What the Postulates Do Not Say

The standard postulates specify how quantum states, transformations, measurements, and composite systems fit together. Given a physical model, they produce probabilities and conditional state assignments with extraordinary accuracy. They are not, however, a microscopic theory of every detector, an algorithm for discovering Hamiltonians, or a unique account of what exists between preparation and observation.

Those limits should not be turned into either dismissal or mystification. The postulates say a great deal, and their open boundaries are different kinds of questions:

• a modeling question asks which Hilbert space, Hamiltonian, environment, or detector describes an experiment;
• a foundational question asks how the formal rules relate to definite events, probability, and physical reality;
• an interpretive proposal adds an explanatory reading or ontology;
• a modified theory changes the dynamics or probability rules and can differ empirically.

This page classifies those layers. Minimal Postulates owns the compact rules, and Assumptions and Scope owns their physical regime.

What the Postulates Do Say

It helps to begin positively. Suppose a preparation gives a density operator $\rho$, an unconditional process is represented by a channel $\mathcal E$, and a later measurement is represented by an instrument $\{\mathcal I_a\}$. The unnormalized outcome branch is

$$\widetilde\rho_a = \mathcal I_a\!\left( \mathcal E(\rho) \right).$$

The postulate package determines

$$p(a) = \operatorname{Tr} \widetilde\rho_a,$$

and, when $p(a)>0$, the conditional output state

$$\rho_a = \frac{ \widetilde\rho_a }{ p(a) }.$$

It also constrains valid states and operations through positivity, normalization, complete positivity, tensor-product composition, and probability consistency. In a closed-system pure-state model, these rules reduce to unitary evolution and the vector Born rule.

What the postulates do not provide in this example is equally specific:

• which physical preparation produces $\rho$;
• which interactions justify $\mathcal E$;
• which device realizes $\mathcal I_a$;
• why outcome $a$ occurs in one individual run;
• what ontological status $\rho$ has;
• whether an approximation used to construct any of these objects is valid.

The mathematical prediction can be complete relative to the model while the modeling or interpretive account remains incomplete.

Three Levels of Question

Level	Typical question	What can answer it
Formal	Given $\rho$ and $E_a$, what is $p(a)$?	Born or trace rule
Modeling	Which $E_a$ describes this detector at this energy and threshold?	apparatus physics, calibration, and measurement theory
Foundational	What makes one record actual in an individual run?	an interpretation, additional ontology, modified dynamics, or a reframing of the question

The levels interact but should not be substituted for one another. Better detector calibration does not by itself select an interpretation. An interpretation does not calculate a missing coupling constant. A valid probability formula does not establish that a laboratory device realizes the assumed POVM.

They Do Not Uniquely Select an Interpretation

The operational core can be used by frameworks that disagree about the meaning of the quantum state, the role of probability, the status of state-update, or the ontology of measurement records. The common probability formula

$$p(a) = \operatorname{Tr}(\rho E_a)$$

does not, by itself, say whether $\rho$ is:

• a complete physical state;
• an incomplete state supplemented by further variables;
• a branch-relative object;
• an agent- or information-relative state assignment;
• a law-like or nomological object;
• one representation of a deeper structure.

These options are not interchangeable philosophical adjectives. A developed interpretation must explain which mathematical objects represent physical structure, what constitutes an event or record, how probabilities are understood, and why the ordinary rules are recovered.

Interpretation versus modified theory

If two proposals map the same preparations and measurements to the same probabilities throughout a stated domain, they are operationally equivalent there even if their ontologies differ. That does not make their explanatory claims identical or unimportant.

If a proposal adds a fundamental term to the dynamics,

$$\frac{d\rho}{dt} = -\frac{i}{\hbar}[H,\rho] +\mathcal C(\rho),$$

and $\mathcal C$ produces observable deviations from unitary quantum mechanics, the proposal is not merely a verbal interpretation of the same theory. It is a modified theory whose parameters can in principle be tested. An equation of the same mathematical form can also appear as an effective open-system equation; whether $\mathcal C$ is fundamental or reduced dynamics is part of the model.

Category boundaries in the literature are not always uniform. The reliable test is to ask which states, dynamics, probability rules, and empirical predictions have changed.

