parameter, parameter by parameter, completing MPN-S2's three open obligations: the timbre space, the tie behavior of the mode selector, and the identifiability analysis. Six functions are given with inputs, codomains and jump sets: dynamics from trauma, tempo and metre from entropy, mode from the dominant register with trauma as a second stage, fragmentation and density an orthogonal pair, harmonic position a walk over the twenty-four consonant triads, and timbre, specified here for the first time. Four results are established: every mode reaching a score is a function of something other than the registers; the DISC-to-timbre map has rank three, profile magnitude the null direction; modal interpolation, alone among three tie rules, is determinate at an exact tie and keeps a frame's music computable from its own state; and the two sets of thirty cognitive-bias entries reconcile exactly. The 31,078 scored rows and 232 annotated frames are outputs of the system under study, so every figure is a property of a proposed map, computed exactly, not a listener measurement.

1. Introduction#

1.1 What this paper does, and where it sits#

MPN-S1 asserts that the transformations professional practice applies to a leitmotif are functions of a psychological state, and that those functions are total, deterministic and inspectable [1]. MPN-S2 establishes what kind of object such a transformation is, and leaves three things to this paper: the timbre space, which it shows to be the only channel the four DISC coordinates reach; the tie behavior of the mode selector; and the completion of the identifiability analysis, which every listening study the programme plans depends on [2].

This paper states each of the six parameter functions, specifies the timbre space, defines the tie rule, completes the identifiability result, and sets out the bias layer in full. Implementing the mapping is MPN-S4's business and choosing the modal table is the author's; what this paper does is state the consequences of each option precisely enough that the choice can be made at a keyboard, and say what an instrument now in the field would have to return for the choice to be made by ear instead.

1.2 The standing caveat, before anything this paper measures#

This is a theory paper, offered for comment and improvement rather than as a finished specification, and the most useful thing a reader can do with it is say where it is wrong. Every figure below is either a theorem, a reading of source, or a design parameter the listening pack is built to settle. Two earlier listener studies exist and both are reported in MPN-S1; their stimuli came from an unseeded generator, and seeding it is what makes a restaged study citable [1].

Section 2.3 does this work in full for the modal question, setting out the four candidate tables, the instrument in the field, and what each of four outcomes costs the theory and the application, and it is not restated there or anywhere else. What belongs here is the count across the six parameters, which this is the one place in the series to state. Three of the six have a listener test specified and built: mode, by the pack's first two parts; the harmonic position, by its third; and timbre, by its fourth [19]. The other three take the discrimination tests section 6 names: dynamics, tempo with metre, and fragmentation with orchestration density, each of which is tested by asking listeners to order cues on the ladder the parameter asserts. Those three carry the same standing today that the first three carried before their parts of the pack were built, and building their parts is the next wave.

Every quantity in this paper is computed from the system's own output. The values of trauma, entropy and the registers in the corpus are values the system produced about itself, so every table below is a proposal, however precisely its numbers are computed. The figures that look like results are properties of a proposed map, which is why each is reproduced by a named script rather than reported.

1.3 Four findings, stated before the detail#

Every mode that reaches a score is a function of something other than the registers, though much else does read them (section 2.3). Five places in the shipped source decide a mode. The name printed on a score comes from a three-way branch on $(\tau + H - 0.5)/2$, a quantity the code calls a Lyapunov exponent and which MPN-S2 renames the imbalance index; that branch is written as a fallback behind a field that sits outside the interface's four declared ones, and a cast keeps the type checker quiet about the access, so the branch wins on every frame. The notated pitches come from a selector fed a hard-coded register triple, which makes the Imaginary dominant for every character in every play and leaves two of the seven modes reachable. The one selector that does read the frame's register triple writes its answer to every stave, and a reader of that field connects it to the score. Two further tables await a caller. MPN-S2 reports the first two of those [2]; what is added here is why the branch always wins, which is the part a repair needs, the discarded selector, and the measurement that the printed name and the notated pitches agree on only 36.0 percent of the state square. The registers do reach the score, through the key, the chord quality and the leitmotif transformation, and section 3 enumerates every reading; each of them reads a single register against a threshold or through a linear term, so which register leads is a comparison the build has yet to make. The modal assertion therefore becomes testable on the build once a register-dependent mode reaches the score, and a listening study run before that would return a null whatever the truth of the assertion.

The timbre channel carries three of DISC's four coordinates (section 2.6). Multidimensional scaling of musical timbre returns three perceptual dimensions, agreed across both cited studies, which disagree only about the acoustic correlate of the third [3], [4]. DISC has four coordinates. Section 2.6 states the map from one to the other and proves that its null direction is profile magnitude: two characters whose DISC scores differ by a constant added to all four produce identical timbre, and they sound the same through this channel to any listener. The loss is exactly one dimension, and the choice the theory gets to make is which direction it is. A listening study that asks a listener to recover a DISC profile from a cue is therefore asking a three-dimensional channel for a fourth coordinate, and this result cancels such a study before it is paid for and specifies the shape study that replaces it.

The tie rule is defined, with one parameter left to the author (section 2.3). MPN-S2 leaves the mode selector's behavior at a tie open and shows that hysteresis and a dead band both fail on it, since both are defined by a gap and both hold their last value when the gap is zero [2]. The interpolation is given here in closed form, and four of its properties are proved in a line each and checked numerically: determinacy at a three-way tie, exact agreement with $\arg\max$ away from the margin, Lipschitz continuity on the closed simplex, and a worst-case notation error of exactly a quarter tone. The tie is a common case: 21 of the 128 frames that reach the simplex sit exactly on one. Under the rounding rule section 2.3 adopts, the quarter tone is a property of the notation path and not of the mapping: the mapping emits cents and the notated pitch is the rounded pre-image.

The modal question is open and under test (section 2.3). Four incompatible register-to-mode tables sit in the corpus, two of them exact inversions of each other on every register, so at most one can be right and all four may be wrong. Section 2.3 sets out what each table claims, where it lives, whether it reaches a score at all, and what would have to be true for each to be right; why the fourth candidate is a different mechanism rather than a fifth table, a jump to a crisis mode per register rather than a dim within a pair; the instrument that is in the field to settle it, described as it is built rather than as a plan [19]; and what each of four outcomes does to the theory and to the application. The question the pack asks is whether listeners converge with each other rather than whether they agree with the theory, because four tables disagree and any one of them taken as the standard would assume the answer. Convergence on any of the four is a result; a spread of answers refutes the premise that there is a table to find.

The bias layer's two sets of thirty reconcile exactly, and reproducibly (section 4.1). Fourteen shared, fifteen implementation-only, sixteen Atlas entries awaiting a musical mapping, fourteen awaiting a domain. Those are MPN-S1's counts [1]; what is new is that a script now derives all four from the two sources rather than restating them, and fails loudly unless the drafted content covers exactly those gaps. The invariant it enforces survives the author striking a drafted row, provided the strike is recorded, so the reconciliation continues to account for all thirty entries and a dropped row shows up as a failure.

1.4 The parameter functions in one place#

For a reader who wants the mapping before its justification:

Eight rows for six parameters, because metre travels with tempo and the harmonic position with trauma, and because the paper counts a parameter by what the theory names rather than by what the implementation packages together.

1.5 Scope#

This paper states the mapping. Implementing it and choosing the modal table are separate pieces of work, owned by MPN-S4 and by the author. The sixteen bias mappings and fourteen domain assignments of section 4 are drafted for the author to strike out what is wrong, and section 4.4 is written so that striking one is a local operation.

1.6 A note on the standing of the quantities#

The quantities this paper maps are dramatic quantities about fictional characters, defined in MPN-S1. Trauma here is the difference between a character in the first scene and the same character after the event the play is about. It is a dramatic quantity throughout, and this paper is a description of a piece of software and of a theory about drama. A paper addressed to music therapists is one the series takes up later.

1.7 Terms used in a particular sense here#

Register, unqualified, means one of Lacan's three, the Real, the Symbolic and the Imaginary, as MPN-S1 defines them; where the musical sense is meant, the paper writes registral or names the octave. DISC is the four-factor personality instrument, Dominance, Influence, Steadiness and Compliance, and a character's DISC profile is four numbers in $[0,1]$. The fourth scale also circulates under the name Conscientiousness; this series writes Compliance throughout, because the wider personality vector of MPN-S1 section 2.3 carries Big Five conscientiousness as a separate coordinate and one vector should not carry two coordinates under a single name. Mode means one of the seven diatonic modes throughout.

Three terms of art are inherited from MPN-S2 and used without re-deriving them. The tripod is the set of three segments from the centre of the register triangle to the midpoints of its sides, which is where two registers lead together. The kite result is MPN-S2's closed form $2\delta - \delta^2$ for the area within $\delta$ of a tie. And a map is Lipschitz if there is a fixed bound on how much its output can change per unit change of its input, which for a musical parameter means the output bends rather than jumps.

The series states its claims as a numbered register of assertions, published in MPN-S1 and in the assertions register [1], [11]. Seven of them are load-bearing here, and each is given below so that a reader meeting the label in the body can resolve it without leaving the page:

One amendment to that register, made on 15 September 2026, changes what this paper may claim. B4 was originally a three-layer claim in which bias, living between speakers, took the harmonic relation between staves rather than any channel on a single stave. That clause is struck. The bias layer is carried in the data rather than rendered, so the whole of the per-stave budget goes to the two rendered layers. Every part of B4 now falls under the budget of two to three, which makes the assertion easier to falsify rather than harder: a measured budget that differs from two to three falsifies it directly, where before a layer could exceed the budget and the assertion could survive by relocating that layer between staves. Sections 4.4 and 5 state where this paper's reasoning changes as a result.

2. The six parameters#

Each subsection states the function, its effective contributions where it takes more than one input, its jump set, and what would show it wrong. Effective contributions are computed at the frame library's dispersions, $\operatorname{sd}(H) = 0.1925$ and $\operatorname{sd}(\tau) = 0.2496$, which are themselves system output and are labeled as such in MPN-S1 [1]; a weight quoted alone is an unexamined claim about a variance [5].

2.1 Dynamics#

The author's decision of 12 September 2026 makes the eight-marking linear form normative [6]. Velocity runs linearly from the threshold of audibility to the ceiling of the instrument,

truncated to an integer rather than rounded, as the shipped module does it with int(), and discretized to the eight conventional markings. The output of $\Phi$ is the marking, not the velocity: the velocity interval is the pre-image, as MPN-S2 section 6.1 sets out. The boundaries are those of the shipped Python module, which is the only implementation carrying all eight:

Two properties follow and both are consequences rather than choices. The bands are unequal, widest in the middle and narrowest at the extremes: the four at the ends of the range are 0.10 wide and the four in the middle 0.15, a ratio of three to two, so the instrument resolves trauma half again as finely at the extremes as it does in the middle. As a property of the band widths, for a character sitting at a uniformly random point inside a band a step of 0.05 in trauma changes the marking one time in two at the extremes and one time in three in the middle, and in the topmost band the marking holds, the ladder having reached its ceiling [12]. And because dynamics is a function of trauma alone, and trauma ratchets, the dynamic marking of a character rises or holds across an act. MPN-S1 states the ratchet as a property of the definition; here is where it becomes audible. Relief comes from a resolution term on trauma, and MPN-S2 carries the design of that relief as an open decision; until it is taken this parameter is monotone by construction.

The path a rendered score actually takes. There is a second implementation and it is the one the application uses. lookupDynamics matches trauma against the reference dictionary's volume_level entries conditioned on trauma, of which there are three, and returns a literal fallback of velocity 72 labeled mf when none matches [12]:

Three things follow and none is a judgment call. The composed function emits three labels, not eight, at three constant velocities, not $20 + 107\tau$. Two of its five intervals are accidents of a lookup that fails rather than bands anyone designed, and they return the same mf as the interval that succeeds, so six tenths of the trauma range collapse to one velocity, which is the observation the commensurability audit makes independently [5]. And the label ppp/pp is not a dynamic marking at all but a display string split on its first space, so a consumer resolving a marking by name finds nothing under it.

The eight-marking linear form is normative and only the Python module implements it. Everything else in this section describes the normative law, which is what the theory says. A test of A7's dynamic ordering runs on a build whose dynamics lookup carries all eight markings of the linear law, because the ordering under test is the eight-marking one.

The jump set is the seven boundaries above. $f_{\text{dynamics}}$ is piecewise constant and the marking is the output, so the parameter steps, and the paper claims a step function for it.

It would be shown wrong if listeners asked to rank cues by the weight the character is carrying returned an ordering other than the ordering of the markings, which is a discrimination test and runs without a verbal anchor.

2.2 Tempo and metre#

Tempo is the one factor of the codomain that is genuinely continuous in the state, and it is continuous only in pieces. The shipped implementation sorts entropy into three bands, strategic at $H \le 0.4$, operational at $0.4 < H \le 0.7$ and crisis above that, looks up a beats-per-minute range for the band, and then places the tempo inside that range in proportion to entropy:

with $(b_{\min}, b_{\max})$ equal to $(40, 60)$, $(80, 100)$ and $(120, 180)$ on the three bands [12]. Within a band the law is therefore affine in the state with a strictly positive slope, and the two discontinuities come from the range changing underneath it rather than from the tempo being constant between them.

What the arithmetic then does is very nearly cancel the continuity it has just established, and the figures are worth having in full:

The slope inside the two lower bands is twenty beats per minute per unit of entropy, so a change of 0.05 in $H$ moves the tempo by one beat per minute, which is the rounding quantum of the output: within those bands the parameter is very nearly inert. The three ranges sit so far apart that of the 141 integer tempi between 40 and 180 only 35 are reachable at all, in the three runs above; the intervals 49 to 87 and 95 to 161 are silent for every character in every play, whatever their state. Against that inertness the jumps are enormous.

So tempo is continuous in form and categorical in effect, and it is worth being careful about why, because the obvious explanation is not quite right and the careful one is more useful.

The within-band variation is not ruled out by the published discrimination thresholds. Drake and Botte give relative just-noticeable differences for tempo of about 6 percent for a single interval and about 3 percent for a six-interval sequence, over interonset intervals from 100 to 1,500 milliseconds with sensitivity best between 300 and 800 [15]. The three bands span 18.2, 6.6 and 10.5 percent of their own midpoints, at interonset intervals of 1364, 659 and 351 milliseconds [12]. Every span clears the single-interval figure and all three clear the sequence figure.

