An Information-Based Framework for Physical Measurability and Reconstructability

Version 0.9 · Preprint / Framework Paper · FORSCH-PROV0420 (number provisional) · as of 6 Oct 2026

This version claims no new fundamental physics. It is an ordering framework for the measurability, reconstructability and falsifiability of physical measurement chains.

0 What changed relative to Version 0.8

Nothing in the work itself. No result, no proof, no figure and no appendix has changed. This version makes the bundle around the work ready for publication; the number is carried along so that work and bundle do not drift apart.

The material is finished. What remains are the seven decisions in VEROEFFENTLICHUNG.md Section 1; all of them belong to the author and none can be derived from the material.

0.1 What had changed relative to Version 0.7

This version changes no result and no proof. It draws the disclosure line explicitly instead of presupposing it.

Substantively unchanged: measurement chain, Theorems 1 and 2, classification A/B/C, numerics, experimental protocol, references, source checklist, licence levels, active/passive assignment.

0.2 What had changed relative to Version 0.6

This version changes no result and no proof. It anchors a cross-track rule that so far existed only as a decision.

Substantively unchanged: measurement chain, Theorems 1 and 2, classification A/B/C, numerics, experimental protocol, references, source checklist, licence levels.

0.3 What had changed relative to Version 0.5

This version changes no result and no proof. It converts the commercial arrangement to a platform-centred one.

Substantively unchanged: measurement chain, Theorems 1 and 2, classification A/B/C, numerics, experimental protocol, references, source checklist.

0.4 What had changed relative to Version 0.4

This version changes no result and no proof. It folds in the source check that version 0.4 listed as the single remaining open task.

What is still outstanding: the choice of edition for fourteen textbooks and three first editions, the edition status of JCGM 100/200 and EMVA 1288, and the access date of the SEP entry. These are itemised in Appendix I.2 and are to be closed before a journal version.

0.5 What had changed relative to Version 0.3

0.6 What had changed relative to Version 0.2

Abstract

This work formulates an information-based ordering framework for physical measurability and reconstructability. The claim is not to introduce new fundamental physics but to organise established concepts from wave theory, electrodynamics, scattering theory, signal theory, digitisation and quantum measurement along a common measurement chain:

$$X_0 \rightarrow H_\text{field} \rightarrow H_\text{mat} \rightarrow H_\text{mix} \rightarrow H_\text{measure} \rightarrow Q \rightarrow Y.$$

The starting point is observational equivalence: if $p(Y\lvert\theta_1)=p(Y\lvert\theta_2)$ for $\theta_1\neq\theta_2$, unique reconstruction from that channel is excluded. This is not new; it is the definition of identifiability of a parametric model, and its local version has been equivalent to regularity of the Fisher information matrix since Rothenberg (1971). The contribution of this work lies in two points.

First, the diagnostic working quantity $K$ is defined as an effective signal-to-noise ratio and derived exactly from the Fisher information for the scalar Gaussian channel: with $K := \theta^2 F(\theta)$ one obtains $K = SNR\cdot C^2$. Second, the product form

$$K = SNR\cdot G(\lvert q\rvert a)\cdot D_\text{mat}\cdot\gamma\cdot R_\text{dir}\cdot C^2$$

receives a testable validity criterion: on a connected product domain it is exact precisely when $\log F$ is additively separable. Stated more sharply via stage efficiencies $\eta_k = F_k/F_{k-1}$, the claim is that each $\eta_k$ depends only on the parameters of its own stage. This is decided for three classes of forward models: multiplicative transmission with single-stage noise and Poisson thinning satisfy the criterion exactly; high-gain EMCCDs and low-SNR one-bit quantisation satisfy it asymptotically; multi-stage noise injection, the crossover between shot and read noise, nonlinearity and saturation violate it.

Ten numerical minimal models with a fixed random seed support the statements, among them a separability map over signal level and read noise and a fully computed optical slit experiment. The scientific load-bearing capacity lies in the falsification conditions.

Keywords: measurability; reconstructability; identifiability; observational equivalence; Fisher information; separability; cascaded systems; wave-vector transfer; coherence; inverse problems; quantum measurement.

1 Introduction

Physical measurements are often described through single terms: signal strength, frequency, wavelength, reflection, resolution, noise, detection. These are necessary but not sufficient. A strong signal may say little about the state sought. A local null signal may arise from destructive interference. A digitally crisp result may rest on coarse quantisation or algorithmic priors. A reflected signal may carry energy while having lost direction and phase.

The central question is therefore not whether a signal exists, but:

Which state information is distinguishable in the signal under the given physical and instrumental conditions?

The contribution is not a new law of nature. It is a framework that orders existing physical descriptions along the question of measurability and — since version 0.2 — states the conditions under which the working quantity used is admissible. Version 0.3 answers that question for the main channel classes.

2 Scope of claim

The model claims no new fundamental physics. It replaces neither Maxwell’s equations, nor quantum mechanics, scattering theory, Fourier optics, information theory, nor the theory of inverse problems. It is an ordering framework whose purpose is to make the measurement question explicit:

1. Which state is to be distinguished?
2. Which physical forward chain generates the signal?
3. Which factors preserve or destroy state information?
4. Which states remain distinguishable in the measurement channel?
5. Which states collapse into the same measurement distribution?

3 Measurement chain

$$X_0 \rightarrow H_\text{field} \rightarrow H_\text{mat} \rightarrow H_\text{mix} \rightarrow H_\text{measure} \rightarrow Q \rightarrow Y. \tag{1}$$

In the linear case $Y = H_\text{total}X_0 + N$ with $H_\text{total}=H_m\cdots H_1$ (2). If $H_\text{total}$ has a null space, different states produce the same signal. In the nonlinear case $Y = F_m(\dots F_1(X_0)) + N$ (3), and local sensitivity decides.

4 State distinguishability

A signal carries state information about $\theta$ if

$$p(Y\lvert\theta_1)\neq p(Y\lvert\theta_2) \quad \text{for} \quad \theta_1\neq\theta_2, \tag{4}$$

and unique reconstruction is excluded if

$$p(Y\lvert\theta_1)=p(Y\lvert\theta_2), \quad \theta_1\neq\theta_2. \tag{5}$$

Provenance of the statement. Equation (5) is the definition of observational equivalence of two parameter values; a model is identifiable if no two distinct parameter values are observationally equivalent. Rothenberg (1971) shows that under weak regularity conditions local identifiability is equivalent to non-singularity of the Fisher information matrix. This work adopts that theorem and applies it to physical forward chains. The contribution lies in the transfer and in the stage decomposition, not in the statement itself.

Three levels must be separated:

In the quantum case the Fisher information is replaced by the quantum Fisher information; Braunstein and Caves (1994) show that the statistical distinguishability of neighbouring states defines a Riemannian metric on the space of density operators.

5 The working quantity $K$

5.1 Definition as effective SNR

$K$ is defined as the effective signal-to-noise ratio of the equivalent Gaussian channel that yields the same Fisher information about $\theta$ as the measurement chain under consideration.

Hence

$$I_\text{eff} = \log_2(1+K) \tag{6}$$

is, in the anchor case, not an approximation but exactly the Shannon capacity of the equivalent channel.

5.2 Anchor case: derivation from the Fisher information

For $Y=s(\theta)+N$, $N\sim\mathcal{N}(0,\sigma^2)$, $F(\theta) = (\partial_\theta s)^2/\sigma^2$ (7). With the dimensionless definition $K := \theta^2 F(\theta)$ and

$$SNR := \frac{s^2}{\sigma^2}, \qquad C := \frac{\theta}{s}\frac{\partial s}{\partial \theta}$$

it follows exactly that

$$K = SNR\cdot C^2. \tag{8}$$

$K$ is therefore the Fisher information with respect to $\ln\theta$, and $C$ is the logarithmic sensitivity (elasticity) of the forward chain. If the further stages act as amplitude-scaling factors $g_i\in[0,1]$ without changing noise power, then $SNR_\text{eff}=SNR\prod_i g_i^2$ and

$$K = SNR\cdot G(\lvert q\rvert a)\cdot D_\text{mat}\cdot \gamma\cdot R_\text{dir}\cdot C^2. \tag{9}$$

5.3 Theorem 1 (separability)

Let $F(\theta; p_1,\dots,p_n)$ be positive and twice continuously differentiable on a connected product domain $P_1\times\dots\times P_n$. Then the following are equivalent:

1. $F = \prod_{i=1}^{n} f_i(p_i)$ for suitable functions $f_i>0$;
2. $\dfrac{\partial^2 \log F}{\partial p_i\,\partial p_j}=0$ for all $i\neq j$. (10)