No-go theorems constrain, but do not choose alone

Bell, Kochen–Specker, and later ontology theorems rule out or constrain specific classes of underlying models under stated assumptions. They narrow the interpretive landscape; they do not convert the bare postulates into one unique ontology.

For example, the Pusey–Barrett–Rudolph theorem constrains a class of $\psi$-epistemic ontological models under a preparation-independence assumption. Its conclusion is stronger than “nothing can be learned about the wavefunction,” but weaker than “one unique form of wavefunction realism has been proved.” Assumptions and the meaning of the ontic state remain part of the theorem’s scope.

They Do Not by Themselves Model Detectors

A POVM or instrument is an operational model of a measurement. It is not yet a microscopic account of the apparatus that realizes it.

Let a system $S$ interact with an apparatus $A$. The apparatus begins in $\sigma_A$, evolves jointly with the system through $U_{SA}$, and is read by pointer projectors $\{Q_a\}$. The joint state before readout is

$$\rho'_{SA} = U_{SA} \left( \rho_S\otimes\sigma_A \right) U_{SA}^\dagger.$$

The pointer probability is

$$p(a) = \operatorname{Tr}_{SA} \left[ \rho'_{SA} \left( I_S\otimes Q_a \right) \right].$$

Because this expression is linear and positive in $\rho_S$, it can be written as

$$p(a) = \operatorname{Tr}_S(\rho_S E_a)$$

for an induced system effect $E_a$. The corresponding outcome operation in this ideal pointer-readout model is

$$\begin{aligned} \mathcal I_a(\rho_S) ={}& \operatorname{Tr}_A \left[ \left( I_S\otimes Q_a \right) \rho'_{SA}\right.\\ &\left.\qquad{}\times \left( I_S\otimes Q_a \right) \right]. \end{aligned}$$

The postulates tell us how to use $E_a$ or $\mathcal I_a$. The apparatus model must supply:

• the active detector degrees of freedom;
• the interaction Hamiltonian and interaction time;
• the ready state $\sigma_A$;
• the pointer observable and readout electronics;
• losses, dark counts, thresholds, dead time, and finite resolution;
• amplification and coupling to uncontrolled environments;
• the calibration mapping from raw records to outcome labels.

Different apparatus models can induce the same POVM but different state disturbance. Conversely, a nominally identical device can induce different effective POVMs in different energy, timing, or noise regimes.

A detector click is not a projector

A projector is a mathematical effect for an ideal sharp event. A detector click is a macroscopic record produced by a device. Connecting the two requires a model and calibration.

For an imperfect two-outcome readout, the effects might be

$$\begin{aligned} E_+ &= (1-\epsilon)P_+ +\epsilon P_-, \\ E_- &= \epsilon P_+ +(1-\epsilon)P_-. \end{aligned}$$

The error parameter $\epsilon$ is not supplied by the Born rule. It must be measured, derived, or bounded from apparatus physics.

Use Generalized Measurements and Instruments for the operational theory and von Neumann Measurement Model for the canonical system–pointer construction.

They Do Not Explain Why a Particular Outcome Occurs

The Born rule gives a probability distribution over possible records. A selective update gives the state to use conditional on a record. Neither statement, by itself, supplies a dynamical account of why one allowed record is the one obtained in an individual run.

Consider an ideal premeasurement interaction:

$$\lvert s_a\rangle \lvert A_0\rangle \longmapsto \lvert s_a\rangle \lvert A_a\rangle.$$

Linearity implies

$$\left( \sum_a c_a\lvert s_a\rangle \right) \lvert A_0\rangle \longmapsto \sum_a c_a \lvert s_a\rangle \lvert A_a\rangle.$$

The right-hand side correlates system alternatives with pointer alternatives. It does not contain one selected term unless the input already had only one nonzero coefficient or an additional nonunitary rule is invoked.

What decoherence changes

Include an environment:

$$\lvert\Psi\rangle = \sum_a c_a \lvert s_a\rangle \lvert A_a\rangle \lvert E_a\rangle.$$

Tracing out the environment gives

$$\begin{aligned} \rho_{SA} ={}& \sum_{a,b} c_a c_b^* \langle E_b\mid E_a\rangle\\ &\quad{}\times \lvert s_a,A_a\rangle \langle s_b,A_b\rvert. \end{aligned}$$

When environmental records are nearly orthogonal,

$$\langle E_b\mid E_a\rangle \approx \delta_{ab},$$

the reduced state is approximately

$$\rho_{SA} \approx \sum_a \lvert c_a\rvert^2 \lvert s_a,A_a\rangle \langle s_a,A_a\rvert.$$

This explains why interference between macroscopically distinct records is locally inaccessible and why a stable record basis emerges. The reduced state supports the same local statistics as a classical mixture, but it is an improper mixture obtained from an entangled state. The unitary global state has not, by this calculation alone, been replaced by one term.

