Full structural declaration and identifiability

01Which information do we want to recover?

A full structural declaration and a complete record of geometric position answer different questions. The first specifies which systems we are examining and how their realizations arise. The second shows what the selected observation channel preserves. Study EXT-01A examines a case in which the equilibrium position is known, while some parameters of the generating structure remain undetermined.

We will answer this for a specific family: two positive coefficients of a declared interaction functional. The goal is to recover the pair. We will first calculate the unique equilibrium, then determine what each observation identifies, and finally develop an ideal mechanical realization that gives meaning to the additional information.

The case source takes Model II — Dominance Non-Identifiability from Minimal Structural Proofs of Projection Non-Identifiability as its starting point and extends the specialization with zero reference positions to all positive weights. Here we develop the complete argument of this research case as a self-contained website study.

Source: EXT-01A, PDF pages 1–2. The goal of parameter recovery is defined before the observation; it is distinct from full FBM regime identity.

02The structure of the family under study

We hold fixed the reference frame, the meaning of the two coordinates and the normalization of the functions. The coordinates u, v and the weights w₁, w₂ are dimensionless. The declared family comprises three distinct elements: the admissible configurations, the interaction law and the relation that selects the equilibrium.

The equality links the two coordinates; the inequalities delimit a closed line segment. The functional Ψ assigns a value to each configuration. Minimizing it over 𝒜 is the explicitly chosen realization relation. Admissibility alone does not specify this choice.

The quadratic formula defines Ψ_θ on all of ℝ²; the minimization problem uses its restriction to 𝒜.

For each θ there is a particular, fully specified functional. When the observer does not know θ, the question is which member of the known family corresponds to the available data. The weights are part of the model’s structure; what is missing is knowledge of their values.

Source: full declaration of the family, PDF pages 1–2. The offsets a and b in the more general form are set to zero here.

03Why the equilibrium is unique

The constraint gives v = 2 − u and 0 ≤ u ≤ 2. The problem therefore reduces to one variable:

The function f is continuous on the closed interval and strictly convex. It therefore has a minimum, and that minimum is unique. The solution of f′ = 0 lies in the interior of the interval because both weights are positive:

Neither inequality binds at the solution: both coordinates are strictly positive. The equality u + v = 2 continues to constrain the admissible variations. We have calculated the position from the declared relations before deciding what to observe.

Source and calculation: PDF pages 2–3. Uniqueness concerns the minimum of this functional on this particular set.

04What recording the sum loses

We now choose the sum of the coordinates as the observation:

This result already follows from the constraint. There is no need to compare a few sample equilibria: G is constant on the entire admissible set. Consequently, for the purpose of recovering the weights, the value 2 does not distinguish any of the candidate positive pairs.

This does not mean that the weights have no effect on the system. They affect the equilibrium we have just calculated. This particular recording channel preserves only a quantity that the structure has already fixed. The insufficiency follows from the relation between constraint and observation, rather than a general rule about all sums of coordinates.

Source: PDF pages 3 and 10–11.

05Position gives the ratio, not the common scale

If we know u^∗, the declared constraint also gives v^∗ = 2 − u^∗. We now know the entire equilibrium position. The formulas from the preceding section give:

Proposition — Common-scale ambiguity. Two positive pairs of weights produce the same equilibrium position if and only if one is a common positive multiple of the other.

Proof. Multiplying both weights by c > 0 leaves both equilibrium formulas unchanged. Conversely, the same u^∗ gives the same ratio w₁/w₂. For a second positive pair (w′₁, w′₂), set c = w′₂/w₂ > 0. Equality of the ratios requires w′₁ = cw₁, while w′₂ = cw₂ already holds. This proves both directions.

Position therefore determines the normalized weights, but leaves their sum s free:

The pairs (3, 1), (6, 2) and (30, 10) all give the position (0.5, 1.5). If we wanted only the ratio, position would suffice. If we had imposed w₁ + w₂ = 1 from the outset, we would also recover the unique pair of those normalized coefficients. The stronger goal here is to recover both absolute weights on the model’s already fixed scale.

