Alternative Formulations

Is the physical Lorentz claim stronger than the algebraic one? One reader says yes: it is “about physics”. Another says no: the physical claim is just the algebraic claim with $\gamma$ filled in. Both are half right, and the loose word “stronger” hides which half.

Two true theorems are logically equivalent in a trivial sense: each implies the other, because both are true. A useful comparison has to say what is being compared: the hypotheses each demands, the conclusions each draws, or the scope of instances each covers.

This chapter gives a vocabulary for comparing two formulations of “the same” idea, so that a disagreement about which is stronger, equivalent or better becomes a question with an answer. It builds on the claim contract (Chapter 1), the cross-system ladder (Chapter 15) and repair (Chapter 27).

Kinds of alternative

Comparing hypotheses: physical versus algebraic

Executed with Z3, with the square root encoded as a relation ($s>0$, $s\cdot s = 1-v^2$, $\gamma s = 1$):

physical hypotheses imply the gamma relation:                  unsat   (the bridge)
gamma relation implies the physical gamma (γ = 1/s):           sat     γ = −1, s = 1, v = 0

So the physical hypotheses imply the algebraic one, but not the other way round, and the counterexample to the reverse is $\gamma=-1$. The set of $(v,\gamma)$ pairs allowed by the algebraic hypothesis is strictly larger than the physical set: it also contains every case with negative $\gamma$. Both theorems are true. The algebraic theorem covers more instances, and the physical one is the special case $\gamma>0$, which is the one physicists mean.

That is the right sense in which the algebraic claim is “stronger”: it asks less of its input. It is also why the engine proves the algebraic theorem first and treats the physical one as a composition, with the gamma bridge supplying the one missing fact (Chapter 20).

Comparing conclusions: two repairs

The refuted claim of Chapters 25 and 27 says $d^2(B,C) = d^2(A,B) + d^2(A,C)$ for all points. Two repairs:

• A (narrow the assumptions): add $AB\cdot AC = 0$.
• B (correct the conclusion): subtract $2\,(AB\cdot AC)$.

B is true for all triples; A is true for the orthogonal ones. When $AB\cdot AC=0$, B reduces to A’s conclusion, so in instance scope A is the restriction of B to orthogonal triples. Yet in logical terms both are true, so each implies the other trivially.

The meaningful comparison is the one the P7 record makes: a vector of trade-offs, not a ranking. A keeps the original conclusion on a smaller domain; B keeps the domain and changes what is concluded. Neither dominates. The engine records not_decided, and it does not count proofs to choose.

Comparing quantification: gamma 1.0.0 and 1.1.0

The SymPy check of the gamma relation substitutes $\gamma = 1/\sqrt{1-v^2}$, so it quantifies only over $v$. The kernel statements quantify over both. The L0 audit reports the pair as declared_mismatch (Chapter 15). The two are equivalent in content (a function of $v$, versus a variable $\gamma$ pinned to that function) and different in contract. This is the weakest kind of alternative: the same mathematics, written so that each tool must be told which form it is checking.

Equivalence needs a proof

The evidence policy states the rule: “Call two formalizations equivalent only after their equivalence obligation has been proved; matching metadata is merely declared alignment.” Two formulations that look alike can differ in scope (the $\gamma<0$ case above). Two that look different can be equivalent: the $(\gamma,v)$ and $k$ forms are, and the equivalence is an algebra exercise, but it is not yet a theorem in a kernel.

Telling formulations apart by observation (not done)

The book’s plan for this chapter includes telling formulations apart by observation. Two formulations that are mathematically inequivalent could still agree on everything measurable. For the Lorentz material, the negative-$\gamma$ extension and the positive-$\gamma$ case differ only by a time reversal, which no clock comparison tests directly.

Formulations that differ in a testable way are what an empirical layer is for. That layer (the MATH06 milestone) has not started, so this chapter can say what telling them apart would require, but not do it:

1. each formulation as a versioned claim with an explicit prediction;
2. a pinned, versioned dataset with an uncertainty model;
3. the separation between prediction and observation that the book’s plan already requires;
4. a report that says which formulation each dataset counts against, in the status vocabulary (empirically_supported, empirically_challenged, with the limits the evidence policy attaches).

What this establishes

Choosing between formulations is a judgment about generality, simplicity and fit to the physical question. The engine can show what each formulation covers, what each assumes, and whether two are provably equivalent. It does not rank them.

No claim is qualified here. The Z3 results above are illustrative. The P7 repair evaluations and the L0 declared-mismatch result are qualified, in Chapter 27 and Chapter 15 respectively; the P7 candidates (pythagorean-coordinate-2d, squared-distance-decomposition-2d, pythagorean-vacuous-premise-2d) are claims from Chapter 27. The $\gamma=-1$ counter-case is a one-line check; no review is recorded.

The comparison of hypotheses used Z3’s relational encoding of the square root. In Lean’s junk-value encoding, the bridge needs $\lvert v\rvert<1$ explicitly (Chapter 11). The conclusion “the algebraic claim is strictly more general” holds in both encodings, but by different routes.

Try it yourself

uv run python content/examples/math/part4/part4_assumption_deletion_demo.py

The last section of the script prints the two physical-versus-algebraic checks.

Seeded error. In the demo, drop $s>0$ from the “gamma relation implies physical gamma” check. It should remain sat (the model $\gamma=-1$, $s=1$ already works), which confirms that positivity of the square root is not what separates the two.

Exercise. For $v = 3/5$, list every real $\gamma$ allowed by the algebraic hypothesis and every $\gamma$ allowed by the physical one. (Answer: algebraic $\gamma = \pm 5/4$; physical $\gamma = 5/4$ only.)

Open questions.

1. “The algebraic claim has strictly larger scope” depends on defining the physical set with $\gamma>0$. If a text defines $\gamma$ with either sign, the claim changes. Which convention does the book adopt, and where is it recorded?
2. The equivalence of the $(\gamma,v)$ and $k$ forms is easy and unproved in a kernel. Should it become a claim, and with which status?
3. For an empirical comparison, who decides what counts as a “different prediction” when two formulations agree on all current measurements?

Sources

No external literature on comparing formulations is cited. In the repository: docs/research/MATH-VAL01-P7-evidence.md, docs/methodology/evidence-policy.md and docs/research/MATH01-R3-evidence.md.