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Mathematical Models and Physical Theories

Why the physical Lorentz claim is a statement about real numbers, not relativity verified: separating mathematical results, modeling assumptions, and experimental evidence.

The case-study claim is named lorentz-interval-invariance-1d-physical. The word “physical” in that name is a trap. What has been proved about it, in two kernels, is a statement about real numbers:

forall t x v gamma in R, |v| < 1 -> gamma = 1/sqrt(1-v^2)
  -> (gamma*(t - v*x))^2 - (gamma*(x - v*t))^2 = t^2 - x^2

Nothing in that sentence mentions a clock, a rod or a light signal. It is true or false of the real numbers whether or not the universe has any Lorentz symmetry. A reader who sees “physical theorem verified” will hear “relativity verified”. The claim’s own limitations say otherwise: it “remains a statement about the mathematical model; it is not empirical validation of relativity.”

This chapter explains why the engine keeps three kinds of result apart, and what that means for a claim with “physical” in its name. It builds on Chapters 1 to 3.

Three kinds of result

The evidence policy states the rule:

A mathematical inconsistency, a disputed modeling assumption, and an experimental disagreement are three distinct result types. Reports and book chapters must keep them distinct.

Result type Example What would change it
Mathematical The boost preserves $t^2-x^2$ for real inputs A kernel rejection or a validated counterexample
Modeling Spacetime is modeled by a 1+1-dimensional quadratic form with $c=1$ A different model, such as a different metric signature or a non-flat geometry
Empirical Observed clocks and signals obey the Lorentz transformation to some precision Data and an uncertainty analysis

The engine can currently produce evidence of the first kind only.

The mathematics and its physical shadow

The mathematical content is the expansion from Chapter 20:

$$ t'^2 - x'^2 = \gamma^2(1-v^2)\,(t^2 - x^2) = t^2 - x^2 . $$
Its role in physics is that $t^2-x^2$ is the squared interval of the quadratic form assumed to describe spacetime. "The interval is invariant under boosts" is the mathematical shadow of "the laws of physics take the same form in all inertial frames". The shadow follows from the model. Whether the model fits nature is a separate question.

The same care applies to the case study’s false control. The transform $t'=t$, $x'=x+vt$ is a Galilean-style shear, and it fails the interval test with a witness. That is a mathematical result about the shear. It is not a statement that Galilean physics is wrong.

Where the model’s assumptions live

The physical claim records four assumptions: a real domain, subluminal speed, the gamma definition and $c=1$. Each cites einstein-1905 as its source. These are the model’s assumptions.

The postulates of special relativity, the principle of relativity and the constancy of the speed of light, are a further layer. They are what would justify choosing this model, and they are empirical premises. They get their own chapter (Chapter 18), and no claim in the project formalizes them.

The model is also deliberately small: one space dimension and one time dimension. Rotations, three spatial dimensions and the Lorentz group are later work (the MATH04 milestone), not covered in this edition.

The empirical statuses are reserved, not used

The status vocabulary (EvidenceStatus) includes empirically_supported and empirically_challenged. A search of src/ finds them defined in models.py and used nowhere else. No adapter produces them, no dataset is ingested, and the empirical milestone (MATH06) has not started. They are reserved vocabulary, so that the evidence policy can say what an empirical result would require: “versioned observations agree with a stated prediction and uncertainty model.” Until then, every status in a MATH01 report is mathematical.

The policy also states what each empirical status would not imply. empirically_supported does not imply “mathematical consistency or final truth”, and empirically_challenged does not say “which mathematical or physical assumption failed”. The second is the Duhem–Quine point in one line: a disagreement with data blames a bundle of assumptions, not one of them.

What this establishes

The accepted report’s summary of the physical claim at version 1.0.0 is computation_verified_within_stated_domain: SymPy, within the stated domain. The repository README’s “Current boundary” section says the same of the R4 revision: the composed theorems are kernel-accepted, and “cross-kernel semantic equivalence is not established.” The report says nothing about relativity as a physical theory, and this chapter does not manufacture such a statement.

What the engine can honestly say today: the model is internally consistent in the specific identities checked, and one tempting alternative, the shear, does not preserve the interval.

What it cannot say: that the model describes nature, that relativity is confirmed or challenged, or anything about unconventional alternatives. On the last point the evidence policy is explicit and symmetrical: “Established and unconventional claims use the same contracts and evidence thresholds.” A claim that contradicts textbook physics gets the same treatment as one that agrees with it. It needs a precise contract, then evidence at the same bar.

Try it yourself

Nothing in this chapter runs beyond the Chapter 20 checks. One command tests its central statement:

grep -rn "EMPIRICALLY_" src/

It should match only models.py. (This edition’s search did. Rerun it before relying on the chapter: adding any empirical adapter would make the chapter’s central claim false.)

Open questions.

  1. Is “physical” the right word in the claim ID? A rename would be a new claim version, with the usual cost of a new fingerprint.
  2. How should the engine represent a model comparison, such as Galilean versus Lorentzian kinematics, where each is internally consistent and only data separates them?
  3. What counts as an operational definition of $c=1$? The choice of units is currently a one-line assumption.

Sources

The physical claim cites Albert Einstein, “Zur Elektrodynamik bewegter Körper”, Annalen der Physik (1905), https://doi.org/10.1002/andp.19053221004. For the model’s history and standard treatment:

  • H. A. Lorentz, “Electromagnetic Phenomena in a System Moving with Any Velocity Smaller than that of Light”, Proceedings of the Royal Netherlands Academy of Arts and Sciences (1904)
  • Hermann Minkowski, “Raum und Zeit”, Physikalische Zeitschrift (1909)
  • Edwin F. Taylor and John Archibald Wheeler, Spacetime Physics: Introduction to Special Relativity (W. H. Freeman, 1992)

The empirical layer will need experimental sources of class scientific_experiment, which the citation conventions already define. None supports any claim in this chapter. In the repository: the evidence policy, docs/methodology/evidence-policy.md.