01Which test can reveal the structure?
In EXT-01A, equilibrium position and an appropriately normalized multiplier identify two unknown weights in a declared family. EXT-01B continues the investigation when the multiplier is not directly available, by considering the response to a known perturbation.
Can we choose a perturbation that will reveal the weights before knowing the very weights we seek?
The question has two parts. First, we must prove that the response contains the required information. Then we must determine whether the test can be selected in advance while preserving the constraint status under examination. Recovery and test selection have different prerequisites.
The analysis begins with the declared structure and realization relation. Position, response and the limits of the test follow from them. The geometric record participates as a particular observation channel for this outcome.
Source: EXT-01B, PDF pages 1–2, 12–13.
02The family and its unknown common scale
The frame, coordinates, normalization and positive weights remain fixed during the test. The reference offsets of the more general family are set to a = b = 0. The subproblem specifies an admissible set, an interaction functional and constrained minimization as the realization relation:
𝒜 = {(u, v) ∈ ℝ2 : u + v = 2, u ≥ 0, v ≥ 0}.
Ψ0(u, v) = ½w1u2 + ½w2v2, w1, w2 > 0.
q0 = arg min(u,v) ∈ 𝒜 Ψ0(u, v), s := w1 + w2.
The functional formulas are defined on ℝ2; minimization uses their restrictions to 𝒜. With v = 2 − u, the second derivative of the initial functional along the admissible variation is s > 0. Its unique minimum lies in the relative interior and gives:
u0 = 2w2/s, v0 = 2w1/s.
w1 = s(2 − u0)/2 = sv0/2, w2 = su0/2.
For known u0 ∈ (0, 2), the initial position determines the relative distribution of the weights. Every compatible pair is parameterized by s > 0. The remaining ambiguity is the common scale.
The goal is to recover the two weights within this known family. Connectivity, the form of the functional and the realization relation are specified. Identifying an unknown structural architecture or full FBM regime identity would be broader questions.
Source: EXT-01B, PDF pages 1–2.
03The constraint determines the choice of perturbation
Adding −r(u + v) changes the functional by the constant −2r on 𝒜. The minimizing position is unchanged:
arg min𝒜[Ψ0 − r(u + v)] = arg min𝒜 Ψ0.
This intervention does not expose the common scale through a change of position. In a mechanical realization, the reaction enforcing the constraint may nevertheless change; unchanged position does not imply that every mechanical quantity is unchanged.
Now choose a known, calibrated generalized load r > 0, conjugate to v and expressed in the same normalization as Ψ0:
Ψr(u, v) = Ψ0(u, v) − rv.
−rv = −2r + ru on 𝒜.
δu + δv = 0.
The term −rv varies along the admissible internal redistribution. The two loads are therefore compared using the declared constraint relation before the output is calculated or plotted. This identifies the channel that can alter the internal realization.
Source: EXT-01B, PDF pages 2–3.
04The response and the interior window
Substituting v = 2 − u reduces minimization under load to one variable:
fr(u) = ½w1u2 + ½w2(2 − u)2 − r(2 − u), 0 ≤ u ≤ 2.
f′r(u) = su − 2w2 + r, f″r(u) = s > 0.
ur = (2w2 − r)/s, vr = (2w1 + r)/s.
Continuity on [0, 2] ensures that a minimum exists, and strict convexity ensures uniqueness. For positive r, the stationary solution is in the relative interior exactly when ur > 0: vr is already positive. Consequently:
0 < r < 2w2 = su0.
Δ := vr − v0 = u0 − ur = r/s.
Within this interval the equality u + v = 2 remains binding, while both unilateral inequalities remain non-binding. The same status holds throughout the loading path from 0 to the chosen r. The interior admissibility class and the unique branch of constrained minima are preserved.
Preservation has this specific meaning here. It does not establish preservation of every component of a full regime signature, such as mode, contextual dominance or history. A strict constrained minimum also does not by itself establish dynamic or full regime stability.
Source: EXT-01B, PDF pages 3–4.
05EXT-01B-1 · Recovery from position and response
Proposition. For fixed, known r > 0, the map from the weights to the data (u0, Δ) is a bijection between:
Θr = {(w1, w2) : w1 > 0, w2 > r/2}.
