01Which declaration supports which conclusion?
The mathematical description of a system needs to distinguish three questions: which configurations are admissible, which relation specifies realization, and what observation preserves. The same admissible set can be accompanied by different functionals. The same equilibrium position can arise from different parameters. And the same observation can correspond to different positions.
Each conclusion needs its own declared data: admissibility, realization and identification do not substitute for one another.
Theory Formalization Re-Audit examines such transitions between definitions and conclusions. Here we follow three specific findings—on dominance, the observation map, and the relation between admissibility and realization—and calculate their consequences in a common family. We also examine a specific correction in the treatment of non-binding inequalities. The discussion leads to EXT-01A, where additional information permits the exact recovery of two parameters.
The comparison uses the mathematical and theoretical manuscripts dated 22 and 25 March 2026, respectively, together with specific passages from the Re-Audit. These manuscripts are distinct versions from the subsequently registered monographs. The route connecting these passages to the example is the website’s present analytical synthesis.
Starting points: mathematical manuscript, §2.1–2.2 · MRA-01-F009, F010 and F013, pages 32–37 · EXT-01A, pages 1–7.
02Admissibility and realization
In the manuscript, the constraints define the admissible set. Separate definitions of the interaction functional and the equilibrium relation follow. The functional therefore already belongs to the mathematical description; we are not introducing it now to remedy an assumed absence from the theory.
Finding MRA-01-F013 calls for avoiding the transition from “admissible” to “realized” without the relation that justifies it. Declaring a set does not, by itself, select one of its points. If the relation under examination is minimization, what is minimized and over which domain must be specified. Another realization relation would require its own formulation.
For this page’s example, we define the dimensionless set:
𝒜 = {(u, v) ∈ ℝ² : u + v = 2, u ≥ 0, v ≥ 0}.
Every point of the closed line segment is admissible. For each pair of positive parameters θ = (w1, w2), we declare the following polynomial functional on all of ℝ² and restrict its minimization to 𝒜:
Ψθ(u, v) = ½w1u² + ½w2v², w1, w2 > 0,
q∗(θ) = arg minq ∈ 𝒜 Ψθ(q).
The final line is the declared realization relation of this particular model. The functional is defined and differentiable on an open neighbourhood of 𝒜, giving the ordinary derivatives a specified domain. The use of a point q∗ rather than a set of solutions will shortly be justified by a uniqueness proof. The frame, the meaning of the coordinates and the scales of the functional remain fixed throughout the comparison.
In the Re-Audit’s final reconstruction, the realization relation is declared separately from admissibility and may take different forms. Scalar variational selection is a special case, rather than a required law for every system. Here we choose it explicitly and prove what holds within it.
Manuscript: Definition 2.3–2.6, printed pages 2–3 / PDF 5–6; Theorem 7.1, printed 9 / PDF 12. Review: MRA-01-F013, pages 36–37; MRA-03-F004–F005, 87–89; MRA-10 §2.3–2.4, 507–508, and §3, 511. The particular family and minimization are developed in EXT-01A, pages 1–3.
03The same admissible set, different equilibria
The constraint gives v = 2 − u, with 0 ≤ u ≤ 2. Minimization reduces to a function of one variable:
fθ(u) = ½w1u² + ½w2(2 − u)²,
f′θ(u) = (w1 + w2)u − 2w2,
f″θ(u) = w1 + w2 > 0.
The function is continuous on the compact interval, so it attains a minimum. It is strictly convex, so the minimum is unique. Setting the first derivative to zero gives:
u∗ = 2w2/(w1 + w2),
v∗ = 2w1/(w1 + w2).
Since the weights are positive, both coordinates lie strictly between 0 and 2. The critical point belongs to the relative interior of 𝒜 and is the unique global minimum. Both inequalities are non-binding at the solution.
| Weights | Functional | Unique equilibrium |
|---|---|---|
| (3, 1) | Ψ = 3u²/2 + v²/2 | (u∗, v∗) = (½, 3/2) |
| (1, 3) | Ψ = u²/2 + 3v²/2 | (u∗, v∗) = (3/2, ½) |
The admissible set is exactly the same. The coefficients of the already declared functional differ, and so does the selected equilibrium. This proves that 𝒜 alone does not determine equilibrium in this family. It is not a counterexample to the sufficiency of the full declaration: when the weights are also given, the realization relation determines the outcome uniquely.
A precise check for non-binding inequalities
For θ = (3, 1), write the equality as e = u + v − 2 = 0 and the inequalities as h1 = −u ≤ 0, h2 = −v ≤ 0. The Lagrangian Ψ + λe + ν1h1 + ν2h2 gives the stationarity conditions:
3u + λ − ν1 = 0, v + λ − ν2 = 0.