Decoherence is therefore crucial physical dynamics, not a cosmetic addition. It addresses interference suppression and preferred robust records. On its own, it does not assert which record is actual in one run or specify how to interpret the resulting branches. Decoherence and the Classical Transition owns the technical mechanism.

Three related foundational questions

Question	Compact statement
Outcome problem	Why is one definite record obtained rather than a superposition of records?
Preferred-structure problem	Which alternatives become stable, record-like, and effectively classical?
Probability problem	Why should branch weights or state amplitudes be interpreted through the Born rule?

Decoherence makes major progress on preferred structure and on the stability of records. An interpretation or modified theory must still state how it handles the outcome and probability questions.

Conditional update starts after the condition

For an instrument branch,

$$\rho_a = \frac{ \mathcal I_a(\rho) }{ \operatorname{Tr}\mathcal I_a(\rho) }.$$

This formula answers:

Given that outcome $a$ is the record available for later predictions, what state should be used?

It does not answer:

Why was $a$, rather than another possible outcome, the record in this run?

Confusing the two questions makes a conditional inference look like a mechanism of event selection.

They Do Not Remove the Need for Hamiltonian Modeling

The dynamics postulate says how a specified Hamiltonian generates evolution. It does not identify the correct Hamiltonian for an unknown physical system.

A model may be organized as

$$H = \sum_j g_j O_j,$$

where the operators $O_j$ represent allowed interactions and the coefficients $g_j$ encode masses, charges, fields, couplings, control amplitudes, or effective parameters. The postulates do not determine:

• which degrees of freedom are retained;
• which operators $O_j$ are allowed by geometry and symmetry;
• the numerical values and scale dependence of $g_j$;
• which terms have been integrated out or neglected;
• the operator domains and boundary conditions;
• whether the system boundary supports a unitary model;
• whether parameters are known independently or inferred from the same data being predicted.

Symmetry can strongly constrain a Hamiltonian, and microscopic theory can derive effective terms. Neither removes the need to state the physical model.

The inverse problem is not automatic

Observed spectra or time traces do not always identify a unique Hamiltonian. Different parameter combinations, hidden degrees of freedom, gauges, effective models, or noise processes can reproduce the same limited data. Hamiltonian learning is an inference problem with identifiability assumptions, not an inversion performed by the postulates.

The system decomposition is also modeled

Writing

$$\mathcal H = \mathcal H_S\otimes\mathcal H_E$$

declares which degrees of freedom count as system and environment. That split determines which partial trace, channel, entanglement, and decoherence statements are meaningful. The abstract tensor-product rule does not uniquely choose the experimentally useful decomposition.

Hamiltonians owns their role as generators and physical model components. Assumptions and Scope owns regime and truncation diagnostics.

They Do Not Specify the Ontology of the Wavefunction

The formal statement is precise:

$$\psi(x) = \langle x\mid\psi\rangle$$

is the position representation of an abstract state. A momentum wavefunction

$$\widetilde\psi(p) = \langle p\mid\psi\rangle$$

represents the same state in a different basis. Global phase is redundant, while relative phase affects interference.

For $N$ particles, a position representation has the form

$$\Psi( \mathbf x_1,\ldots,\mathbf x_N ),$$

which is a function on configuration space. These facts constrain any ontological account, but they do not alone decide whether the quantum state is a physical field, information about deeper variables, a law-like object, a branching structure, or something else.

Representation facts are not ontology

The position-space function changes under a basis transformation while physical predictions remain invariant. Therefore a claim that $\psi(\mathbf x)$ is literally a field in ordinary three-dimensional space needs more content than the coordinate formula. Conversely, calling the wavefunction “only information” does not explain whose information, about what underlying possibilities, or why interference follows the Hilbert-space rules.