D = {(u0, Δ) : 0 < Δ < u0 < 2}.
The inverse first recovers the common scale and then the two weights:
s = r/Δ.
w1 = r(2 − u0)/(2Δ), w2 = ru0/(2Δ).
Proof and complete verification of the inverse
For θ ∈ Θr, we have 0 < r < 2w2. Since Δ = r/s and u0 = 2w2/s, it follows that 0 < Δ < u0 < 2. The relation s = r/Δ and the initial-position formulas require precisely the unique weight values above.
Conversely, take any data in D and define the weights by these formulas. Since 2 − u0 > 0, we have w1 > 0; since u0/Δ > 1, we have w2 > r/2. Furthermore:
w1 + w2 = r/Δ.
2w2/(w1 + w2) = u0.
r/(w1 + w2) = Δ.
The forward map returns exactly the original data. The inverse is therefore defined on all of D and recovery is unique. ∎
In the ideal elastic interpretation of EXT-01A, with a consistent normalization of the quantities, the interior stationarity relations become:
w1ur = ρr, w2vr − r = ρr.
w2vr − w1ur = r.
It is under this declared elastic realization that the weights can be interpreted as corresponding stiffnesses. At the same load, a larger s gives a smaller internal displacement. The known load scale can thus replace direct access to the multiplier for this particular recovery problem.
Source: EXT-01B, PDF pages 4–6.
06One initial position, two different responses
The pairs (3, 1) and (6, 2) have the same relative distribution and initial position. Apply the same known r = 1:
| (w1, w2) | s | (u0, v0) | r | (ur, vr) | Δ |
|---|---|---|---|---|---|
| (3, 1) | 4 | (0.5, 1.5) | 1 | (0.25, 1.75) | 0.25 |
| (6, 2) | 8 | (0.5, 1.5) | 1 | (0.375, 1.625) | 0.125 |
Both solutions are interior: their load thresholds are 2 and 4 respectively. For the first measurement, s = 1/0.25 = 4 and the inverse gives w1 = 3, w2 = 1. For the second, s = 1/0.125 = 8 and we recover 6 and 2.
An informative intervention also needs an informative observation: the sum ur + vr remains 2 in both cases.
If only this sum is recorded, the new test still does not distinguish the pairs. Obtaining Δ requires access to the internal displacement of a coordinate relative to its initial value.
Source: EXT-01B, PDF pages 6.
07A known command and a known mechanical load
Recovery assumes that the actual r is known. If only a command rcmd is known and its mechanical effect includes an unknown factor γ, another ambiguity appears:
r = γrcmd, γ > 0.
Δ = γrcmd/s.
(w1, w2, γ) ↦ (cw1, cw2, cγ), c > 0.
The transformation multiplies both s and γ by the same positive number. It preserves the initial ratio and γ/s, and therefore returns the same response to each identical command within the interior window. Even the threshold expressed in command units, su0/γ, remains unchanged.
A known command identifies γ/s here without separating the two quantities. Calibration supplies the external scale reference used by the inverse. This is additional information for the test, rather than automatically a new primitive.
Source: EXT-01B, PDF pages 6–7.
08EXT-01B-2 · Existence for each system and a common choice
Every particular positive pair has a nonempty window 0 < r < 2w2. Before the test, however, if only u0 is known, every s > 0 remains compatible with the initial position. Preservation requires r < su0.
Proposition. Without a positive lower bound on s, there is no common preselected r > 0 that preserves ur > 0 for every pair compatible with the same u0 ∈ (0, 2).
Proof that no common positive perturbation exists
Take any r > 0. Choose 0 < s < r/u0 and construct the positive weights w1 = s(2 − u0)/2 and w2 = su0/2. They give the same initial position, but r > su0 = 2w2. The selected load exceeds the interior window. Every preselected positive r therefore fails for some compatible pair. ∎
In the following quantifiers, θ ranges over weights with the same u0. The essential distinction is:
∀θ ∃rθ > 0 : rθ < 2w2.
¬∃r > 0 ∀θ : r < 2w2.
The result concerns this family of preselected loads. It does not exclude adaptive testing with additional monitoring or other information channels.
A justified lower bound s ≥ s− > 0 makes a common choice possible:
0 < r < u0s− ≤ u0s.
