What We Can Conclude

Thirty chapters ago, the book asked whether “Lorentz boosts preserve the interval” is true, and answered that it depends on which sentence you mean. This chapter collects what the book can now say, in the vocabulary of the earlier chapters: the bounded findings, the obligations still open, and the question of whether any of it is new. It adds no new capability.

What has been established

In the mathematics (kernel- or computation-checked)

Claim	Strongest recorded standing	Where
Algebraic Lorentz interval invariance (γ²(1−v²)=1)	formally_verified in Lean and Rocq; Z3 unsat (computation); hosted dossier source_claim_qualified for the project-owned source claim	Ch. 20
Gamma bridge (γ = 1/√(1−v²), |v|<1 ⟹ γ²(1−v²)=1)	formally_verified in Lean and Rocq (1.0.0 baseline); hosted dossier formal_statement_verified	Ch. 11, 12, 20
Composed physical Lorentz theorem (1.1.0)	Kernel-accepted in Lean and Rocq (R4 source/audit bundle); not migrated into the accepted report, which still records 1.0.0 as computation_verified_within_stated_domain (SymPy only)	Ch. 20
The shear t'=t, x'=x+vt preserves the interval	refuted_by_valid_counterexample (SymPy and Z3, residual −1/4)	Ch. 5
Coordinate Pythagorean theorem	formal_statement_verified (Lean and Rocq, SymPy and Z3 support)	Ch. 25
Unconditioned Pythagorean claim	refuted_by_valid_counterexample (residual −2)	Ch. 25
Two repairs of it (A: add orthogonality; B: add the cross term)	mathematically_qualified_candidate, neither selected (not_decided)	Ch. 27
A third repair (premise 0 = 1)	rejected_vacuous despite kernel acceptance	Ch. 27

About process (engine behaviours that held)

• Evidence is bound to an exact claim fingerprint, and a changed claim does not inherit old evidence (Ch. 1).
• Missing tools are not_installed / not_evaluated, never a pass (Ch. 3, 13, 14).
• Placeholders, unexpected axioms, and kernel rejections are caught by three independent guards (Ch. 3, 11).
• Conclusions are recomputed by fixed precedence from the records, with no tool vote and no score (Ch. 6).
• Failures, including the project’s own failed hosted runs and a corrected first implementation (P5-R1), are preserved (Ch. 8, 25, 28).

About an external record (P8)

For one pinned external Lean formalization record, the engine established a vector, not a verdict: the challenge wrapper has a sorry (interface scope), the separate solution theorem is kernel-accepted with a standard axiom set, the executed Comparator accepts the configured mapping, the catalogue and documentation disagree, statement correspondence is qualified, and proof digestion and historical correspondence are not_evaluated. Neither dismissal nor full verification is justified (Ch. 28).

What has not been established

Item	Standing
Cross-kernel semantic equivalence of Lean’s and Rocq’s reals	not_established (L3, Ch. 12/15)
Historical Euclid I.47	not_evaluated; the modern theorem does not qualify it (Ch. 25)
Fidelity of the Euclid transcription to the page, and of the page to the Greek	not_evaluated (Ch. 25)
Any claim-bound Isabelle or SageMath result	Illustrative Docker checks only; no engine EvidenceResult (Ch. 13, 14)
Human proof digestion, anywhere	not_evaluated
Independent reproduction beyond the project’s own control	not_evaluated (Ch. 29)
Whether either repair is “right,” or accepted by anyone	not_decided / not_claimed (Ch. 27)
Anything empirical about relativity	No evidence layer exists (Ch. 4, 18)
Three-dimensional Lorentz group, tensors, curvature, field equations	No engine work (old-outline chapters 23–28, not written)
Any published criticism of any theory	None ingested (Ch. 23)
Experimental evidence	None (old-outline chapter 34, not written)

Open obligations, in rough order of cost

1. Report migration (R4). Move the accepted MATH01 report to physical claim 1.1.0, so the report and the kernel proofs say the same thing.
2. Human review. A person reads the proofs, the transcription, and these drafts. This is the cheapest high-value step and nothing else substitutes for it.
3. Bibliography. Entries now exist for the historical sources (Einstein, Lorentz, Minkowski, Heath), the tools and the Cannon background, but most were added as grounding and have not been read closely. Still missing: the reproducibility literature, the paper whose theorem P8 formalizes, and specific documentation pages for the tool behaviours the book relies on.
4. Turn illustrative tool runs into evidence. Rocq and Isabelle compiled in Docker examples, and Sage ran illustrative computations (Ch. 12-14). Add the adapter and EvidenceResult records, and decide which repaired sources may be promoted through a versioned evidence update.
5. Close the SymPy domain gap. The recorded |v|<1 domain is not enforced by SymPy (Ch. 9); change the status wording or the domain field in a new claim version.
6. Independent reproduction (R4). One outside party following the guide (Ch. 29).
7. Cross-kernel correspondence. P9’s hosted record reports structurally_consistent for the algebraic pair; formal equivalence remains not_established, and the gamma-bridge coverage case is inconclusive. P10-A is a local-only abstract-ring equivalent_under_mapping certificate. P10-B is local-only concrete correspondence under mapping for the designated 1.1.0 gamma and physical targets. No concrete real-model equivalence is established.
8. A real challenge case. Choose a published criticism and run the anatomy of Chapter 23 on it, with the faithful-reconstruction rules.
9. The empirical layer and Part III’s later chapters. Large, separate programmes.

The novelty question

Is any of this new? The record answers what it can.

• The project’s landscape note says Physlib already contains a Lorentz-group definition, boosts, and rotations in Lean, so MATH01’s 1+1 identity “is a qualification example, not a claim to have newly formalized Lorentz invariance.”
• A targeted search found no equally direct maintained Rocq or Isabelle project for this exact theorem, which the note itself calls “a search result, not proof that no such formalization exists.”
• The P8 source note disclaims novelty for the conjecture, the formalization, the Comparator, and MathVet, and describes the project’s contribution as “a hypothesis-driven method, not a claim of novelty or acceptance.”

So this book makes no novelty claim. If a claim is later made, it would have to be for the method (evidence-first contracts, bounded conclusions, preserved failures) and would need a systematic literature search of the kind the landscape note lists as an immediate obligation. The mathematics is textbook.

What the book can responsibly say

A sentence that survives every status above: In a small, fully specified model, a Lorentz boost preserves the interval (checked by two proof kernels), a tempting alternative does not (checked by two solvers), and for a handful of related statements the engine can say precisely what was checked, by what, and what remains open. It does not say relativity is true, that anyone’s criticism fails, or that the formalizations mean what their prose says.

What this establishes

The project’s evidence is strong on a narrow band and says nothing outside it. That is the intended result: the method’s value is the discipline about which band a sentence belongs to. A reader who leaves able to ask of any mathematical claim “which claim, which version, which tool, which assumptions, and what remains open?” has what the book set out to give.

Human review of the book itself: none is recorded.

Try it yourself

Re-derive this chapter’s tables from the repository: the accepted report (artifacts/math01/report.md), the R4 bundle (artifacts/math01-r4/), and the P6, P7 and P8 evidence records. A discrepancy between this chapter and those records means this chapter is wrong.

Open questions.

1. Is “no novelty claim” the right stance, or does the method deserve a claim that a literature search would test?
2. Which of the nine open obligations should block publication of a first edition, and which can follow it?
3. If a reviewer found a flaw in the shared statement of the algebraic theorem, how many rows of the first table would change?

Sources

The earlier chapters and, through them, docs/research/MATH01-status.md, docs/research/formalization-landscape.md and the P1 to P8 evidence records.