From machine‑epsilon to Planck‑length: How symmetry and representation shape computational boundaries
In computational mathematics, we often encounter limits to what we can calculate with precision. These limits arise not just from hardware constraints, but from fundamental mathematical structures.
Scaffold: A discrete mathematical structure that supports computation by defining the granularity of possible state changes.
Just as physical scaffolding has discrete steps, computational scaffolds have minimum "step sizes" that limit how finely we can move through a state space. This presentation examines these scaffolds through group theory, revealing deep connections between computational precision and mathematical symmetry.
When we modify a system's state, we perform what mathematicians call a group action.
A reference configuration S exists in a state space. A group G acts on this space via the mapping S → g·S where g∈G.
What engineers call a "perturbation Δ" is mathematically an element g∈G of this group.
ℝn (real numbers) under addition. Allows theoretically infinite precision.εℤn (integer multiples of ε). Has a minimum "step size" ε.Consider moving a particle in 3D space:
A representation ρ : G → GL(V) maps group elements into linear transformations on a vector space V.
In computation, V is our finite address space (memory).
The smallest non-identity element g∈G whose representation ρ(g) remains distinguishable from the identity defines the machine-epsilon ε.
Formally: ε = inf{ ||g|| : g∈G, g≠e, ρ(g)≠ρ(e) }
In IEEE-754 double precision:
A common misconception is that more memory (RAM) automatically means finer computational precision. Group theory shows why this fails:
Extra RAM enlarges dim(V), but if ker(ρ) already contains all g with ||g|| < ε, then:
ker(ρnew) = ker(ρold) despite dim(Vnew) > dim(Vold)
Warning: Adding more bits to a representation doesn't necessarily improve precision if the representation's kernel (elements mapped to identity) remains unchanged.
Doubling memory from 32 to 64 bits while keeping the same fixed-point format (e.g., 16 bits after decimal) only allows representing larger numbers, not more precise ones. The scaffold granularity (2-16) remains unchanged.
The integers ℤ form a group under addition. We can enumerate non-zero integers in different ways, creating three different "alphabets":
α₁: …, -3, -2, -1, 1, 2, 3, … (standard ordering) α₂: 1, 2, 3, …, -1, -2, -3, … (positives first) α₃: 1, -1, 2, -2, 3, -3, … (alternating)
Each of these provides a different way to traverse the same group. The canonical map φi between any two alphabets is the unique bijective order-preserving automorphism fixed by the chosen sign policy.
Different programming languages enumerate error codes using different schemas:
| Alphabet | Example System | Characteristic |
|---|---|---|
| α₁ | POSIX error codes | Positive/negative have distinct meanings |
| α₂ | HTTP status codes | All success codes first (2xx), then error codes (4xx, 5xx) |
| α₃ | Some binary protocols | Alternating to maximize Hamming distance between adjacent codes |
Computational failures often occur when we hit the limits of our scaffold. In the language of group cohomology, this manifests as an obstruction class [ω] ≠ 0.
In group cohomology, an obstruction class represents a mathematical barrier to extending a representation from a subgroup to the full group while preserving essential properties.
When your calculation hits machine-epsilon, you're encountering the computational shadow of such an obstruction.
When computing f(x) = √(x² + 1) - x for large x:
f(x) = 1/(√(x² + 1) + x) to avoid the obstructionWhen simulating orbital dynamics:
Theorem: Let S be a state space and Aut(S) its automorphism group under composition. Then the set of all permissible perturbations forms a subgroup G ≤ Aut(S).
Proof:
g,h∈G then applying g followed by h is the composite h∘g. By physical closure, this composite is also a permissible perturbation, so h∘g∈G.e∈G.g∈G has an inverse g-1∈G that undoes its effect.Aut(S) is associative, so (f∘g)∘h = f∘(g∘h) for all f,g,h∈G.Therefore, G satisfies all group axioms. ∎
Theorem: The set εℤn = {εk | k∈ℤn} with addition as the group operation forms a discrete subgroup of ℝn.
Proof:
εj, εk ∈ εℤn, their sum ε(j+k) ∈ εℤn since j+k ∈ ℤn.0 = ε·0 ∈ εℤn.εk ∈ εℤn, its additive inverse -εk = ε(-k) ∈ εℤn.ℝn.r = ε/2. For any distinct εj, εk ∈ εℤn, the open balls B(εj,r) and B(εk,r) are disjoint. This is because ||εj - εk|| = ε||j-k|| ≥ ε for j ≠ k, which exceeds the combined radii 2r = ε. Hence, εℤn has the discrete topology.Therefore, εℤn is a discrete subgroup of ℝn. ∎
Theorem: There exist unique order-preserving bijections between the three enumeration schemes α₁, α₂, and α₃ of the non-zero integers ℤ∖{0}.
Proof:
φ₁ = id (identity map) as the trivial bijection from α₁ to itself.φ₂ : α₁ → α₂:
φ₂(k) = k for k > 0φ₂(−k) = M+k where M is the position after all positive integersφ₃ : α₁ → α₃:
k to position 2k−1 if k > 0−k to position 2k if k > 0The remaining bijections α₂ → α₃ can be derived as composites of the above. ∎
Theorem: The computational precision limit is exactly characterized by the kernel of the representation ρ : G → GL(V).
Proof:
g,h∈G are computationally distinguishable if and only if their representations ρ(g) and ρ(h) differ in at least one bit in the finite address space V.g and h are distinguishable if and only if ρ(g⁻¹h) ≠ ρ(e), or g⁻¹h ∉ ker ρ.g∈ker ρ, we have ρ(g) = ρ(e), meaning g is computationally indistinguishable from the identity element.ker ρ affects at least one bit in the representation and is therefore detectable.g with minimal norm such that g ∉ ker ρ.Therefore, the limit of computational precision is precisely characterized by ker ρ. ∎
This theoretical framework has concrete applications in: