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INDEPENDENT STUDY · EXT-01C

When does a new examination add independent information?

From permitted variations to the rank criterion: unique identification of weights, unresolved directions and the limits of recovery.

Rank criterion · Proof and examplesSite study · 5 November 2026Panagiotis Salikopoulos

01The question after 01A and 01B

An additional observation is informative when it constrains the unknowns through a relation that is not already contained in the previous relations. In the declared family studied here, the constraints and interaction functions are known; their fixed positive weights are unknown.

01A studies recovery using an additional known reaction. 01B studies the initial configuration and response to a known load, together with the margin for preserving the required admissibility class. 01C develops an independence criterion for the relations governing the specified unknowns.

The structure determines the permitted variations and, through them, the relations obtainable from equilibrium configurations.

Independence here is algebraic. It does not mean statistical independence, an independent laboratory, or certification of a physical arrangement. Repetition can improve the assessment of variability without contributing a new independent equation.

Source: EXT-01C, §1 and §5, PDF pages 1 and 6–8.

02The declared family

The frame, coordinates, units and admissible set are known:

𝒜={q∈ℝn:Aq=bCq≤d}rankA=m<n\mathcal A=\{q\in\mathbb R^n:Aq=b,\ Cq\le d\},\qquad \operatorname{rank}A=m<n.

The functions φ1, …, φp are known, differentiable and convex on a suitable convex domain containing 𝒜. The p ≥ 1 weights are unknown, strictly positive, and common to all examinations. For examination j, the generalized load fj is known and calibrated:

w∈(0,∞)pΨw,j(q)=∑i=1pwiφi(q)−fjTqw\in(0,\infty)^p,\qquad \Psi_{w,j}(q)=\sum_{i=1}^p w_i\phi_i(q)-f_j^{\mathsf T}q.

Every positively weighted sum is assumed strictly convex on 𝒜. The data consist of exact, full equilibrium configurations qj, with all declared inequalities strictly inactive:

Aqj=b,Cqj<dj=1,…,NAq_j=b,\qquad Cq_j<d,\qquad j=1,\ldots,N.

At least one positive vector of weights is assumed to explain every observed equilibrium through this minimization problem. These hypotheses define the particular family covered by the result. They are not universal assumptions for all FBM systems.

Source: §2, PDF pages 2–3.

03From constraints to available relations

Let the columns of Z form a basis of ker A. The permitted small variations at these interior configurations are:

Z∈ℝn×(n−m)δq=ZξZ\in\mathbb R^{n\times(n-m)},\qquad \delta q=Z\xi.

Z describes the directions permitted by the constraint structure. The projection used here to express stationarity is not the FBM observation map. Because the inequalities remain strictly inactive, sufficiently small variations in both directions along ker A are admissible. At equilibrium:

ZT(∑i=1pwi∇φi(qj)−fj)=0Z^{\mathsf T}\left(\sum_{i=1}^p w_i\nabla\phi_i(q_j)-f_j\right)=0.

Define a matrix and vector for each examination and stack the relations:

Bj=[ZT∇φ1(qj)⋯ZT∇φp(qj)]βj=ZTfjB_j=\begin{bmatrix}Z^{\mathsf T}\nabla\phi_1(q_j)&\cdots&Z^{\mathsf T}\nabla\phi_p(q_j)\end{bmatrix},\qquad \beta_j=Z^{\mathsf T}f_j.
B=[B1⋮BN],β=[β1⋮βN]Bw=βB=\begin{bmatrix}B_1\\ \vdots\\ B_N\end{bmatrix},\qquad \beta=\begin{bmatrix}\beta_1\\ \vdots\\ \beta_N\end{bmatrix},\qquad Bw=\beta.

The order of construction matters: declared constraints, permitted variations, equilibrium relations, and equations for the weights. Rank evaluates this structurally derived matrix.

Source: §3, PDF pages 3–4.

04EXT-01C-1 · The criterion for unique identification

Theorem EXT-01C-1. Under the declared family hypotheses, and assuming at least one positive compatible solution, the p weights are uniquely determined by the declared data if and only if:

rankB=p\operatorname{rank}B=p.

The existence of a positive compatible solution is essential. Full rank alone does not establish that the equations are consistent or that their solution is positive.

The conclusion concerns weights within the known family. A unique solution does not establish that the selected constraints and functions correctly describe a physical mechanical object.

Source: §4, PDF pages 4–6; §9, PDF pages 12–13.

05Why the criterion is necessary and sufficient

Sufficiency. If B has full column rank, Bw = β has at most one solution. The assumption of positive compatibility ensures that this solution exists and belongs to the declared family.

Necessity. If rank B < p, there is a nonzero z in the kernel of B. Given a positive compatible solution w*, sufficiently small |t| gives:

w(t)=w*+tz>0Bw(t)=βw(t)=w^*+tz>0,\qquad Bw(t)=\beta.