Proof (sketch). (1) $\Rightarrow$ (2): $\log F = \sum_i \log f_i(p_i)$; each summand depends on exactly one variable, so all mixed second derivatives vanish. (2) $\Rightarrow$ (1): if all mixed second derivatives of a $C^2$ function vanish on a connected product domain, then $\partial\log F/\partial p_i$ is a function of $p_i$ alone; integration yields $\log F = \sum_i h_i(p_i) + \text{const}$. $\square$

The criterion is therefore not sufficient or necessary but equivalent to the product form. It is invariant under coordinatewise reparametrisations $p_i\mapsto\varphi_i(p_i)$, but not under mixing transformations. Whether, for a given model, a mixing reparametrisation exists that makes $\log F$ separable is a boundary-value problem of the same kind as the orthogonalisation of Cox and Reid (1987); it remains open.

5.4 Theorem 2 (stage formulation)

For a chain $\theta \to X_0 \to X_1 \to \dots \to X_n$ with Fisher informations $F_k$ about $\theta$, the telescoping identity always holds:

$$F_n = F_0 \prod_{k=1}^{n} \eta_k, \qquad \eta_k := \frac{F_k}{F_{k-1}} \in [0,1]. \tag{11}$$

The bound $\eta_k\le 1$ follows from the data-processing inequality for Fisher information (Zamir 1998). Identity (11) is therefore trivial and by itself carries no content. The substantive claim of the framework is:

The stage efficiency $\eta_k$ depends only on the parameters $p_k$ of its own stage — not on upstream parameters $p_{<k}$ and not on the signal level.

Only in this form is the product form a testable physical statement about channel structure. It is equivalent to Theorem 1 applied to the parameter groups of the individual stages.

5.5 Demarcation from neighbouring concepts

5.6 Classification of forward models

This three-class assignment rests on the literature strands searched — data-processing inequality, cascaded systems, quantisation, detector noise — and claims no completeness over all conceivable channel classes.

The general reason for class C fits in one line: for $Y\sim\mathcal{N}(s(\theta),\sigma^2(\theta))$,

$$F = \frac{(\partial_\theta s)^2}{\sigma^2} + \frac{(\partial_\theta\sigma^2)^2}{2\sigma^4}. \tag{12}$$

The second term is absent from $K=SNR\cdot C^2$. As soon as one parameter acts on both signal and noise, it contributes to both terms and the factors are no longer independently calibratable.

Convention for the excess noise factor. Robbins and Hadwen (2003) define the excess noise factor as an amplitude quantity with $F_\mathrm{ENF}\to\sqrt2$ at high gain; other works, such as Mortensen et al. (2010), already quote the squared quantity. This work consistently carries the squared quantity $\mathrm{ENF}^2$ as a variance factor: the variance grows by $\mathrm{ENF}^2$, the Fisher information falls by $1/\mathrm{ENF}^2$. For a high-gain EMCCD this gives $\mathrm{ENF}^2\to 2$ and the factor $1/2$ in class B.

6 Direction and signed flux

Direction is determined relative to an axis, aperture, source, detector or interface. For a wave with wave vector $k$ and observation axis $e$:

$$c_e = \frac{k\cdot e}{\lvert k\rvert}. \tag{13}$$

For two counter-propagating components along $e$ the signed direction balance is

$$\Phi_e = \frac{\lvert A_+\rvert^2-\lvert A_-\rvert^2}{\lvert A_+\rvert^2+\lvert A_-\rvert^2}, \qquad K_\Phi = K\cdot\Phi_e. \tag{14}$$

$\Phi_e$ is not a new quantity but the net flux normalised to total power. With reflection coefficient $\Gamma=A_-/A_+$ and standing-wave ratio $s=(1+\lvert\Gamma\rvert)/(1-\lvert\Gamma\rvert)$, exactly

$$\Phi_e = \frac{1-\lvert\Gamma\rvert^2}{1+\lvert\Gamma\rvert^2} = \frac{2s}{1+s^2}. \tag{15}$$

$\Phi_e$ is thus a rescaled VSWR reading, directly measurable in any RF laboratory (checked numerically in 14.5).

7 Resolution and wave-vector transfer

$$q = k_\text{out}-k_\text{in}, \qquad \lvert q\rvert a\gtrsim 1, \qquad \lvert q\rvert = 2k\sin(\Theta/2). \tag{16}$$

Near-field methods, structured illumination, nonlinearity, repeated measurements or priors extend this bound. They are not violations of the model but additional information channels that must be entered into the chain explicitly.