The lower bound has not already solved identification. It supplies enough prior information to choose a test that can solve it.
Source: EXT-01B, PDF pages 7–9.
09EXT-01B-3 · Preservation and distinction under error
Compare two positive scales s1 ≠ s2 with the same initial position u0. In this subproblem, the initial position and calibrated load are treated as exactly known. The stated uncertainty concerns displacement as the final datum:
Δ̃ = Δ + e, |e| ≤ ε, ε ≥ 0.
Ii = [r/si − ε, r/si + ε], i ∈ {1, 2}.
I1 ∩ I2 = ∅ ⇔ r|1/s1 − 1/s2| > 2ε.
If displacement is obtained from two uncertain position records, ε includes their combined uncertainty. Uncertainty in the initial position used as a shared parameter, or in the load, would require a separate development of the error model.
The compatible measurement intervals must be disjoint to guarantee distinction for every allowed error. Equality at their endpoints is insufficient because a common measurement remains possible. Both solutions must also stay interior:
smin := min(s1, s2), smax := max(s1, s2).
2ε / |1/s1 − 1/s2| < r < u0smin.
Proposition. For this pair and this error model, the common open window is nonempty if and only if:
2ε < u0(1 − smin/smax).
Proof of the necessary and sufficient condition
Distinction requires r > 2ε/|1/s1 − 1/s2|. Preservation requires r < u0smin. Such a positive r exists exactly when the lower endpoint is strictly below the upper endpoint. The scales are positive and distinct, so:
|1/s1 − 1/s2| = 1/smin − 1/smax > 0.
Multiplying the endpoint inequality by this positive number gives 2ε < u0(1 − smin/smax). Every step is reversible, making the condition both necessary and sufficient. ∎
For the numerical pair s1 = 4, s2 = 8 and u0 = 0.5, the condition becomes ε < 0.125 and the window is 16ε < r < 2. With ε = 0.02, this is 0.32 < r < 2: r = 1 preserves and distinguishes both candidates. This calculation is a numerical application of the proposition in the present study.
As the two scales approach one another, the available response difference before the boundary decreases. The exact bijective recovery in EXT-01B-1 and guaranteed distinction with positive error are different results.
Source: EXT-01B, PDF pages 9–10.
10The boundary, saturation and path information
For every r ≥ 0, strict convexity still gives a unique constrained minimum. The complete solution is:
ur∗ = max{0, (2w2 − r)/s}, vr∗ = 2 − ur∗.
r ≥ 2w2 ⇒ (ur∗, vr∗) = (0, 2).
When r ≥ 2w2, the derivative at u = 0 is nonnegative and increases with u, so the minimum lies at the left endpoint. At the threshold, activation of u ≥ 0 is weak; beyond it, the corresponding multiplier is positive. Reaching the boundary does not mean that existence or uniqueness of equilibrium has been lost here.
If only the initial position and final position (0, 2) at a given load r are known, every scale in the following range is compatible:
0 < s ≤ r/u0.
For (3, 1) and (6, 2), the thresholds are 2 and 4. A load r = 5 gives the same final position for both. A stronger perturbation can therefore merge the endpoint records again.
Different information is available if the whole loading path is monitored and the exact first load at which the boundary is reached is known:
rb = su0, s = rb/u0.
This protocol uses the boundary event to recover scale in the ideal model. It has, however, reached the boundary of the initial interior class. Recording only the final position, recording the path and requiring interior preservation must be specified separately.
Source: EXT-01B, PDF pages 10–11.
11Explore the declared example
The diagram computes responses for the two specific candidates (3, 1) and (6, 2), with their shared initial position (0.5, 1.5). Varying r lets you examine interior distinction, arrival at the two thresholds and coincidence of the final positions. The parameter ε refers to the error model in the preceding proposition.
The curves are calculated solutions of this model. They are not experimental records and do not replace the proofs for the whole declared family. The common preservation–distinction window concerns these two candidate weight pairs, a known calibrated r and the specified error bound.
Source: EXT-01B, PDF pages 6, 9–11.