Three qualifications belong with that, and they matter more than the comparison. The thresholds come from a two-interval forced choice on isochronous sequences with one dimension varying, which is a best case for a listener; clearing them means a difference is not ruled out, not that it would be heard in a score where several parameters move at once. The operational band clears the 6 percent figure by 0.6 points, which is inside the precision of the source. And the strategic band's 1364 milliseconds lies outside the range where the thresholds were best measured, so 6 percent is not the applicable figure there and the applicable one is probably larger.

What the comparison does establish, and what section 3 needs, is a ratio. Taking 6 percent as the threshold, about 3.0, 1.1 and 1.8 just-noticeable differences fit inside the three bands, while the mapping emits 9, 7 and 19 distinct integer tempi in them. Inside a band the mapping emits roughly five times as many levels as a listener could resolve. That is the sharpest statement of the difference between what the mapping emits and what could be recovered from it that this paper can make, and section 3's counts should be read against it.

What makes the parameter categorical in effect, then, is proportion rather than inaudibility: the steps between bands are five and eleven times the spans within them, so a listener attending to speed hears three plateaux with cliffs between, and the gradient inside each plateau is real, is probably at the edge of resolution, and is swamped. Section 6 records that as a prediction rather than asserting it.

Metre travels with the tempo, and travels badly. It is looked up from entropy against four conditions, below 0.3, 0.3 to 0.5, 0.6 to 0.8 and above 0.8, giving common time, waltz time, an irregular 5/4 or 7/8, and free metre [12]. Those conditions do not cover 0.5 to 0.6, where no entry matches and the lookup returns its literal fallback of 4/4, the metre of the most ordered characters in the play. Metre as shipped therefore runs 4/4, 3/4, 4/4, irregular, free as disorder increases, and a character at $H = 0.55$ is notated like one at $H = 0.10$. That is an implementation defect and not a statement of the theory, reported here because the defect falls exactly on the meaning this section asserts.

The two jump sets are disjoint as well. Tempo steps at 0.4 and 0.7 and metre at 0.3, 0.5, 0.6 and 0.8, and six boundaries partition the unit interval into seven cells, which direct enumeration confirms. Every boundary in one parameter falls at a point where the other holds steady. The offset is what two independently written lookup tables happen to produce, and aligning them is a decision the theory has yet to take.

The claim underneath all of this is that disorder in the subject's symbolic organization shows as instability of pulse rather than as speed. It is worth separating from the more obvious claim it is often confused with: this is not that agitated characters play faster, and a character can be scattered and slow. The implementation asserts the claim and then works against it, because the signal it produces is dominated by speed, a step of 40 and then of 68 beats per minute, while metre, where instability of pulse would have to live, has a hole in the middle of its range and returns the most ordered metre there. The jump set is $\{0.4, 0.7\}$ for tempo and $\{0.3, 0.5, 0.6, 0.8\}$ for metre, and the section would be shown wrong if listeners heard the band changes as changes of speed rather than of stability. On the current build that test would most likely fail, and section 6 records the prediction in that direction rather than in the one that would flatter the mapping.

2.3 Mode, and what happens at a tie#

Assertion A4 selects the mode from the dominant register, with trauma as a second-stage switch. Which register takes which mode is the author's to settle, and this paper puts the four options in the corpus on the page so that the choice is made from a list rather than from a search:

This paper leaves the choice to the author and equips it. What follows does three things in order. It states what each of the four claims, where it lives, whether it reaches a score, and what would have to be true of the world for it to be right. It describes the instrument that is now in the field to settle it. And it traces each of four possible outcomes through to the documents that would change and the work that would open or close. Governing all three is a prior fact this section establishes: every mode that reaches a score is a function of something other than the registers, so the instrument that tests A4 today is the listening pack rather than the application, and what puts A4 under test on a build is a register-dependent mode reaching the score.

The four candidates, one at a time#

A choice between four tables is cheap to take only if what each one claims is on the page beside what would have to be true for it to be right. The four differ in more than their assignments. They differ in whether they reach a score at all, and one of them differs in mechanism rather than in assignment, which takes it off the axis the other three sit on.

One, the live pitch table, leitmotif_transformation_rules.ts:63-91. It claims that the Real takes Dorian and dims to Aeolian above trauma 0.6, the Symbolic takes Lydian and dims to Mixolydian, and the Imaginary takes Phrygian and dims to Locrian. It reaches a score, and it is the only one of the four that reaches the notated pitches, through getModalTransformation on the composer path. Its input is a literal triple, because composeMelody substitutes one before calling it, so on the build as it stands the table is live and its input is a constant and every character in every play is notated in Phrygian or in Locrian. For this table to be right, three things would have to hold. Brightness would have to run with the register in the order Symbolic, Real, Imaginary, since Lydian is the brightest of the seven modes by raised degrees against lowered and Phrygian the second darkest. The second stage would have to be a dim within a pair, one degree flattened and the character of the scale preserved, rather than a change of kind. And the Imaginary would have to take a trauma partner, which the author's decision of 12 September 2026 rules against [6]. The first of those is a musical claim about what the three registers sound like and is exactly what Part A of the listening pack asks. The third is a conflict on the face of the corpus: the shipped table gives the Imaginary a trauma partner and the decision assigns it none.

Two, the reference dictionary, mpn_reference_data.ts:1034-1102. It claims that the Real takes Phrygian or Locrian, the Symbolic takes Ionian, and the Imaginary takes Lydian or the whole-tone scale. It is consulted by lookupMode, which is called on every frame and which is the only function in the application that sorts the frame's actual register triple; its answer is written into every stave's musicParams, which a reader of that field connects to the score. It is an exact inversion of the live table on every one of the three registers: where the live table gives the Symbolic the brightest mode and the Imaginary the darkest, this one gives the Imaginary the brightest and the Real the darkest. At most one of the two can be right, and both may be wrong, and that is the most useful thing that can be said about the pair before anyone listens. For this table to be right, brightness would have to run Imaginary, Symbolic, Real, and the Real would have to be the darkest register, which is a defensible reading and not an obviously worse one than the live table's, the Real being in MPN-S1's terms what resists symbolization. Two mechanical facts sit against it whatever any listener says. Two of its three entries are slash-joined display pairs rather than mode names, produced by splitting a display label on its first space, so a consumer resolving a mode by name looks up phrygian/locrian or lydian/whole-tone and comes back empty. And its Imaginary entry names Lydian while the formula beside it is the six-degree whole-tone scale, so a name and a formula disagree inside one module, and the cardinality constraint stated later in this section makes the interpolation undefined on this table rather than merely awkward.

Three, lookupModeName in psychometric_calculus.ts:109-113, awaiting a caller. It claims that the Real takes Phrygian, the Symbolic Ionian and the Imaginary Lydian, in one stage. It is consulted by rsiToMode, whose callers today are its own tests; a MODES constant in the same module is declared and awaits its first read. It is the reference dictionary's assignment with the alternatives stripped out and the trauma stage dropped, which makes it the cleanest statement of the inverted reading and the only one of the four that is a bare function of the registers. For it to be right, the inverted brightness ordering would have to hold and the trauma switch would have to be inert: mode would be a function of the register triple alone. That is a stronger claim than the reference dictionary makes and a more easily falsified one, because it predicts that the same register at any trauma takes the same scale, which a listener can be asked about directly.

Four is not a fifth option on the same axis, and it needs its own treatment. The eight synthetic raters run in September 2026 are not a table. At maximum extremity, six of the eight moved the Symbolic to the semitone above the tonic rather than to a dimmer partner of its reference scale, which fits none of the four tables in the corpus [6]. Read as a table it is an oddity. Read as a mechanism it is something else and more interesting. The semitone above the tonic is the flat second, and the flat second is the degree that distinguishes Phrygian and Locrian from every other mode in the set. A rater who moves the Symbolic there at maximum trauma is not dimming Lydian to Mixolydian, one degree flattened inside a bright pair. They are jumping to a different kind of scale. So what the raters propose is a jump to a crisis mode per register rather than a dim within a pair, and that changes A4's second stage rather than A4's table.

That distinction is worth holding because it decides which of two questions an instrument has to ask. A table question is answered by asking which scale goes with which predicament. A mechanism question is answered by asking whether that answer changes when the predicament is at its worst, and by how far it changes. This is why every situation in Part A of the pack carries a second question, whether the respondent's choice changes if the situation is at its most extreme, on the same seven items plus "no change". Three answers to that question are distinguishable in the returns and they are answers to three different mechanisms. A respondent who names one scale for a predicament and a neighbouring one at extremity is describing a dim. A respondent who names a scale from the other end of the brightness ordering is describing a jump. A respondent who names the same scale at extremity is describing a single-stage mechanism, which is the third candidate's claim. All three of those are distinct from the happiest-to-saddest ranking the pack asks first, which is why the ranking and the situations are separate questions rather than one.

Two limits on the fourth candidate belong with it. The raters reported the Symbolic only, so even taken at face value it is one register of three and speaks to the Symbolic alone at extremity; the pack asks all four situations of every respondent for that reason. And they were synthetic, which is the whole reason the matter was not settled at a desk and was put instead to people who score drama.

Where the modes actually come from#

Five places in the shipped source decide a mode. They are set out in full here because the claim this section rests on is a negative one, and a negative claim about an implementation is worth only as much as the enumeration behind it. The enumeration is s3_modes.py, which searches every non-test source file for lines returning or assigning one of the seven mode names or a scale formula, rather than for the names of functions that might do so, which is the distinction the claim turns on [12], [13].

One: the name printed on the score. The rendered frame takes its mode from (output.global as any)?.mode || (lyapunov < 0 ? 'Ionian' : lyapunov < 0.1 ? 'Lydian' : 'Phrygian'), where lyapunov is $(\tau + H - 0.5)/2$, a quantity the code names for an exponent it is not [12]. The first operand looks like a deference to whatever the orchestrator decided. It is not, and the reason is worth stating exactly, because it is the whole of the repair. The OrchestratorOutput.global interface declares four fields, tempo, time signature, key and dynamics, and the object built at the return site carries the same four; mode sits outside that set. The expression is written with an as any cast, which is what keeps the type checker quiet about an access to a field outside the declared four. So the first operand evaluates to undefined on every frame and the branch always wins. The printed mode is therefore a three-way function of $\tau + H$ alone: Ionian below 0.5, Lydian to 0.7, Phrygian above. MPN-S2 section 6.5 reports that this branch is what reaches the score [2]; what is added here is why it always wins, and that is the part a repair needs, because adding the field without removing the cast would change nothing.

Two: the mode the pitches are built from. composeMelody opens by writing const rsi = { real: 0.33, symbolic: 0.33, imaginary: 0.34 }; and passes that literal, not the state, into getModalTransformation [12]. Since 0.34 is strictly the largest, the dominant register is the Imaginary for every character in every play, whatever their state. The table getModalTransformation holds is:

Of the seven modes the module defines, two are reachable on this path and five become reachable with the repair, Ionian among them. MPN-S2 section 6.5 reports this as well [2].

Three: the one selector that reads the registers, and what becomes of its answer. lookupMode is reached on every frame, from psychometricToMusical at score_orchestrator.ts:267, and it is the only function in the application that sorts the frame's actual register triple and chooses on it. Its result is written into every stave's musicParams at score_orchestrator.ts:360, where it awaits a reader: the string musicParams occurs with .mode at that one site in the source [12]. The claim is about the mode field and not about the assignment. The mutated stave is returned at :363 and the object is read downstream, at score_exporter.ts:81, :84, :88 and :89, which take instrumentFamily, timbre, dynamic and articulation from it. The mode sits outside those four: the exporter writes mode: 'ionian' as a literal at :73, under the comment TODO: extract from params. The orchestrator's own output interface carries the same four, OrchestratorOutput.global declaring tempo, time signature, key and dynamics at :61-70 and the output object supplying those four at :440-445, which is why the page's (output.global as any)?.mode at page.tsx:488 is undefined on every frame and the Lyapunov branch behind it always wins [12]. So the register-dependent mode is computed on every frame and held. Two of the three strings it would emit are slash-joined display pairs rather than mode names, phrygian/locrian and lydian/whole-tone, produced by splitting a display label on its first space, so a consumer resolving a mode by name would come back empty on either.

Four and five: two tables awaiting a caller. rsiToMode in psychometric_calculus.ts is called from its own tests, and lookupModeName, which it consults, is referenced inside it; that table gives the Real to Phrygian, the Symbolic to Ionian and the Imaginary to Lydian, which is an inversion of the live pitch table on every one of the three registers. lookupAllParams awaits a caller as well, and lookupModeScale, which reads a fourth table out of the reference dictionary, is referenced inside it; in that table the Imaginary's entry is named Lydian and its formula is the six-degree whole-tone scale, so a name and a formula disagree inside one module. A MODES constant in psychometric_calculus.ts is declared and awaits its first read. MPN-S1's report of four incompatible register-to-mode tables, two of them exact inversions, is confirmed here at path and line, with two corrections: there are five decision points rather than four once the Lyapunov branch is counted, and the table the documentation describes is the one awaiting a caller.

The trauma switch, against the decision of 12 September. The pitch table gives all three registers a trauma partner, the Imaginary included, and the author's decision rules that the Imaginary has no partner [6]. The conflict is reported, not resolved: A4's second stage is already one of the questions the listening pack puts [6]. If the synthetic raters' reading holds, that the switch is a jump to a crisis mode per register rather than a dim within a pair, then decision and code are both wrong and A4's mechanism changes rather than its table.

The printed name and the notated pitches disagree, measurably. The name is a function of $\tau + H$ and the pitches of $\tau$, so a score's header and its accidentals need not describe the same mode. Over the $(\tau, H)$ square they agree on 36.0 percent of states; on the rest a score labeled Ionian or Lydian is notated in Phrygian, or one labeled Phrygian is notated in Locrian [12].

What follows for A4. The assertion is testable in principle and tested by Part A of the listening pack: a study run instead on stimuli from the current build would measure a branch on $\tau + H$ against a constant register triple and return a null whatever the truth of A4. The repair belongs to MPN-S4 and it is small: give OrchestratorOutput.global a mode field, remove the cast, and pass the frame's register triple into composeMelody instead of the literal.

Why the pack answers this and a study on the application waits on the repair#

This governs everything that follows and it is the reason the question goes to synthesized stimuli rather than to a rendered score. No mode that reaches a score is a function of the registers on the current build. A study run on stimuli rendered by that build would be putting a three-way branch on $\tau + H$, and a register triple that is a literal, in front of listeners; it would return a null whatever the truth of A4, and the null would look like a verdict on the assertion rather than on the wiring. The pack's stimuli do not come from the build. They are synthesized directly from the seven scale formulas by a script that takes a list of semitone offsets above the tonic and renders a melody on them, so the chain from a mode to a sound contains no orchestrator, no composer and no register substitution [19]. That is why a modal study on the application waits on a register-dependent mode reaching the score, while a modal study in the pack waits on nothing.

