mathguard — Math-Heavy Optimization for AI Code

lemmaly makes you pick the right classical algorithm. mathguard kicks in when the classical algorithm is already optimal but mathematics gives a better bound — usually by accepting bounded approximation, exploiting structure, or moving to a smarter algebraic space.

The model knows these techniques. It almost never proposes them spontaneously. mathguard fixes that.

Violating the letter of these rules is violating the spirit of the skill. A Bloom filter where the caller assumed exact answers is a production incident, not an optimization.

When to Use This Skill

Use mathguard when:

• Working with large-scale data (n ≥ 10⁶): similarity search, deduplication, top-K / heavy-hitters, streaming analytics, cardinality estimation, embeddings, recommender systems.
• Doing signal/image processing, polynomial or big-integer arithmetic, convolution, graph distance, computational geometry, randomized algorithms.
• The classical O(n log n) is already the floor and you need an asymptotic win (Bloom filter, HyperLogLog, Count-Min Sketch, MinHash/LSH, FFT/NTT, Johnson-Lindenstrauss projection, sweep line, kd-tree/BVH, fast exponentiation, monoid parallel reduction, amortized potential method).
• Loaded after lemmaly has confirmed the classical answer is not enough.

Do not use mathguard when:

• The caller needs exact answers (auth, billing, dedup-for-correctness, primary keys).
• n is small (n < 10⁴) and the path is not hot.
• The bottleneck is I/O, not CPU/memory.

The Iron Law

NO APPROXIMATE STRUCTURE WITHOUT WRITTEN ε/δ AND EXPLICIT CALLER ACCEPTANCE

Probabilistic data structures (Bloom, HyperLogLog, Count-Min, MinHash/LSH, t-digest), randomized projections (JL), and lossy transforms (floating FFT) all change the answer's meaning. Before proposing one:

1. Write the error parameter the caller will see (false-positive rate, relative error, distortion bound).
2. Identify the caller and state, in one sentence, that they tolerate this kind of wrong answer.
3. If you cannot identify the caller, or they need exact (auth checks, billing, dedup keys, deduplication for correctness, anything that flows into a primary key), DO NOT propose the approximate structure. Keep classical, or escalate to a sharded/streaming exact design.

This rule has saved more incidents than any other in this skill. Do not soften it.

Non-negotiable rules

1. Declare exact vs approximate up front. Before suggesting a math-level technique, state:

   • mode: exact or mode: approximate
   • If approximate: the error parameter (ε, δ, false-positive rate) and a sentence on whether the caller can tolerate it.
   • If the caller needs exact and there is no exact win, say so and stop — do not silently degrade to approximate.

2. Cite the technique by name. Never describe a probabilistic or numerical trick in vague terms. Name it: Bloom filter, HyperLogLog, Count-Min Sketch, MinHash + LSH, Johnson–Lindenstrauss projection, FFT, NTT, fast exponentiation, Karatsuba, Strassen, sweep line, kd-tree, BVH, union-find with path compression, Floyd's cycle detection, Boyer-Moore majority, reservoir sampling, Knuth shuffle, Aho-Corasick, suffix automaton, segment tree with lazy propagation, Fenwick tree, monoid scan / parallel prefix. A named technique is auditable; "a smart approximation" is not.

3. State the trade you are making. Every math-level optimization buys something at a cost. In one line:

   • Buys: space, time, wall-clock, parallelism.
   • Costs: accuracy ε=?, code complexity, dependency, non-determinism, numerical stability.
   • If the cost is invisible to the caller, write "callers see no change".

4. Justify the asymptotic win. Do not propose a math technique without a one-line bound argument:

   • "HyperLogLog: count uniques in O(log log n) bits at standard error 1.04/√m."
   • "FFT: polynomial multiplication O(n log n) vs schoolbook O(n²)."
   • "JL projection: preserves pairwise distances within (1±ε) using O(log n / ε²) dimensions."
   • "Sweep line: rectangle overlap from O(n²) pair checks to O(n log n) events." No bound, no proposal.

