Mathematics on Programmer.ie: Modern AI programming

The Claim Before the Proof

Tue, 20 Oct 2026 00:00:00 +0100

“Lorentz boosts preserve the interval.” Is that true?

It depends on which sentence you mean.

If $\gamma$ is any real number with $\gamma^2(1-v^2)=1$, yes, by a three-line expansion.
If $\gamma$ is defined as $1/\sqrt{1-v^2}$, it takes an extra argument, and it needs $\lvert v\rvert<1$ so that the square root is real and nonzero.
If “preserve” means that the physical world obeys the formula, no theorem settles it.

One sentence, three claims. A tool that reports “verified” for the sentence has told you nothing until you know which claim it checked. The design record behind the engine puts it plainly: the same sentence “can acquire materially different meanings when translated to SymPy, Z3, Lean, Rocq, or Isabelle,” and “a proof may also be correct under a stronger assumption than the prose disclosed.”

Definitions and Assumptions

Tue, 20 Oct 2026 00:00:00 +0100

Which of these does the algebraic Lorentz theorem need: $\lvert v\rvert<1$, $\gamma>0$, $c=1$, or $\gamma=1/\sqrt{1-v^2}$?

None of them. The Lean statement takes four real numbers and one hypothesis, $\gamma^2(1-v^2)=1$:

theorem lorentzIntervalInvariantAlgebraic
    (t x v gamma : ℝ)
    (hgamma : gamma ^ 2 * (1 - v ^ 2) = 1) : ...

A reader who “knows” that boosts need $\lvert v\rvert<1$ may assume the theorem uses it. It does not. The opposite mistake is worse: the physical theorem does need $\lvert v\rvert<1$, because without it $1-v^2$ can be zero or negative, and then $\sqrt{1-v^2}$ is not a real number. Whether a condition carries weight is a fact about one specific claim, and the claim contract from Chapter 1 is where that fact is written down.

What Counts as Proof?

Tue, 20 Oct 2026 00:00:00 +0100

Here are four artifacts. A reader might call each of them “proved”:

A paragraph of algebra that expands the squares.
A SymPy call that simplifies a residual to 0.
A Z3 run that returns unsat on the negation.
A Lean file that compiles.

The fourth is the dangerous one. This file compiles:

import Mathlib

theorem intentionallyIncomplete (x : ℝ) : x = x := by
  sorry

Lean accepts it with a warning, because sorry inserts an axiom (sorryAx) that proves anything. The repository keeps this file as formal/lean/fixtures/Incomplete.lean for exactly that reason: its header says the adapter “must reject this placeholder even if Lean permits it with a warning.” It compiled is not it was proved.

Mathematical Models and Physical Theories

Tue, 20 Oct 2026 00:00:00 +0100

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.”

Counterexamples

Tue, 20 Oct 2026 00:00:00 +0100

You can test a false universal claim many times and watch it pass. Take the shear

$$ t' = t, \qquad x' = x + vt, $$

and the claim that it preserves $t^2-x^2$. At $t=0$ it does, since $x'=x$. At $v=0$ it does. At $x=-vt/2$ it does too, because then $x' = vt/2$ and $x'^2 = x^2$. A reader who tries a few convenient inputs may conclude that the claim holds. The residual explains why it seems to:

Uncertainty Without Confidence Theatre

Tue, 20 Oct 2026 00:00:00 +0100

Imagine a dashboard that says “Algebraic Lorentz theorem: 94% verified.” What would the number mean? That 94% of tools agreed? That a model is 94% sure? That 94% of the proof is done? Each reading is wrong for this kind of claim, because a theorem is either checked or it is not. A number invites the reader to treat a missing check as a small deduction.

The design record (ADR 0001) rejects the idea outright. It lists “majority voting among tools” as a rejected alternative, because “a mistranslation can make every tool agree on the wrong proposition.”

AI as Proposer, Not Authority

Tue, 20 Oct 2026 00:00:00 +0100

A language model is asked to prove the algebraic Lorentz theorem. It produces fluent, confident Lean:

theorem lorentzIntervalInvariantAlgebraic ... := by
  simp
  ring_nf
  done

Three things can happen next. The proof is right. The proof is wrong, and Lean rejects it. Or the model “explains” that the proof works, and nobody runs it. In the third case a polished paragraph has been mistaken for a result.

