Dentatus ARD: from adjustment to threshold

01The Dentatus question

Two successive articulator settings produce similar traces. Is that enough to say that the guides interacted in the same way? Conversely, if the second trace becomes straighter, is that difference enough to locate a mechanical transition?

In Dentatus ARD as Frame-Bound Mechanical System (April 2026), the starting point is the set of relations imposed by the frame. Posterior guidance, anterior guidance and lateral coupling jointly constrain what the system can do. Dominance concerns which relation plays the determining role in a particular mechanical configuration. A trace is one way of investigating that configuration, provided we know what the recording captures.

A setting tells us what we changed on the instrument. A threshold claim also requires us to show which structural distinction we crossed.

We first follow the eleven relations in the original paper. We then examine a separate Atlas entry and develop a worked example of how a threshold can be located within an interval. The example belongs to this site study of 6 November 2026.

02The three relations of the frame

The paper declares the system as follows:

S_D=(F_D,\Theta,C,A,D,G,R)

(1)

F_D is the Dentatus frame and Θ denotes its parameters. C is the constraint family, A the admissible set, D the dominance structure, G the projection and R the realized regime. The distinction matters in practice: knowing a value in Θ does not by itself tell us which constraint governs the response.

Posterior relation: The configuration x must belong to the set defined by the posterior guide at setting θ_c.
$C_c:x\in\Gamma_c(\theta_c)$ (2)
Anterior relation: The same x must also be compatible with anterior guidance at setting θ_i.
$C_i:x\in\Sigma_i(\theta_i)$ (3)
Lateral coupling: The coupling equation restricts the combinations of coordinates that can occur together. Satisfying just one of the three relations does not establish joint admissibility.
$C_b:\psi_b(x;\theta_b)=0$ (4)

The curves and surfaces have a specific mechanical role here: they express constraints of the declared frame. They are distinct from the observed trace y. Confusing the two would turn the record into an assumption about the mechanism that produced it.

Original paper: §§1–3.3, p. 1 · relations (1)–(4).

03From admissibility to realization

Relation (5) expresses joint admissibility. Relation (6) describes equilibrium selection among the admissible configurations:

A(\Theta)=\{x\mid c_i(x)\leq 0,\;\forall c_i\in C^*\}

(5)

x^*=\arg\min_{x\in A(\Theta)}\phi(x;\Theta,w)

(6)

A configuration x satisfying (5) is a candidate. To be selected under (6), it must also minimize φ for the specified Θ and w. The paper does not specify C^∗, φ or w in sufficient detail for a numerical calculation. We retain its notation x^∗; if several minimizers exist, the expression alone does not choose one of them.

D=D(\Theta)

(7)

R=x^*(\Theta,w)

(8)

Relation (7) expresses how dominance depends on the parameter configuration. Relation (8) uses the selected x^∗ as local notation for R. There is a limit to that reading: neither a guide being in contact nor a recorded position establishes complete regime identity without a further criterion. The Re-audit distinguishes admissibility from realization, and a change in weights from a change in dominance.

Original paper: §§4–7, p. 2 · relations (5)–(8).

04What would count as a transition?

G specifies what is retained as an observation of the mechanical state. The next two relations belong together:

G:X\to Y,\qquad y=G(x^*)

(9)

G(x_1)=G(x_2)\not\Rightarrow x_1=x_2

(10)

If two records coincide, (10) does not license identifying the underlying states. Nor does it establish that they differ. For a fixed G, two different exact output values rule out identical inputs. A difference between measured traces that lies within recording uncertainty does not establish the same conclusion. Even a clearly resolved difference does not identify which relation changed or whether the dominance class changed.

\Delta\Theta\Rightarrow\Delta D\Rightarrow\Delta R

(11)

The paper’s claim is that a small change in settings may lead to an abrupt reordering. Relation (11) describes that sequence when it occurs. It does not say that every turn of an adjustment screw changes the regime. In §11 the paper lists abrupt transitions, switching of dominance, instability near a boundary and nonlinear response as consequences of interest on the bench. It gives neither a measurement series nor a transition value.

Original paper: §§8–11, p. 2, and §§12–13, p. 3.

05A separate Atlas reading

Section 6.2 of the earlier Atlas Ultra Monograph manuscript (25 March 2026) describes the region in which a straight Dentatus trace appears as a ridge: in the Atlas interpretation, the margin around that state is narrow. The shape of the trace and the margin over which it persists are two different properties.

