Section 2.1, paragraph 2The abstraction excludes scale-adaptive regimes, which move towards TC0
Informal interpretation Motivation, interpretation or scope; not a formal claim.
1 Paper text
We emphasise that this is not a statement about floating-point arithmetic per se. The abstraction intentionally excludes scale-adaptive regimes where thresholds/biases grow with fan-in or where weights shrink with dimension (e.g. or normalisations), which move the analysis toward threshold-circuit classes such as . This class allows constant-depth, polynomial-size circuits with general threshold gates, including majority gates; unlike , it can compute parity.
Generated from paper_jmlr.tex, lines 318–319.
2 Explain this
TC0 means constant-depth, polynomial-size circuits with general threshold gates, including majority gates. It can compute parity, so constant depth alone cannot support the paper's lower bound.
The result is about fixed scale, not about floating-point formats as such. What breaks the argument is scale adaptation: thresholds or biases that grow with the fan-in, or weights that shrink like or (as in many normalisation schemes). Those regimes behave like threshold circuits (), which can count and hence compute parity.
3 How this is formalised
An interpretive paragraph. Its precise counterpart is the scope remark that follows, where Lean checks the boundary example: a majority gate violates the fixed-scale validity condition once its fan-in exceeds (Remark 2.2 (model scope)).
| Paper anchor | c-not-floating (website anchor; the manuscript has no label here) |
|---|---|
| Displayed as | Section 2.1, paragraph 2 |
| Claim | The abstraction excludes scale-adaptive regimes, which move towards TC0 |
| Coverage category | Informal interpretation |
| Statement source | paper_jmlr.tex lines 318–319; SHA-256 of the normalised block source 1ff774847c3d248d… |
| Depends on | Remark 2.2 (model scope) |
| Explanations | explain/c-not-floating, formal/c-not-floating |
| Build evidence | fresh isolated rebuild, audit report |
4 Lean statement and source
No Lean statement corresponds to this claim. This is interpretation or motivation, not a formal mathematical claim.
Build evidence: fresh isolated rebuild of all 38 bundled modules with Lean (version 4.26.0, started 2026-10-10T15:21:56Z, finished 2026-10-10T15:27:00Z: 38/38 modules compiled, 1588 axiom reports, no errors. See Build for the commands and the independent audit of 2026-10-10 (dated evidence).