The instrument, and it exists#

Listening pack version 2 is built and published, its stimuli are final and hashed, and Part A is the part that bears on A4 [19]. What follows describes the instrument that is in the field, not a plan for one.

What a respondent hears. Seven items, labeled A to G. Every one of them is the same eight-bar melody, at the same tempo of 100 beats per minute, in the same key, over the same low pedal, a C two octaves below middle C sounding once a bar, and on the same neutral tone, which is an additive voice of five partials under one fixed envelope. Only the scale changes, and every other property is held identical across the files, which the page states before the first item. The melody is twenty-two notes across the eight bars, an arch that rises to the octave and returns, and it is published in scale degrees in the manifest so that a reader can check that no degree is omitted or favoured. The octave is held at twelve semitones in every item, so the tonic and the octave are fixed points and every judgment a respondent makes is about the five degrees between them on which the seven modes disagree. The seven scales are the seven diatonic modes. No random number is drawn anywhere in the generation, the set is a pure function of one script, every file carries a SHA-256 in the manifest, and the pack ships the audio rather than a promise to regenerate it.

What they are asked. Two questions, in that order, and the page instructs that each part stands as answered once a later one is heard. First, a ranking: take six of the seven, A to F, and put them in order from the one that sounds happiest to the one that sounds saddest, with a separate box for ties and for any item the respondent leaves unranked. The wording is careful about which question it is asking, and says so in its own words: what the music sounds like it is expressing rather than how it makes the listener feel, those being two different questions. The seventh item is held out of the ranking, and it is the one scale of the seven whose fifth is diminished. Second, four predicaments: four situations a character can be in, written as a director would describe them and, on the therapist path, as a colleague would; the situation wording is the whole of the difference between paths and every stimulus is identical across them, so a therapist's Part A answers and a composer's are directly comparable and are pooled. For each situation, listening to all seven, the respondent gives four things. A pick, from A to G, or "none of them fits", or "unable to tell". A confidence, on a slider from 0 to 100 that starts unset and is recorded as unset until it is touched, so that a lean, a certainty and an untouched control are three distinguishable states. A reason in free text, prompted to say what it is about the sound rather than what the scale is called. And an answer to whether the pick changes if the situation is at its most extreme, on the same seven plus "no change" and "unable to tell", which is the question that separates a dim from a jump.

The four predicaments are the registers, written without naming them. The first is something that has happened to the character and stays, resisting every explanation, a fact that holds its shape and comes back exactly as it was however they try to account for it. The second is a world held in place by role, rule, duty and position, where who the character is is where they stand, and what is at stake is whether that order holds and where they sit inside it. The third is a character living through a picture of themselves and measuring themselves constantly against other people, with vanity, rivalry, admiration, and a version of themselves they keep reaching for. Those are the Real, the Symbolic and the Imaginary as MPN-S1 defines them, put into a director's language so that a respondent can answer without holding a theory [1]. The fourth stands outside the three: a character whose attention rises and falls on a fixed cycle running independently of what is happening around them, so that events arrive early or late with respect to it. It is a decoy, answering to a register the triple leaves out. Since "none of them fits" is on offer for every situation, a respondent who gives the fourth a scale as confidently as the other three has said something about the instrument rather than about the table, and the confidence slider is what makes that visible.

What the pack holds back, and why. Three things are held back from every respondent, and the pack says on its face which three and why. The scale names. What the theory predicts. And what the earlier panel of language models said. The stated reason is that a respondent who knew would answer from that knowledge, and the pack is built to collect an answer from the sound. How each item was built and why it is in the set are held back on the same ground. The key that decodes the labels is held in the corpus rather than shipped in the pack, and a returned file is read against that key. Presentation order is one permutation drawn once from seed 20260914 and identical for every respondent, and that is the whole of the comparability argument: one respondent's item and another's under the same label are the same item, so two answers can be put side by side with no further assumption. Every respondent gets the same order, and a change to the order opens a new wave that is pooled on its own [20].

What the analysis does with the returns. A response arrives as one plain text file, or as the same text pasted into a message, and that file is the whole of what is collected. It is filed verbatim under a code assigned in the order responses are received, never reused and never reassigned. The returned file is filed verbatim, typos and open quotes included; a repair needed before it will parse goes in a sibling file and the reason goes in the register [20]. The intake script reads the directory, joins each answer to the key, and writes one summary per respondent and one pooled summary. For Part A specifically it tallies rather than scores. It computes, over those rankings that are six distinct letters from A to F, each item's mean position in the happiest-first order, and it reports how many rankings were usable of how many respondents attempted the part. It tallies the picks per situation, one count per item picked. It reports the confidences, and it reports a confidence set against an empty pick as a contradiction rather than dropping it silently. Free text is tallied as answered or not and then read by a person. And it prints, in its own words, that convergence is read off the spread of those picks rather than off any key, because the key carries rows for the other parts and none for Part A. An abandoned part makes a response partial, and a partial response is filed, registered and scored exactly like a complete one, on the stated ground that most of the value of the pack survives an abandoned part and all of it depends on the response being filed. Five conditions make a response unusable and one of them is about the content of an answer: the path is unestablished, the response is empty, the labels are untrustworthy, the respondent was told what the items were before answering, or an earlier part was revised after a later one was heard and the respondent says so. A response that contradicts the theory is usable, and so is a response of "unable to tell" throughout.

What counts as an answer, and this is the one thing that decides everything. The question is whether listeners converge with each other rather than whether they agree with the theory. Four tables disagree and two of them are exact inversions, so any one of the four taken as the standard against which a respondent is marked would assume the answer the pack exists to get. Convergence on any of the four is a result, including convergence on a table the theory sets aside, because it establishes that there is a table to find and says which one it is. Convergence on a fifth pairing is a result and a sharper one, because it would mean the registers pair with scales in a way this corpus has yet to write down, which is a finding about the world rather than about the corpus. And a spread of answers refutes the premise. If expert listeners diverge from each other about which scale goes with which predicament, then A4 is a statement about a fact that the panel finds no evidence of.

The panel, stated honestly. This is a small expert panel, and it is the right instrument for settling a design parameter and the wrong instrument for a citable perceptual claim. The pack runs without pre-registration, counterbalancing or a power calculation, and it says so in its own documentation rather than leaving a reader to notice. The author's decision of 12 September 2026 sets the number at five people [6]. Every figure the pack returns is a design parameter rather than a perceptual result, and this paper reads them as design parameters throughout. Part R of the pack asks respondents themselves what a citable study would require, in four places: the minimum panel size they would accept, the statistic they would want reported, what is missing from the pack, and and what they make of a pack that presents one fixed order to every respondent. That is the instrument asking to be replaced by a better one, and the answers to Part R are the specification for its replacement. Two listener studies already exist in this corpus, and their stimuli came from an unseeded system, which is what a restaged study repairs by seeding [1]. These stimuli are reproducible, and that is the one methodological property this pack has that its predecessors lacked.

What each outcome does to the theory and to the application#

Four outcomes are possible and each is traced through here, because a question is only worth asking if the answers are worth having and that has to be shown rather than asserted.

One: listeners converge on the live pitch table. A4 stands as written, mechanism and assignment both: the dominant register selects the mode and trauma dims within a pair. The theory holds as stated and this section's tables hold as printed. What changes is the standing of the repair. The two plumbing changes stop being housekeeping and become the two changes that turn a warranted table into a rendered one, so they are what makes A4 testable on a build, and the mode channel becomes the first parameter in the whole mapping with listener warrant behind it. A standing constraint on the implementation, that a modal table is built under A4 once the second-stage question returns, is satisfied on this outcome with the live table as the one to implement [23]. The open question about the Imaginary's trauma partner closes in the direction that the code is right and the decision is wrong, since the live table gives the Imaginary a partner and a panel converging on that table is endorsing a table that gives it one; the record would then note a decision overturned by a return rather than by an argument, which would be the first such entry in the corpus.

Two: listeners converge on its inversion, the reference dictionary. A4's mechanism stands and its table is inverted. This is the most expensive of the four outcomes for the corpus, because the live assignment is what every document carrying a modal example carries. This section's own worked material reads from it, the two worked frames of section 3 read from it, and the pack's own Part B blends Dorian against Lydian because that is the live table's Real against Symbolic tie, so those stimuli would be answering a question about the wrong pair even though the properties they test survive. Every document carrying the live table needs correcting and the list is nameable: MPN-S1's statement of A4 and its section 6.3, the assertions register entry for A4, section 2.3 and section 3 of this paper, and the unified design framework's modal observation [1], [11], [24]. The application's repair changes too. The two plumbing changes are still correct, but the table they carry is replaced, and implementing the replacement takes one further step, because the dictionary as it stands carries two slash-joined labels that are display strings rather than mode names, and a six-degree whole-tone entry that the cardinality constraint below rules inadmissible, which leaves the interpolation undefined on it. So this outcome opens new work, a clean seven-degree statement of the inverted table, and that is a table question rather than a plumbing one, which puts it with the author before the implementation starts. The question about the Imaginary's trauma partner stays open on this outcome: the form of the reference dictionary that reaches lookupMode carries a single mode per register, so whether the Imaginary has a partner stays live, and its content changes from a conflict between code and decision to a gap between a table and an assertion.

Three: listeners converge on the synthetic raters' mechanism. A4's second stage changes rather than its table. The switch is a jump to a crisis mode per register and not a dim within a pair, and the assignment below trauma 0.6 may stand untouched by the finding, which is why this outcome is compatible with either of the first two on the assignment question and settles a different axis. Three things move with it. The relief mechanism MPN-S2 leaves open moves, because section 2.1 names it as what supplies relief from a monotone dynamic marking, and a second stage that jumps rather than dims is a different object to make releasable; the release has to be designed against a jump. The trauma switch moves: the surface at $\tau = 0.6$ stays a jump surface, so this section's jump-set result survives untouched, but what the surface switches between becomes a distant mode rather than a neighbouring one, so the semitone that the live table moves at that surface becomes a larger interval and the size of the discontinuity has to be recomputed against whatever crisis mode each register takes. And the trauma partner for the Imaginary is settled against the decision of 12 September: if the mechanism is a jump to a crisis mode per register, then every register has a crisis mode, the Imaginary included, and a decision assigning the Imaginary no partner is a statement about a mechanism other than the one under test. The documents that change are MPN-S1 section 6.3 and the assertions register entry for A4, both of which carry the mechanism [1], [11]; the decision record needs an entry against the 12 September ruling [6]; and section 1.4's parameter table keeps its jump set for mode and changes its description of the second stage. What opens is a new drafting task, the crisis mode per register, which is three assignments rather than three pairs and which this corpus has yet to write down.

Four: the answers spread. This is the outcome to be most careful about, because it says the least and is the easiest to overstate in either direction. It leaves the mapping's correctness untouched and it is a result about the kind of claim A4 makes rather than about the mapping. A4 is then a convention with an author rather than a fact about listeners, and the honest consequence is that the mode channel carries a convention rather than a finding. A convention has real uses and this paper already labels one: section 2.6 reads the spectral centroid as the task-against-people contrast and says in the table itself that this is a convention. What A4 asks of it is more than a convention carries, which is that the register-to-mode pairing is consequence rather than taste. On this outcome A4 is restated as a convention with a named author and a stated arbitrariness, in the form section 2.6 already uses, and MPN-S1's entry for A4 and the assertions register both change to say so [1], [11]. The implementation's work list is unchanged, which is the sharpest single consequence: the two plumbing changes remain correct and stop being urgent, because the table they would carry is a convention and the convention runs on the build as well without them.

And something survives outcome four, which is why it is worth stating carefully. The interpolation, the Lipschitz result and the jump set are properties of the rule and not of the table. The weight $w_k$ is defined on any pair of modes of equal cardinality and mentions no mode by name. The proof that the rule is determinate at a three-way tie is independent of which modes sit at the vertices. The exact agreement with $\arg\max$ outside the margin is a property of the weight function alone, since a register $\delta$ below the leader has weight exactly zero whatever mode it names. The Lipschitz constant is proportional to $1/\delta$ for the same reason. And the jump set is the single surface $\tau = 0.6$ and that surface alone, because it is the table that switches there while the simplex map runs continuously, so the shape of the result holds for whichever table switches. Change the table and every one of those results holds in statement and in proof; what changes is the size of the bend at the surface, which depends on how far apart the two modes of a pair are. A panel whose answers spread therefore takes A4's content and leaves this section's mathematics intact. It also leaves the cardinality constraint standing, which is what rules the reference dictionary's whole-tone entry inadmissible whatever any listener says, since that constraint is about the arithmetic of a weighted mean over degrees rather than about anybody's ear.

All four outcomes leave this section's mathematics untouched. That is by construction and it is worth saying plainly: this section's formal content stands whatever the answer, and its assignment content turns entirely on it.

The tie, and how the code disposes of it by accident#

The tie is not a corner case. On the frame library, 21 of the 128 states that reach the simplex sit exactly on one, 16.4 percent, and a further 104 frames produce no state at all because the analyzer returns $(0,0,0)$, 44.8 percent of all 232 [12]. So a sixth of the states that reach the simplex take their mode from whatever the comparison happens to do, and none of the selectors records a rule.

Those two counts correct figures this programme has carried since MPN-S2, and the correction is worth stating because its cause is a parser and not an instrument. Every script that read the frame library matched the annotation field with a non-greedy pattern that stops at the first occurrence of the quote character and reads backslash escapes literally. Seven of the 232 annotations contain an escaped apostrophe, so on those seven the pattern returned a truncated string, and on three of them the truncation removed the one register keyword the annotation carried, which the analyzer then read as a degenerate triple. One of the three, Comparison of authenticity. The actor's fake emotion feels more Real than Hamlet's true drive, is pure Real once read in full. The figures carried until then were 107 degenerate frames and 125 on the simplex; the corrected figures are 104 degenerate, 128 on the simplex and 101 at a vertex. The tie count of 21 and the single interior point are unchanged, so every result that rests on the tie stands. s3_frames.py is now the only reader of the library and the other scripts call it; run directly, it prints the comparison. The corrections to MPN-S2 and to the register simplex figure are issued separately [14].

lookupMode and getModalTransformation both build the array in the order Real, Symbolic, Imaginary, sort it descending, and take the first element. Array.prototype.sort has been required to be stable since ES2019, so on a tie the earlier element survives and the effective rule is Real over Symbolic over Imaginary. The test-only rsiToMode initializes its answer to 'symbolic' and replaces it on a strict inequality against both others, so a tie there yields Symbolic instead. The two disagree on the three-way tie and on every two-way tie involving the Real, and in both cases the behavior is a side effect of how a comparison was written rather than a decision anybody took. On the current build both selectors' answers are held rather than rendered, so the disagreement between them takes effect the moment the two plumbing repairs land.