The engine’s position is that a model’s text is evidence in none of these cases. Only the first two produce a verdict, and that verdict is the kernel’s, not the model’s. This chapter describes the typed boundary that marks model output as an unverified candidate and gives it no path to becoming evidence. It builds on the status ladder of Chapter 3 and the bounded conclusions of Chapter 6.

Provenance and Reproducibility

Tue, 20 Oct 2026 00:00:00 +0100

A theorem-prover result that exists only on one laptop is hard to trust. Here are four ordinary ways it fails to travel:

The proof depended on a Mathlib version nobody wrote down.
The source file hashes differently on Windows and Linux because of line endings.
A report was edited by hand “just to fix a typo” after the run.
A stale result file from an earlier run is still in the output directory and gets read as current.

The repository has a defence against each, and this chapter walks through them: what every result records, how a fresh run is compared with the committed one, and what has still not been reproduced. It builds on claims (Chapter 1) and the evidence statuses (Chapter 3). The project also keeps its own failed runs on record, because a reproducibility story with no failures in it is probably missing some.

Symbolic Algebra with SymPy

Tue, 20 Oct 2026 00:00:00 +0100

The accepted report says the physical Lorentz claim is computation_verified_within_stated_domain, and the recorded domain is “real t, x, v with |v| < 1 and c = 1.” A reader naturally assumes SymPy checked the condition $\lvert v\rvert<1$.

It did not. Executed:

gamma = 1/sqrt(1 - v^2)       (v declared real)
simplify(gamma^2 (1 - v^2) - 1)               -> 0
same expression with v = 2    (so 1 - v^2 = -3) -> 0

The residual is zero at $v=2$, where $\lvert v\rvert<1$ is false and the square root is not real. SymPy’s sqrt(-3) is the imaginary number $i\sqrt3$, its square is $-3$, and the algebra closes anyway. The simplifier proves an identity of expressions; it has no idea what range of $v$ the physics wants. The adapter’s own recorded limitation says as much: “SymPy’s simplifier does not itself prove that physical side condition.”

Adversarial Math From First Principles

Mon, 09 Nov 2026 00:00:00 +0000

A mathematical answer is only as trustworthy as the claim being tested, the assumptions behind it, and the evidence supporting it.

AI can generate mathematical arguments, construct proofs, identify patterns, and propose solutions. But a convincing explanation is not necessarily a correct proof, and a formally verified theorem may not establish what its author intended.

Adversarial Math From First Principles explores a different approach: mathematics in which AI proposes, independent tools check, counterexamples challenge, and every conclusion remains connected to the evidence that supports it.

Constraints and Models with Z3

Tue, 20 Oct 2026 00:00:00 +0100

A solver says unsat. What has been shown?

Not “the theorem is true.” What has been shown is that one specific formula has no model: the conjunction of the recorded assumptions and the negation of the goal. If that formula is the wrong encoding of the claim, unsat is a true fact about the wrong formula. The evidence policy states the rule: “Interpret Z3 unsat only as unsatisfiability of the recorded negation and assumptions.”

Lean 4 and Mathlib

Tue, 20 Oct 2026 00:00:00 +0100

Put $v=2$ into $\gamma^2(1-v^2)$ with $\gamma = 1/\sqrt{1-v^2}$. What do you get?

SymPy says $1$ (Chapter 9). Lean says $0$. Re-executed on 2026-11-09 with Lean 4.35.0 and Mathlib v4.35.0, using the example from Chapter 2:

Real.sqrt (1 - 2^2) = 0                      (1 - 2^2 = -3, sqrt of a negative)
(1 / Real.sqrt (1 - 2^2))^2 * (1 - 2^2) = 0  (gamma = 1/0 = 0)

Lean has a junk value for division by zero and for the square root of a negative number: both are $0$. SymPy instead takes $\sqrt{-3}$ to be the imaginary $i\sqrt3$, and the algebra closes. Two tools disagree about the same expression at the same point, because each defines the square root of a negative number differently. Neither is wrong; they are different encodings. It is a small, concrete case of the cross-system problem that Chapter 15 takes on.

Rocq as an Independent Kernel

Tue, 20 Oct 2026 00:00:00 +0100

Two kernels accepted a theorem about the square root. Does that show the theorem is right?

