lem·ma /ˈlemə/ — a small result proved on the way to a larger one. Find the right one and everything after it gets easier.
A San Francisco growth strategy firm. We partner with founders and leadership teams to turn ambitious goals into clear priorities, executable systems, and measurable results — across positioning, acquisition, brand, partnerships, and AI-enabled operations.
The premise
No single channel, campaign, or tactic produces outsized growth on its own. We orchestrate the whole go-to-market — positioning, acquisition, product, brand, partnerships, pricing, and the operations underneath — and find the combination that compounds for your business.
$450M+ in ad spend managed — across Meta, Google, Apple Search Ads, TikTok, LinkedIn, and whatever else the mix calls for.
B2B and B2C — enterprise, SaaS, marketplace, and consumer businesses.
Senior-led and selective — scoped around one problem at a time.
Where we enter
We work with founders and leadership teams at inflection points — where the opportunity is clear but the path to capturing it isn't yet. Outside of those moments, you probably don't need us.
Demand is outrunning your ability to deliver it.
The product works, but the market can't yet see what it's worth.
A go-to-market motion needs to be built, or reset from scratch.
Growth has become everyone's job, which means it's no one's.
Focus areas
Most companies at an inflection point don't need more activity. They need to know which of these is currently holding the rest back — and what to do about it this quarter.
Sharpen who you're for, what you're worth to them, and how you reach them repeatably.
Data-driven user acquisition, conversion, retention, and the math that tells you when to spend.
A differentiated story, plus a creative testing system that keeps producing after we leave.
Channel, reseller, and platform partnerships that open growth you can't buy on an auction.
Applying AI to growth workflows, reporting, and decisions — where it earns its place, not everywhere.
The numbers investors will test, and a story your traction actually supports.
The approach
Most growth problems are not marketing problems.
They are clarity, sequencing, ownership, or operating-system problems that happen to show up in the growth numbers. So we start with a diagnosis, not a deliverable — the constraint is rarely where the symptom appears. We find it, test the smallest honest version of the fix, and only then spend behind it. What you keep at the end is a system your team can run and a number that moved, not a deck.
How engagements start
The first conversation is just a conversation. What follows is a fixed scope for a fixed fee with an end date — not a retainer that renews by default. We'd rather tell you the problem is smaller than you thought than sell you a longer engagement.
Sixty minutes with the founder or leadership team. You describe the goal and what's in the way; we tell you plainly whether we're the right people for it.
A structured review of your numbers, your funnel, and how your team sees the bottleneck — often independently, because where the answers differ is the finding.
A scoped sprint against one named problem, with defined deliverables, a fixed fee, and a decision point at the end. Extend only if there's a reason to.
On the name
Butterfly Lemma captures the idea that complex things often share a hidden underlying structure — and that people coming from very different backgrounds, viewpoints, and disciplines can still arrive at the same deeper truth.
I also love the butterfly itself. It suggests transformation, and it echoes Ray Bradbury's butterfly: a small change creating outsized impact. I've seen that with students, where one intervention changes a trajectory, and with organizations, where one strategic shift unlocks growth.
There's a personal reason too. The lemma was first proved by Hans Zassenhaus, a student of Emil Artin — one of my favorite mathematicians, and someone whose Armenian roots make the connection especially meaningful to me.
Mihran Vardanyan · Founder · San Francisco
Tell us the growth problem you'd solve today if you could only pick one. If we're not the right fit, we'll say so and point you to someone who is.
We work primarily through trusted introductions and take on a limited number of companies at a time.