12What is recovered and what remains open
| Datum or requirement | Role within the declared model |
|---|---|
| Initial position | Determines the relative distribution of the weights. |
| Calibrated load and interior Δ | Recover the common scale and hence the two weights. |
| Positive lower bound on s | Permits a common advance choice of an interior load. |
| Bound ε and two candidate scales | Determine whether a common window of guaranteed distinction and preservation exists. |
| Exact threshold recorded along the path | Recovers scale through arrival at the boundary, using a different protocol. |
The results assume positive weights, this particular quadratic form, zero reference offsets, the declared 𝒜 and the chosen minimization relation. Different connectivity, unknown interaction laws, additional loads or another realization relation require a new formulation and analysis.
This mathematical case develops FBM autonomously. Mapping it to a real mechanical or bench arrangement remains a separate research task. No application to a particular articulator, new empirical confirmation or full recovery of regime identity is established here.
Structure determines the response and the margin within which that response can reveal the unknown weights of the declared family.
Source: EXT-01B, PDF pages 11–13.
13Foundations, Re-audit and new mathematical development
The following connections organize the present study. The historical foundation supplies the objects, the Re-audit clarifies their roles and the scope of claims, and EXT-01B proves its particular new propositions.
- Model II and the original mathematical starting point →
EXT-01B identifies Model II of Minimal Structural Proofs, specialized to a = b = 0, as its starting point. The family and constant observation precede the new test.
- EXT-01A · Additional information →
Position and the normalized multiplier provide the first inverse relation. EXT-01B examines a calibrated perturbation as another way to access the common scale.
- Admissibility and realization →
The set 𝒜 specifies compatible configurations. The functional Ψ and constrained minimization determine the solution. Interior-class preservation concerns a particular constraint status.
- The observation channel →
The recorded sum stays constant. Internal displacement, calibration and recording the threshold supply different data, with different proofs of sufficiency.
- Weights and full regime identity →
Unique recovery of two numerical weights does not identify every component of a general regime. The relevant assumptions remain explicit.
Source EXT-01B also refers to the general load-addition relation ΦB = ΦA + ΔΦload in the earlier Atlas material. The particular choice −rv and the three propositions belong to the new EXT-01B development. This historical reference does not attribute the proofs to the Atlas and does not constitute bench validation.
For the precise Re-audit definitions, follow the source documentation in the corresponding study. The route to the canonical monographs gives access to the historical foundation without retrospectively attributing the EXT-01B results to them.
Source: EXT-01B, PDF pages 1, 4, 12–13.
14Exact source and further reading
FBM‑EXT‑01B — Δηλωμένη διαταραχή, ανάκτηση κοινής κλίμακας και παράθυρο διατήρησης της admissibility class.
A 13-page research source in PDF format. This bilingual website study was prepared on 3 November 2026. It develops the assumptions, proofs and implications of this particular source, adding an interactive illustration and a numerical application of the error bound. The original research text retains its own identity; the present development is not presented as a new monograph.
Open the research source ↗| Study content | PDF pages in EXT-01B |
|---|---|
| Starting point, complete family and initial scale | 1–2 |
| Choice of perturbation, response and preservation | 2–4 |
| EXT-01B-1, mechanical interpretation and numerical pair | 4–6 |
| Unknown calibration | 6–7 |
| EXT-01B-2 and a positive lower bound | 7–9 |
| EXT-01B-3 and a common window under error | 9–10 |
| Boundary, endpoint and loading history | 10–11 |
| Synthesis and scope of claims | 11–13 |
Page numbers refer to the sequence of pages in the PDF. The English version is an edited rendering of the same development. Formulas use consistent notation, and the additional calculations follow explicitly from the displayed relations.
Transcription note: the inequalities on page 1 are written explicitly here as u ≥ 0, v ≥ 0; together with u + v = 2, these are equivalent to uv ≥ 0. The common-scaling triple on page 6 is typeset inconsistently. Here it is written (cw1, cw2, cγ), with direct verification that it preserves γ/s and Δ = γrcmd/s. The original PDF is retained unchanged.
- All mathematical cases →
EXT-01B retains its own source and connects to the other cases through their declared mathematical questions.
- Mathematical Monograph →
The historical foundation and canonical identity of the monograph. The source of the new propositions on this page is EXT-01B.
- Research index →
Autonomous works, editions and separate analyses within the broader FBM field.
Continue · EXT-01C: independent relations and unique identification →