Only if both are about the same square root. Lean’s Real.sqrt comes from Mathlib, built on one construction of the real numbers. Rocq’s sqrt comes from its standard library, built on another. If the two constructions differed on some input, both kernels could accept their own statement and still disagree about the physics. The project has not proved that they agree. The accepted report says so, and attaches the conclusion semantic_equivalence_not_established to the gamma pair.

Isabelle/HOL and Countermodels

Tue, 20 Oct 2026 00:00:00 +0100

The first time the project’s Isabelle theory met a real Isabelle, it did not build. Three defects appeared, one after another:

*** Malformed command syntax (line 5)
*** Undefined method: "ring" (line 18)
*** Inner lexical error at "… = t^2 - x^2" (line 19)

The first is a portability problem: the UTF-8 text cartouche did not parse in this image, while the portable text \<open>...\<close> form does. The second had been predicted in an earlier draft of this chapter: ring is not a built-in Isabelle/HOL proof method. The third is a typographic ellipsis character where Isabelle expects its special ... fact. These are real failures of the candidate file, not failures of the mathematics.

SageMath and Scientific Computation

Tue, 20 Oct 2026 00:00:00 +0100

The engine’s host probe says SageMath is absent:

sagemath available: False | reason: none of sage was found on PATH

And yet this chapter shows SageMath output. That is not a contradiction. The probe checks the host PATH; the run below happened inside the official Docker image. Being available in a container is not the same as being integrated into the engine. The engine’s sagemath.probe.v1 registry entry still says “SageMath capability only; no evidence”, and the manual sagemath-lane remains unrun in the hosted record.

The Cross-System Claim Contract

Tue, 20 Oct 2026 00:00:00 +0100

Here is what the engine reports when it compares three tools’ versions of the gamma-relation claim at version 1.1.0 (executed):

gamma-sympy-v1 <-> gamma-lean4-v1   declared_mismatch   ['quantifiers']
gamma-sympy-v1 <-> gamma-rocq-v1    declared_mismatch   ['quantifiers']
gamma-lean4-v1 <-> gamma-rocq-v1    semantic_equivalence_not_established

The SymPy check and the Lean check are both of “the same” gamma claim, and the engine says their contracts do not match. The reason is concrete. The SymPy binding substitutes the defining equation $\gamma = 1/\sqrt{1-v^2}$ into the expression, so it quantifies only over $v$. The Lean and Rocq statements quantify over $\gamma$ as a variable and take the definition as a hypothesis. The recorded obligation reads: “gamma was eliminated by substituting the defining equation … this binding quantifies only v.”

The Investigation Engine

Tue, 20 Oct 2026 00:00:00 +0100

The first version of almost any verification tool is a single script. Here it would be run_math01: import SymPy, call Z3, shell out to lean, print the results. It works. Then someone wants a second claim, and copying the script duplicates the placeholder screening. A third claim needs Rocq; a fourth needs a notebook. Each copy decides afresh what “success” means, and sooner or later one of them reports a missing tool as a pass.

Events and Reference Frames

Tue, 20 Oct 2026 00:00:00 +0100

Two events, $A$ and $B$, happen at the same time in a frame $S$: $A$ at $(t,x)=(0,0)$ and $B$ at $(0,1)$. Are they simultaneous?

The question has no answer that does not depend on the frame. Take a second frame $S'$ moving at $v=3/5$ relative to $S$ (so $\gamma=5/4$, in units where $c=1$). Executed with exact rationals:

$$ t'_A = 0, \qquad t'_B = \gamma(0 - v\cdot1) = \tfrac54\cdot\left(-\tfrac35\right) = -\tfrac34 . $$

In $S'$, $B$ happens three quarters of a time unit before $A$. Both frames are right. Simultaneity belongs to a frame’s way of assigning time coordinates, not to the events. What both frames agree on is the interval:

The Postulates and Their Roles

Tue, 20 Oct 2026 00:00:00 +0100

“Special relativity follows from two postulates.” That sentence is a slogan, and a derivation that takes it literally will not go through. A closer count of what a standard derivation uses looks like this:

The principle of relativity: the laws of physics take the same form in all inertial frames.
Constancy of the speed of light: light travels at the same speed $c$ in all inertial frames.
Homogeneity of space and time, so that the transformation between frames is linear.
Isotropy, so that there is no preferred direction along the line.
A synchronization convention (Chapter 17).
Reciprocity: if $S'$ moves at $v$ relative to $S$, then $S$ moves at $-v$ relative to $S'$, and the transformation depends only on $v$.