At the proven minimum (½, 3/2), take λ = −3/2 and ν1 = ν2 = 0. Both equations are satisfied, the inequalities are strict, and complementarity νihi = 0 holds. A non-binding inequality may have a negative value; its corresponding multiplier is zero.
In Definition 2.6 of the March manuscript, the displayed equilibrium operator includes h as a block to be set to zero. Requiring h = 0 here would impose u = v = 0, contradicting u + v = 2. This is the specific problem addressed by the distinction between equalities, inequalities and complementarity in MRA-03-F001. The correction has a calculable consequence: it allows the non-binding interior solution already proved above.
Original formulation: Definition 2.6, printed page 3 / PDF 6. Correction: MRA-03-F001, pages 85–86. Substituting these particular numbers is a new development of this page; it is not presented as a general proof of the sufficiency of KKT conditions.
Full calculation for the family: EXT-01A, pages 2–4. The comparison between the two members is used here to make the distinction in MRA-01-F013 calculable.
04Observation as a specific map
Having defined how equilibrium arises, we need to specify what is recorded. Consider two maps sharing the domain 𝒜:
G : 𝒜 → ℝ, G(u, v) = u + v,
H : 𝒜 → ℝ, H(u, v) = u.
G is constant: G(q) = 2 for every q ∈ 𝒜. It therefore does not distinguish the two different equilibria in the table. This is a specific proof of non-injectivity: we have supplied different elements of the domain with the same image.
H, by contrast, is injective on 𝒜. If u is known, the declared constraint determines v = 2 − u uniquely. The same arrangement can therefore have one channel that loses position information and another that determines it. The informational property belongs to the map with its particular domain. The existence of one non-injective example does not prove that all admissible observation maps are non-injective.
We also need to distinguish two objects appearing in the original texts. In the theoretical manuscript, Π maps trajectories to geometric expression. In the mathematical manuscript, P in Definition 5.1 is an orthogonal projector onto the admissible tangent subspace, used in the stability analysis. The properties of the latter do not automatically transfer to the former.
MRA-01-F010 distinguishes a general observation map from the specific case of a linear projector. The word “projection” is not sufficient to assume linearity or the identity P² = P. The latter requires a suitable domain and codomain for the composition to be defined. G and H here take values in the real numbers, not in 𝒜; applying them again to their output has not even been defined.
Geometry retains its role: coordinates represent the admissible set and enable the calculation of equilibria. Whether a record recovers the position or the generating parameter is a subsequent, separate question. H recovers position on 𝒜, but we will see that the composition θ ↦ H(q∗(θ)) does not recover the entire weight pair.
Original formulations: theoretical manuscript, §4.1–4.2, printed 7 / PDF 13; mathematical manuscript, Definition 5.1, printed 7 / PDF 10. Review of type and scope: MRA-01-F010, pages 33–34; MRA-10 §2.7, 509–510. The constant record G belongs to EXT-01A, pages 1 and 3. The explicit comparison of G and H is explanatory material developed for this page.
05Numerical weights and general dominance
In this family, the weights are positive coefficients of two specified terms. We can distinguish their common scale s from their normalized distribution:
s = w1 + w2 > 0,
p1 = w1/s, p2 = w2/s,
p1, p2 > 0, p1 + p2 = 1.
The pair (p1, p2) belongs to the relative interior of the numerical simplex with two components. The equilibria become u∗ = 2p2 and v∗ = 2p1. Thus the full position determines the normalized distribution exactly.
For every c > 0, the pairs θ and cθ have the same distribution and the same equilibrium. The converse also holds: the same u∗ implies the same ratio w1/w2 = (2 − u∗)/u∗, so the two positive pairs differ only by a common positive multiplier. The fibres of the position map are the positive rays {cθ : c > 0}.
The original mathematical manuscript defines normalized numerical components pi from declared nonnegative scores Di with a positive total sum, and defines a regime selector σ separately. The theoretical manuscript describes dominance as an organisation of relative influence and hierarchy. Comparing these levels does not permit us to treat every numerical vector as a full description of the general relation. The normalization of wi in our example is declared separately; we do not identify the coefficients of Ψ with the scores Di of Definition 2.7 without a specific correspondence.
MRA-01-F009 sets this boundary: without additional assumptions, a general dominance relation cannot be identified with a numerical vector in the simplex. The normalization above is fully defined for our family. It does not prove that every general priority relation, every interaction context or every regime identity takes this form. The later reconstruction defines dominance within a declared comparison context, without a universal requirement of total ordering, and distinguishes a change in numerical weights from a change in qualitative class.
Why a numerical weight cannot represent every priority relation
As a separate logical example, consider two labels a and b and the relation R = {(a, a), (b, b)}. It is reflexive and transitive, while a and b remain incomparable. An attempt to represent it exactly by numbers under the rule xRy if and only if r(x) ≥ r(y), for all x, y ∈ {a, b}, would fail: two real numbers are always comparable. Equal numbers would give both relations, rather than incomparability.