What the theory should do is a separate question. MPN-S2 leaves it open and offers three candidate rules, ruling out two of them on grounds this paper adopts rather than re-derives [2]. Hysteresis holds the previous frame's register until the new leader exceeds it by a margin: it is defined by a gap, so it holds its last value when the gap is zero, and on a scene's first frame it needs a previous register supplied to it. A dead band freezes the selector while the gap is small: the same two properties. Both also make $\Phi$ a function on $\mathcal{P} \times \mathcal{P}$ rather than on $\mathcal{P}$, which costs A11's decomposability, the property that lets a frame's music be computed from that frame's state alone. Interpolation is the one of the three that is determinate at an exact tie and on a first frame, and the one that leaves decomposability intact.

Modal interpolation, defined#

The choice among the three rules is the author's; what this paper owes is the interpolation itself, defined so that it works on any pair of modes, and that is what follows. The idea is simple. Where two registers are close, the mode is not chosen between them: the degrees on which the two candidate modes differ are bent proportionally, so a character poised between the Real and the Symbolic sounds poised rather than sounding like whichever register won by a thousandth. Let $x = (r,s,i)$, let $m = \max x$, and fix a margin $\delta$. Give the mode of register $k$ the weight

normalize to sum to one, and sound each scale degree at the weighted mean of the degrees the candidate modes assign it. The leader always has weight 1 and a register $\delta$ below it has weight exactly 0, so candidates enter and leave the set continuously; a weighting without that property jumps at the margin.

Four properties follow, each with a one-line proof and each checked numerically as well [12]. The weights are defined at every point of the simplex, so the rule is total and the three-way tie is determinate: at the barycentre the weights are equal and the result is the mean of the three modes, and at a two-way tie the third register has weight 0 and the result is the midpoint. If $x_k \le m - \delta$ then $w_k = 0$, so wherever the gap exceeds $\delta$ the leader alone survives and the rule is $\arg\max$ exactly; enumeration finds exact agreement on all 180,183 such states. And $w$ is a composition of 1-Lipschitz maps scaled by $1/\delta$ over a normalizer bounded below by 1, since the leader's weight is identically 1, so the map is Lipschitz on the closed simplex with a constant proportional to $1/\delta$. Numerically, at $\delta = 0.05$ and a grid step of $1/1200$ the largest change in any degree between adjacent states is 0.0476 semitones, halving with the step; the $\arg\max$ selector jumps a full semitone across the tripod at every resolution. The Lipschitz property holds for every $\delta > 0$, and the constant grows as $\delta$ shrinks.

The margin $\delta$ is the register gap below which two modes blend, and it is the author's to fix. Every property above holds for any positive value; what changes with it is this:

The Lipschitz constant grows as $1/\delta$, so a small margin bends fast over a small region and a large one bends gently over a large one; the worst-case notation error is a quarter tone at every margin, because it is attained at the tie itself. That quarter tone is a property of the notation path and not of the mapping, for the reason the next paragraphs give. Ahead of a microtonal renderer, the practical way to choose is to hear one blended pair at $\delta = 0.05$ against $\delta = 0.20$ on a synthesizer that takes cents, which runs without notation. What a larger margin buys is a gentler bend rather than more reach. Outside the margin the rule is $\arg\max$ exactly, so the region on which the registers reach the mode continuously is the neighbourhood of the tripod; by MPN-S2's kite result its area is $2\delta - \delta^2$, which is 9.75 percent at $\delta = 0.05$ and goes to zero with $\delta$ [2], [12]. The interpolation removes the discontinuity and leaves the channel as categorical as it was.

The cost is stated rather than buried. Interpolated degrees are microtonal, so this parameter's codomain is no longer the seven diatonic modes but a continuum containing them, and any renderer targeting standard notation must round. The worst case is exactly a quarter tone, reached at a two-way tie between modes whose degrees differ by a semitone, and rounding there reintroduces the jump the interpolation removed. The interpolation is what the theory means. What a score can print is a separate question, and the six paragraphs that follow settle it.

$\Phi$ emits cents, and the rounding belongs to the renderer. Under the author's decision of 14 September 2026, $\Phi$ emits the blended scale degree as a continuous value in cents, and the notated pitch is the rounded pre-image of that value, a property of the notation renderer rather than of the mapping [17]. An earlier statement of this section held that the implementation owed a rounding rule. The rounding sits in the renderer, so the debt sits there too, and what a renderer does with 3.46 semitones above the tonic is a setting on that renderer. The argument is one precedent and it is already in this paper: section 2.1 settles the identical question for dynamics in a single sentence, that the output of $\Phi$ is the marking and the velocity interval is the pre-image, as MPN-S2 section 6.1 sets out. Doing the same for mode makes the two channels consistent rather than introducing a new rule into either, and that is the whole of the argument.

The jump set stays where the proof puts it, and that is the point of the rule. It remains exactly what this section proves it to be, the single surface $\tau = 0.6$ and that surface alone. Rounding inside $\Phi$ would have put a jump on every rounding boundary inside the margin, a set that grows with $\delta$ and that would take an enumeration of its own. Section 1.4's parameter table stands as printed for the mode row, and the pre-image rule is what keeps it standing.

What rounding inside $\Phi$ would have cost, measured rather than argued. Rounding inside $\Phi$ would leave a scale different from $\arg\max$ on 0.64 percent of the whole simplex at $\delta = 0.05$ and on 3.05 percent at $\delta = 0.20$ [16]. Everywhere else it snaps every blended degree back onto the winning mode. Each figure is the product of two measured columns: the share of the simplex inside the margin, 9.21 percent at $\delta = 0.05$ and 35.36 percent at $\delta = 0.20$ on a grid of 20,301 points at a step of $1/200$, against the closed form's 9.75 and 36.00, which is the cross-check that the grid is computing the margin this paper describes; and the share of points inside the margin at which the rounded blend is a different scale from the one $\arg\max$ would have chosen, 6.96 percent and 8.62 percent. So rounding inside $\Phi$ would go well past degrading the interpolation that this section chose over hysteresis and a dead band and spent its longest passage defending. It very nearly deletes it.

The rounding convention decides the scale exactly where the interpolation exists. At an exact tie the weights are one half each, so every degree on which the two modes differ by a semitone lands on a half. On the live table's Real against Symbolic tie, Dorian against Lydian, the blend is 0, 2, 3.5, 5.5, 7, 9, 10.5. Round half up returns 0, 2, 4, 6, 7, 9, 11, which is Lydian exactly, one of the two inputs. Round half to even returns 0, 2, 4, 6, 7, 9, 10, which is none of the seven, and it is what a Python implementation gets by writing round [16]. One convention returns an input and the other returns a scale that is not in the table, and both do so at precisely the states the interpolation was chosen for: this section's own count puts 21 of the 128 frames that reach the simplex exactly on a tie. It is not a detail to leave to a language.

Where the convention question goes, since it survives the decision. It moves to the notation renderer. A renderer printing a score from the continuous value still chooses between half up and half to even, faces the same two outcomes at the same states, and must declare which it chose on the face of the score. What the pre-image rule buys is that the choice sits with one renderer rather than gating the four papers of this path, and that the audio path carries the exact value regardless, so the $\delta$ experiment runs today on a synthesizer that takes cents. Part B of the listening pack is that experiment. Its seven items are the two pure inputs, the exact blend at the tie, the two near-tie blends at $\delta = 0.05$ and $\delta = 0.20$ separated by 17.4 cents, and both roundings of the tie as separate items, so the panel's answer about the convention falls out of the same seven items that answer about $\delta$ [19]. Two of those seven carry the same scale, because rounding half up at an exact tie returns one of the two inputs exactly, which makes that pair a result and a catch trial at once.

A consequence that reaches past notation. Under the pre-image rule $\Phi$ ceases to discretize mode at all [21]. Mode therefore joins the channels with no stated discretization anywhere in this programme, and the consequence is that no perturbation budget can ever be written for that channel, a budget being a bound stated against a quantum and this channel now having none. Any later proposal writing a bias or an influence principle into mode must supply a discretization or state its bound in cents. Section 4's layer is the obvious candidate and it is the reason the point is recorded here.

What the pre-image rule costs, printed rather than implied. It costs the notation path its fidelity inside the margin, and the cost is exactly the quarter tone above: notation and audio disagree by up to a quarter tone, only inside the margin, on the share of the simplex the table above gives, 9.8 percent at $\delta = 0.05$ and 36.0 percent at $\delta = 0.20$. Anyone reading a score while hearing the audio hears the difference there and it is not a rendering artifact but a stated property of the two paths. It costs the mode channel any future perturbation budget. And it moves an unresolved convention onto a renderer that has no specification yet, which improves where the problem sits without solving it.

Two standing constraints bind any future choice under A4. Cardinality: interpolation is defined degree by degree, so every mode in a table must have the same number of degrees, which holds on the live table's six seven-degree modes and fails on the reference dictionary, whose Imaginary entry is a six-degree whole-tone scale against seven-degree modes elsewhere, making interpolation undefined there rather than merely awkward. And the ruling of 12 September 2026: the Imaginary takes no trauma partner, so any table giving it one is outside A4's admissible space as that ruling stands [6]. Within those, A4 may reassign the modes freely.

Two statements about the jump set need care. The map is continuous at the simplex boundary: the weight map is Lipschitz on the closed simplex, and crossing a face where a register reaches zero moves every degree by at most 0.0196 semitones at a step of $1/2000$ [12]. But the table itself switches at $\tau = 0.6$, so the candidate modes change discontinuously in trauma whatever the interpolation does on the simplex, and on the live table that switch moves a degree by a full semitone. The jump set of the mode parameter under interpolation is therefore the single surface $\tau = 0.6$ and that surface alone. That surface is a property of A4's second stage rather than of the tie rule, and making the second stage continuous is what would remove it, which is a proposal for another paper.

2.4 Fragmentation and orchestration density#

MPN-S1's amendment to A8 gives the pair as a direction and its orthogonal complement [1]:

Effective contributions at the library dispersions: density is 24.8 percent entropy to 75.2 percent trauma; fragmentation is 64.3 to 35.7 [5]. Both ladders are at even fifths, 0.2, 0.4, 0.6 and 0.8, declared in advance rather than fitted, giving five stages each.

MPN-S2's Theorem 1 establishes what the superseded pair cost and the amended pair buys. For two weighted sums of the same inputs with every weight at least $\varepsilon$, the correlation stands at or above $\sin(2 \arctan(\varepsilon / (1 - \varepsilon)))$ [2]. The superseded pair, $0.6H + 0.4\tau$ against $0.7\tau + 0.3H$, has every weight at least 0.3, so it stood at $+0.7241$ or above whatever any corpus contained: the collinearity the A8 amendment responds to was a property of the weights and not a fact about the plays. On the 232 annotated frames it realizes $+0.9150$. What buys the separation in the amended pair is the negative weight on trauma in the fragmentation term, which takes the second vector out of the positive quadrant where Theorem 1's bound lives, and that negative weight is a musical claim with a direction: weight makes a theme more fully stated. The adopted pair is exactly orthogonal away from the clip, and the clip binds on 21 of those frames, 20 strictly and one on the boundary. On the library as a whole the adopted pair realizes $+0.0571$ with the clip applied and $+0.0031$ with it removed, the second figure being what the theorem predicts once the library's own dispersion and its trauma-entropy correlation of $+0.2802$ are substituted into it. One caveat belongs with those numbers. Restricted to the 211 frames where the clip is slack the correlation rises to $+0.2072$, because conditioning on being off the clip is conditioning on a half-plane defined by both variables. Orthogonality is a property of the pair over the whole state space rather than of any subsample selected by the clip. A character under great weight whose account still holds gets the theme whole and thick, which is what the second worked frame in section 3 is chosen to show.

None of this is implemented. The shipped code still computes fragmentationScore = (entropy * 0.6) + (trauma * 0.4) with cuts at 0.25, 0.5, 0.75 and 0.9, and an orchestration intensity = (trauma * 0.7) + (entropy * 0.3) with cuts at 0.2, 0.4, 0.6 and 0.85 [12]: the superseded pair, the one MPN-S1 amends and Theorem 1 bounds below at $+0.72$, on two ladders that are unevenly cut and unequal to each other. At the library's dispersions those nominal weights are effectively 53.6 percent entropy to 46.4 percent trauma, and 75.2 percent trauma to 24.8 percent entropy [5], [12]. Two qualifications belong with them. On the composer path entropy is pinned at 0.5, so the two reduce to $0.3 + 0.4\tau$ and $0.7\tau + 0.15$, and each is a function of trauma alone while that stands [5]. And the amended forms above are what this paper specifies and what the implementation owes.

The two jump sets are the eight ladder boundaries, four each, and both ladders are piecewise constant, so each quantity steps in the output however continuously it moves in the state. It would be shown wrong if listeners ordered the five fragmentation stages by how much of a theme they had heard in an order other than the ladder's, which is the test A7 already carries and which also tests B5. There is a sharper version of that test available now that the two quantities are orthogonal: present pairs of cues that differ in fragmentation and agree in density, and pairs that differ in density and agree in fragmentation, and ask which pair differs more. Under the superseded pair those two conditions were barely distinguishable, because the quantities moved together; under the amended pair they are independent, so a listener who hears the two pairs as differing by the same amount is telling you the two ladders are one ladder to the ear whatever the arithmetic says.

2.5 The harmonic operator#

The harmonic codomain is the 24 consonant triads under the neo-Riemannian transition algebra generated by $P$, $L$ and $R$, which the sister series sets out in full [7]. What matters for the mapping is that this codomain has a metric: the Cayley graph distance under the three generators, which is the only thing in the theory that makes the size of a harmonic move a number [2].

That structure is checked here rather than cited, because everything in this subsection depends on it [12]. The three generators are involutions with no fixed points, the action on the 24 triads is transitive, and the Cayley graph has diameter 5 and radius 5, which confirms the sister series' figure independently. The distance distribution over all 576 ordered pairs is 24 at distance 0, then 72, 144, 192, 120 and 24. The last of those is worth noticing: from any triad there is exactly one triad at distance 5, its antipode, and from C major that antipode is B flat minor. The hexatonic pole, $PLP$ of C major, which is G sharp minor, sits at distance 3 and is not the farthest point.

