01The continuation of mathematical research
Mathematical development examines what follows from a declared structure: which realization it allows, which differences an observation distinguishes, and what additional information a specified question requires. Geometric representations and parameters acquire their meaning within that declaration.
The Mathematical Monograph provides the foundational level of the corpus. The Re-audit examines and clarifies the relationship between declared objects, assumptions and conclusions. The following cases develop particular mathematical problems against this background, each with its own complete specification.
Each case stands on its own: its result is governed by its assumptions and proof. A mechanical application additionally requires its own substantiated correspondence.
The route foundation → Re-audit → further development expresses the relationship of the present study to its background. Each connection identifies the particular element it uses. Source editions remain distinct from subsequent development and from the website’s editorial treatment.
Exact locators refer to the mathematical manuscript v1.0 of 22 March 2026. Its page numbers are not attributed to the canonical DOI edition without edition verification.
02The explicit counterexample in Model II
Model II provides a minimal construction in which the entire calculation can be carried out. We use the zero-offset case a = b = 0, with positive weights:
𝒜 = {(u, v) ∈ ℝ2 : u + v = 2, u ≥ 0, v ≥ 0}, w1, w2 > 0,
Ψw(u, v) = ½w1u2 + ½w2v2, q∗(w) ∈ arg minq ∈ 𝒜 Ψw(q).
The polynomial formula defines Ψ on all of ℝ2; minimization is restricted to 𝒜. Setting v = 2 − u, with u ∈ [0, 2], gives:
fw(u) = ½w1u2 + ½w2(2 − u)2,
f′w(u) = (w1 + w2)u − 2w2, f″w(u) = w1 + w2 > 0.
Continuity on the closed interval gives existence of a minimum, and strict convexity gives uniqueness. The stationary point lies in the interior because the weights are positive. Thus:
q∗(w) = (2w2/(w1 + w2), 2w1/(w1 + w2)).
q∗(3, 1) = (½, 3/2), q∗(1, 3) = (3/2, ½).
The equilibria differ while sharing the same admissible set. But if the observation is G(u, v) = u + v, then G = 2 throughout 𝒜. The two equilibria have exactly the same image. We have a specific witness to non-injectivity of observation, while each forward problem remains single-valued.
On the Non-Identifiability of Constraint-Defined Systems from Geometric Observations describes the corresponding fibre Xy = {x : G(x) = y}. Section 3 begins by assuming such a distinct pair; here the pair has been calculated. In Model II, the fibre of the value 2 within 𝒜 is all of 𝒜. Compatibility with that value does not select a unique equilibrium or weight pair.
The construction proves that non-uniqueness can occur in declared constraint systems. It does not prove that every observation map is non-injective.
Sources: Model II, §A.2, PDF 1–2; §3, PDF 1–2, and §§4–6, PDF 2. The convexity check is developed here for the explicitly selected case a = b = 0.
03What the Re-Audit clarifies
The counterexample makes the Re-Audit’s distinctions concrete. The w1, w2 are coefficients of a known functional. The general dominance structure is not, by definition, identified with a numerical pair. The title “Dominance Non-Identifiability” in Model II describes the weighting of that particular construction.
Likewise, the position q∗ does not, on its own, constitute a full regime identity. Both coordinates are positive at both equilibria: the inequalities are non-binding. The change in weights and position does not establish a change in active set or full regime here.
Finally, declared realization and inverse identification are different problems. Model II solves the former and demonstrates failure of uniqueness for the latter under G. This property is checked through the values of the specific map, rather than the name “projection”.
Sources: MRA-01-F009–F010, 32–34; MRA-05 D7, 185; MRA-10 §§2.4, 2.6–2.7, 508–510. The connection between these particular passages is the present analytical synthesis.
04The EXT-01 source series
The entries below provide access to each complete original source, together with its question, scope and specific locators. EXT-01A, EXT-01B and EXT-01C have detailed studies on the website. EXT-01D–E enter here as standalone research sources, with their proofs in the corresponding PDF.
Complete structural declaration and additional information
The equilibrium position identifies the ratio of the two positive weights. An independently available, appropriately normalized reaction supplies the common scale. The result concerns the specified known family of functionals and constraints.
The distinction between admissibility and realization determines what must be declared. The normalization and mechanical meaning of the multiplier determine when the additional value can be used.
Family and equilibrium: PDF 1–3; common scale: 3–4; inverse and information requirements: 4–8; ideal mechanical interpretation: 9–18.
Original source title: FBM-EXT-01A — Πλήρης δομική δήλωση και προέλευση της πρόσθετης πληροφορίας
Complete original source ↗Declared perturbation and preservation margin
The initial configuration and response to a known calibrated load identify the weights before the boundary becomes binding. The source distinguishes the existence of a suitable perturbation for each system from the ability to choose one in advance for every compatible candidate. It also examines the preservation–separation window under a declared error bound.
Preserving the interior admissibility class and unique branch of minima is distinct from full regime stability. This scope connects to the distinction between realization and structural signature.
Declaration and response: PDF 1–4; EXT-01B-1: 4–5; calibration: 6–7; EXT-01B-2: 7–8; error window, EXT-01B-3: 9–10; boundary saturation and history information: 10–11.