7.1 Rayleigh and Sparrow as thresholds on $d'$

The classical resolution criteria of Rayleigh, Sparrow and Abbe are formulated for incoherent point sources and impose a fixed threshold: resolved or not. The Fisher information knows no such threshold; it falls continuously. Both descriptions can be reconciled rather than played off against each other.

For two point sources at separation $\Delta$, locally $d'^2 \approx F(\Delta)\,\Delta^2$, and $d'$ is the normalised separability of detection theory. A classical criterion is then the statement that $d'$ reaches a particular value:

The gain of this reformulation is practical: the classical criteria know nothing of photon count. They say the same for an image with ten photons and one with ten million. The Fisher form says how many photons are needed for two states to become distinguishable at a given aperture — and that is the question actually asked in design.

8 Material response and information loss

$R+T+A=1$ is an energy balance, not an information balance. A mirror preserves directed information; a matte surface may return a similar amount of light while diffusing direction and phase. $D_\text{mat}$ describes how much of the signal structure usable for the state question arrives ordered at the detector.

9 Interference, cancellation and standing waves

$$\Psi = \sum_j A_j \exp[i(k_j\cdot x-\omega_j t+\phi_j)], \qquad \lvert\Psi\rvert^2 = A_1^2+A_2^2+2A_1A_2\cos(\Delta k\cdot x-\Delta\omega t+\Delta\phi). \tag{17}$$

Destructive interference means that field contributions add vectorially to zero at the point considered. A local null signal is not a global proof of absence.

10 Linear superposition, linear time variance and nonlinear mixing

Morgenthaler (1958) solves Maxwell’s equations for media with time-dependent permittivity and permeability and shows that the fields remain linear when the medium parameters do not respond to field strength, yet a monochromatic wave undergoes a frequency deviation. Parametric amplifiers, modulators, switched plasmas and time-modulated metamaterials (Galiffi et al. 2022) belong to this class. With $Y=\alpha X+\beta X^2+\gamma_3X^3+\dots$, by contrast, mixing products $f_1\pm f_2$ and intermodulation $2f_1-f_2$ arise.

11 Electromagnetic core

$$\nabla\cdot D=\rho,\quad \nabla\cdot B=0,\quad \nabla\times E=-\partial_t B,\quad \nabla\times H=J+\partial_t D. \tag{18}$$

In the linear, isotropic, local limit $D=\varepsilon E$, $B=\mu H$, $J=\sigma E$. In anisotropic, dispersive, nonlinear or time-dependent media, $\varepsilon$, $\mu$, $\sigma$ must be modelled as tensors, frequency functions, convolution kernels or nonlinear response functions. In this section $\mu$ denotes permeability only; the coherence factor is $\gamma$.

12 Digitisation

If $Q$ is pure post-processing of the same analogue signal, the data-processing inequality holds:

$$I(X;Y_\text{digital}) \le I(X;Y_\text{analog}). \tag{19}$$

Digitisation creates no physical state information that is contained neither in the analogue signal nor in additional data, priors or further measurements.

13 Quantum measurement

$$p(y\lvert\theta)=\operatorname{Tr}(\rho_\theta E_y), \qquad E_y\ge 0,\quad \sum_y E_y=I. \tag{20}$$

The corresponding estimation bound is the quantum Cramér–Rao bound (Helstrom 1969); the distinguishability metric is formulated by Braunstein and Caves (1994), with $F_Q=\max_\text{POVM} F_C$. An accessible overview for applications is given by Paris (2009).

The quantum Fisher information is monotone under completely positive, trace-preserving maps; this is the quantum form of the data-processing inequality of Section 12 and the reason $\eta_k\le 1$ also holds in the quantum case. For estimation bounds under noise there is a general framework (Escher, de Matos Filho & Davidovich 2011); under dephasing the attainable scaling falls back asymptotically to $1/\sqrt N$ and the Heisenberg advantage is lost. Decoherence means that phase relations become practically inaccessible through coupling to detector or environment; one then considers $\rho_S=\operatorname{Tr}_E(\rho_{SE})$. All purifications of a reduced state are related by unitaries on the ancilla (Hughston, Jozsa & Wootters 1993): infinitely many total states yield the same local statistics. That is the quantum case of Equation (5).