5. Forbid math cargo-culting. Do not introduce these techniques when:

   • n is small enough that a linear scan finishes in microseconds (n < ~10⁴ unless it is a hot path).
   • The problem is I/O-bound — the math win disappears behind network/disk.
   • Exact answers are required and no exact technique exists.
   • The team will not maintain it (write that down: "team familiarity: ?").

The pre-proposal protocol

Before suggesting a math-level technique, your message must contain — in this order:

1. The classical floor — what is the best non-mathy algorithm and its Big-O? ("Hash join is O(n+m); we're already there.")
2. Why classical is not enough — n too large, space blows up, real-time deadline, etc.
3. The math technique — named (rule 2).
4. Exact or approximate — with ε if approximate (rule 1).
5. The new bound — with one-line derivation (rule 4).
6. The trade — buys/costs (rule 3).
7. When NOT to use this — at least one disqualifier.
8. The code or pseudocode.

If any of 1–7 is missing, do not propose the technique.

Playbook — math technique → problem → win → caveat

Sketches and probabilistic structures (massive data, approximate)

Problem	Classical	Math technique	Win	Caveat
Membership: "have I seen this key?" at scale	Set<id>, O(n) space	Bloom filter	O(n) bits at chosen ε false-positive	False positives only; cannot remove (use Cuckoo if needed)
Count distinct values in a stream	Set to count, O(unique) space	HyperLogLog	O(log log n) bits, ~1% relative error	Approximate; cannot list elements
Top-K / heavy hitters in a stream	full counter, O(unique) space	Count-Min Sketch + heap	O(log(1/δ)·1/ε) space	Overestimates; choose ε,δ deliberately
Document / set similarity at scale	full Jaccard, O(n·m)	MinHash + LSH	Sub-linear ANN query	Tunes recall vs precision; param search
k-NN in high-dim vectors	brute O(n·d)	JL projection → HNSW / IVF	O(log n) per query, (1±ε) distortion	Index build cost; recall < 1
Reservoir of size k from a stream of unknown length	buffer all, O(n) space	Reservoir sampling	O(k) space, uniform sample	Single-pass only
Find majority element	counter map	Boyer-Moore majority vote	O(1) space, O(n) time	Requires majority exists; verify pass
Quantiles in a stream	sort, O(n log n)	t-digest / GK	O(1/ε) space, ε-accurate quantiles	Approximate

Fast arithmetic / transforms (numeric and combinatorial)

Problem	Classical	Math technique	Win	Caveat
Multiply two polynomials / big integers	O(n²)	FFT / NTT / Karatsuba	O(n log n)	Floating FFT loses precision — use NTT for integers
Convolution of two signals	O(n·m)	FFT-based convolution	O((n+m) log(n+m))	Numerical noise at very small magnitudes
pow(a, b) mod p, b large	O(b) multiplications	Fast exponentiation (square-and-multiply)	O(log b)	Watch for overflow inside; use modular arithmetic
GCD of large integers	repeated subtraction	Euclidean algorithm	O(log min)	Standard; AI sometimes still writes the subtraction loop
Matrix multiplication, n large	O(n³)	Strassen (then Coppersmith-Winograd family)	O(n^2.81)	High constant; only wins for very large dense
Solving Ax=b for sparse A	O(n³) dense	Conjugate gradient / sparse LU	O(nnz · iterations)	Numerical conditioning matters
Modular inverse	brute force	Extended Euclidean or Fermat when p prime	O(log p)	p must be prime for Fermat

Dimensionality reduction and linear algebra

Problem	Classical	Math technique	Win	Caveat
Similarity in d-dim, d large	O(n·d) brute	JL projection to k = O(log n / ε²)	O(n·k) at (1±ε) distortion	Random; verify on validation set
Recommender from rating matrix	iterate full matrix	Truncated SVD / matrix factorization	O(k·(n+m)) for rank-k	Choose k; refresh strategy
Document-term similarity	TF-IDF O(n·m)	LSA via SVD	rank-k approximation	Latent dims are not interpretable
PCA on n samples in d dims	O(n·d²)	Randomized SVD	O(n·d·k) for rank-k	Randomized; set oversampling

Geometry (spatial queries)

