"If a company raises $5 million in Series A at a $25 million post-money valuation and exits at $500 million seven years later, the annualized return to the Series A investor exceeds 40% before dilution."
This is a clean mathematical result: the stated scenario produces an annualized return of about 53.4%, comfortably above the claimed 40% threshold.
What Was Claimed?
The claim describes a specific Series A investment scenario and asks whether the annual return to the investor would exceed 40%. Anyone who has heard VC investors talk about "target returns" or "fund hurdle rates" has likely encountered numbers in this range — the claim is checking whether this particular deal structure actually clears that bar.
What Did We Find?
The math here is straightforward. When a Series A investor puts in $5 million at a $25 million post-money valuation, they own exactly 20% of the company (5 ÷ 25 = 20%). If the company later exits at $500 million and the investor still holds that same 20% stake, their share of the exit proceeds is $100 million.
That turns a $5 million investment into $100 million — a 20× return (called a MOIC, or multiple on invested capital). Spread over 7 years, the annualized rate of return works out to 20^(1/7) − 1, which equals approximately 53.41% per year.
Two independent calculation methods — one using exact symbolic algebra (sympy) and one using a numerical equation-solver (scipy) — both produced 53.41%, agreeing to better than one part in ten trillion.
The claim's 40% threshold is therefore exceeded by about 13 percentage points. The claim is true.
What Should You Keep In Mind?
The single most important caveat is the phrase "before dilution." In the real world, a startup that raises a Series A will almost certainly raise additional rounds of financing before it exits. Each new round typically dilutes earlier investors — their percentage ownership shrinks as new shares are issued. If the Series A investor is diluted by 20% in each of three subsequent rounds, their stake would fall from 20% to roughly 10.24%, cutting their exit proceeds to about $51 million and reducing the annualized return to about 39.5% — just below the 40% threshold.
So the claim is mathematically true for the clean hypothetical it describes, but the "before dilution" qualifier does real work here. It is not a throwaway caveat.
How Was This Verified?
This proof involves no empirical sources — it is pure mathematics, and the computation is machine-checked. You can read the structured proof report, review the full verification audit, or re-run the proof yourself with a standard Python installation (requires sympy and scipy).
detailed evidence
Evidence Summary
All facts are Type A (computed):
| ID | Description | Value |
|---|---|---|
| A1 | Series A ownership stake | 20.00% |
| A2 | Exit proceeds (20% × $500M, no dilution) | $100,000,000 |
| A3 | MOIC (multiple on invested capital) | 20.00× |
| A4 | CAGR via sympy exact symbolic solve | 53.4127% |
Proof Logic
| Method | CAGR | Agreement |
|---|---|---|
| sympy exact symbolic solve | 53.4127% | — |
| scipy brentq NPV=0 numerical solve | 53.4127% | Δ = 1.8 × 10⁻¹³ ✓ |
Conclusion
The claim is PROVED. Under the stated no-dilution scenario: - Ownership: 20% - Exit proceeds: $100M - MOIC: 20× - CAGR: 53.41% (> 40% threshold by 13.4 percentage points)
audit trail
Subject: Annualized return (CAGR) to a Series A investor who invests $5M at a $25M post-money valuation, holds for 7 years, and exits at a $500M company valuation.
Formal specification: CAGR > 0.40 (40%)
Formalization scope: The claim specifies "before dilution" — this proof models a scenario with no intermediate financing rounds. The investor retains the full 20% ownership stake (= $5M / $25M) from investment through exit. If typical dilution of ~20% per subsequent round is modeled over 7 years (3 rounds), CAGR falls to approximately 39.5% — below the 40% threshold. The "before dilution" qualifier is therefore load-bearing.
CAGR is defined as MOIC^(1/years) − 1, which equals IRR for a two-cashflow investment (initial outlay, single exit).
Cite this proof
Proof Engine. (2026). Claim Verification: “If a company raises $5 million in Series A at a $25 million post-money valuation and exits at $500 million seven years later, the annualized return to the Series A investor exceeds 40% before dilution.” — Proved. https://proofengine.info/proofs/if-a-company-raises-5-million-in-series-a-at-a-25-million-post-money-valuation/
Proof Engine. "Claim Verification: “If a company raises $5 million in Series A at a $25 million post-money valuation and exits at $500 million seven years later, the annualized return to the Series A investor exceeds 40% before dilution.” — Proved." 2026. https://proofengine.info/proofs/if-a-company-raises-5-million-in-series-a-at-a-25-million-post-money-valuation/.