The density operator creates a related distinction. A reduced density operator can be mixed because of entanglement even when the global state is pure. Treating every mixed state as classical ignorance about a hidden pure state can therefore misrepresent its composition and correlations.

Tomography does not settle ontology

Quantum-state tomography estimates the state that best accounts for measured statistics within a chosen model. It demonstrates that the state has reconstructible operational content. It does not, without additional argument, decide the state’s ultimate metaphysical category.

Likewise, no-go theorems should be stated with their assumptions. PBR-type results constrain overlap between ontic-state distributions for distinct quantum states under preparation independence. They are relevant to wavefunction ontology, but they do not replace an interpretation comparison.

Wavefunctions as Representations owns the coordinate mathematics. The ontology question belongs to foundations once those formal facts are fixed.

They Do Not Make Approximation Choices Automatic

Perturbation theory, semiclassical fields, two-level truncations, Born–Oppenheimer separation, rotating-wave approximations, Markov limits, continuum limits, and detector coarse-graining are not consequences of the bare postulates.

The formalism can calculate with an effective model

$$H_{\mathrm{eff}} = P H P+\Delta H,$$

but it does not choose the projector $P$, derive every correction $\Delta H$, or certify an error tolerance. Those require scale separation, physical input, convergence tests, and comparison with data.

An approximation can also change the apparent conceptual problem. Tracing out an environment produces irreversible reduced dynamics from reversible global dynamics. Treating a pointer as classical suppresses the very system–apparatus superposition that motivates the measurement problem. Approximations may be excellent for prediction while being inappropriate as premises in an ontological argument.

The validation workflow belongs to Assumptions and Scope and to the canonical method page for the approximation being used.

Worked Example: Layers in a Stern–Gerlach Experiment

Consider a spin-$1/2$ particle with spatial wavepacket $\lvert\varphi\rangle$ and spin state

$$\lvert\chi\rangle = \alpha\lvert+z\rangle +\beta\lvert-z\rangle.$$

A Stern–Gerlach magnet does not measure spin by applying the Born rule directly to an abstract two-level system. A physical model includes spatial motion and an inhomogeneous magnetic field, for example

$$H = \frac{\mathbf p^2}{2m} -\boldsymbol\mu\cdot \mathbf B(\mathbf r).$$

Under an idealized interaction, the outgoing state becomes

$$\begin{aligned} \lvert\varphi\rangle \lvert\chi\rangle \longmapsto{}& \alpha \lvert\varphi_+\rangle \lvert+z\rangle\\ &+ \beta \lvert\varphi_-\rangle \lvert-z\rangle, \end{aligned}$$

where the spatial packets separate. A downstream detector turns the two paths into records.

What each layer contributes

1. Postulates: given effects $E_+$ and $E_-$, compute record probabilities.
2. Hamiltonian model: derive whether the field gradient separates the packets and estimate their overlap.
3. Detector model: account for acceptance, efficiency, path misclassification, dark counts, and state disturbance.
4. Approximation analysis: justify the nonrelativistic treatment, prescribed field, paraxial motion, and neglect of other spin couplings.
5. Foundations: explain how the final quantum description relates to one definite record and what the quantum state represents.

For a simple symmetric readout error $\epsilon$,

$$\begin{aligned} E_+ &= (1-\epsilon)P_+ +\epsilon P_-, \\ E_- &= \epsilon P_+ +(1-\epsilon)P_-. \end{aligned}$$

If the separated packets are otherwise ideal, the positive-record probability is

$$p(+) = (1-\epsilon) \lvert\alpha\rvert^2 +\epsilon \lvert\beta\rvert^2.$$

The postulates evaluate this formula. They do not provide $\epsilon$, derive the magnet geometry, prove the packet-separation approximation, or explain why the positive rather than negative record occurs in one trial.

This layered account is not a weakness. It is how mature physical theories connect general structure to specific experiments.

Labeling Claims Correctly

Claim type	Example	What supports it
Definition	a POVM is a normalized family of positive effects	mathematical convention and consistency
Theorem	unitary evolution preserves inner products	proof from stated assumptions
Model assumption	the atom is an isolated two-level system	physical argument and regime
Approximation	off-resonant levels contribute below one percent	expansion, bound, convergence, or data
Calibration claim	detector efficiency is $\eta$	experimental procedure and uncertainty analysis
Interpretive claim	the quantum state is ontic, epistemic, relational, or law-like	a developed foundations framework
Modified-theory claim	fundamental stochastic dynamics replaces exact unitarity	explicit equations and experimental bounds

Calling every item a “postulate” hides where evidence and judgment enter. Calling every unresolved item “mere philosophy” hides real logical and empirical distinctions.

