5.5.1. Finite Path First🔗

TorchLean deliberately separates two questions:

1. What happens on the ordinary finite path, where every operation produces a finite float32 value?

2. What happens for every IEEE-754 corner case, including NaN, Inf, subnormal behavior, signed zero, and backend-specific library behavior?

The FP32 model used in proofs is aimed at the first question. It is the right object for margin transfer theorems, forward error bounds, and verifier enclosure inflation. The executable IEEE32Exec model is the right object for bit level behavior and for checking what the runtime does. Bridge theorems and explicit assumptions connect the two where the finite side conditions are available.

This is a practical mathematical choice. If every robustness theorem had to carry the full IEEE state machine directly, most theorem statements would become harder to read than the property they are trying to prove.

The finite path includes normal values and subnormals. It excludes computations whose executable path produces NaN or infinity. For example, a small underflowed subnormal output may still be a finite value covered by a finite-path theorem, while 0/0, overflow to Inf, or sqrt of a negative finite value are handled by the executable IEEE32Exec layer and total toReal? bridge, not by pretending they are ordinary real numbers.

5.5.2. Proof Float32 Model🔗

FP32 is the clean proof model for finite-path error budgets. IEEE32Exec is the executable bit model for special values and edge cases. Bridge theorems connect them when the computation remains finite.

In TorchLean, FP32 serves as the model for binary32-style rounding on the finite path.

More concretely:

• TorchLean.Floats.FP32 in NN.Floats.FP32 API is the canonical float32 semantics for proofs: a rounded real model instantiated to binary radix, binary32 style exponent (fexp32), and round-to-nearest-even (rnd32).

• It is smaller than full executable IEEE semantics by design. It does not try to represent NaN, Inf, or signed-zero behavior directly.

• For the executable layer, IEEE32Exec is used, with bridge theorems to relate it to FP32 and to say when the executable computation agrees with the proof model.

Executable IEEE-754-style float32 starts at the IEEE32Exec semantics API. Bridge and soundness theorems connect that executable view to the standardized FP32 proof semantics API, which is still the model we prefer for clean paper-level claims.

This division matters because paper claims usually want the cleanest theorem statement first:

• the proof model gives a theorem about rounding error and interval enclosure,

• the executable model gives a checkable implementation account,

• and the bridge theorems say when the executable and proof views coincide on the finite path.

So the relevant claim is never just "Float32 is close." It is a statement with an interpretation map:

\operatorname{IEEE32Exec} \xrightarrow{\operatorname{toReal}} \mathbb{R} \quad\text{and}\quad \operatorname{FP32} \xrightarrow{\operatorname{toReal}} \mathbb{R}

plus hypotheses saying the values are finite and the rounded executable path agrees with the proof model being cited.

5.5.3. Where The Approximation Story Becomes Concrete🔗

The repository turns this picture into named theorems in three recurring settings: layerwise error bounds, small network-level bounds for common MLP families, and verifier side statements saying that real valued enclosures can be widened enough to remain sound for float32 execution.

Theorem names that tend to matter first in this area:

• approxT_linear_fp32: float aware forward error for y = Wx + b

• approxT_tanhMlp3_fp32 / approxT_reluMlp2_fp32: float aware forward error for common MLP patterns

• fp32_le_of_real_le_sub_margin / fp32_ge_of_real_ge_add_margin: margin lemmas for lifting real-spec inequalities

• ibpBound_contains_reluMlp2_fp32: inflate a real IBP/CROWN box to cover FP32 execution of the two-layer ReLU MLP fragment

Behind those statements sit more generic rounded-runtime lemmas for operators and linear algebra. Those are the engine room for the more visible theorems:

• they keep the algebra of rounding explicit,

• they make it easy to reuse the same proof pattern across many operators,

• and they let the FP32 semantics guide stay focused on the model rather than every proof detail.

5.5.4. One Transfer Pattern🔗

Most theorems in this area follow a repeatable proof pattern:

1. Prove or compute a real-spec enclosure or inequality with a margin.

2. Use approxT_* to obtain an explicit eps such that the float32 semantics is within eps.

3. Apply a margin lemma like fp32_le_of_real_le_sub_margin or fp32_ge_of_real_ge_add_margin to transfer the property to FP32 execution.

The kind of theorem TorchLean aims at looks like this:

import NN.Proofs.RuntimeApprox.FP32.Layers
import NN.Proofs.RuntimeApprox.FP32.MLP
import NN.Proofs.RuntimeApprox.FP32.CROWN

open NN.Proofs.RuntimeApprox.FP32

-- If the real valued network stays above a threshold by more than `eps`,
-- then the FP32-rounded execution still stays above the threshold.
#check approxT_linear_fp32
#check approxT_reluMlp2_fp32
#check approxT_tanhMlp3_fp32
#check ibpBound_contains_reluMlp2_fp32
#check fp32_le_of_real_le_sub_margin
#check fp32_ge_of_real_ge_add_margin

That is the shape to keep in mind when reading the declarations. One theorem gives the approximation error, and another theorem uses that approximation error to preserve the property one actually cares about.