14 Numerical minimal models

Figures from werkzeuge/FORSCH-PROV0420_minimalmodelle_v0.2_2026-08-26_numerik.py (seed 20260826, models M1–M6) and …_v0.3_2026-10-05_numerik.py (seed 20261005, models M7–M10). Raw results: ergebnisse.json and ergebnisse_v0.3.json.

14.1 M1 — Gaussian channel, Fisher information, Cramér–Rao

Model $Y=A\theta+N$, $A=1.0$, $\sigma=0.5$, $\theta_0=1.0$, $n=200{,}000$.

14.2 M2 — Null space and conditioning continuum

$H=\begin{pmatrix}1&1&0\\0&0&1\end{pmatrix}$, $\sigma=0.1$: Fisher eigenvalues $\{0;100;200\}$, null-space direction $(0.707;-0.707;0)$. Replacing the exact null space by a small singular value $\epsilon$, the Cramér–Rao bound grows as $1/\epsilon$ (from 0.1 at $\epsilon=1$ to $\approx 1.0\cdot10^{5}$ at $\epsilon=10^{-6}$).

What decides is the singular-value spectrum relative to the noise level — the Picard condition of inverse-problem theory (Bertero & Boccacci 1998).

14.3 M3 — Two-sensor bearing model

$\tau=d\sin\theta/v$, $v=343$ m/s, $\sigma_\tau=2$ µs, $n=40{,}000$ per point.

Median relative deviation 0.20 %, maximum 3.4 %.

14.4 M4 — Breakdown of the product form under gain-dependent noise

$\sigma^2(g)=\sigma_0^2(1+\eta g^2)$, $\sigma_0=0.2$, $\eta=0.05$. The true Fisher information saturates at 500; the product form calibrated at $g_0=1$ grows as $g^2$.

14.5 M5 — Signed flux balance and standing-wave ratio

Identity (15) checked over $\lvert\Gamma\rvert\in[0;0.999]$; maximum deviation $2.2\cdot10^{-16}$ (machine precision).

14.6 M7 — Stage efficiency in a cascade

Poisson source $\theta=1000$ photoelectrons, three transmission stages with nominal $\eta=(0.80;\,0.60;\,0.50)$, read noise $\sigma_r$ at the end of the chain. Measured: $\eta_k^\text{eff}=F_k/F_{k-1}$.

This is Theorem 2 in numbers. Without read noise, $\eta_k^\text{eff}$ is exactly the nominal transmission of its own stage and the product form holds. With read noise, $\eta_3$ depends on how much signal stages 1 and 2 left behind — the stage efficiency is then no longer a stage property, and the factors cannot be calibrated separately.

14.7 M8 — Separability map

Mixed second derivative $\partial^2\log F/\partial\ln\lambda\,\partial\ln\sigma_r$ over signal level and read noise, grid 120 × 120 over $\lambda\in[1;10^5]$ $\mathrm{e}^-$ and $\sigma_r\in[0.1;300]$ $\mathrm{e}^-$.

At a threshold of 0.05, 58 % of the parameter area considered is separable. The structure is instructive: separability holds in the two asymptotic regimes — clearly shot-limited or clearly read-limited — and breaks in the crossover band $\lambda\approx\sigma_r^2$. The product form is therefore not an either/or but a property of the operating point. In practice: factorisation is admissible where one noise contribution clearly dominates.

14.8 M9 — Qubit under dephasing

State $\rho=\tfrac12(I+r(\cos\varphi\,\sigma_x+\sin\varphi\,\sigma_y))$ with $r=e^{-\Gamma t}$, $\Gamma=1$.

Two lessons for the framework. First, non-distinguishability in the quantum case is measurement-relative: at $\varphi-\alpha=0$, $F_C=0$ although $F_Q>0$ — the information is present in the state and inaccessible in the chosen channel. Second, $\log F_Q(\omega)=2\log t-2\Gamma t$ with $\partial^2/\partial t\,\partial\Gamma=-2\neq 0$ shows that the separability criterion is violated in the parameters $(t,\Gamma)$ but satisfied in $(t, D=e^{-\Gamma t})$. Admissibility of the product form depends on the choice of parametrisation.

14.9 M10 — Slit experiment, fully computed

Single slit, $a_0=150$ µm, $\lambda=650$ nm, $L=1.00$ m, pixel 1.55 µm, 4096 pixels, read noise 3 $\mathrm{e}^-$, conversion 1 $\mathrm{e}^-$/ADU, peak signal 4000 $\mathrm{e}^-$, pixel integration with 16 nodes.