It remains to show that these solutions actually produce the same equilibria. Projected stationarity and the fact that every feasible difference q − qj belongs to ker A imply:

∇Ψw(t),j(qj)T(q−qj)=0∀q∈𝒜\nabla\Psi_{w(t),j}(q_j)^{\mathsf T}(q-q_j)=0\qquad\text{for every }q\in\mathcal A.

Convexity gives:

Ψw(t),j(q)≥Ψw(t),j(qj)+∇Ψw(t),j(qj)T(q−qj)=Ψw(t),j(qj)\Psi_{w(t),j}(q)\ge\Psi_{w(t),j}(q_j)+\nabla\Psi_{w(t),j}(q_j)^{\mathsf T}(q-q_j)=\Psi_{w(t),j}(q_j).

Thus qj is a global constrained minimum for each such w(t), and strict convexity makes it unique. Different positive weights explain the same recorded equilibria. The ambiguity therefore concerns realizations within the declared model, rather than merely formal solutions of an auxiliary system.

Source: §4, PDF pages 4–6.

06What remains indistinguishable

The compatible weight set is:

𝒲data={w>0:Bw=β}\mathcal W_{\mathrm{data}}=\{w>0:Bw=\beta\}.

Near any positive compatible solution its dimension is p − rank B. This counts unresolved coefficient directions, not the number of distinct regimes.

If all loads vanish, a nonzero positive w* with Bw* = 0 rules out full column rank. The same ambiguity follows directly from the functional:

Ψcw=cΨwarg min𝒜Ψcw=arg min𝒜Ψwc>0\Psi_{cw}=c\Psi_w,\qquad \arg\min_{\mathcal A}\Psi_{cw}=\arg\min_{\mathcal A}\Psi_w,\qquad c>0.

If an additional known normalization, such as weights summing to 1, is part of the model, it must be included explicitly. The set of independent unknowns then changes.

Source: §4, PDF page 6; §6, PDF pages 8–9.

07The recovery routes in 01A and 01B

Consider u + v = 2 with u, v ≥ 0 and strictly positive interior coordinates:

φ1(u,v)=12u2φ2(u,v)=12v2Z=[1−1]\phi_1(u,v)=\frac12u^2,\qquad \phi_2(u,v)=\frac12v^2,\qquad Z=\begin{bmatrix}1\\-1\end{bmatrix}.

An initial unloaded equilibrium supplies only one independent relation:

[u0−v0]w=0w1w2=v0u0\begin{bmatrix}u_0&-v_0\end{bmatrix}w=0,\qquad \frac{w_1}{w_2}=\frac{v_0}{u_0}.

In 01B, a known load adds the term −rv. The second interior equilibrium and displacement Δ give:

B=[u0−v0ur−vr]β=[0−r]Δ=u0−ur=vr−v0Δ=rw1+w2B=\begin{bmatrix}u_0&-v_0\\u_r&-v_r\end{bmatrix},\qquad \beta=\begin{bmatrix}0\\-r\end{bmatrix},\qquad \Delta=u_0-u_r=v_r-v_0=\frac{r}{w_1+w_2}.
detB=−2Δw1=rv02Δ,w2=ru02Δ\det B=-2\Delta,\qquad w_1=\frac{rv_0}{2\Delta},\qquad w_2=\frac{ru_0}{2\Delta}.

For r > 0 while the solution remains interior, the new relation removes the common-scale ambiguity.

In 01A, the known reaction ρ0, with the fixed sign convention and constraint normalization, supplies a different information system:

[u000v0]w=[ρ0ρ0]\begin{bmatrix}u_0&0\\0&v_0\end{bmatrix}w=\begin{bmatrix}\rho_0\\\rho_0\end{bmatrix}.

This matrix has rank 2 when u0, v0 > 0. Measured reactions are additional observations: they are not automatically included in the B defined in §3, whose construction eliminates equality-constraint reactions. Both routes use independent relations, but their declared data differ.

Source: §5, PDF pages 6–8. The distinction between the two information systems is made explicit here.

08A worked numerical example

For illustration, choose w1 = 3, w2 = 1 and r = 1 in consistent declared units of the idealized model. This is an instructional calculation from the source equations, not a measurement. The condition 0 < r < 2w2 holds.

The initial and perturbed configurations are:

(u0,v0)=(12,32)(ur,vr)=(14,74)Δ=14(u_0,v_0)=\left(\frac12,\frac32\right),\qquad(u_r,v_r)=\left(\frac14,\frac74\right),\qquad\Delta=\frac14.
B=[12−3214−74]detB=−12,β=[0−1]B=\begin{bmatrix}\frac12&-\frac32\\[2pt]\frac14&-\frac74\end{bmatrix},\qquad\det B=-\frac12,\qquad\beta=\begin{bmatrix}0\\-1\end{bmatrix}.