Where to Study These Questions

Question	Canonical direction
What are the standard operational rules?	Minimal Postulates, Born Rule, and State Update Rule
How do apparatus interactions induce measurements?	von Neumann Measurement Model
How are noisy probabilities and disturbances represented?	Generalized Measurements and Instruments
What does environmental decoherence explain?	Decoherence and the Classical Transition
How did the measurement problem arise?	Measurement Problem as Historical Problem
Which Hamiltonian and approximation are appropriate?	the relevant physical volume, Hamiltonians, and Assumptions and Scope
Which interpretation or ontology should be adopted?	the dedicated Foundations and Interpretations treatment

Ordinary calculations do not need to wait for unanimous interpretation. Foundations work, in turn, must respect the operational, mathematical, and experimental details rather than replacing them with slogans.

Common Mistakes

• Treating a preferred interpretation as one of the standard postulates.
• Saying all interpretations are identical without checking whether dynamics or predictions change.
• Saying a no-go theorem proves a unique ontology while omitting its assumptions.
• Treating the Born rule as a microscopic detector model.
• Identifying a detector click with a projector without an apparatus model.
• Using a selective update as an explanation of why its conditioning event occurred.
• Saying decoherence is collapse or that it has no foundational relevance.
• Treating an improper reduced mixture as ordinary ignorance about a pre-existing local pure state.
• Assuming symmetry or quantization uniquely determines every Hamiltonian coefficient.
• Treating a reconstructed wavefunction as proof of one particular ontology.
• Using a classical pointer approximation to argue that unitary dynamics automatically produced a single outcome.
• Forgetting that successful approximation and complete explanation are different standards.

References

• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995.
• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.
• J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, 2nd ed., Cambridge University Press, 2004.
• P. Busch, P. Lahti, J.-P. Pellonpää, and K. Ylinen, Quantum Measurement, Springer, 2016.
• M. Schlosshauer, “Decoherence, the Measurement Problem, and Interpretations of Quantum Mechanics,” Reviews of Modern Physics 76, 1267–1305 (2005).
• T. Maudlin, “Three Measurement Problems,” Topoi 14, 7–15 (1995).
• M. F. Pusey, J. Barrett, and T. Rudolph, “On the Reality of the Quantum State,” Nature Physics 8 (2012), DOI: 10.1038/nphys2309.
• D. Wallace, The Emergent Multiverse: Quantum Theory According to the Everett Interpretation, Oxford University Press, 2012.
• T. Norsen, Foundations of Quantum Mechanics: An Exploration of the Physical Meaning of Quantum Theory, Springer, 2017.

Exercises

1. Classify each statement as a postulate-level rule, model assumption, approximation, calibration claim, interpretive claim, or modified-theory claim:

   • $p(a)=\operatorname{Tr}(\rho E_a)$;
   • the detector efficiency is $94\%$;
   • the atom is accurately two-level for the next $10\,\mu\mathrm{s}$;
   • the quantum state is a complete physical state;
   • a fundamental stochastic term modifies Schrödinger evolution;
   • counter-rotating terms are negligible.

Solution

The trace rule is a postulate-level probability rule once $\rho$ and $E_a$ are specified. The detector efficiency is a calibration claim. The two-level description is a model assumption with a stated time regime. The completeness of the quantum state is an interpretive claim. A fundamental stochastic term is a modified-theory claim because it changes the dynamics. Neglecting counter-rotating terms is an approximation that requires a scale and timescale analysis.

2. In the system–apparatus model, explain why the pointer probabilities define a POVM on the system even before writing an explicit formula for each $E_a$.

Solution

For fixed $\sigma_A$, $U_{SA}$, and pointer projectors $Q_a$, the map

$$\rho_S \longmapsto p(a)$$

is linear because tensoring, unitary conjugation, multiplication, and the trace are linear. It is nonnegative for every positive $\rho_S$ because it is the probability of a positive pointer effect.

In finite dimension, trace duality therefore gives a unique operator $E_a$ such that

$$p(a) = \operatorname{Tr}_S(\rho_S E_a).$$

Positivity of the functional implies $E_a\geq0$. Since $\sum_aQ_a=I_A$, the probabilities sum to one for every normalized state, so

$$\sum_aE_a = I_S.$$

Thus the induced operators form a POVM.

3. For a two-outcome ideal premeasurement,

$$\begin{aligned} & \left( \alpha\lvert0\rangle +\beta\lvert1\rangle \right) \lvert A_0\rangle \\ &\quad\longmapsto \alpha\lvert0\rangle\lvert A_0'\rangle \\ &\qquad{}+ \beta\lvert1\rangle\lvert A_1'\rangle, \end{aligned}$$

derive the reduced system state when the pointer states are orthogonal. Why is this not automatically a proper preparation mixture?

Solution

The joint density operator contains diagonal terms and cross terms. Taking the pointer trace multiplies the cross terms by $\langle A_1'\mid A_0'\rangle$ and its conjugate. Orthogonality removes them, giving

$$\rho_S = \lvert\alpha\rvert^2 \lvert0\rangle\langle0\rvert + \lvert\beta\rvert^2 \lvert1\rangle\langle1\rvert.$$

This reduced state is an improper mixture: it arises because $S$ is entangled with the pointer. The global state remains pure. A proper preparation mixture would describe classical randomization among locally prepared states. The two have the same statistics on $S$ alone but different global correlations.

4. Suppose decoherence makes $\langle E_1\mid E_0\rangle$ negligibly small in a measurement model. State two questions this answers and one it does not answer by itself.

Solution

It answers why interference between the two records is negligible for observers without coherent access to the environment, and it helps identify a stable record structure selected by the interaction.

By itself, it does not state why one record rather than the other is actual in an individual run. The global unitary state can still contain both correlated terms.

5. A spin-only Hamiltonian

$$H_{\mathrm{spin}} = -\boldsymbol\mu\cdot\mathbf B$$

is proposed as a complete model of a Stern–Gerlach beam splitter. What is missing?

Solution

A beam splitter requires spatial motion and a field gradient. The model needs at least a motional Hilbert space and a Hamiltonian such as

$$H = \frac{\mathbf p^2}{2m} -\boldsymbol\mu\cdot \mathbf B(\mathbf r).$$

It also needs an initial wavepacket, magnet geometry, interaction time, free propagation, and a detector model. A uniform-field spin Hamiltonian can describe spin precession but cannot by itself predict spatial beam separation.

6. Why does the existence of both $\psi(x)$ and $\widetilde\psi(p)$ caution against identifying one coordinate wavefunction with ontology without further argument?

Solution

Both functions are representations of the same abstract state:

$$\psi(x) = \langle x\mid\psi\rangle, \qquad \widetilde\psi(p) = \langle p\mid\psi\rangle.$$

They are related by a unitary Fourier transform and give the same physical predictions when observables are translated consistently. A literal ontological reading of one representation must therefore explain why that representation is privileged or how the invariant abstract structure is realized. The formalism alone supplies no such privilege.

7. A theorem rules out a class of $\psi$-epistemic models under preparation independence. Why is “the theorem proves the wavefunction is the unique real physical field” too strong?

Solution

The conclusion is conditional on the theorem’s framework and assumptions, including how independently prepared systems are represented ontically. It rules out or constrains a specified class of overlapping ontic-state models.

It does not select one unique ontology from every remaining possibility, prove that the coordinate-space function is a literal field, or decide among all interpretations that treat the quantum state as ontic, relational, law-like, or otherwise structurally significant. The correct conclusion must match the scope of the theorem.

8. A proposed collapse model agrees with ordinary quantum mechanics for microscopic systems but predicts mass-dependent loss of interference for large superpositions. Is it merely an interpretation? What would make the claim scientific?

Solution

Because the model changes the dynamics and predicts deviations in principle, it is a modified physical theory rather than merely a reinterpretation of unchanged operational rules.

The claim becomes testable when the modified equation, parameters, scaling with mass or size, noise assumptions, and experimental observable are stated. Interference, heating, spontaneous-radiation, or mechanical-superposition experiments can then constrain the parameter region. Agreement in a limited microscopic regime does not make the theories globally identical.