With this setup the slit width is estimable to about 36 nm, i.e. $2.4\cdot10^{-4}$ of its value. The decisive point for the experimental programme: accuracy is limited by photon statistics and model fidelity, not by pixel size.

For empirical testing of the separability criterion, the dependence on exposure and read noise serves:

At high exposure $F$ is essentially proportional to exposure and independent of read noise — the product form holds. At low exposure the two parameters couple: the drop from $\sigma_r=1$ to 30 is a factor of 8.4 there, but only 1.09 at tenfold exposure. Exactly this mixed dependence is measurable in the laboratory and is the test plan of Section 19.

15 Falsifiability

16 The Vienna lineage and the philosophy of measurability

The framework stands in a tradition that begins in Vienna and leads via operationalism to modern model-based measurement theory. This placement is not decoration: it decides what the criterion may claim.

Position of the framework. It is a model-based, information-metric measurement theory. From Mach it takes economy, from Boltzmann model pluralism, from Carnap gradation. It avoids operationalism because it does not define meaning via operations but only excludes what no channel can distinguish. The classical objection to Mach — atoms were not directly observable and exist nonetheless — does not apply: atoms were identifiable through suitable channels, such as Brownian motion. Non-identifiability is channel-relative, not absolute. That sentence is the philosophical point of the framework and at the same time its safeguard against any positivist overreach.

17 Demarcation from observability theory and neighbouring frameworks

Clean demarcation. Kalman and Hermann–Krener observability answer whether states are distinguishable from noise-free output trajectories of a dynamical system. The present framework asks how much information about $\theta$ each physical stage of a static or quasi-static measurement chain transmits under noise, and whether that transmission factorises stagewise. In the linear-Gaussian case the rank condition and the regularity of the Fisher matrix coincide; this equivalence is cited here as known and not presented as a result.

New relative to all of these are three points: the assignment of stages to physical sub-theories, the criterion for admissibility of the product form, and the treatment of the quantum stage with the distinction between POVM-bound and attainable information.

18 Real-world examples

19 Experimental protocol: optical slit experiment

The protocol is designed to be feasible in a workshop with 3D printing, a laser diode and a camera sensor, without laboratory equipment. It tests two things: the structure-transfer factor $G(\lvert q\rvert a)$ and the separability criterion.

19.1 Setup

19.2 Calibration before measuring

1. Photon-transfer curve per EMVA 1288: determine conversion gain $g$, read noise $\sigma_r$, linearity and full well. The result is the noise model $\sigma^2(\mu)=\mu/g+\sigma_r^2$ that carries every subsequent Fisher computation.
2. Linearity: use RAW only; gamma correction and JPEG processing turn the model into a nonlinear characteristic and violate the criterion in class C.
3. Pixel scale: check against the data sheet and additionally against a grating of known constant.
4. Dark frames at the same temperature and exposure time.
5. System MTF per ISO 12233:2024 — explicitly with LED illumination, not with the laser: the standard presupposes incoherent imaging; under coherent illumination different cut-off frequencies apply and edges show ringing.

19.3 Measurement series

19.4 Evaluation

1. Set up the forward model $\mu_k(a)$ as the pixel integral of $\operatorname{sinc}^2$, not the point formula.
2. Estimate $a$ per frame by maximum likelihood; compare the empirical variance against $1/F(a)$ (Cramér–Rao efficiency). The simulation in 14.9 expects $\sigma_a\approx 36$ nm at $a=150$ µm.
3. From R2 × R3, record the surface $\log F(\ln\lambda, \ln\sigma_r)$ and form the mixed second differences. Expectation per 14.7: near zero in the shot- and read-limited regimes, clearly non-zero in the crossover.
4. From R1, plot the structure-transfer factor as the ratio of measured to ideal contrast over $\lvert q\rvert a$.

19.5 Expected results and falsification

The framework is supported if the mixed derivative differs significantly from zero in the crossover and not in the asymptotes. Theorem 2 would be refuted if stage efficiencies stayed constant across all operating points although the photon-transfer curve shows multi-stage noise injection.

19.6 Laser safety

Anyone who alters or removes the optics of a classified laser module must reclassify and counts as a manufacturer under the standard. Exact limits are to be taken from the standard’s tables, not from secondary sources.

20 Discussion

The utility of the framework lies in systematically separating signal, energy, information, direction, material response and decision. A bright image is not automatically information-rich. A null signal is not automatically absence. A digital reconstruction is not automatically physically unique.