The recovery formulas return exactly w1 = 3 and w2 = 1. The first row gives w1 = 3w2, and the second determines w2 = 1.

Repeating only the initial configuration would repeat the row (1/2, −3/2), retaining rank 1. Additional information comes from the different known load and its response within the same required admissibility class.

Additional instructional calculation for this site study, from the relations in §5, PDF pages 6–7.

09A different constraint organization

Now take a new admissible set: u = v = q, with q ∈ ℝ and no unilateral limits. The previous conditions u, v ≥ 0 do not carry over. The construction describes two idealized linear elastic channels sharing a common displacement, with f ∈ ℝ:

Ψf(u,v)=12w1u2+12w2v2−fuu=v=qq*=fw1+w2\Psi_f(u,v)=\frac12w_1u^2+\frac12w_2v^2-fu,\qquad u=v=q,\qquad q^*=\frac{f}{w_1+w_2}.

Each examination produces a row (qj, qj). The two columns remain identical regardless of the number of loads. A known nonzero load and the corresponding configuration identify the sum of the weights, not its distribution:

w1+w2=fq*rankB≤1w_1+w_2=\frac{f}{q^*},\qquad\operatorname{rank}B\le1.

The pairs (3, 1) and (2, 2) produce the same relation q* = f/4 for every load in this family. At f = 4 both give q* = 1.

Declared relation and informationWhat is identifiedWhat remains unknown
u + v = 2, initial unloaded equilibriumWeight ratioCommon scale
u + v = 2, initial configuration and response to a known load, under the interior conditionsBoth weightsNo weight ambiguity within the family
u = v, nonzero total load and common displacementSum of weightsDistribution between channels

Changing the constraint relation changes the permitted variations and the available information relations. Recoverable information depends on that structural organization. The elastic-channel interpretation is idealized; it is not assigned to a particular articulator or to an already validated bench arrangement.

Source: §7, PDF pages 9–11.

10Uniqueness, preservation and accuracy

Full rank concerns exact data that already satisfy the hypotheses. Acquiring those data without leaving the required class remains a separate question. In the 01B example, the interior margin is 0 < r < 2w2; the unknown w2 prevents this pointwise condition from becoming a common examination choice without additional information.

Uniqueness does not ensure low sensitivity to error. Set s = w1 + w2 and take r to be exactly known. If the only error considered in the following calculation affects displacement:

Δ~=Δ+e|e|≤ε<Δs^=rΔ~\widetilde\Delta=\Delta+e,\qquad|e|\le\varepsilon<\Delta,\qquad\widehat s=\frac{r}{\widetilde\Delta}.
|s^−s|s=|e||Δ+e|≤εΔ−ε\frac{|\widehat s-s|}{s}=\frac{|e|}{|\Delta+e|}\le\frac{\varepsilon}{\Delta-\varepsilon}.

This bound concerns the recovered sum under the specified error model. It is not a general bound for every coefficient or for errors in all data. As the response becomes small relative to uncertainty, recovery becomes sensitive even though exact rank may remain full.

At a binding unilateral boundary, the present proof’s hypotheses cease to hold. Appropriate active-constraint and complementarity analysis is then required. EXT-01D is the next research source on this pathway.

Source: §8, PDF pages 11–12.

11Foundations, Re-audit and independent development

Constrained admissibility and the interaction functional provide the theoretical background. The exact linear dependence on unknown positive weights, convexity conditions, known loads and rank criterion constitute the declared development of 01C. The new result is not retrospectively attributed to the historical monograph.

In continuity with the Re-audit, constraint and multiplier normalization, numerical variation of weights, dominance class and complete regime identity remain distinct. Identifying weights does not identify all of these concepts or validate the model for a physical object.

Geometric configuration has a precise role as data within the declared structure. Its ability to distinguish unknowns is examined through the relations permitted by that structure.

Source: introduction, §2, §5 and §9, especially PDF page 13.

12Sources and reading pathways

FBM-EXT-01C — Ανεξάρτητη δομική πληροφορία και μοναδικός προσδιορισμός των βαρών.

Greek research source, 13 PDF pages. This bilingual site study was developed on . It preserves the source’s hypotheses and scope, develops the proof steps, and adds the instructional numerical example in §8. The original PDF remains unchanged and retains its own identity.

Open the research source ↗
Part of this studyExact source location
Independent information§1, PDF page 1
Family and hypotheses§2, PDF pages 2–3
Projected stationarity and matrix§3, PDF pages 3–4
Theorem, proof and compatible weights§4, PDF pages 4–6
Connection with 01A and 01B§5, PDF pages 6–8
Common scale§6, PDF pages 8–9
Common displacement§7, PDF pages 9–11
Preservation and error§8, PDF pages 11–12
Scope of the conclusion§9, PDF pages 12–13

Locations refer to the PDF page order. The English version presents the same development with identical notation and equations.