In the Dentatus table of §11.5, D3 denotes threshold onset and D6 a state beyond the stable margin. D3 and D6 are Atlas entry identifiers, not values of the function D(Θ). They suggest a useful question to ask of a record: when does a change begin, and when does the state cease to persist? The table does not calibrate (11) or locate a threshold on a particular instrument.

Separate source: Atlas Ultra, §§6.2 and 11.5. These locators refer to the March manuscript. The later Atlas monograph v1.0 has its own identity in the catalogue.

06How closely can a threshold be located?

An analytical example developed for this study. Specify a path of settings Θ(s), with a dimensionless coordinate s from 0 to 1 and a fixed direction of examination. Let α and β be two distinct classes with a declared criterion that distinguishes them independently of trace shape. Write d(s) for the label returned by that criterion. It is not the paper’s entire dominance structure D(Θ).

Assume exactly one switch along the path: d(s) = α for s < τ and d(s) = β for s ≥ τ. If we establish d(a) = α and d(b) = β at exactly known positions a < b, then a < τ ≤ b. The label α at a requires the switch to lie ahead. The label β at b means that the switch has already occurred, or occurs precisely at b.

This is the tightest interval supplied by those two observations: every τ in (a, b] produces both labels. The open left endpoint and closed right endpoint follow from the convention d(τ) = β. A test at the midpoint can exclude half the interval if it yields a determinate label.

WORKED ILLUSTRATIONOne further examination at the midpoint

Here s is a dimensionless parameter on a hypothetical path from 0 to 1. Assume a single switch in the label: d(s) = α for s < τ and d(s) = β for s ≥ τ. The initial labels, α at 1/2 and β at 3/4, are taken as independently established.

Open circle: endpoint excluded. Filled circle: endpoint included.

Compatible threshold valuesτ ∈ (1/2, 3/4]Only the two initial labels are available so far.

Label α: τ ∈ (5/8, 3/4]. Label α at 5/8 excludes that point: the switch has not yet occurred there.

Label β: τ ∈ (1/2, 5/8]. Label β at 5/8 includes that point: the switch occurs there or has already taken place.

Inconclusive: τ ∈ (1/2, 3/4]. An inconclusive result rules out none of the original interval.

The values are positions on the hypothetical path, not angles or measurements of the Dentatus ARD.

If there are multiple switches, if the direction of approach affects the response, or if α and β cannot be distinguished reliably, the same procedure does not locate a unique physical threshold.

07What could be tested at the bench?

Before choosing a step size, we need to decide what will count as evidence that the determining mechanical role has changed. For instance, observing that a component remains in contact establishes something about contact. It does not, on its own, establish that this component governs the response.

Declare the path. Choose a path of settings and record the instrument, starting state, support, loading, direction of approach and settings held fixed. This establishes which observations can be compared.
Separate the kinds of evidence. State in advance what would distinguish the two candidate organizations and how it would be checked: for instance, contact or reaction evidence with declared uncertainty and a reason it can distinguish the candidates. Record the trace alongside it. If both explanations still fit the evidence, leave the classification open.
Check persistence. Use repetitions and a separate reverse sweep to examine whether the same bounds can be reproduced. If the two directions give different results, retain separate paths rather than combining their limits into a mean “threshold.”

This is a research proposal. The operational dominance criterion on a particular ARD, its independent verification and the range over which a result would be repeatable remain open.

08Sources and reading pathways

Panagiotis Salikopoulos. Dentatus ARD as Frame-Bound Mechanical System. April 2026 · research note, 3 pages.

Page 1 · §§1–3.3. Purpose, system declaration and three constraint relations, (1)–(4).
Page 2 · §§4–7. Admissibility, equilibrium selection, dominance and the local notation for R, (5)–(8).
Page 2 · §§8–11. Projection, limits of inverse inference, transition and bench consequences, (9)–(11).
Page 3 · §§12–13. The author’s structural interpretation and conclusion.

Read the original paper ↗

Frame-Bound Mechanics: A Structural Theory of Articulator Systems. Atlas Ultra Monograph — Final Edition. 25 March 2026. §6.2, “Dentatus as ridge manifold”; §11.5, “Canonical system tables”, Dentatus entries D3 and D6.

March Atlas Ultra manuscript ↗ · Canonical Atlas monograph in the catalogue →

Site study · 6 November 2026. The proof of the interval for τ, the interactive example and the proposed bench question belong to this development. The original paper remains a work in its own right.

Dentatus in the catalogue →
The original research note within the collection of articulator studies.
Observation and structural inference →
The Re-audit’s conditions on what an observation map can distinguish.