The parameter, in the absolute form. Chord position is

around the $LR$ Hamiltonian cycle from a fixed origin, and the realized chord walks to the target by a shortest word, one generator per bar. The constant is $k_{\max} = 23$ and the next subsection says why that value and no other. The parameter is a function on $\mathcal{P}$ alone: it reads no previous state and no previous chord. Shortest words are not unique and the tie-break is lexicographic with $P$ before $L$ before $R$, which the sister series fixes and this paper adopts unchanged so that two implementations agree.

What the relative form cost, kept as a record of why the form changed. An earlier form of this proposal read differently: a state change of size $\delta$ in the trauma coordinate moved the chord $\operatorname{round}(k_{\max}\delta)$ positions along the alternating chain from the chord already sounding. Two words in that sentence were doing more than they looked. A state change was a function of two states, and the realized chord was the one the previous bar left behind. So that parameter, unlike every other in section 2, was not a function on $\mathcal{P}$ at all: it was a function on $\mathcal{P} \times \mathcal{P}$ and on the chord already sounding. Those were exactly the two dependencies section 2.3 uses to rule out hysteresis and a dead band for the mode selector, so the paper carried a rule it disqualified elsewhere, and it had the first-frame problem too, since a scene's opening bar has no previous chord and no previous state. In a therapy session the first frame is the opening of every session, which is where that defect would have been felt. The question was then put, without being computed, whether the parameter could be re-expressed as a function of the current state alone. It can, and the absolute form above is that re-expression [16], [17]. A11's decomposability, the property that lets a frame's music be computed from that frame's state alone, now holds of six parameters of six rather than five.

The alternating chain is well defined, which was not obvious. Alternating $L$ and $R$ from any triad closes after 24 steps having visited every triad exactly once, so the $LR$ chain is a Hamiltonian cycle and "position along the chain" is an integer modulo 24. From C major the first eight positions are C, E minor, G, B minor, D, F sharp minor, A, C sharp minor, which is the circle of fifths with each relative minor interleaved. The other two alternations do not have this property and could not carry the parameter: $PL$ closes after 6 triads and $PR$ after 8, which are the hexatonic and octatonic cycles.

$k_{\max} = 23$, and why no other value. The constant was open from the first draft of this paper and the author's decision of 14 September 2026 closes it at 23 [17]. The constraints were computable and are set out first, because the value follows from them rather than from a preference. Under the relative form, a change of $\delta = 1$, a character destroyed over a play, moves $k_{\max}$ positions and can reach $k_{\max} + 1$ distinct chords:

Two constraints are real and a third, which an earlier draft of this paragraph asserted, is not. At 24 and above the map becomes non-injective by an artifact of the modulus rather than by any fiber of the theory, so $k_{\max} \le 23$. At the other end, the antipode of a triad sits at chain offset 11 from twelve of the 24 triads and offset 13 from the other twelve, so no value below 11 lets any trauma trajectory reach a chord at the graph's full diameter, and no value below 13 lets every trajectory do so. The claim an earlier draft made, that $k_{\max} = 5$ makes a full sweep span the diameter, is false: five chain positions cost three generator steps, and the greatest distance reachable at $k_{\max} = 5$ is 4, which the table's own row said while the sentence beside it said otherwise.

So the defensible range is 13 to 23 if the theory wants a destroyed character to be able to arrive at the far side of the harmonic space, 11 to 23 if it will accept that happening from half the starting chords, and 1 to 23 if the reach is left open.

Reach, which is where the two forms part company and where 23 is forced. The two forms agree on the size of every move by construction. They do not agree on what can ever be visited. The relative form reaches all 24 triads at any $k_{\max}$ by accumulating moves; the absolute form reaches $\min(k_{\max} + 1, 24)$ from its fixed origin [16]:

Three reasons, and which of them is decisive. First, 23 is the only value at which the absolute and the relative forms have the same reachable set, as the table shows: below it the absolute form is a strictly smaller mapping than the one this paper published, and at 24 and above it wraps by the artifact of the modulus already ruled out. Second, 23 sits at the top of this section's own defensible range of 13 to 23, so a character destroyed over a play can arrive at the far side of the harmonic space from every starting triad rather than from half of them. Third, 23 is the largest value that stays injective. The first is decisive. It is not a musical preference for 23 over 22; it is what makes the re-expression possible at all, and the other two reasons would not have chosen a value on their own. The constant is therefore load-bearing in a way it was not under the relative form, where any value in the range would have served, and that is part of what the change costs.

What the absolute form fixes. The harmonic parameter becomes a function on $\mathcal{P}$ alone, reading no previous state and no previous chord. That removes both of the dependencies section 2.3 uses to rule out hysteresis and a dead band for the mode selector, so this paper no longer carries a rule it disqualifies elsewhere. It restores A11's decomposability for the one parameter of six that lacked it. And it loses the first-frame problem.

What the constant leaves standing, and the constant should be read with this beside it. Chain position and Cayley distance still rise and fall independently, so the parameter is as far from monotone in the metric at 23 as at any other value. The musical judgment about how far a chord should travel when a character is destroyed stays with the author; what the paper now records is that at 23 a full sweep traverses the whole cycle, and that traversing the whole cycle is a different thing from traveling far. The next paragraphs set that out in full.

A defect in the proposal, found in checking it, and it survives the change of form. The proposal, in either form, says that the size of the quantity driving the parameter sets the size of a harmonic move, and cites the Cayley metric as what makes that size a number. Measured in that metric it does something other than what it says. Chain position and Cayley distance rise and fall independently, because a cycle of 24 is embedded in a graph of diameter 5 and folds back on itself. An earlier statement of this made the point from three data points, that four positions cost four generator steps, five cost three and twenty-three cost one. Three points understate how badly the function behaves, and the full table from C major replaces them [16]:

Read downward the column rises and falls. The greatest distance any chain position buys is reached at 13 and the function falls away on both sides of it. Position 17 and position 23 both give distance 1, so at three quarters of the range and at the full sweep alike the character ends one generator from where they started. So a larger change in $f$ can produce a smaller harmonic move. Fixing $k_{\max}$ at 23 leaves this standing and makes the worst case plainest.

Two repairs are available and each has a price. Define the move in the metric instead of on the chain, taking the target to be a triad at Cayley distance $\operatorname{round}(5 f(\text{state}))$, the graph's diameter of 5 standing in for the cycle's 23: monotone by construction, and it names a set of chords rather than one, since from C major there are 3 triads at distance 1, 6 at distance 2, 8 at distance 3, 5 at distance 4 and 1 at distance 5, so a selection rule goes with it. Or keep the chain and withdraw the claim that the move size is metric, in which case the Cayley distance is decoration and the parameter is a walk along a fixed cycle. The first is what the theory appears to mean and the second is what its present wording describes. The choice decides whether the harmonic parameter carries a magnitude or an index, and it is the author's, and it is unchanged by the change of form, because what the absolute form settles is decomposability and not magnitude [16]. Part C of the listening pack is the test: ten triad pairs at five Cayley distances, each item one chord, a short gap and another, with the respondent asked to rate how far the second chord feels from the first on a scale from next door to as far as it gets, and asked to judge by ear rather than by naming the chords [19]. If rated distance tracks graph distance, the metric carries a magnitude and the first repair is available. If rated distance runs free of it, the second repair is the honest one, the metric is decoration, and the parameter is a walk along a fixed cycle. The ten pairs are deliberately unevenly spaced, for the reason the table above gives.

One loose end, and it is the author's. The function inside the harmonic parameter is named by the author. The decision of 14 September fixes the form and the constant and leaves the naming of $f(\text{state})$ to that decision, and this paper leaves it open rather than adopting the one candidate the live implementation offers [17]. That candidate is the tension term $0.9r + 0.1H$ that section 3 enumerates as the one live linear reading of a register, and three facts put its adoption to the author rather than into this paper. It is a function of the Real and of entropy rather than of trauma, so adopting it changes what section 1.4's table says the harmonic position is a function of. At the frame library's dispersions its nominal 90 to 10 is effectively 94.9 to 5.1, so it is very nearly the Real alone [5]. And on the 164 frames of 232 whose annotation section 3 counts as carrying a Real keyword nowhere, the Real is exactly zero, so tension reduces to $0.1H$ in $[0, 0.1]$, which under $\operatorname{round}(23 f)$ confines the chord to three of the 24 triads on roughly seven frames in ten. Section 7 carries the naming of $f$ as open and as the author's.

What this paper adopts, and what it leaves aside. The affective reading of the individual generators, that $P$ darkens and $L$ elevates, is a convention awaiting its experiment, and the foundations paper of the sister series says so in its own words [8]. What is adopted is the metric, which is a statement about the graph rather than about a listener.

Jump set and falsification. The jump set is the 24 chain positions of $\operatorname{round}(23 f)$, which is a quantum of $1/23 = 0.0435$ in $f$ and is writable now that the constant is fixed [21]. It would be shown wrong if listeners did not rank pairs of chords by the Cayley distance between them, which is a discrimination test on the metric alone and needs no affective vocabulary. That test is worth running before either repair is chosen, because if the metric is not audible as a magnitude then the second repair is the honest one.

2.6 Timbre, which the theory has never specified#

This is the parameter MPN-S2 assigns to this paper, and it is the one where the specification changes what the theory can claim.

What the timbre literature supplies. Multidimensional scaling of musical timbre asks listeners to rate the dissimilarity of pairs of sounds and recovers the small number of perceptual dimensions those judgments imply, then looks for acoustic quantities that correlate with each. McAdams and colleagues, working with synthesized instrument tones, find three shared dimensions with correlates log rise time, spectral centroid and degree of spectral variation, together with instrument-specific specificities, attributes of a particular instrument that lie on no shared dimension at all [3]. The specificities matter for what follows, because they are categorical and are what an instrument-family label carries. A confirmatory study using synthetic tones designed to isolate the candidates also finds three dimensions: it confirms attack time on a logarithmic scale, spectral centroid, and spectrum fine structure modeled as even-harmonic attenuation, and finds spectral flux only weakly salient and strongly context-dependent, contributing little when attack time and centroid vary concurrently [4]. So the number of dimensions is agreed at three. What the two studies disagree about is the acoustic correlate of the third: spectral variation in the earlier, spectrum fine structure in the later. The dimension is not in dispute; its correlate is. There is also a categorical residue in both, the instrument specificities.

The space. MPN-S2 assigns the timbre space to this paper and shows it to be the one channel the four DISC coordinates reach [2]. Take

with $F$ a finite set of instrument families carrying the specificities, and the three continuous coordinates the log attack time, the spectral centroid, and the third dimension, which is named as the third dimension rather than as one of its candidate correlates because the two studies disagree about which correlate it is. Writing it that way keeps the disagreement visible in the specification instead of burying it in a choice. One number the space needs comes from a measurement rather than from either study or from this paper: how finely a listener resolves distance in the three continuous coordinates, which decides how many characters the channel separates and therefore what any capacity claim made on it is worth, and which Part D of the listening pack is built to fix [19].

The map from DISC. Write the four DISC coordinates in an orthonormal contrast basis, a scaled Hadamard matrix, verified orthonormal:

Two are the instrument's own axes: $h_1$ its pace contrast, D and I against S and C, and $h_3$ its task-against-people contrast, D and C against I and S. $h_2$ is a contrast the instrument leaves unnamed, and $h_0$ is the height of the whole profile rather than a contrast. The assignment:

The result, which is a constraint rather than a design. The map has rank three, the timbre space being three-dimensional on both cited studies, against a domain of dimension four, so its nullity is exactly one. Its null direction is $h_0$. Two characters whose DISC profiles differ only by a constant added to all four coordinates produce identical timbre. Verified on two worked pairs [12]: $(0.3, 0.3, 0.3, 0.3)$ and $(0.7, 0.7, 0.7, 0.7)$ both give contrasts $(0, 0, 0)$, and $(0.7, 0.2, 0.5, 0.1)$ and the same profile raised by 0.2 both give $(0.15, 0.45, 0.05)$. The first pair is the extreme case, a character with no shape at all at two different heights; the second is an ordinary profile moved bodily up the scale. Each pair sounds identical through this channel to any listener. This is a property of the arithmetic rather than a defect of the assignment. Any map from four coordinates into three dimensions has a null direction, and the question a theory gets to answer is which one; the choice made here is that profile shape is audible and profile magnitude is the direction given up, which is how the DISC instrument is read in the first place. The presence of a null direction is fixed rather than chosen: a three-dimensional channel carries three of four coordinates, and the loss is exactly one dimension. The claim is conditional on the rule for choosing among $F$'s members, which is specified together with the enumeration of $F$ and which could in principle read magnitude. On the shipped build it reads shape: discToInstrument takes $\arg\max$ over the four DISC coordinates, which is invariant under adding a constant to all four [12], so the null direction survives into the instrument family as well. Any future rule for $F$ preserves that, or the result above weakens.

The consequence for the programme is direct and it is the kind of result worth having early. A listening study that asks a listener to recover a character's DISC profile from a cue is asking a three-dimensional channel for a fourth coordinate, and would return a partial failure that read as a failure of the listeners or of the mapping when it is a failure of the question. What such a study can ask is whether listeners discriminate characters differing in profile shape, holding magnitude constant or varying it deliberately as a control, and it should be designed to that from the start. Section 2.6 therefore cancels one study and specifies its replacement, which is the most useful thing a result of this kind can do.

The partition of the timbre channel, which MPN-S2 section 6.1 assigns to this paper, follows $F$: the three coordinates are continuous here, so the jump set is written once $F$ and its selection rule are fixed, and the partition of $T$ is written with them. Section 7 records it as outstanding.

One debt carried forward, and a narrower result in its place. Assertion B4 puts the number of independent quantities a single stave can carry at two to three, calls it an estimate rather than a measured figure, and assigns this paper either its derivation or a recovery study [1]. The timbre literature answers a different question. B4 counts independent state quantities carried by a stave, of which timbre is one channel among several; the scaling literature counts perceptual dimensions within the timbre channel. Those are different quantities and the agreement between "two to three" and "three" is a coincidence of numbers. What the result does give is narrower and still worth having: an upper bound of three on the independent quantities the timbre channel alone can carry, measured rather than estimated, for one channel of the several. B4 remains an estimate and the recovery study discharges it.