Relative to version 0.2 the contribution shifts a second time. There, $K$ was derived but the validity of the product form was open. Here it is decided for the main channel classes, and the result is less comfortable than the product form suggests: factorisability is not a property of the system but of the operating point. The same camera is separable in bright light and not at dusk. That is not a weakness of the presentation but a statement about measurement chains, and it is measurable.

The mathematical core remains the measurement distribution $p(Y\lvert\theta)$. $K$ is a diagnostic tool for engineering use, not the theoretical core.

21 Limits

First, this is not a new fundamental theory. Second, the core statement is not new but adopted. Third, outside the Gaussian anchor case $K$ is an approximation. Fourth, the classification in Section 5.6 is not exhaustive; it covers the channel classes examined. Fifth, distinguishability in principle does not guarantee practical reconstructability. Sixth, information here means physical state distinguishability, not meaning. Seventh, all ten minimal models are simulations with known forward models; the experiment of Section 19 is planned, not performed. Eighth, the literature search was limited: “not found in the literature” does not mean “does not exist”. Ninth, not every reference has been verified at the work itself; Appendix I flags the open entries individually.

22 Conclusion

Measurability is distinguishable state dependence within a real physical measurement chain. The starting point — $p(Y\lvert\theta_1)=p(Y\lvert\theta_2)$ excludes unique reconstruction — is the observational equivalence of identifiability theory and is adopted here, not claimed. The own contribution is the stage decomposition of a physical measurement chain and the criterion stating when it may be collapsed multiplicatively.

The shortest version: measurability is distinguishable state dependence — and the product form holds exactly as far as its factors do not secretly contain one another. Whether they do is decided by the operating point, not by the instrument.

Appendix A — Symbol table

Appendix B — Validity ranges of central formulas

Appendix C — Reproduction

cd werkzeuge
python3 FORSCH-PROV0420_minimalmodelle_v0.2_2026-08-26_numerik.py   # M1-M6, seed 20260826
python3 FORSCH-PROV0420_minimalmodelle_v0.3_2026-10-05_numerik.py   # M7-M10, seed 20261005

Dependencies: Python 3, numpy, matplotlib. The scripts write ergebnisse.json and ergebnisse_v0.3.json and all figures to werk/abb/. Every number in Section 14 comes from these files.

Appendix D — References

Every entry carrying an identifier (DOI, ISBN, bibcode) was confirmed for this version with the publisher, the registration agency’s record, or the full text. Entries without an identifier are standard textbooks whose edition is still to be fixed; they are listed as open in Appendix I.

• Abbe, E. (1873). Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung [Contributions to the theory of the microscope and microscopic perception]. Archiv für mikroskopische Anatomie, 9(1), 413–468.
• Amari, S. (2001). Information geometry on hierarchy of probability distributions. IEEE Transactions on Information Theory, 47(5), 1701–1711. DOI: 10.1109/18.930911
• Amari, S. (2016). Information Geometry and Its Applications. Springer, Applied Mathematical Sciences 194. DOI: 10.1007/978-4-431-55978-8
• Aspden, R. S., Padgett, M. J., & Spalding, G. C. (2016). Video recording true single-photon double-slit interference. American Journal of Physics, 84(9), 671–677. DOI: 10.1119/1.4955173
• Bellman, R., & Åström, K. J. (1970). On structural identifiability. Mathematical Biosciences, 7(3–4), 329–339. DOI: 10.1016/0025-5564(70)90132-X
• Berreman, D. W. (1972). Optics in stratified and anisotropic media: 4×4-matrix formulation. JOSA, 62(4), 502–510. DOI: 10.1364/JOSA.62.000502
• Bertero, M., & Boccacci, P. (1998). Introduction to Inverse Problems in Imaging. IOP Publishing, Bristol. ISBN 978-0-7503-0435-1.
• Boltzmann, L. (1905). Populäre Schriften. Barth, Leipzig. — picture theory; several models for the same phenomena.
• Born, M., & Wolf, E. Principles of Optics. Cambridge University Press.
• Boyd, R. W. Nonlinear Optics. Academic Press.
• Braunstein, S. L., & Caves, C. M. (1994). Statistical distance and the geometry of quantum states. Physical Review Letters, 72(22), 3439–3443. DOI: 10.1103/PhysRevLett.72.3439
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Appendix E — Errata and version history

Appendix F — What was not found in the literature

The following list belongs to the integrity of the contribution. It states what could not be found after reviewing the strands data-processing inequality, cascaded systems and detective quantum efficiency, quantisation, detector noise, parameter orthogonality, information geometry, observability theory and structural identifiability. The search comprised roughly two dozen targeted queries; it was not exhaustive and does not replace a systematic review. “Not found” does not mean “does not exist”. Every statement in this appendix is qualified accordingly in the text.