Full study · From structure to the choice of a test →
Original source title: FBM-EXT-01B — Δηλωμένη διαταραχή, ανάκτηση κοινής κλίμακας και παράθυρο διατήρησης της admissibility class
Complete original source ↗Independent relations and unique identification
For known affine constraints, known differentiable convex interaction functions, known calibrated loads and exact interior equilibria, the source constructs a linear system for the unknown positive weights. Under its stated assumptions of strict convexity and existence of compatible weights, uniqueness is equivalent to full column rank. A second organization with common displacement shows why total load may identify only the sum of the weights.
The matrix rank concerns the declared coefficients. Numerical weighting remains distinct from general dominance and full regime identity. Exact uniqueness and robustness to error are examined separately.
Family and assumptions: PDF 2–3; EXT-01C-1: 4–6; relationship to EXT-01A/B: 6–8; common scale: 8–9; different constraint organization: 9–11; preservation and accuracy limits: 11–13.
Full study · When does a new examination add independent information? →
Original source title: FBM-EXT-01C — Ανεξάρτητη δομική πληροφορία και μοναδικός προσδιορισμός των βαρών
Complete original source ↗Binding boundaries and compatible weights
The analysis extends to equilibria on binding boundaries. Bidirectionally admissible variations provide equations, while one-sided variations provide inequalities. Their combination determines the compatible weight set. The source gives a uniqueness criterion through the cone of feasible weight variations and an example in which two boundary tests isolate a parameter without full rank of the equations.
The binding state and reaction information have different content. Active informational inequalities in weight space are distinct from the mechanical binding set.
Model II at the boundary and reaction: PDF 3–5; general family: 5–6; EXT-01D-1: 7; EXT-01D-2: 8–9; uniqueness from two boundaries: 9–12; finite uncertainty: 12–13.
Original source title: FBM-EXT-01D — Δεσμευμένοι περιορισμοί, αντιδράσεις και γεωμετρία των συμβατών βαρών
Complete original source ↗Partial identification and indistinguishable directions
Ambiguity in the weights can leave a predetermined mechanical quantity fully identified. The source gives a criterion for linear quantities and, within a compact convex class with strictly convex functionals, characterizes when two specializations have the same equilibrium for every linear load: their functionals differ by a constant on the shared admissible set. It contrasts this persistent ambiguity with cases in which an appropriate additional test suffices.
The sufficiency of observation is examined relative to a particular target. Identical complete positional response does not imply identical internal reaction distribution. If constraints change, transferring a conclusion requires evidence of what is preserved.
Target and EXT-01E-1: PDF 1–4; family and EXT-01E-2: 4–6; common displacement: 7–9; nonlinear example and EXT-01E-3: 10–13; changed admissibility and transfer: 13–15.
Original source title: FBM-EXT-01E — Επιμέρους μηχανικός προσδιορισμός, δομικά αδιάκριτες κατευθύνσεις και όρια επέκτασης των εξετάσεων
Complete original source ↗05How the cases are connected
The sources state specific relationships between their results. EXT-01B continues the known family of EXT-01A with a load. EXT-01C considers a broader class of interior equilibria. EXT-01D includes binding boundaries and recovers the previous criterion as a special case. EXT-01E examines what can be identified even when several compatible weights remain.
This continuity also permits independent reading: each text specifies its own object, additional assumptions and conclusion. Convex analysis, first-order conditions and rank retain their established mathematical roles. The particular applications and extensions are attributed to the corresponding research text.
Revisiting an earlier work may use one of these results after its hypotheses have been checked against that work’s own declaration. The original source is retained and new development is added separately. The research works catalogue gathers these autonomous starting points.
06Sources and editions
The complete original EXT-01A–E sources are linked above. The following references document the mathematical starting point and the specific intermediate clarifications.
Minimal Structural Proofs of Projection Non-Identifiability.
Three-page text. §A.2, PDF 1–2: Model II and the two numerical pairs. The present development is explicitly restricted to zero offsets and positive weights.
Open Minimal Structural Proofs ↗On the Non-Identifiability of Constraint-Defined Systems from Geometric Observations.
Salikopoulos Panagiotis, 15 April 2026, 3 pages. §3, PDF 1–2: the assumption of different states with a shared image; §§4–6, PDF 2: fibres and compatibility without unique identification. No DOI or version number is stated.
Open the identifiability manuscript ↗Frame-Bound Mechanics — Structural Theory of Articulated Systems.
Mathematical manuscript Final Manuscript for Repository Publication, Version 1.0, March 22, 2026, file frame_bound_mechanics.pdf. Definitions 2.3–2.8: printed pages 2–3 / PDF 5–6; Definition 5.1: printed 7 / PDF 10; Theorem 7.1: printed 9 / PDF 12. The accompanying proof of Theorem 7.1 uses admissibility together with the interaction functional. The canonical mathematical edition retains its own identity.
Open the mathematical manuscript ↗Theory Formalization Re-Audit.
Working edition of 553 pages. MRA-01-F009–F010: 32–34; F013: 36–37; MRA-05 D5: 182, D7: 185; MRA-10 §2.4: 508, §§2.6–2.7: 509–510. These passages support the distinctions developed in the website’s related study.
Re-audit study and detailed sources →Theory Formalization Deepening & Expansion.
Research direction text of 66 pages. PDF 53–55: predetermined target and augmentation of observation; 60–66: precision and documented relationships between theoretical, mathematical and mechanical sources.
PDF numbers follow each file’s page sequence. Re-audit locators refer to the particular working edition. Including a source does not automatically constitute independent validation of all its claims.