Where the budget stands after the amendment of 15 September 2026. Two points belong here because they change what may be said about the budget rather than what the timbre result says. The budget now goes entirely to the two rendered layers: the between-stave clause has been struck from B4, so every part of that assertion falls under the per-stave budget of two to three, and a measured budget that differs from two to three falsifies B4 directly instead of relocating a layer out of its way. And the budget applies per stave, so it is independent of how many characters a score carries; what caps the cast is the timbre channel's own capacity, which is the resolution number above and is a separate question from the budget, settled by the discrimination experiment that measures it. On the interaction side of the programme, the per-person measures of Layer 1 now number six rather than five, backchannel having been named a measure in its own right rather than a parameter of the overlap row, and it is measured differently according to how the material arrives: lexically on script text, where the words are exact and the speaker labels are given, and from timing on recorded material, where a detector is running in any case. One measure with two definitions carries two settings rather than one, and an output must declare which definition produced a number, because a count taken lexically and a count taken from timing are two different quantities, and the declaration is what keeps a surface from putting them side by side.

3. What survives into a score#

MPN-S2 computes a coarse bound on the information in one frame, states which state dimensions reach which parameters, and leaves the fiber structure open pending the timbre space [2]. With the space specified in section 2.6 that analysis can be completed. What is new here is the completion and the arithmetic, not the shape of the table: MPN-S2 section 6.4 already states that the two register degrees of freedom reach only mode and that the four DISC coordinates reach only timbre, and MPN-S2 section 6.5 states the stronger fact, confirmed and sharpened in section 2.3 above, that on the composer path the register triple is a constant, so the mode is the one parameter the registers reach once a register-dependent mode reaches the score rather than today. They do reach other parameters now, and this section sets out which.

The state has eight degrees of freedom [2]. Taking them in turn:

One row of that table is conditional on the function the author names. Section 2.5 re-expresses the harmonic position as $\operatorname{round}(23 f(\text{state}))$ and leaves $f$ open, so which state coordinates reach the harmonic parameter is open with $f$. The row above lists it against trauma because trauma is what the earlier relative form read. Under the absolute form it is a function on $\mathcal{P}$ and the author fixes which part of $\mathcal{P}$; section 1.4's table says so.

The register plane. Under the selector as MPN-S1 and MPN-S2 describe it, mode takes the dominant register, so the simplex reaches the output only through $\arg\max$ and every state in one cell produces the same mode: a character at $(0.9, 0.05, 0.05)$ and one at $(0.4, 0.35, 0.25)$ are indistinguishable here. The interpolation of section 2.3 removes the discontinuity but not the collapse: outside the margin it reproduces $\arg\max$ exactly, and the region where it does anything else has area $2\delta - \delta^2$, under a tenth of the simplex at $\delta = 0.05$.

DISC loses exactly one dimension. Section 2.6.

Counting. Multiplying the cell counts together, three tempo bands by eight markings by three modal cells by five fragmentation stages by five density levels, gives 1,800 frames and about eleven bits. That is a loose upper bound: dynamics, density and fragmentation are three functions of the same two coordinates and move together, and tempo and metre are two functions of one of them. Enumerating the reachable output tuples over the $(\tau, H)$ square on a grid of 1,201 by 1,201 gives the figures below, the three modal cells included as a factor and timbre excluded [12]:

The shipped rows use the shipped laws throughout, three dynamic labels rather than eight and the superseded A8 pair with its uneven ladders. They are lower for the reason one would expect: a lookup with holes emits fewer distinct things than one without.

Four cautions attach to every figure. They count emission over the whole $(\tau, H)$ square rather than over the states a play produces, and the corpus reaches only part of it; on the composer path entropy is pinned at 0.5, so the shipped fragmentation stage there is a function of trauma alone and the shipped rows overstate what that path emits [5]. They count emission rather than recovery: the larger figures need tempo resolved to the beat per minute, and the whole span inside the two lower bands is eight and six beats per minute. They take the modal factor as three, which brackets both readings: the shipped one selects mode from $\tau + H$, and the specified one has a continuum for its codomain, so three is a bracket rather than an equality. And they exclude the harmonic coordinate, which the table above lists as reached. The reason previously given, that section 2.5 left $k_{\max}$ unbounded and the number of reachable triads with it, is superseded: $k_{\max}$ is fixed at 23 and the codomain is 24 triads exactly. The counts stand as computed here, and the exclusion now rests on a different ground. Adding a factor of 24 would move every figure in the table above, and that recomputation belongs with the script that produced them rather than with a multiplication written into a sentence, because the harmonic factor moves with the others once $f$ is named, and naming $f$ is the author's. MPN-S2's own bound of 4,032 includes 24 triads as a factor where this one holds the harmonic coordinate aside.

Two frames, all the way through. None of this is much use without cases, so here are two real ones, taken from the shipped library with the trauma, entropy and annotation the author wrote for them, and the register triple the shipped analyzer reads from that annotation [12]. They are chosen by stated criteria rather than picked for effect: the first is the highest-trauma frame that reaches the simplex without sitting at a vertex, and the second the lowest-trauma frame with a register above 0.6. Between them they cover a tie and a vertex, both sides of the trauma switch, both ends of the dynamic range, and one register threshold that fires against four that stay below the cut.

Frame 1: ANAGNORISIS, Oedipus. Trauma 1.0, entropy 0.1, annotation "Total Symbolic Collapse. The Ego (Eye) gives way before the vision of the Real." The analyzer reads $(\tfrac13, \tfrac13, \tfrac13)$: the barycentre, an exact three-way tie, and the single interior point in the whole library.

Frame 2: Othello/Desdemona, Othello. Trauma 0.1, entropy 0.3, annotation "The origin of love in narrative." The analyzer reads $(0, 0, 1)$, pure Imaginary.

Five things are visible in that pair that a general statement conveys less sharply. Each is derived from the table by the script rather than written beside it, so the two stay in step.

The printed name and the notated pitches disagree on both frames. Frame 1 is headed Phrygian and notated in Locrian; frame 2 is headed Ionian and notated in Phrygian. On frame 1 the printed name happens to coincide with the head of what the registers would have given, phrygian/locrian, which makes the mechanism look as though it is working. It is a coincidence of two unrelated functions: the branch that produced Phrygian read trauma and entropy and no register, and it produced Phrygian because $\tau + H = 1.10$ is above 0.7. That is exactly how a defect of this kind survives a spot check.

The register thresholds are silent where the registers are most interesting. Frame 1 is the barycentre, the one state in the library where all three registers are equal, which is where A4 has the most to say and where MPN-S2's tripod result bites. It is also the frame where no key threshold fires, because no register reaches 0.6, so the key falls through to its default of C Major. Frame 2 is pure Imaginary, one threshold fires, and the key is E Major. At most one threshold can ever fire, since at most one register can exceed 0.6 on a simplex.

The dynamic law agrees on a label and not on a velocity. Frame 1 is fff either way, but at velocity 127 under the normative law against 118 in the build. Frame 2 is pp under the normative law and ppp/pp in the build, at velocity 30 both. Agreement on a label here is agreement over a much wider band: the shipped lookup has three labels where eight are normative, so ppp/pp covers what the normative law splits into two markings and two velocities.

The A8 amendment moves fragmentation and leaves density alone, on both frames. Frame 1: fragmentation rung 0 specified against rung 1 shipped, density rung 3 against rung 3. Frame 2: fragmentation rung 1 against rung 0, density rung 0 against rung 0. That is what taking the trauma weight negative is for. Oedipus at the moment of recognition is at the top of the density ladder under either pair, because he is carrying everything the play has; the amended pair drops his fragmentation to zero, because his account of what has happened is at that moment complete. Whether that is the right reading is a question for a composer's ear, and it is printed here so it can be disagreed with.

Timbre awaits its instrument in both columns. The DISC coordinates come from a rating instrument, and inferDISC returns null until one exists, so every consumer handles a null profile [12]. The previous implementation banded four values off average trauma and filled the band with Math.random(), fabricating four of the nine state components per character; that is withdrawn, and a declared null replaces it. The one channel section 2.6 analyses is the one channel both frames exercise once a DISC instrument supplies its input.

The bound that matters is not the count but the reach. Of the eight state dimensions, two are recoverable as quantities, two reach a channel that is categorical under $\arg\max$ and continuous within the margin under interpolation, and four reach a channel that carries three of them and leaves their magnitude aside. That is the honest ceiling, and it belongs in front of any listening study rather than after it is analyzed.

What the registers reach on the build as it stands. A search for mode names supports a conclusion about modes and not one about registers, so the claim here rests on a wider search: every line in the source that reads a register component, enumerated [12]. The registers reach the leitmotif transformation, through selectTransformation, which is called with the frame's actual triple and rewrites every pitch in the melody; they reach the key, which unlike mode is a declared field of the output; and they reach the tension, and through it the chord type and the harmony readout. What they do not reach is the mode.

That enumeration turns up a further finding of its own. Every live register reading takes a single component in isolation. None of the six compares the three against each other. Five are thresholds at 0.6, two in selectTransformation and three in the key branch, and on the simplex at most one component can exceed 0.6 so they are well defined; the sixth is linear, $\text{tension} = 0.9r + 0.1H$, which reaches the harmony readout as a continuous number and a five-way chord type through cuts at 0.2, 0.4, 0.6 and 0.8 [12]. Two things must be said about that coefficient together, and the pair of them is what is true of the system [5]. At the library's dispersions the nominal 90 to 10 is effectively 94.9 to 5.1, because the Real varies more than entropy does; and on the 164 frames of 232 where the analysis text carries no Real keyword the Real is exactly zero, tension reduces to $0.1H$, which lies in $[0, 0.1]$, and the chord quality is major7 by arithmetic rather than by theory [12]. The one comparison between components in the build is lookupMode's sort, and its answer is held rather than rendered. So a state at $(0.59, 0.21, 0.20)$ and one at $(0.34, 0.33, 0.33)$ take the same branch on every threshold and differ on the chord type only through $r$. What reaches a score is the value of individual registers, and on the 68 frames where the Real is non-zero it is chiefly the Real; on the other 164 the register contribution is zero. Which register leads is what reaches the score once the mode printed on it is computed from the register triple. That is a weaker quantity than A4 is about and a different one from the competition A3 asserts.

4. The bias layer#

What the layer is, before any count in this section is read. The bias layer renders no music in version one. It is a Layer 3 detector whose output is a marked textual proposal beside the Layer 0 turn that prompted it, which is the author's decision of 14 September 2026 on the layer's channel question, taken as the null among five candidates [17], [24]. This is a statement of what the layer is in the version being built and not a deferral of a device or of a channel, and it belongs before section 4.1 because every count below reads differently under it. The thirty devices set out in this section are therefore a specification awaiting a renderer, and those are the words to use for them. The alternatives mislead. Deferred implies a date, and what governs here is the reopening condition below. Proposals is the word section 4.4 already uses for their evidential status, which is a different matter. And struck would say the reconciliation invariant of section 4.1 had stopped accounting for all thirty, which it has not. The phrase stands unsoftened.

The reopening condition, which replaces a permanence clause. The option as put to the author carried a permanence clause, and the decision replaces it with a stated condition: a measured detection rate from the two detector trials the programme has scheduled [17], [24]. The first pilots the layer on choice-supportive bias, which section 4.2 below names as the natural pilot because it is the only mapping checkable against earlier material note for note without a listener at all. The second plants the frame-local biases in generated dialogue and scores detection per turn, and scores the historical ones as passage pairs. If the detector is shown to find planted moves at a usable rate, the channel question reopens on measured evidence, which is the evidence the trials supply; and if it reopens, the candidate is the fourth stave rather than articulation and voicing, that being the option a later revision of this paper is likeliest to reclaim for a parameter of its own, and a channel a later specification reclaims costs more than waiting does, because its devices are then specified twice.

What the layer's status costs, recorded rather than softened. It defers for a second time a capability the author asked for twice. It leaves the whole of this section, thirty devices, sixteen mappings, fourteen domains and fourteen coordinates, as a specification whose route to evidence runs through the detection rate the two trials measure. And it changes why section 3 excludes the layer from the identifiability counts, which section 4.4 states.

MPN-S1 settles the classification at the author's Cognitive Bias Atlas [9], thirty entries in four domains, and records that the Atlas and the reference implementation carry different thirties [1]. The author's decision of 12 September 2026 assigns the missing content to this paper: it is drafted here, and the author strikes out what is wrong [6].

4.1 The two sets of thirty, reconciled reproducibly#

MPN-S1 gives four counts from name matching. A script now derives all four from the Atlas and the implementation rather than restating them, and asserts that the drafted content covers exactly the gaps, so that a change to either source fails the script rather than silently invalidating the paper.

The implementation carries 29 distinct traits across 30 entries because framing appears twice, positively and negatively framed. The strict reading is the one used, because it is the one that has to be defended; the loose reading adds optimism against the planning fallacy, liking against the halo effect and cognitive dissonance against choice-supportive bias, and the argument holds on either reading.

4.2 The thirty mappings: sixteen drafted, fourteen inherited#

Each names the musical device, the domain and, in fifteen of the sixteen, the state coordinate the device rides on, together with one clause giving the reason the device follows from the bias, so that a strike can be judged on the row rather than by opening a script. A bias here is a modulation of something the state already determines.

The exception is CB-004, survivorship, and it is named rather than absorbed. That row rides on fragmentation, which is a $\Phi$ output on the very channel the device writes to, rather than on a state coordinate [21]. A row that rides on the output it perturbs is a different object from a row that rides on the state, and the difference matters for a strike: strike survivorship and what is left is a channel with one fewer thing written to it rather than a stated hole in a state coordinate.

How to strike one. Delete the row, delete the entry of the same id from DRAFT_MAPPINGS in 05_DATA/03_generators/s3_bias_reconcile.py, and add a line | CB-0nn | why | to 08_PAPERS/MPN-S3-STRIKES.md. The script reads that file, and its invariant is that every Atlas entry awaiting a musical mapping is covered by a drafted mapping or by a recorded strike, so a strike recorded with its reason keeps the check green and an unrecorded strike makes it fail. That is the intended behavior in both directions: the reconciliation continues to account for all thirty entries, and a dropped row shows up as a failure. The paper's argument stands independently of any single row, and the rows stand independently of each other, with the single exception noted under Survivorship below. A domain marked with a dagger is itself drafted in section 4.3 and can be struck separately; a mapping and its domain are struck separately in either order.

The fourteen inherited mappings, with coordinates supplied. The sixteen above are drafted here. The fourteen inherited from the reference implementation named no coordinate at all, and the table below supplies one for each in the same six columns, so that the two read as one table of thirty [22]. A domain marked with a dagger is drafted in section 4.3 and is strikeable separately, on the same terms as the sixteen. A row marked undecided carries the drafter's flagged alternative, because the choice there is genuinely open and a flagged row is easier to strike than a defended one.