The first two are the famous ones. The others are routinely used in textbook derivations without being called postulates. Which of them a given proof uses, and which it quietly assumes, is exactly what a claim contract exists to record. (The list is general background, not a repository record.)

The Lorentz Transformation

Tue, 20 Oct 2026 00:00:00 +0100

The standard boost is

$$ t' = \gamma(t-vx), \qquad x' = \gamma(x-vt), \qquad \gamma = \frac{1}{\sqrt{1-v^2}} . $$

Why $\gamma$, and why that $\gamma$? A reader can check that the formula preserves the interval (Chapter 20), but that shows only that the formula is consistent with one requirement. It does not show that the formula is forced. The shear $t'=t$, $x'=x+vt$ is simple and plausible, and wrong. Something has to say why the right answer is this one.

The Invariant Interval

Tue, 20 Oct 2026 00:00:00 +0100

What are we actually proving?

A Lorentz boost in one space and one time dimension is often introduced with a formula and a promise: the formula preserves the interval between events. That sentence compresses several different mathematical claims. Before any proof assistant is asked to check one of them, we have to say which one we mean.

This chapter separates four claims.

An algebraic identity. For real numbers $t,x,v,\gamma$ satisfying $\gamma^2(1-v^2)=1$, the boost $t'=\gamma(t-vx)$, $x'=\gamma(x-vt)$ preserves $t^2-x^2$.
A gamma bridge. For real $v$ with $\lvert v\rvert<1$ and $\gamma=1/\sqrt{1-v^2}$, the hypothesis of the algebraic identity holds.
A physical composition. Combining the two gives the interval-preserving boost with the Lorentz factor defined by the positive square root.
A deliberately false control. The shear $t'=t$, $x'=x+vt$ does not preserve the interval. A rational witness refutes it.

The mathematical question is not whether special relativity is true. Special relativity is a well-tested physical theory. The question here is narrower and more useful for a book about verification: what exactly do the formal artifacts establish, under which assumptions, and where does the verification stop?

Time Dilation and Length Contraction

Tue, 20 Oct 2026 00:00:00 +0100

“Moving clocks run slow.” Let frame $S'$ move at $v=3/5$ relative to $S$.

Clock $C'$ rests in $S'$. Its reading $\tau=4$ happens at $S$-coordinates $(t,x)=(5,3)$. So $S$ sees $C'$ advance $4$ while its own time advances $5$: $C'$ is slow in $S$.
Clock $C$ rests in $S$ at $x=0$. Its reading $4$ happens at $S'$-coordinates $(5,-3)$. So $S'$ sees $C$ advance $4$ while its own time advances $5$: $C$ is slow in $S'$.

(Executed.) Each frame says the other’s clock is slow. If that were a contradiction, relativity would be inconsistent. It is not. The two observations compare different pairs of events, and “slow” has a meaning only once you say which pair. This chapter is about saying which.

Velocity, Light Cones, and Causality

Tue, 20 Oct 2026 00:00:00 +0100

Can a boost ever make an effect come before its cause? Take two events with $\Delta t=1$ and $\Delta x=2$, and boost by $v=3/5$ ($\gamma=5/4$). Executed:

$$ \Delta t' = \gamma(\Delta t - v\,\Delta x) = \tfrac54\left(1 - \tfrac65\right) = -\tfrac14 . $$

The order reversed: event $B$ is before $A$ in the new frame. If $A$ could cause $B$, causality would depend on the frame, which is intolerable. Now take $\Delta t=2$, $\Delta x=1$: $\Delta t' = \tfrac54\left(2-\tfrac35\right) = \tfrac74 > 0$, and the order is intact. The difference is whether $\lvert\Delta x\rvert < \Delta t$. This chapter explains why that is the right dividing line.

Anatomy of a Challenge

Tue, 20 Oct 2026 00:00:00 +0100

A hypothetical critic writes:

“Lorentz boosts cannot preserve the interval, because γ blows up as v approaches the speed of light.”