@misc{proofengine_if_a_company_raises_5_million_in_series_a_at_a_25_million_post_money_valuation,
title = {Claim Verification: “If a company raises $5 million in Series A at a $25 million post-money valuation and exits at $500 million seven years later, the annualized return to the Series A investor exceeds 40\% before dilution.” — Proved},
author = {{Proof Engine}},
year = {2026},
url = {https://proofengine.info/proofs/if-a-company-raises-5-million-in-series-a-at-a-25-million-post-money-valuation/},
note = {Verdict: PROVED. Generated by proof-engine v1.11.0},
}
TY - DATA TI - Claim Verification: “If a company raises $5 million in Series A at a $25 million post-money valuation and exits at $500 million seven years later, the annualized return to the Series A investor exceeds 40% before dilution.” — Proved AU - Proof Engine PY - 2026 UR - https://proofengine.info/proofs/if-a-company-raises-5-million-in-series-a-at-a-25-million-post-money-valuation/ N1 - Verdict: PROVED. Generated by proof-engine v1.11.0 ER -
View proof source
This is the proof.py that produced the verdict above. Every fact traces to code below. (This proof has not yet been minted to Zenodo; the source here is the working copy from this repository.)
"""
Proof: Series A IRR on $5M → $500M exit in 7 years (no dilution)
Claim: If a company raises $5 million in Series A at a $25 million post-money
valuation and exits at $500 million seven years later, the annualized return
to the Series A investor exceeds 40% before dilution.
Generated: 2026-04-08
"""
import os
import sys
PROOF_ENGINE_ROOT = os.environ.get("PROOF_ENGINE_ROOT")
if not PROOF_ENGINE_ROOT:
_d = os.path.dirname(os.path.abspath(__file__))
while _d != os.path.dirname(_d):
if os.path.isdir(os.path.join(_d, "proof-engine", "skills", "proof-engine", "scripts")):
PROOF_ENGINE_ROOT = os.path.join(_d, "proof-engine", "skills", "proof-engine")
break
_d = os.path.dirname(_d)
if not PROOF_ENGINE_ROOT:
raise RuntimeError("PROOF_ENGINE_ROOT not set and skill dir not found via walk-up from proof.py")
sys.path.insert(0, PROOF_ENGINE_ROOT)
import sympy as sp
from scipy.optimize import brentq
from scripts.computations import compare, explain_calc, cross_check, emit_proof_summary
# =============================================================================
# 1. CLAIM INTERPRETATION (Rule 4)
# =============================================================================
CLAIM_NATURAL = (
"If a company raises $5 million in Series A at a $25 million post-money "
"valuation and exits at $500 million seven years later, the annualized "
"return to the Series A investor exceeds 40% before dilution."
)
CLAIM_FORMAL = {
"subject": "Annualized return (CAGR) to Series A investor in described scenario",
"property": "CAGR over 7-year hold period, before dilution",
"operator": ">",
"operator_note": (
"'Before dilution' means the investor retains the full ownership stake "
"from Series A through exit — no intermediate rounds are modeled. "
"Ownership = invested / post-money = $5M / $25M = 20.0%. "
"Exit proceeds = ownership × exit_value = 20% × $500M = $100M. "
"MOIC = exit_proceeds / invested = $100M / $5M = 20x. "
"CAGR = MOIC^(1/years) - 1 = 20^(1/7) - 1. "
"Threshold is 0.40 (40%). The claim is TRUE if CAGR > 0.40."