What no row of the fourteen rides on, and why that is a statement rather than an omission. No row rides on the Real. The Real in this calculus is what resists symbolization, and a cognitive bias is by definition an operation of the account rather than a failure to have one, so the Real is the wrong register for all fourteen; the drafted sixteen put exactly one entry there, CB-023, and the clause it carries is that ignoring information is a hole in a specific place rather than silence, which is a refusal at intake and not a distortion in processing. No row rides on fragmentation or on density either. Density is $0.3H + 0.7\tau$, which at the frame library's dispersions is three quarters trauma, so a row on density is a row on trauma with a weaker statement of itself. Fragmentation is clipped at zero on a fifth of the annotated frames, so a proposal conditioned on it is conditioned on a quantity that is constant over a large region of the state space. If the author wants a row on fragmentation, CB-021 is the candidate, a blind spot being a hole in the account [22].

Loading the fourteen takes the layer from twelve signatures and four collisions to eighteen and eight. Four of the eight collisions are created by this drafting and are strikeable with the rows that create them, which is the price of supplying coordinates at all: the collisions were always latent and were invisible only because half the table was blank.

CB-017, framing, remapped. The inherited device for framing sets the mode, and no channel allocation is available on which setting the mode is admissible [21]. It is remapped to the same pitch classes restated in a different registral disposition and spacing, open and high against close and low, with mode, key, tempo, metre, dynamic marking and instrument all held [22]. Every $\Phi$ parameter is held, so the device writes outside everything section 2 determines, which is the property that makes the remap admissible.

Two costs come with it and both belong here rather than in a note. First, it costs the frame-local classification. A restatement is audible only against the statement it restates, so CB-017 moves out of the frame-local class into the historical one: the per-turn score becomes a passage-pair score for that row, and the layer's detector partition moves from twenty-two frame-local and eight historical to twenty-one and nine. That is a real loss to the cheapest test the layer has. Second, it rests on a sub-channel a later revision specifies. The octave placement, the spacing and the voicing of the pitch set the leitmotif transformation produces are open here, which makes the sub-channel free today and fragile tomorrow: a later revision specifying voicing takes it back, and puts CB-017 one specification away from a collision with a parameter the theory owns.

Four of the sixteen drafted rows are worth more than a clause, because they are the ones a composer is most likely to argue with and because the argument in each case is about the device rather than about the bias.

Survivorship is the only mapping that rides on fragmentation rather than on a state coordinate, and it has to: the bias is reasoning from what remains, which is audible only if the listener heard what was dropped. It therefore requires the theme to have been stated whole earlier in the scene, and it is the one mapping in the set that is not local to a frame.

Groupthink and in-group bias are both textural and both social, and they are distinguished deliberately rather than by accident of vocabulary. Social proof, which is in the influence layer and not here, is joining: a voice that was not playing begins to play what the others play. Groupthink is absorption: a voice that was playing something else is drawn into the unison mid-phrase, so what the listener hears is a dissent that stops. In-group bias is a third thing, and it concerns two textures rather than one: it is two groups whose tuning or articulation are incompatible, so the disagreement is about the terms of playing rather than about the content. If the author strikes one of those three the other two need rewording, because each is defined partly by contrast with the others; that is the one coupling in the table and section 4.4 declares it.

Choice-supportive bias is the natural pilot for the whole layer, being the only mapping that restates earlier material with a change and so the only one checkable against that earlier statement note for note. Every other mapping asks a listener to recognize a device; this one lets an analyst compare two passages of the same score and say whether the restatement is more favorable than the original, without a listener at all. If the layer is to be piloted cheaply, it should be piloted here.

Zero-risk bias and information bias both ride on entropy and both concern voicing density, and they pull in opposite directions: the first closes the voicing, the second thickens it. A character high in both should sound contradictory, and if that never happens in practice the two are not independent.

4.3 The fourteen drafted domains#

Assigned on the Atlas's own definitions and nothing else: Perception filters the input, Memory holds the account, Decision calculates risk, Social conforms to the network.

Completed, the Atlas distributes as Perception 8, Decision 10, Social 6, Memory 6, totalling thirty. The drafted fourteen are more weighted toward Memory than the catalogd sixteen, which move from 2 of 16 to 4 of 14, 12.5 percent to 28.6; Decision shifts by one entry's worth, 31.3 percent to 35.7. That is a fact about the expanded reference rather than about the mind: the entries left undomained are disproportionately about what is recalled after the fact.

Two assignments are genuinely arguable and are flagged rather than defended, because the author is asked to strike what is wrong and a defended assignment is harder to strike than a flagged one.

Status quo bias could be read as Perception, on the ground that what is currently the case is what gets attended to and everything else has to be imagined. It is placed in Decision because the Atlas's own definition makes it a preference for the current state, and a preference is a weighting applied at the point of choosing rather than a filter applied at the point of noticing. That is a fine distinction and the Atlas's four domains force it.

Outcome bias could be read as Decision, since what it distorts is the evaluation of decisions. It is placed in Memory because the evaluation happens afterwards, against the record of what happened, and Memory is the Atlas's storage layer: the bias is in what the record is allowed to contain, not in the calculation performed on it.

The two turn on the same distinction, between a distortion at the moment of acting and a distortion in what is available to act on. If the author strikes either, the other should be reconsidered with it, because a reading of the Atlas's domains that rescues one will usually move the other.

4.4 What the drafting rests on, and how to strike it#

The drafting has to be strikeable one entry at a time. Three properties make it so.

Each mapping is independent of every other. No device is defined by reference to another bias's device, so removing one leaves the remaining fifteen intact. The two exceptions are declared: survivorship depends on the theme having been stated, and groupthink, in-group bias and social proof are defined by contrast with each other, so striking one of those three requires the other two to be reworded. Note that social proof sits in the influence layer of section 5 rather than among the sixteen, so that coupling crosses a layer boundary and striking it is not a strike operation on this table.

Fifteen of the sixteen name the state coordinate they ride on, so a struck mapping leaves a stated hole rather than an undefined one: the coordinate is still mapped, the bias simply no longer modulates it. CB-004, survivorship, is the named exception, for the reason section 4.2 gives, and striking it leaves a channel rather than a coordinate.

The claim about the inherited entries, qualified by the class the enumeration covered. A claim about where a state coordinate reaches a bias entry is worth exactly as much as the enumeration behind it, which is the rule every enumerative claim in this paper follows. As shipped, the reconciliation script reads a five-field JSON projection and assigns each inherited entry a null coordinate at the record-building line, working from the projection rather than from the reference implementation, so that script is evidence about the projection rather than about the code [21], [22]. The claim is therefore stated from a wider enumeration and is narrower and stronger for it. A search of the checkout of 14 September 2026 for values rather than for function names finds a state coordinate attached to a cognitive bias in exactly three places, and all three attach to the list as a whole rather than to a single entry. They are a trauma gate that suppresses the whole bias list above trauma 0.5 and treats every member alike; a wiring of confirmation bias into the mode selector that overrides the register triple and misses on every frame; and a visualization that gives six named channels a common amplitude in trauma and a common spread in entropy while emitting text alone. The claim reaches source text and data committed as source, and it is a statement about one checkout on one date; a runtime dependence, an unshipped branch or a build artifact is a separate enumeration.

One field set aside rather than used. The projection carries a strength in $[0.75, 0.95]$ on all thirty entries, and the script leaves it unread [21], [22]. It is recorded here and set aside, with the reason: it is a constant band whose derivation is still to be stated, so using it would put a weight into the layer ahead of the argument for it.

And none of the sixteen is load-bearing for anything else in the series. No assertion depends on a particular bias mapping, and B1 asserts only that the layer exists and is not predicted by the state.

Why section 3 excludes the layer. The identifiability result of section 3 counts the rendered channels, and the bias layer sits outside them. The reason given for that exclusion used to be B4, on the ground that the layer was carried between staves rather than on one [11]. That ground is superseded. B4's between-stave clause was struck on 15 September 2026, so the assertion's whole content is per-stave and the per-stave budget applies to every layer under it. The exclusion holds for a plainer reason and the section says which: an identifiability count counts what a listener hears, and the layer is carried in the data and rendered as text. Section 3's tuple and bit figures stand either way.

Assertion B1 is tested once the layer renders. B1 asserts that the bias layer exists and carries variance the state leaves unexplained. Testing it needs a rendered bias that an analyst or a listener can find in a score, and version one carries the layer as text. The sentence above, that every assertion stands independently of any particular bias mapping, holds unaltered. What changes is the register entry for B1, which should record that its test is now gated on the same reopening condition as the channel, a measured detection rate from the two detector trials [11], [14], [17].

What the drafting rests on is a reading of the Atlas. These are proposals by a reader of it rather than findings, and the honest status of all thirty mappings, the fourteen inherited as well as the sixteen drafted here, is that each awaits the listening test that establishes what it carries.

5. The influence layer#

This layer renders as data too, and for a different reason from the bias layer's. The decision of 14 September 2026 answers the bias layer's channel allocation, and takes the null [17]. The influence layer's channel allocation is a question still to be put, so its six devices are carried in version one as a specification awaiting one. The practical consequence is the same and it should be stated in the same words: the six below are a specification awaiting a renderer, and this section's account of what distinguishes the layer is an account of a specification. The difference matters in one place, and it is where a future channel would come from. If the bias layer's channel question reopens on a measured detection rate, what reopens is the bias layer's channel; the influence layer's is a separate question, and the three structural differences below are why it takes an argument of its own rather than inheriting that one.

The author's decision of 12 September 2026 separates Cialdini's six principles of influence [10] from the bias index: a technique applied to a subject and a distortion in a subject are different objects [6]. MPN-S1 section 7.2 gives the reason the merge went unnoticed, that both sets happen to number thirty, and this section adopts that rather than restating it [1]. All six are present in the implementation and five of them are implementation-only, with authority the one that also matches an Atlas entry.

These six are read out of the implementation rather than drafted, and like everything in section 4 they are proposals, each awaiting the listening test that establishes whether it is audible as the principle it names. Where the bias mappings carry a state coordinate, these carry the interlocutor instead, because an influence principle is indexed to the speaker applying it, which is itself a reason the layer is separate. They are listed under the implementation's own bias- identifiers, which is provenance rather than classification: the separation is from the Atlas index, not from the file they happen to live in.

The layer differs from the bias layer in three ways that the theory should state, because the merge happened silently and the difference is structural rather than one of emphasis.

It has a source and a target. A bias is a property of a subject; an influence principle is something one character does to another. Its magnitude is therefore indexed by an ordered pair and not by a person, and a scoring system with one index per character has nowhere to put it. That is the same structure B3 gives the dyadic biases, so the machinery already exists and the influence layer should use it rather than inventing a second one.

It is intentional, and biases are not. A character can be subject to anchoring without anyone anchoring them: the anchor may be a number they read. Reciprocity reaches a character when another character invokes it. A system that distinguishes a distortion a character arrives with from a technique another character applies to them is representing the drama, because in drama the second is usually the event and the first is usually the condition.

Its musical devices are directed. Call and response has a caller. An accelerando under scarcity is imposed by the speaker on the scene's pulse; it expresses the speaker's act, and written into the listener's stave it says the wrong thing. Where a bias modulates a character's own stave, an influence principle modulates the relation between two staves. That relation is a real object in the drama and it is a channel this theory has yet to allocate: since 15 September 2026 every assertion in the register is per-stave, so an influence principle wanting a between-stave channel argues for it against the per-stave budget rather than inheriting it.

Authority sits in both sets and should stay in both, under two readings this paper distinguishes. Authority bias is the subject's disposition to over-weight a source, which is CB-012 and stays in the Atlas; the authority principle is the deployment of status cues by a speaker, which joins the influence layer. They are two different objects, they have different indices, one is a condition and the other an act, and the shared name is why they were conflated in the first place. A drama can carry either without the other: a character may over-weight every authority they meet without anyone in the play deploying status, and a character may deploy status all evening on people immune to it. The musical consequence is that the two want different devices, and the implementation currently has one.

6. What would show the mapping wrong#

The mapping can fail in six ways, and they are worth separating because each names a different instrument and because the cheapest of them can be run before the expensive ones and would settle a good deal. A note on order: three of the six run on a build whose score carries the mapping this paper specifies, and running them earlier would produce nulls that mean nothing. The ordering constraint is stated with each.

The parameter orderings could fail: if listeners rank cues by dynamic marking, fragmentation stage or density level in an order other than the one the ladders assert, the ladders are private codes. A discrimination test that runs without verbal anchors, and the cheapest in the series.

The modal assignment could fail, which is A4 and which MPN-S1 already costs. Section 2.3 adds a prior question that is settled by reading source: the assignment under test has to be one the application implements, and at present every mode that reaches a score is a function of something other than a register. A listening study run on the current build would be putting a branch on $\tau + H$ in front of listeners and calling the null a verdict on A4. A study on the registers more broadly is in a different position: they do reach the score, and section 3 states how: five readings are thresholds at 0.6 on single components, and the sixth, the tension term, reads the Real linearly and reaches the harmony readout as a continuous number and the chord type through cuts at 0.2, 0.4, 0.6 and 0.8. What such a study could test is a threshold and one linear term, where A3 asserts an ordering. The order of work is therefore fixed: the repair in section 2.3 comes before a study on the application.

A study on the application is not the only kind, and Part A of the listening pack is the other kind. Because the pack's stimuli are synthesized from the scale formulas rather than rendered by the build, Part A runs ahead of the repair, and A4 can be put to listeners now. Section 2.3 describes the instrument and traces the four outcomes. The one point to repeat here, because this is the section about failure, is what failure means for this test. Listeners disagreeing with the theory's table is a result rather than a failure. A4 fails as a claim about listeners if their answers spread, and the section states what survives in that case, which is the whole of the interpolation result and the whole of the assignment's standing as a convention.

Interpolation could fail musically. Section 2.3 adopts it because it is the only option determinate at a tie; if musicians hear an interpolated mode as out of tune rather than as intermediate, the adoption is wrong and the theory must accept a state-dependent selector and lose decomposability. Part B of the listening pack is that test, and it carries a second question with it: whether the two margins are distinguishable at a separation of 17.4 cents, which is what decides $\delta$. If listeners separate them, a small margin is doing audible work and $\delta$ can be small; if they do not, a small margin buys nothing and the larger one is the honest choice [19].