The recurring scalar pattern is:

|y^{\mathrm{fp32}} - y^{\mathbb R}| \le \varepsilon

and

y^{\mathbb R} \ge t + \varepsilon

therefore

y^{\mathrm{fp32}} \ge t

For classifier margins, the same idea is applied componentwise: the true logit may move down, the competing logit may move up, and the certified real margin must pay for both movements.

For a vector output, the same pattern is componentwise:

\forall i,\quad \left| y^{fp32}_i - y^{real}_i \right| \le \varepsilon_i

and a classifier margin proof normally spends that budget on the true class and the competing classes before concluding that the argmax is unchanged.

5.5.5. Template Statement🔗

The most useful mental model here is a transfer lemma with explicit side conditions:

Assume a real valued network output stays above a threshold by margin δ.

Assume the corresponding FP32 execution stays within error ε.

If ε < δ, then the FP32 execution still stays above the threshold.

TorchLean splits that argument into reusable pieces instead of reproving it from scratch every time:

• one theorem gives the forward approximation error,

• one theorem packages verifier side enclosure inflation,

• and one margin lemma moves the property across the ε gap.

That is why this theorem family matters beyond small MLP examples. The same pattern shows up again in layerwise proofs, in verifier side corollaries, and in the kinds of robustness statements one would actually want to cite in a paper.

In theorem statements, this usually appears as three named ingredients:

• a real-valued property or enclosure;

• an approximation relation such as approxT_*;

• a margin lemma such as fp32_ge_of_real_ge_add_margin.

5.5.6. A Concrete Linear Layer Walkthrough🔗

The linear theorem is the smallest useful example because it already has the same ingredients as a network:

1. real tensors and real parameters define the ideal layer y = Wx + b;

2. FP32 tensors and FP32 parameters define the rounded layer;

3. approxT_linear_fp32 supplies the output error budget;

4. a verifier or margin lemma spends that budget against the property we want.

import NN.Proofs.RuntimeApprox.FP32.Layers

open NN.Proofs.RuntimeApprox.FP32

#check approxT_linear_fp32
#check fp32_le_of_real_le_sub_margin
#check fp32_ge_of_real_ge_add_margin

The important point is that the result is not "linear layers are close" as prose. The theorem names the input approximation, parameter approximation, output tolerance, and scalar semantics. Larger MLP and CROWN statements reuse that shape.

5.5.7. How The FP32 Theorems Fit The Tensor Implementation🔗

The theorem names above are not isolated scalar lemmas. They sit on top of the same tensor and layer specifications used elsewhere in TorchLean:

• LinearSpec ℝ inDim outDim is the real valued layer specification.

• LinearSpec FP32.R inDim outDim is the rounded-runtime layer at the proof scalar.

• Tensor R s is still a TorchLean tensor; only the scalar semantics changed.

• approxT is the relation saying the interpreted runtime tensor is within a tolerance of the real tensor, componentwise in the relevant norm.

The linear theorem is therefore a real implementation bridge, not just a scalar exercise:

import NN.Proofs.RuntimeApprox.FP32.Layers

open NN.Proofs.RuntimeApprox.FP32

-- Real linear layer and FP32 linear layer are related by an explicit output budget.
#check approxT_linear_fp32

The MLP theorems compose that layer theorem with rounded activations:

import NN.Proofs.RuntimeApprox.FP32.MLP

open NN.Proofs.RuntimeApprox.FP32

-- Whole-network approximation statements for common examples.
#check approxT_reluMlp2_fp32
#check approxT_tanhMlp3_fp32

Then the CROWN/IBP theorem widens a real valued verification box so it also contains the interpreted FP32 output:

import NN.Proofs.RuntimeApprox.FP32.CROWN

open NN.Proofs.RuntimeApprox.FP32

#check ibpBound_contains_reluMlp2_fp32

That is the intended user workflow: train or import a model, lower or specify its tensor computation, prove/check a real valued enclosure, and then spend an explicit FP32 error budget to keep the checked claim connected to rounded execution.

5.5.8. Proof Model Versus IEEE Execution🔗

The rounded-real FP32 model has a narrower, reusable analysis interface. The bit-level IEEE32Exec model handles NaN, infinity, signed zeros, and executable binary32 behavior. The bridge theorems connect the two on the finite path.

5.5.9. What People Usually Cite🔗

Usually the right answer is:

• FP32 for the proof model,

• IEEE32Exec for executable bit behavior,

• and NN.Proofs.RuntimeApprox source tree for the theorem family that turns real bounds into float aware bounds.

For the float model split, read Floating-Point Semantics. For how these bounds plug into tools, read Verification.

5.5.10. References🔗

• IEEE 754-2019 standard: https://standards.ieee.org/standard/754-2019.html

• Goldberg, "What Every Computer Scientist Should Know About Floating-Point Arithmetic", https://dl.acm.org/doi/10.1145/103162.103163

• Flocq: https://flocq.gitlabpages.inria.fr/

• Higham, Accuracy and Stability of Numerical Algorithms (2nd ed., SIAM, 2002), the standard numerical analysis reference for forward error and stability arguments of the kind TorchLean packages into margin-transfer lemmas.