Appendix G — Track assignment and commercial use

This work is run platform-centred. The platform syntagma-science is where everything converges; the open research channel feeds it rather than being the destination.

G.1 The two tracks

Tools run exclusively on the platform. No tool is deposited on Zenodo, not even a preview or demonstration build — what is deposited there is the preprint with figures, scripts and raw results. Anyone who wants to compute rather than read comes to the platform.

G.2 Licence levels

Following the licence and platform scheme LIZ-PROV0001.

G.3 Product classes

G.4 What this means for publication

The open preprint is a publication and reach channel, not an end in itself: it substantiates the method openly so that the platform rests on something checkable. Every deposit of the preprint carries the reference to syntagma-science. Both tracks are maintained; neither replaces the other. The identifiers FORSCH-PROV0420 and SYN-PROV0002 are provisional and are to be checked against the register before publication.

G.5 Active and passive versions

The assignment follows the cross-track rule REGEL-PROV0001 (version 1.0 of 6 Oct 2026, text in regelwerk/). It governs all projects, not only this one.

Only the passive version goes on the deposit. The active version runs exclusively on the platform.

The test, in this order: does it have an operating surface that computes on input? If yes, active. If no: is it read, viewed, or run once for verification? If yes, passive.

The computation scripts of Appendix C are passive and go on the deposit. A fixed-seed script that regenerates the numbers of Section 14 is reproduction material, not an application. Without it this work could not be checked, and checkability is the reason for depositing at all. Anyone who wants to verify the numbers here does not need MESSWERK; the scripts suffice. MESSWERK removes the setup work and computes on someone else’s measurement chain rather than on the examples of this work — that is the difference the licence rests on.

Never on the deposit: preview, demonstration and reduced builds of the tool, the tool’s source including its surface, offering and pricing material, and internal files. Either the full tool on the platform or none at all.

G.6 Self-protection and the disclosure line

The line between what lies open and what stays with the author follows the cross-track rule REGEL-PROV0002 (version 1.0 of 6 Oct 2026, text in regelwerk/). It is named here because this work sits beside a paid offering, and the question would otherwise go unstated.

The line runs at the subject, not the format: the method this work claims and substantiates is public. Without evidence it would be advertising, not research. The product sold alongside it is protected.

Checkability outranks protection. Where a detail is needed to keep the claim refutable, it is published, even where a third party benefits. A framework that cannot be refuted supports no offering.

A browser tool, deliberately open. MESSWERK runs as a single HTML file in the browser and therefore ships its computing core to everyone who opens the page. “In the browser” and “core source internal” exclude each other; REGEL-PROV0002 accordingly requires a deliberate choice per tool. For MESSWERK the choice has been made and is recorded in syntagma-science/ANGEBOT.md: the universal edition stays open on the platform page. The licence then carries not the computing core but maintenance and version tracking, own branding and multi-seat operation, training, the measurement-chain audit and design against a required accuracy. This is not a fallback but the decision to weigh the funnel higher than the protection of a few hundred lines of computation code that are derivable from this work anyway.

Appendix H — Publication path

H.1 Order

Platform first, deposit second, journals last.

H.2 The Zenodo bundle is prepared separately

The working bundle and the publication bundle are not the same thing. The folder zenodo-upload/ holds exactly what belongs on the deposit and nothing else: preprint in four formats per language, figures, computation scripts, raw results, licence, metadata record, read-me. It does not hold tool builds, offering and pricing material, internal handovers or checklists. The bundle’s metadata name the platform as the related identifier.

H.3 Journals and repositories

H.4 Recommended split

A single work covering mathematics, experiment and philosophy of science fits no single journal. Splitting it into three submissions is more workable, with the bilingual monograph as the common, openly accessible deposit:

Appendix I — Source checklist

Status verified means the entry was confirmed for this version against the publisher’s record, the standards register, the full text, or the registration agency’s entry. open means the entry is still not individually substantiated and is to be confirmed before a journal version. The check was carried out on 6 Oct 2026.

I.1 Verified

I.2 Still open

I.3 What the check changed in the work