7. Working dictionary: do not mistake names for identifications
This dictionary separates a definition, its role in the argument, and the extra inference one must not make from the definition alone. Standard terms have checked references. The IUT watchlist below consists of questions for the primary-paper audit, not substitute definitions: familiar-sounding names in IUT need their own exact source locators.
Standard terms and logical distinctions
| Term | Working definition or exact scope | Why we need it; a false shortcut to avoid |
|---|---|---|
| Radical \(R\) | \(\operatorname{rad}(abc)=\prod_{p\mid abc}p\) for a primitive triple | Measures prime support, not the magnitude of \(abc\); 01, G1 |
| \(p\)-adic valuation \(v_p(n)\) | The exponent of \(p\) in the factorization of a nonzero integer \(n\), extended to rationals by subtraction | \(v_p(p^m)=m\), but the radical records only whether \(m>0\); 01 |
| Place | An equivalence class of archimedean or nonarchimedean absolute values on a number field; calculations choose normalized representatives | A numerical comparison valid at one place need not hold at another; Goldfeld §2 |
| Height \(H\) | For a primitive positive triple, \(H([a:b:c])=\max(a,b,c)=c\); number-field heights combine all places | Do not replace a projective height with the size of one chosen coordinate in an arbitrary field; 01, G1 |
| Weak abc | \((\lvert ABC\rvert)^{1/3}\ll_\varepsilon \operatorname{rad}(ABC)^{1+\varepsilon}\) for coprime nonzero \(A+B+C=0\) | A geometric-mean bound is not literally the bound on \(\max(\lvert A\rvert,\lvert B\rvert,\lvert C\rvert)\) in ordinary abc; Goldfeld §1 |
| Model/minimal discriminant | A Weierstrass model has a discriminant; a local minimal model minimizes its \(p\)-adic valuation | A worked model calculation does not compute the minimal discriminant; Goldfeld §3 |
| Conductor \(N_E\) | A product \(\prod_p p^{f_p}\) with exponents determined by reduction type of an elliptic curve \(E\) | It is not literally \(R=\operatorname{rad}(abc)\) for every Frey model; Goldfeld §4 |
| Frey--Hellegouarch curve | For \(a+b=c\), \(y^2=x(x-a)(x+b)\) is an elliptic curve associated to the triple | Its invariants make a conditional bridge to abc, not a proof of Szpiro; 01, G1 |
| Étale fundamental group | For connected \(X\) and geometric base point \(\bar x\), \(\pi_1(X,\bar x)=\operatorname{Aut}(F_{\bar x})\) for the fiber functor on finite étale covers | Recovering a particular geometric object from group data requires additional hypotheses and reconstruction, not just this definition; Stacks §58.6 |
| Implication \(P\Rightarrow Q\) | A derivation of \(Q\) from a stated input \(P\) | A checked formal proof of this implication is not a checked proof of \(P\) or necessarily a faithful translation of the paper's \(P\); see the project map |
| Copy, isomorphism, label | A copy can carry corresponding structure without being literally the same object; a label names an object without prescribing its arithmetic value | Write the specific map and structures it preserves before comparing quantities in different copies; this is general logic, not a verdict on IUT |
IUT-specific terms: source-audit questions
These are reading prompts until the actual definitions and allowed morphisms are extracted from the primary papers. A quick analogy should always carry this warning. Read the primary-paper route for checked locators; the Lean vocabulary describes typed counterparts, not a replacement for the papers' definitions.
| Term | What the reader must determine from the cited paper | Dangerous unstated inference |
|---|---|---|
| Initial \(\Theta\)-data | Which elliptic curve, base fields, primes, and auxiliary choices are fixed? | That a later theater automatically retains every original identification |
| Hodge theater | Which categories and arithmetic structures are present in a single theater? | That two theaters are the very same structured object |
| Frobenioid | What category is reconstructed, from what data, and up to what equivalence? | That an abstract reconstruction has fixed numerical coordinates |
| Theta-link / log-link | What is the domain, codomain, and exact preserved structure of each link? | That a transport preserves operations not included in its definition |
| Log-shell | What subset or object is being measured, at which places, with what normalization? | That it has the same value after an arbitrary relabeling |
| Log-theta-lattice | What are its nodes and the two sorts of connections? | That walking a diagram automatically yields a numerical inequality |
| Indeterminacy | Which group/action or range of choices is explicitly permitted? | That choosing one representative is canonical or comparison-safe |
| Mono-anabelian reconstruction | Which invariants recover which kind of arithmetic data? | That a recovered copy is identical to the source in all senses |
For every row, complete the seven-item definition gate in 02, then cite a theorem that uses it in the claim graph. The open cells are deliberate: learning the source's permission to compare is harder, and more important, than memorizing its terminology.