),
"threshold": 0.40,
}
# =============================================================================
# 2. COMPUTATIONS (Type A — pure math)
# =============================================================================
INVESTED = 5_000_000 # $5M Series A investment
POST_MONEY = 25_000_000 # $25M post-money valuation
EXIT_VALUE = 500_000_000 # $500M exit valuation
HOLD_YEARS = 7 # 7-year hold period
ownership = explain_calc(
"INVESTED / POST_MONEY",
{"INVESTED": INVESTED, "POST_MONEY": POST_MONEY},
label="A1: Series A ownership stake",
)
exit_proceeds = explain_calc(
"ownership * EXIT_VALUE",
{"ownership": ownership, "EXIT_VALUE": EXIT_VALUE},
label="A2: Exit proceeds (no dilution)",
)
moic = explain_calc(
"exit_proceeds / INVESTED",
{"exit_proceeds": exit_proceeds, "INVESTED": INVESTED},
label="A3: MOIC (multiple on invested capital)",
)
# Primary computation: sympy exact symbolic solve
r_sym = sp.Symbol("r", positive=True)
cagr_sympy = float(
sp.solve(sp.Eq((1 + r_sym) ** HOLD_YEARS, moic), r_sym)[0]
)
print(f"A4: CAGR (sympy exact symbolic): {cagr_sympy:.8f} = {cagr_sympy * 100:.6f}%")
# Cross-check: scipy brentq — independent NPV=0 solve
def npv_func(r):
return -INVESTED + exit_proceeds / (1 + r) ** HOLD_YEARS
cagr_scipy = brentq(npv_func, 0.001, 5.0, xtol=1e-12)
print(f"A4-xcheck: CAGR (scipy brentq NPV=0): {cagr_scipy:.8f} = {cagr_scipy * 100:.6f}%")
cross_check(
cagr_sympy, cagr_scipy,
tolerance=1e-8,
mode="absolute",
label="A4 cross-check: sympy vs scipy CAGR agreement",
)
# =============================================================================
# 3. VERDICT
# =============================================================================
verdict_holds = compare(
cagr_sympy, ">", CLAIM_FORMAL["threshold"],
label="CAGR > 40% threshold",
)
VERDICT = "PROVED" if verdict_holds else "DISPROVED"
# =============================================================================
# 4. ADVERSARIAL CHECKS (Rule 5)
# =============================================================================
adversarial_checks = [
{
"description": "Post-money vs pre-money valuation interpretation",
"verification_performed": (
"Confirmed 'post-money valuation' is the standard VC term meaning valuation "
"AFTER the investment closes. Pre-money = $25M - $5M = $20M. "
"Investor ownership = $5M / $25M = 20.0%. No alternative interpretation "
"of 'post-money' exists in standard VC practice."
),
"breaks_proof": False,
},
{
"description": "Effect of dilution if 'before dilution' caveat removed",
"verification_performed": (
"Modeled 3 subsequent rounds at 20% dilution each: ownership → "
"20% × 0.80^3 ≈ 10.24%. Exit proceeds → $51.2M. MOIC → 10.24x. "
"CAGR → 10.24^(1/7) - 1 ≈ 39.5% < 40%. "
"The 'before dilution' qualifier is therefore load-bearing: "
"typical dilution would push the IRR near or below 40%. "
"The claim is mathematically correct for the stated (no-dilution) scenario."
),
"breaks_proof": False,
},
{
"description": "Is CAGR the correct measure for 'annualized return'?",
"verification_performed": (
"CAGR (compound annual growth rate) is the standard measure of annualized "
"return for a single lump-sum investment with a single exit. It is equivalent "
"to IRR when there are only two cash flows (initial investment and final exit). "
"No alternative annualized return formula would change the result for this "
"simple two-cashflow scenario."
),
"breaks_proof": False,
},
]
# =============================================================================
# 5. FACT REGISTRY
# =============================================================================
FACT_REGISTRY = {
"A1": {"label": "ownership — Series A ownership stake", "method": None, "result": None},
"A2": {"label": "exit_proceeds — exit proceeds with no dilution", "method": None, "result": None},
"A3": {"label": "moic — multiple on invested capital", "method": None, "result": None},
"A4": {"label": "cagr_sympy — annualized return (CAGR) via sympy exact solve", "method": None, "result": None},
}
# =============================================================================
# 6. JSON SUMMARY
# =============================================================================
if __name__ == "__main__":
summary = {
"claim_natural": CLAIM_NATURAL,
"claim_formal": CLAIM_FORMAL,
"fact_registry": FACT_REGISTRY,
"cross_checks": [
{
"description": "sympy exact symbolic vs scipy brentq NPV=0",
"value_a": round(cagr_sympy, 8),
"value_b": round(cagr_scipy, 8),
"tolerance": 1e-8,
"passed": abs(cagr_sympy - cagr_scipy) <= 1e-8,
}
],
"adversarial_checks": adversarial_checks,
"verdict": VERDICT,
"key_results": {
"ownership_pct": round(ownership * 100, 4),
"moic": round(moic, 6),
"cagr_pct": round(cagr_sympy * 100, 4),
"threshold_pct": CLAIM_FORMAL["threshold"] * 100,
"claim_holds": cagr_sympy > CLAIM_FORMAL["threshold"],
},
"generator": {
"name": "proof-engine",
"version": "1.11.0",
"repo": "https://github.com/yaniv-golan/proof-engine",
"generated_at": "2026-04-08",
},
}
emit_proof_summary(summary)
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