Is that true? The question cannot be answered until someone says what the sentence claims. Here are three readings:

Reading	Statement	Status by inspection
R1	As $v\to1$, the transformed coordinates $t'$, $x'$ are unbounded	True for fixed $(t,x)$ with $t\neq x$ (for example $t=1$, $x=0$ gives $t'=\gamma\to\infty$)
R2	For some $\lvert v\rvert<1$, the interval is not preserved because $\gamma$ is large	False: the algebraic theorem (Chapter 20) holds for every $\gamma$ with $\gamma^2(1-v^2)=1$
R3	The boost is undefined at $v=1$	True, and the claim contracts exclude it with the hypothesis $\lvert v\rvert<1$

A check by hand: at $v=3/5$, $\gamma=5/4$; at $v=12/13$, $\gamma=13/5$. For $(t,x)=(1,0)$ the boosted coordinates are $(5/4, -3/4)$ and $(13/5, -12/5)$. The coordinates grow, while the intervals, $25/16 - 9/16 = 1$ and $169/25 - 144/25 = 1$, do not change.

Hidden-Assumption Audits

Tue, 20 Oct 2026 00:00:00 +0100

A theorem is stated with four hypotheses. Which of them does the proof actually use?

The question matters in both directions. A hypothesis the proof does not use is a needless restriction. One the proof quietly relies on without stating it is a hole. Reading the proof is one way to find out. A mechanical test is another, and it catches cases a reader misses.

This chapter gives a repeatable test for which premises a conclusion really depends on, and a catalogue of the related audits the engine already runs. It builds on assumptions (Chapter 2), solvers (Chapter 10) and the cross-system ladder (Chapter 15).

From a Source Page to a Checked Theorem

Tue, 20 Oct 2026 00:00:00 +0100

Someone says: “Euclid proved the Pythagorean theorem, Lean has now proved it, so Euclid is verified.” Every part of that sentence slides between two things.

What Euclid wrote is a statement about squares on sides, in synthetic geometry.
What Lean checked is a statement about six real numbers and a polynomial identity.

A kernel’s acceptance cannot tell you whether those are the same claim. This chapter keeps them apart on purpose, and the engine records the gap between them as data.

Alternative Formulations

Tue, 20 Oct 2026 00:00:00 +0100

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.

Repairing a Refuted Claim

Tue, 20 Oct 2026 00:00:00 +0100

Start from the claim refuted in Chapter 25:

pythagorean-all-triangles-2d@1.0.0
d²(B,C) = d²(A,B) + d²(A,C)   for every three real points A, B, C

Its witness is $A=(0,0)$, $B=(1,0)$, $C=(1,1)$, with residual $-2$. The tempting move is to patch the claim until that witness stops being a counterexample. The engine tests three patches, and one of them is a fix in name only.

Given a claim already refuted by a validated counterexample (Chapter 5), the engine can record proposed repairs as new claims with an explicit record of what changed, test whether each repair’s premise can be satisfied, test a patch for overfitting, and compare the results as a set of trade-offs with no automatic winner. This chapter builds on the negative control of Chapter 25 and the kernel evidence of Chapter 20.

The `sorry` That Wasn't the Proof

Tue, 20 Oct 2026 00:00:00 +0100

Here are six facts about one Lean formalization of an action theorem related to Cannon’s conjecture, all true at the pinned commit:

the challenge file ends in sorry
the catalogue omits the result
the documentation says the formalization proves the theorem
the separate solution theorem passes the Lean kernel
the Comparator accepts the mapping
the human reviewer says the statement looks plausible but was not carefully checked

A reader who sees only the first fact concludes that the proof is incomplete. A reader who sees only the fourth concludes that it is done. Both are reading one record as if it were the whole story.

Independent Reproduction

Tue, 20 Oct 2026 00:00:00 +0100

“Reproduced” can mean at least six different things:

I ran it again and got the same output.
A fresh build on my machine produced the same hashes.
A CI server I do not control ran it and it passed.
Someone else downloaded the bundle and verified it.
Someone else, with different tools, re-derived the result.
Someone else wrote their own implementation from the specification.

The first is a weak check and the last a strong one, and they are not interchangeable. A report that uses the word without saying which is making a stronger claim than its author may realize.

What We Can Conclude

Tue, 20 Oct 2026 00:00:00 +0100

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).