The example explains why a numerical representation requires checking which properties it preserves. It proposes no definition of dominance for FBM and does not claim that these particular labels are mechanical constraints.
Original formulations: Definition 2.7–2.8, printed 3 / PDF 6; §2.5, printed 4 / PDF 10, and §8.1, printed 15 / PDF 21. Review: MRA-01-F009, pages 32–33; MRA-05 Definition D5, 182, and D7, 185; MRA-10 §3, 511. The family’s ratio and common scale: EXT-01A, pages 3–4 and 12–15. The incomparability example is a new explanatory example.
06From the ratio to full recovery
EXT-01A proceeds from this ambiguity. It defines the target as recovering the two weights of the known family, and adds information concerning the declared minimization. With the exact normalization e(u, v) = u + v − 2, the Lagrangian is:
ℒ = Ψθ + λ(u + v − 2).
At the interior minimum, stationarity gives w1u∗ + λ∗ = 0 and w2v∗ + λ∗ = 0. Setting ρ∗ = −λ∗, we obtain:
ρ∗ = w1u∗ = w2v∗ = 2w1w2/(w1 + w2) > 0.
The map Γ(θ) = (u∗, ρ∗) from (0, ∞)² to (0, 2) × (0, ∞) has the inverse:
w1 = ρ/u, w2 = ρ/(2 − u).
For every 0 < u < 2 and ρ > 0, these formulas give positive weights. Substituting them into the equilibrium and multiplier formulas returns exactly (u, ρ). This proves both uniqueness and the existence of a preimage for every element of the declared codomain.
The pairs (3, 1) and (6, 2), for example, give the same position (½, 3/2), but ρ∗ = 3/2 and 3, respectively. The additional information distinguishes the scale that position left undetermined. The multiplier follows from the already declared functional and constraint; access to its value is a different matter from its existence within the model.
The full development, including the mechanical realization and the conditions on the additional information, appears in EXT-01A: position and multiplier. Recovery concerns the parameter pair, rather than the complete identity of an arbitrary regime.
Source and full inverse relation: EXT-01A, pages 4–7. Placing this proof within the route through the Re-Audit is a present connection between the sources.
07Four connections that change the reading
The example allows us to read the findings as specific obligations of proof. The Re-Audit is not used here as a substitute for the original source: we keep the formulation, the review and the present computational consequence distinct.
| Transition under review | Re-Audit finding | Consequence in the family |
|---|---|---|
| From the admissible set to equilibrium | MRA-01-F013, pp. 36–37: admissibility is not sufficient for realization or selection. | The shared 𝒜 accommodates different Ψ. The declared minimization and strict convexity give the particular unique equilibrium. |
| From an inequality to setting it to zero | MRA-03-F001, pp. 85–86: the inequality retains its sign condition and complementarity condition. | At (½, 3/2), the hi are negative and the νi are zero. Requiring h = 0 would incorrectly exclude the solution. |
| From the word “projection” to a property of a map | MRA-01-F010, pp. 33–34: a specific type of map is needed, with separate justification of non-injectivity. | G is constant on 𝒜. H is injective on 𝒜, but θ ↦ H(q∗(θ)) leaves the common scale ambiguous. |
| From general dominance to a numerical simplex | MRA-01-F009, pp. 32–33: numerical representation requires additional specification. | The pi represent the relative coefficients of the particular Ψ. They neither recover the common scale nor define full regime identity on their own. |
These distinctions work together: first declare the family and realization relation, then the domain and target of observation, and finally the particular identity to be recovered. Only then does the question of whether additional information is sufficient have a precise meaning.
The original statements and exact versions are recorded in the sources. The table’s connections are analytical, rather than a claim about the historical order in which the works developed.
08When a multiplier acquires mechanical meaning
A numerical multiplier is not called a force simply because it appears in a Lagrangian. We need to know what the coordinates, functional, constraint and units represent. EXT-01A supplies an explicit idealized interpretation: two linear elastic elements in series, with no additional force at the intermediate connection, known unloaded positions and an imposed total extension.
There, the weights correspond to stiffnesses after a declared nondimensionalization. The nonnegative domain of extensions is stated separately to match 𝒜; it does not follow from linear elasticity. The common force T is related to the multiplier through T = k0ℓ0ρ, with known positive units k0 and ℓ0.
If the same constraint is rewritten as ẽ = αe, with α > 0, the corresponding multiplier becomes λ̃ = λ/α. The product λ̃∇ẽ = λ∇e is preserved, rather than the multiplier’s bare numerical value. Recovery requires the known normalization and a reaction value independently available for the same realization. Calculating it from candidate weights cannot subsequently provide independent confirmation of those same weights.