Problem	Classical	Math technique	Win	Caveat
Range / nearest-neighbor in 2D-3D	O(n) per query	kd-tree / R-tree / BVH	O(log n) per query	Degrades in high d; use ANN instead
Rectangle / interval overlap pairs	O(n²) pair check	Sweep line + active set (BBST)	O((n+k) log n)	k = output size; segment tree variant exists
Polygon point-in-polygon at scale	O(n·v)	BSP / monotone decomposition / R-tree	O(log v) per query after build	Build cost
Convex hull of n points	O(n²) gift wrap	Graham scan / Andrew's monotone chain	O(n log n)	Numerical robustness for collinear
Closest pair of points	O(n²)	Divide and conquer	O(n log n)	Carefully merge across the strip

Graph and algebraic tricks

Problem	Classical	Math technique	Win	Caveat
Connected components under merges	recompute BFS each merge	Union-Find with path compression + rank	α(n) ≈ O(1) per op amortized	Inverse Ackermann is effectively constant
Range sum / update on array	O(n) per query	Fenwick tree	O(log n) per op	Inclusive ranges; off-by-one risk
Range query with monoid (sum/min/max/gcd)	O(n) per query	Segment tree (with lazy if range updates)	O(log n)	More code than Fenwick; more general
LCA in a tree, many queries	O(n) per query	Binary lifting or Euler tour + RMQ	O(log n) or O(1) per query	Preprocessing cost
Shortest path on DAG	Dijkstra	Topo sort + relax	O(V+E)	Only works on DAG
Detect cycle in linked list	hash visited	Floyd's tortoise and hare	O(1) space	Same big-O time, dramatic space win
Parallel reduction over n items	sequential fold	Monoid + parallel scan	O(n/p + log p) on p cores	Operation must be associative; verify it

Amortized and online algorithms

Problem	Classical	Math technique	Win	Caveat
"Dynamic array push is expensive"	per-op O(n) on resize	Amortized analysis (doubling)	O(1) amortized	This is what ArrayList / vec already do; just defend it
Streaming median	re-sort	Two heaps (max-heap + min-heap)	O(log n) per insert	Maintain size invariant
Online interval scheduling	re-sort by deadline	Greedy with priority queue	O(log n) per arrival	Specific objective; check problem fit
Sliding-window max	O(n·k)	Monotonic deque	O(n) total	Window invariant subtle to maintain

Canonical example — counting distinct users

Problem. Count unique users seen across a 24-hour event stream. ~2B events/day, ~50M unique users. Reported on a dashboard, ±2% is acceptable.

Without the protocol — silent OOM, or worse, silent billing error

// "Just use a Set" — silently OOMs the box at ~50M strings
const seen = new Set<string>();
for await (const event of stream) {
  seen.add(event.userId);
}
return seen.size; // exact, but the process died at row 41M

Or worse — proposed with a HyperLogLog "for performance" but plugged into the billing pipeline, which keys off the result. Billing then sees 49.7M instead of 50.0M users and a fraction never get charged.

With the protocol — auditable HLL

// Classical floor: O(unique) memory for an exact Set. At 50M strings × ~50B each, ~2.5GB.
// Why classical is not enough: dashboard box has 512MB and refreshes every minute.
// Technique: HyperLogLog (HLL).
// Mode: approximate. ε ≈ 1.04/√m. With m=2^14 registers → ~0.8% relative error.
// Trade: buys O(log log n)-bit space (~12KB); costs ±0.8% on the displayed count.
// When NOT to use: anything that flows into billing, primary keys, or per-user actions.
// Caller acceptance: confirmed — dashboard product owner accepts ±2%, written in PR.

import { createHLL } from 'hyperloglog-lite';
const hll = createHLL({ precision: 14 });
for await (const event of stream) {
  hll.add(event.userId);
}
return hll.estimate(); // 49.6M ± 0.4M; dashboard reads ~50M

The first version is not "no HLL" — it is "HLL without writing down ε and who tolerates it." The second is identical in technique but auditable: ε is in the comment, the caller is named, the disqualifier (billing) is explicit.