Source: Proposition EXT-01A-1, PDF pages 3–4; relative weighting and common scale, pages 12–15.

06Position and multiplier: the full inverse relation

We use the precisely specified form h(u, v) = u + v − 2 and the augmented functional:

Here, the stationarity conditions, together with the declared equality, suffice to determine the solution whose uniqueness we have already proved. The inequalities do not bind at equilibrium. Defining ρ^∗ = −λ^∗, we obtain:

Proposition — Bijective augmentation. The map Γ_aug from positive weights to the position–multiplier pair is one-to-one and onto the declared data space:

Proof of uniqueness. The stationarity equations give the inverse relation directly. The denominators are positive. Two pairs of weights with the same data (u^∗, ρ^∗) must therefore coincide.

Proof that the entire data space is covered. Take any u ∈ (0, 2) and ρ > 0. Set w₁ = ρ/u and w₂ = ρ/(2 − u). These are positive, and their sum is 2ρ/[u(2 − u)]. Substitution into the formulas gives u^∗ = u and ρ^∗ = ρ. Every admissible data pair therefore comes from exactly one pair of weights.

The multiplier arises from the declared realization relation. The proof establishes what we can recover if we have access to its relevant value; whether this information is available is a separate question.

Source: Proposition EXT-01A-2 and proof, PDF pages 4–7.

07A declared mechanical realization

Consider two ideal linear elastic elements in series. Their unloaded positions are known, no additional force acts on the intermediate connector, and we examine static equilibrium under an imposed total extension d > 0. The extensions x₁, x₂ are measured from the unloaded positions, and the declared domain is x₁ ≥ 0, x₂ ≥ 0. Nonnegative extension is a declared restriction of the admissible domain; it does not follow from linear elasticity.

The k_i are stiffnesses with units of force per length, the x_i have units of length, and U has units of energy. The equilibrium minimizes U subject to these constraints. Using fixed positive reference units of length ℓ₀ and stiffness k₀, set:

We now have an exact correspondence with the declared dimensionless family. An admissible internal variation satisfies δx₂ = −δx₁. At an interior equilibrium:

T is the common magnitude of the tensile force and the reaction maintaining the total extension. The stiffer element extends less: k₁/k₂ = x₂^∗/x₁^∗. If both stiffnesses double, the distribution of extensions remains the same and T doubles.

This is an ideal mechanical interpretation of the construction. The correspondence between weights and stiffnesses follows from the declared elastic energy; it does not automatically apply to every interaction functional or to a particular articulator.

Source: mechanical realization and nondimensionalization, PDF pages 10–12; force and scale, pages 13–16. The nonnegative domain of the extensions is explicitly declared here to make the correspondence with 𝒜 complete.

08Three cases: what does each measurement distinguish?

Choose ℓ₀ = 1 mm, k₀ = 1 N/mm and d = 2 mm. The following numbers are calculated from the model and are hypothetical examples, not experimental measurements.

The sum of the extensions is 2 mm in all three cases. Position distinguishes A from C, but not A from B. Force distinguishes A from B, but not A from C. Position together with force distinguishes all three, while the preceding proof covers the entire declared family of positive weights.

For example, the simultaneously available values x₁^∗ = 0.5 mm, x₂^∗ = 1.5 mm and T = 3 N give:

For case A, the normalized weights are 0.75 and 0.25. These describe the relative stiffness of the two terms. They are not percentages of force distribution. In the series arrangement, with no additional force at the intermediate connector, both elements transmit the same T.

Source: numerical example, PDF pages 16–17; interpretation of relative weighting, pages 12–13.

09Normalization and additional information

The numerical value of a multiplier depends on how the constraint is written. If the same equality is written as h̃ = αh, with α > 0, the equivalent stationarity condition uses λ̃ = λ/α. The geometric condition is unchanged, but the numerical coefficient changes.

A “multiplier with known normalization” therefore means that we know the exact form of the constraint, the scale of the functional and the units. It does not mean dividing the multiplier by its magnitude or converting it into a percentage; that would discard the information about the common scale.