The timbre map could fail. Section 2.6 makes two predictions that are directly testable and that pull in opposite directions, which is what makes the pair worth running together. First, characters differing only in DISC magnitude are indistinguishable on this channel: that is a null prediction, and a null prediction is unusual in this programme and therefore worth having, because confirming it costs little and disconfirming it would mean the map has a different null direction from the one claimed, which is a fact about the map and not about the listeners. Second, characters differing in the pace contrast are distinguished by attack time: a positive prediction on a dimension both cited studies agree about. A design that varies magnitude and shape orthogonally tests both at once. Both tests run ahead of any change to the application, because the timbre channel is built with the rating instrument, so both are run on purpose-built stimuli rather than on the Conductor's output, and that is a virtue here rather than a limitation. Part D of the listening pack is built on exactly that principle, eight same-or-different pairs with pitch, length and loudness held so that color is the only thing that could differ, and two of the eight genuinely identical as a floor on a respondent's reliability [19].

The rhythmic claim could fail, and section 2.2 predicts that on the present implementation it will. Entropy is supposed to be heard as instability of pulse rather than as speed, yet the implementation moves tempo by 40 and then 68 beats per minute and hands metre a lookup with a gap in it. The test is a two-alternative forced choice, speed against steadiness, and the prediction recorded here is that listeners will answer speed. That is a prediction against the mapping, and satisfying it needs the metre lookup repaired and the tempo ranges narrowed, which is the implementation's work.

The bias layer could fail as a layer, which is B1: if the bias vector proves largely predicted by the state vector, it is a view of the first layer and not a second one, and thirty new quantities collapse into a restatement of nine. MPN-S1 section 7.1 settled the list of thirty; what section 4.1 adds is that the reconciliation is reproducible and the domains complete, so the list to rate is fixed rather than reconstructed each time. That test runs once the layer renders. Section 4 states the layer's status, and B1's test is gated on the same reopening condition as the channel, a measured detection rate from the two detector trials, so this is the one of the six failure modes whose instrument is built by that reopening [17].

7. What this paper decides and what it proves#

Decided here. The eight-marking dynamics. The rounding pre-image rule: $\Phi$ emits cents, the notated pitch is the rounded pre-image, and the rounding convention belongs to the notation renderer, which is what keeps the mode parameter's jump set at the single surface $\tau = 0.6$. $k_{\max} = 23$ with the absolute form, which makes the harmonic parameter a function on $\mathcal{P}$ alone. The bias layer's status in version one, a Layer 3 detector emitting a marked textual proposal, with its thirty devices a specification for a renderer and a reopening condition rather than a permanence clause. The definition of modal interpolation; the choice of interpolation over hysteresis and a dead band remains the author's, and section 2.3 sets out what each costs. Interpolation is the one of the three determinate at a tie, the alternatives holding their last value on 21 of the 128 frames that reach the simplex, 16.4 percent of those and 9.1 percent of all 232, with the weight stated, four properties checked, and the margin $\delta$ left to the author. The constraint that a table under A4 may reassign modes freely and holds every scale to one cardinality. The timbre space, its three coordinates and the contrast map from DISC, with profile magnitude as the declared null direction. The influence layer's separation from the bias index.

Proved here. That the absolute and the relative forms of the harmonic parameter have the same reachable set at exactly one value of $k_{\max}$, namely 23, the absolute form reaching $\min(k_{\max}+1, 24)$ triads and the relative form reaching all 24 at every value, which is what makes the re-expression possible rather than merely desirable. That chain position against Cayley distance is non-monotone across the whole range of 23 positions, with positions 17 and 23 both at distance 1. That rounding inside $\Phi$ would leave a scale different from $\arg\max$ on 0.64 percent of the simplex at $\delta = 0.05$ and 3.05 percent at $\delta = 0.20$, so it would very nearly delete the interpolation rather than degrade it, and that at an exact tie half up returns one of the two inputs while half to even returns none of the seven modes. That the neo-Riemannian Cayley graph on the 24 triads has diameter and radius 5, computed from the generators rather than cited, which confirms the sister series' figure independently; that the $LR$ alternation is a Hamiltonian cycle, so position along the alternating chain is well defined, while $PL$ and $PR$ generate shorter cycles and are set aside for that reason. That the DISC-to-timbre map has rank three against a domain of dimension four, so its nullity is exactly one, and that the null direction is profile magnitude. That the interpolation of section 2.3 is determinate, reduces to $\arg\max$ away from the margin, and is Lipschitz on the simplex. Together with MPN-S2 section 6.4, which supplies the reach table, these complete the identifiability bound of section 3.

8. How every number in this paper is reproduced#

Every numerical claim in this paper is produced by one of fourteen scripts in 05_DATA/03_generators/, none is transcribed, and each aborts if the lines of the implementation it reads have changed. Eleven are this paper's own and three are shared with the blocking-numbers note and with the listening pack, marked as shared below. 05_DATA/03_generators/S3-GENERATORS-README.md, committed beside them, says how to run them; the eleven need MPN_REPO set to the Conductor working tree and that alone, and the three shared ones read the tables directly rather than the implementation at all, which is why the figures they supply are unaffected by the state of the build.

9. References#

On the paths in these entries. An entry that gives a path in backticks names an artefact in the MPN working corpus rather than a page on this site. That corpus is held privately, so such a path resolves inside the working corpus rather than from this page. 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. Keeping the citation is what discloses the dependency.

A citation of the form psychometric_calculus.ts:211 names a file and a line in the MPN Conductor source tree, read at the working tree of 13 September 2026, which is the tree every implementation claim in this paper is pinned to and the same tree MPN-S2 section 6.5 cites.

The rest of the series is published in this working group: MPN-S1, the theory; MPN-S2, the formal apparatus; MPN-S5, the dialogue use cases; and MPN-S6, the expression aid. Where an entry below also gives a working filename such as S1-mckenney-lacan-theory.md, that is the corpus copy of the same document.

[1] J. McKenney, "The McKenney-Lacan psychometric calculus: a theory of musical representation for psychological state," MPN-S1, 08_PAPERS/S1-mckenney-lacan-theory.md.

[2] J. McKenney, "The formal apparatus of the McKenney-Lacan psychometric calculus," MPN-S2, 08_PAPERS/S2-mathematics.md. Cited for the state space, the identifiability bound this paper completes, the register instrument audit, and the three candidate tie rules.

[3] S. McAdams, S. Winsberg, S. Donnadieu, G. De Soete and J. Krimphoff, "Perceptual scaling of synthesized musical timbres: common dimensions, specificities, and latent subject classes," Psychological Research, vol. 58, pp. 177-192, 1995. Three shared dimensions with acoustic correlates log rise time, spectral centroid and degree of spectral variation, plus instrument specificities.

[4] A. Caclin, S. McAdams, B. K. Smith and S. Winsberg, "Acoustic correlates of timbre space dimensions: a confirmatory study using synthetic tones," Journal of the Acoustical Society of America, vol. 118, no. 1, pp. 471-482, 2005. Confirms a three-dimensional space with correlates log attack time, spectral centroid and spectrum fine structure modelled as even-harmonic attenuation; finds spectral flux only weakly salient and strongly context-dependent, contributing little when attack time and centroid vary concurrently.

[5] "Commensurability audit," MPN-AUDIT-01, 08_PAPERS/COMMENSURABILITY-AUDIT.md. Cited for the effective contributions and for the rule that no coefficient appears without one.

[6] "Decision log, 12 September 2026," MPN-DECISIONS-01, 08_PAPERS/DECISION-LOG-2026-09-12.md. Cited for the eight-marking dynamics being normative, for the ruling that the Imaginary takes no trauma partner, for the assignment of the bias drafting to this paper, for the separation of the influence principles from the bias index, for the synthetic raters' reading of the second stage, and for the panel size of five.

[7] "Musical psychometric notation: the notation system," MPN-2, 08_PAPERS/MPN-2-notation.md, section 3.6. Cited for the neo-Riemannian transition algebra, the Cayley metric, the alternating chain and the tie-break on shortest words.

[8] "Musical psychometric notation: foundations," MPN-1, 08_PAPERS/MPN-1-foundations.md, correction 12. Cited for the statement that the affective reading of the PLR operators is a convention without experimental evidence.

[9] J. McKenney, "The cognitive bias atlas: critical infrastructure of the mind," Unified Psychometric Field Theory, volume XV. The classification of record: thirty entries, four domains, sixteen catalogued and fourteen in the expanded reference.

[10] R. B. Cialdini, Influence: The Psychology of Persuasion. Cited for the six principles that are separated from the bias index.

[11] J. McKenney, "The McKenney assertions: a register for examination," 08_PAPERS/ASSERTIONS-REGISTER.md. Cited for A3, A4, A7, A8, A11, B1, B3, B4 and B5, and for the amendment of 15 September 2026 striking B4's between-stave clause.

[12] MPN Conductor reference implementation, cited at path and line. src/components/mpn-lab/psychometric_calculus.ts:170-187 for entropyToRhythm, :109-113 and :192-199 for two tables awaiting a caller, and :296-320 for psychometricToMusical; src/components/mpn-lab/mpn_reference_lookup.ts:31-49, :110-120, :125-145, :150-176, :220-242, :261-278 and :344 for checkCondition, the metre, tempo and dynamics lookups, lookupMode and the remainder lookupAllParams, which awaits a caller; src/components/mpn-lab/mpn_reference_data.ts:248-406, :664-742 and :1034-1102 for the metre, tempo, dynamics and mode entries; src/components/mpn-lab/score_orchestrator.ts:61-70, :267, :321, :360 and :440-445 for the output interface, the two live calls, the mode assignment that awaits a reader, and the output object carrying the four declared fields; src/components/mpn-lab/score_exporter.ts:73 for the literal mode: 'ionian' written under the comment TODO: extract from params, and :81, :84, :88 and :89 for the four fields the exporter does take from musicParams, which do not include the mode; src/components/mpn-lab/GeniusComposer.ts:161, :180 and :188 for the hard-coded register triple; src/lib/leitmotif_transformation_rules.ts:63-107 and :209-250 for the modal selector at :63-91, the scale formulas at :96-107 and the shipped A8 pair; src/components/mpn-lab/leitmotif_generator.ts:243-266 and src/components/mpn-lab/psychometric_calculus.ts:320, :356-361 for the register readings that do reach a score; src/app/mpn-conductor/page.tsx:353 and :488 for the Lyapunov quantity and the branch that sets the printed mode; and ml/psychoscore_v2/models/mckenney_lacan_calculus.py:58-90 for the velocity law and the eight-marking discretisation.

[13] "The implementation audit behind S3: every path and line," MPN-NOTE-03, 08_PAPERS/MPN-NOTE-03-implementation-audit.md. Carries the call chains, the extracted evidence, and the account of the withdrawn call-graph tool.

[14] "Register and audit amendments arising from S3," MPN-S3-AMENDMENTS, 08_PAPERS/MPN-S3-AMENDMENTS.md. Replacement text for A4, A8 and B4, two corrections to the commensurability audit, and the correction of the degenerate-frame count against MPN-S2 and the register simplex figure.

[15] C. Drake and M.-C. Botte, "Tempo sensitivity in auditory sequences: evidence for a multiple-look model," Perception and Psychophysics, vol. 54, no. 3, pp. 277-286, 1993. Relative just-noticeable differences for tempo of about 6 per cent for a single interval and about 3 per cent for a six-interval sequence, with interonset intervals from 100 to 1,500 milliseconds and best sensitivity between 300 and 800.

[16] "Two blocking numbers: the rounding rule and $k_{\max}$," MPN-NOTE-05, 08_PAPERS/MPN-NOTE-05-blocking-numbers.md, 14 September 2026, with 05_DATA/03_generators/s7_blocking_numbers.py and its committed output. Cited for the 0.64 and 3.05 per cent figures, the margin cross-check against the closed form, the two roundings at the tie, the absolute-form reach table, and the full chain-position against Cayley-distance table.

[17] "Decision log, 14 September 2026," MPN-DECISIONS-2026-09-14, 08_PAPERS/DECISION-LOG-2026-09-14.md. The bias layer's channel taken as the null with a reopening condition; and the rounding pre-image rule with $k_{\max} = 23$ and the absolute form.

[18] "Decision log, 15 September 2026," 08_PAPERS/DECISION-LOG-2026-09-15.md. Cited for the amendment of B4 striking its between-stave clause, and for backchannel being named a sixth Layer 1 measure with a definition that depends on the ingest class, lexical on script text and timing-derived on recorded material.

[19] "Listening pack version 2: what it asks, who it goes to, and how to read what comes back," MPN-PACK-02, 14 September 2026, with the published pack, 05_DATA/03_generators/s7_listening_stimuli.py, 05_DATA/03_generators/stimuli/MANIFEST.json, 05_DATA/03_generators/stimuli/DISPLAY.json and 05_DATA/03_generators/stimuli/ANSWER-KEY-v2.json. Cited for Part A's seven items and two questions, Part B's seven items and the 17.4 cent separation, Part C's ten triad pairs, Part D's eight pairs and six voices, Part R's four method questions, the blinding, the fixed order drawn from seed 20260914, and the statement that the panel is the right instrument for a design parameter and the wrong one for a citable perceptual claim.

[20] "Intake protocol: what happens to a returned file," MPN-PACK-02-INTAKE, with 05_DATA/03_generators/s8_intake.py. Cited for the filing rule, the codes, the register, the completeness and unusability conditions, the pooling rule on the build and wording hashes, and the statement that Part A is tallied rather than scored because the key holds no row for it.

[21] "Arbitration of the unified design framework," 08_PAPERS/ARBITRATION-DESIGN-R7.md, 14 September 2026. Cited for the finding that the mapping now discretises mode nowhere and therefore admits no perturbation budget on that channel, for the inadmissibility of a bias device that sets the mode, and for the reading of the reconciliation script's coordinate field.

[22] "The fourteen inherited coordinates," MPN-S3-BL1-DRAFT, and "CB-017, the remap," MPN-S3-BL2-DRAFT, both 14 September 2026. The fourteen coordinates, the three flagged rows, the search behind the negative claim, and the remap of framing with its two costs.

[23] "Arbitration of S3," 08_PAPERS/ARBITRATION-S3.md, 13 September 2026. Cited for the constraint that no modal table may be implemented under A4 until the second-stage question returns.

[24] "The unified design framework," MPN-DESIGN-01, 08_PAPERS/DESIGN-UNIFIED-FRAMEWORK.md. Cited for the Layer 3 detector, for the two detector trials, for the bias layer's channel question, and for the capacity table Part D of the pack closes.