Output discipline

Code that uses a math-level technique must include:

• One comment naming the technique with a doc link or one-line citation.
• The exact error parameters chosen (ε, δ, bits, dimensions, etc.) and why those values.
• A measured or asymptotic justification next to the chosen parameters.
• An exact-mode fallback path, if the caller might need it.

When to escalate or redirect

• The bottleneck is I/O, not CPU/memory → go back to lemmaly rule 4; math will not help.
• You need bit-exact reproducibility → avoid floating FFT, randomized projections, and probabilistic structures.
• The result is consumed by a downstream system that assumes exact → keep classical or wrap with a validation pass.
• You need a correctness proof (not just a bound) → load invariant-guard after picking the technique.

Rationalizations to watch for

Excuse	Reality
"A set works — I'll flag the memory issue in a comment."	Noticing the problem is not solving it. If memory is the budget, ship the structure that respects it.
"Probabilistic structures sound fancy / academic."	Cloudflare runs Bloom filters in the request path. Redis ships HyperLogLog. These are production-tested, not academic.
"Approximate is risky — I'll do exact and let it OOM later."	Silent OOM at 3am is riskier than a stated 0.81% error. State the ε, pick parameters, ship.
"I'll just shard the set across machines."	Sharding multiplies your infra cost; HLL solves it in 12KB on one box. Ask whether you actually need exact.
"FFT is overkill for this."	True 99% of the time. But state the n. At n ≥ ~64 for polynomial mult, schoolbook is already losing.
"JL projection feels too lossy for embeddings."	At ε = 0.1, JL preserves pairwise distances within 10%. For ANN this is almost always fine — measure recall, do not eyeball.

Red flags — STOP

• Proposing a probabilistic structure without stating ε and δ.
• Saying "we can use FFT here" without writing the n at which FFT actually beats schoolbook.
• Using JSON.parse(JSON.stringify(...)) to deep-clone when structuredClone exists, then claiming it as an optimization.
• Recommending Strassen on a 100×100 matrix.
• Switching to approximate output without the caller having agreed to it.
• Naming a technique you cannot derive the bound for.
• Math optimization where n is small and not on a hot path.
• "Should be O(log n) on average" with no average-case argument.

Verification checklist

Before shipping code that uses a math-level technique:

• The technique is named (no "a smart approximation").
• If approximate: ε and δ (or the equivalent error parameter) are written in code or in the PR description.
• The caller has been identified and their tolerance for that error is stated.
• A one-line bound derivation is present (asymptotic or measured).
• At least one disqualifier ("when NOT to use this") is documented.
• An exact-mode fallback exists, OR a one-line note explains why exact is impossible.
• If randomized: the seed strategy is documented (fixed for reproducibility, or stated as non-deterministic).
• Downstream consumers that assume exactness (joins on this value, billing, auth, primary keys) have been audited.

Cannot check every box? The technique is not ready to ship. Keep classical, or stop and ask.

Limitations

• Not for exact-required pipelines. Any system where the result is a primary key, dedup key, billing input, or auth decision is out of scope — keep classical.
• Assumes representative inputs. ε/δ bounds are average-case or high-probability; adversarial inputs can blow past them. State the threat model.
• Library quality varies. Bloom / HLL / MinHash implementations differ in seed strategy, hash function, and memory layout — pick a maintained library and pin the version.
• Numerical stability. Floating FFT, randomized SVD, and JL projection accumulate float error; for combinatorial exactness use NTT or exact integer variants.
• Team-familiarity risk. A technique nobody can debug at 3 a.m. is a liability — write the maintainer note next to the trade-off.
• Not a profiler. mathguard tells you which asymptotic ceiling you can break; it does not measure constant factors. Benchmark before claiming a wall-clock win.

The thesis, in one line

When classical algorithms hit their floor, mathematics still has another floor below. mathguard makes the model reach for it instead of accepting the first answer.

Related Skills

• lemmaly — gateway; pick the classical algorithm first before reaching for math.
• invariant-guard — for stating ε-bounds as part of the postcondition of an approximate algorithm.
• complexity-cuts — when baseline code already exists and the bottleneck is CPU/memory, not approximation.