How physics proofs work

A physics proof establishes what follows from stated assumptions. Experiments test how well those assumptions and their predictions describe nature. A complete deduction can remain valid even when its physical model fails in a particular regime.

Four kinds of physics item

Postulate

An adopted physical assumption, with a stated formulation and scope. The Born rule, for example, assigns detection probabilities from a quantum state. Experiments can support it, but agreement with its predictions supplies no deductive proof that it holds universally. Postulates need clear explanations and checked sources; they have no fabricated proof section.

An assumption can belong to a whole framework or to a restricted model. An ideal spring's force law assumes force is proportional to displacement. Its useful range must be stated; it makes no claim about every spring at every extension.

Physics theorem

A precise conclusion with a complete proof from mathematical material and explicit physical premises. The standard of deduction is the same as in mathematics. The physical assumptions and interpretation determine its scope.

For an ideal spring, let x be displacement, m a constant positive mass, k a positive spring constant, and dots time derivatives. Assume Newton's equation mẍ = −kx. The energy E = mẋ²/2 + kx²/2 satisfies Ė = ẋ(mẍ + kx) = 0 by the chain rule. Energy conservation follows exactly within this model. Applying it to a real spring requires the model's assumptions, including negligible damping.

Experiment

A record of an actual reported experiment: preparation, apparatus, procedure, observations, uncertainty and interpretation, with sources. Mathematics justifies the analysis; assumptions about the apparatus justify how readings are interpreted. The measured outcome must come from observations.

Timing a spring's oscillations can estimate its period. A source must report enough about timing, calibration and uncertainty to support the claimed estimate. A calculated period or a simulated trace supplies a prediction, not a new measurement.

Thought experiment

An idealized setup with a precise conclusion and a complete conditional argument. In this library it has the same proof requirements and dependency permissions as a physics theorem. Its hypothetical setup gives it a separate name.

Imagine two identical ideal springs carrying masses m and 4m. In the undamped model, substituting x(t) = A cos(ωt + φ) into mẍ = −kx gives ω² = k/m for nonzero amplitude A. Here ω is angular frequency and φ is phase. The period T = 2π/ω therefore doubles for the heavier mass. This deduction supplies a comparison to test; imagining it supplies no measured periods. An assumed future measurement must be labelled hypothetical.

What a dependency means

A prerequisite supplies something an item needs: a definition, an assumption, an earlier result, a measurement procedure or an empirical premise. Each dependency must say what it supplies. A source that is merely relevant belongs in a citation or contextual link.

Read each row as supplier → consumer. “Allowed” requires a specific necessary role and an acyclic chain.
Supplier ↓ / Consumer →	Mathematics	Postulate	Physics theorem	Thought experiment	Experiment
Mathematics	Allowed	Allowed	Allowed	Allowed	Allowed
Postulate	Excluded	Allowed	Allowed	Allowed	Allowed
Physics theorem	Excluded	Allowed	Allowed	Allowed	Allowed
Thought experiment	Excluded	Allowed	Allowed	Allowed	Allowed
Experiment	Excluded	Allowed	Allowed	Allowed	Allowed

Mathematics supplies language and proof tools

Mathematical axioms, definitions, theorems, lemmas, propositions and corollaries retain their ordinary roles. Calculus supplies the spring proof. Vector spaces and inner products supply language for quantum postulates. Probability and statistics supply experimental analysis.

Physics definitions, examples and remarks follow the same boundary: they may use mathematics. Mathematical items cannot depend on physical assumptions, results or observations, even indirectly. A mathematical theorem about an equation proves a claim under mathematical hypotheses; applying that equation to nature introduces a physical premise. Shared mathematics keeps its canonical identity.

Physics prerequisites have different roles

Postulates supply physical assumptions for theorems and thought experiments, and apparatus assumptions for experiments. One postulate may also need another to formulate its terms.
Physics theorems and thought experiments supply established conditional results to either proof class. They can also supply predictions or apparatus analysis to experiments, or context needed to formulate a postulate. Their assumptions travel with every use.
Experiments can supply calibration or preparation to later experiments, operational definitions to postulates, and qualified empirical premises to either proof class. A calibration procedure needed to state a postulate does not prove that postulate.

For example, a theorem may turn a measured length interval into a calculated volume interval under an assumed geometry. The deduction must retain the measurement conditions and uncertainty. Exact algebra cannot make the input exact. Draft or planned mathematics must retain its stated proof status; a needed but unfinished supplier leaves a conditional reliance or an open obligation.

Evidence links are separate

“Supports,” “tests,” “challenges,” “replicates” and “motivates” record relationships outside the prerequisite graph. A postulate can guide an experiment while the experiment supports the postulate. Those links can form a cycle. Prerequisites cannot: an argument cannot rely on its own conclusion through a chain of earlier items.

An experiment that tests a prediction is linked as evidence. It becomes a proof prerequisite only when a specific observed result is used as a premise, with its conditions, statistical assumptions and uncertainty carried into later conclusions.

Double slit: prediction, observation, inference

In a coherent two-path experiment, individual detections are localized. The accumulated distribution can show interference. Let a and b be the complex amplitudes for the two paths at a detector position, including preparation weights. Under the Born rule, the coherent model predicts a density proportional to |a + b|².

|a + b|² = |a|² + |b|² + 2 Re(a b̄)

Re means real part; b̄ is the complex conjugate of b. The last term produces interference.

A simple independent-particle path model instead adds the two path probabilities, giving a density proportional to |a|² + |b|². This comparison assumes that opening both paths leaves the separate path distributions and apparatus behavior otherwise unchanged. Both densities must be normalized over the detection region.

Controlled data can challenge that specified classical particle model and support the quantum prediction. They cannot logically prove all quantum postulates or invalidate all classical mechanics at every microscopic scale. Classical waves also interfere, so fringes alone do not rule out classical electromagnetism. The experiment tests a model together with its preparation and apparatus assumptions.

Finite samples fluctuate. An unclear histogram need not contradict the coherent model. Loss of coherence, phase drift, background and detector resolution can also reduce visibility. Even an ideal density of zero at one point does not imply zero probability for a detector bin of finite width; the model must predict the probability across the bin. Statistical rejection requires a stated test. A p-value is not the probability that a theory is false.

How to read a claim

Ask which framework is assumed, what is defined, what is deduced, and what was measured. Check the units, mathematical types and regime of validity. Equal units do not make two quantities identical across theories: a classical position and a quantum position operator need an explicit correspondence. Review of a proof concerns its argument; review of an experiment concerns faithful reporting and qualified inference. Neither guarantees a model's universal applicability.