In the ideal elastic arrangement, the relation T = k₀ℓ₀ρ has been declared. The reaction must be independently available for the same realization. If we first choose candidate stiffnesses, calculate the force from them and then use that force to “confirm” those same stiffnesses, the procedure is circular.

The proof of bijectivity establishes the mathematical sufficiency of the data. It does not establish that a sensor already supplies the correct quantity or that measuring it leaves every relevant structural property unchanged.

Source: normalization and independent information, PDF pages 6–7 and 17–18. The identity λ̃∇h̃ = λ∇h is written here as a direct explanation of the change of scale.

10The scope of unique recovery

The exact inverse relation contains the terms 1/u^∗ and 1/(2 − u^∗). Near the ends of the interval, a small uncertainty in position can strongly affect the estimate of a weight. Exact uniqueness does not by itself provide a quantitative guarantee of robustness to error.

The result assumes the known family: positive weights, a specific quadratic form, zero reference positions, declared connectivity and constraint. The mechanical interpretation also assumes linear elasticity and no additional force at the intermediate connector. Unknown unloaded positions, different laws or additional loads change the problem. A different nonlinear law can agree with the same position and force at one point.

Recovering the two weights does not recover every possible mechanical structure. Nor do different numerical values of the weights automatically establish different regimes: that requires the full definition of structural identity and the relevant dominance class. The parameter subproblem has been declared separately from the outset.

This is a proved case with a specific domain. Extending it to uncertain data, other families or full regime identity requires further declarations and proofs. Connecting it to actual records requires a precise correspondence between the arrangement, the model and the observed quantities.

Source: PDF pages 1, 7–9, 13 and 18. Methodological starting point: distinguishing access, specialization and foundations, PDF pages 49–56; correspondence between levels, pages 62–66.

11The connection to the mathematical Re-audit

The mathematical Re-audit examines the connection between declared objects and the conclusions attributed to them. EXT-01A provides a concrete application of three related distinctions. The correspondence below is a present reading of the two sources.

• Admissibility and realization →

  The set 𝒜 declares compatible configurations. The functional Ψ_θ and the explicit minimization relation determine which equilibrium is under study. Recovery therefore starts from a complete family, with each element’s role specified.

• Observation map and proved indistinguishability →

  The inability to distinguish the weights through G is not inferred simply from the word projection. It follows from G|_𝒜 = 2. The augmented map is examined separately, with its own complete proof.

• Numerical weights and dominance structure →

  The weight pair and its normalized ratio belong to this particular family. General dominance structure and full regime identity require their own definitions. Recovering the pair does not replace that broader question.

Correspondence with MRA-01-F013, F010 and F009 of Theory Formalization Re-Audit. The new Re-audit study develops the passages, versions and scope of these connections.

This case belongs to the autonomous mathematical development of FBM. The foundations supply the objects; the Re-audit clarifies their roles and assumptions; this case develops and proves its own specific result. Its mathematical validity does not depend on correspondence to an articulator.

12Connections, sources and further research

• Model II and its mathematical continuation →

  The original counterexample is studied independently. EXT-01A develops the same zero-offset family for all positive weights and examines the additional information required for recovery.

• Declared structure, admissibility and realization →

  The admissible set constrains the configurations. The interaction functional and declared minimization select the equilibrium within it. Their roles remain distinct.

• From the exact model to evidence-supported interpretation →

  Actual application requires declared conditions, units, uncertainty and documented access to the reaction. Agreement within this particular model does not replace that work.

The next research connection may start from a mathematical question, a theoretical statement or a mechanical record. First, the relevant sources and their correspondence are identified. We then distinguish whether a fuller specialization, a new proof or a specific mechanical examination is required. A connection formulated today must be identified as a present synthesis, without attributing it retrospectively to an earlier work.

The origin in Model II is attributed to source EXT-01A. The link to the Mathematical Monograph provides a route into the wider corpus; the proofs and page references given here are supported by this specific research PDF.