The arrangement is a hypothetical mechanical model. Connecting it to a particular articulator or to Atlas observations requires its own evidence. An observation of increased resistance is not sufficient for numerical identification with ρ. Evidence-supported interpretation begins with an exact correspondence between the actual arrangement and the declared model.
Mechanical interpretation and limits: EXT-01A, pages 8 and 10–18 · full development of the elastic realization.
09What the review completes and what it opens
In its closing passages, the Re-Audit records completion of a first diagnostic and reconstructive cycle. The reconstructed core v0.1 is described as an audited draft awaiting independent mathematical review. This differs from completing all proofs or issuing a new canonical monograph. The value of the particular findings is that they clarify which declaration is missing, which conclusion needs to be restricted, and which step requires a new proof.
In the present study, the result is complete for the declared positive quadratic family: we calculated a unique equilibrium, proved what G and H distinguish, identified the common-scale fibres and supplied a bijective augmentation. These do not constitute a universal theorem for identifying every FBM structure.
Further work can examine measurement uncertainty, different functionals or a different target identity. In each case, the full realization relation and the identification question need to be stated anew. The exact inverse relation does not imply good numerical behaviour near u = 0 and u = 2.
Review status: MRA-10, page 505; final closure, pages 551–553. Limits of exact recovery: EXT-01A, pages 7–9 and 18.
10Sources and further study
Panagiotis Salikopoulos. Frame-Bound Mechanics — Structural Theory of Articulated Systems.
Manuscript internally designated Final Manuscript for Repository Publication, Version 1.0, March 22, 2026. File: frame_bound_mechanics.pdf. Locators here refer to this version: Definition 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. In the condensed chain of Theorem 7.1, the accompanying proof requires the admissible set together with the interaction functional; we do not treat the former as producing the latter on its own. These locators are not transferred to the distinct DOI edition of 29 April 2026.
Open the mathematical manuscript ↗Panagiotis Salikopoulos. Frame-Bound Mechanics — Structural Theory of Articulated Systems. Theoretical Monograph / Zenodo Edition.
Manuscript dated 25 March 2026, file frame_bound_mechanics_theory_publication.pdf, 44 PDF pages. No version number is assigned here. The passages used are §2.5, printed page 4 / PDF 10; §4.1–4.2, printed 7 / PDF 13; and §8.1, printed 15 / PDF 21. Section 4.2 states that Π is non-injective; the specific proof on this page concerns its own explicitly defined G and H.
Open the theoretical manuscript ↗Theory Formalization Re-Audit.
Internal research and reconstruction file. The working edition used here has 553 pages. The findings used are MRA-01-F009, pages 32–33; MRA-01-F010, 33–34; MRA-01-F013, 36–37; MRA-03-F001, 85–86; and MRA-03-F004–F005, 87–89. Later formulations appear in MRA-05, Definition D5, page 182, and D7, 185; and in MRA-10, page 505, §2.3–2.7, 507–510, and §3, 511. The final closure appears on pages 551–553. Numbers refer to the pagination of this working edition. The status of each finding is distinguished from later development and from publication status.
FBM‑EXT‑01A — Πλήρης δομική δήλωση και προέλευση της πρόσθετης πληροφορίας.
Research source PDF, 18 pages: family and equilibrium 1–3; common scale 3–4; multiplier and full inverse 4–7; interpretation and limits 7–18. The source attributes its starting point to Model II — Dominance Non-Identifiability in Minimal Structural Proofs of Projection Non-Identifiability. The proof for the entire family of positive weights is the new development of EXT-01A.
EXT-01A study and detailed source guide →The published mathematical monograph is recorded separately as Frame-Bound Mechanics:Formal Mathematical Core Of Articulated Systems, v1.0, 29 April 2026, DOI 10.5281/zenodo.19886914. The bibliographic identity concerns the monograph’s record; the specific locators on this page refer to the editions above.
- Autonomous mathematical development →
Cases EXT-01A–E retain their own declarations and connect to the relevant foundational objects and Re-audit distinctions.
- Continue to EXT-01A →
From the distinction between admissibility, realization and observation to the full recovery proof and the declared elastic realization.
- Return to the declared relations →
Coordinates, constraints, admissible variation and local mathematical conclusions.
- Read the Hanau case →
An independent reading of the original paper: admissibility, realization relation, dominance and observation within its own declaration.
A further autonomous case: EXT-01B
A declared perturbation separates admissibility, realization and observation in a concrete inverse problem. The new study proves what can be recovered and distinguishes preservation of the interior active set from full regime identity. This is a present connection to the Re-audit distinctions, not an attribution of the new proofs to the historical source.
Read EXT-01B →