Applying the Mutual Zero-Knowledge Proof (ZKP) Framework to Startup Growth

Disclaimer: This note is not about pure computational complexity but rather meta-semantically leverages its theoretical framework to materialize startup earnings growth.

1. Theorem of Startup Success

At HITSERIES CAPITAL, operated by TANAAKK, the fundamental truth of startups is defined as follows:

A startup (Prover) that possesses the ability to succeed (a product) may not be immediately recognized as successul It will be rejected by the majority at first. However, to a early buyer (Verifier) with strong evaluation capabilities, the elements of success are evident and easily verifiable. An Accredited Verifier can confirm the Prover's truth which is essential for success without requiring any information disclosure regarding the proof problem (Zero Knowledge Proof).

1-1. Basics of ZKP and the Concept of Mutual ZKP

Zero-Knowledge Proof (ZKP) is a cryptographic method where a Prover demonstrates to a Verifier that a statement is true without revealing any information beyond the truth of the statement itself. ZKP has three key properties:

• Completeness: If the statement is true, the Verifier will accept the proof.
• Soundness: If the statement is false, the Verifier will not be convinced.
• Zero-Knowledge: The Verifier learns nothing beyond the fact that the statement is true.

Traditional ZKP involves a one-way proof. For example, if a mathematics student (Prover) discovers a new axiom, they can convince a professor (Verifier) of its truth without revealing the proof method.

In Zero Knowledge Proof techniques, verification protocols such as the four-color theorem/tricolor diagram approach allow a 99.999% confidence level with a single trial, 99.99999999% with two trials, and 99.9999999999999% with three trials. In practical scenarios, two interactions are sufficient for a Verifier to be highly confident that the Prover holds a valid theory.

2. Applying ZKP Concepts to Startup Products

This framework mirrors how a startup (Prover) that has just released a prototype convinces Verifiers such as early adopters, corporations, and venture capitalists to invest, purchase, or adopt their product. The Verifier cannot access complete proof of the product’s future success but must make decisions based on limited interactions.

2-1. ZKP Allows Deals to Close in Two Conversations

In practice, two interactions are often sufficient for a startup (Prover) to convince a corporate buyer or VC (Verifier) that its product solves an unknown problem (NP-complete). If the Verifier has sufficient evaluation capability, they can validate the proof without gaining additional knowledge.

This implies that due diligence (DD) is not necessary for truly successful venture investments—it is often performed as a ceremonial formality rather than a substantive verification.

2-2. Differences Between Theoretical and Real-World ZKP Applications

In theoretical ZKP, the Verifier is assumed to have absolute evaluation capability. However, in real-world scenarios, a Verifier’s ability to properly assess the proof is not guaranteed.

For example, in the credit card industry, institutions like VISA, Mastercard, and AMEX serve as Trusted Third Parties, ensuring the buyer’s creditworthiness so that merchants do not bear the verification risk. In emerging markets like startups, however, no equivalent trusted entity exists to outsource the verification process. This necessitates an alternative approach: Mutual ZKP.

In Mutual ZKP:

• The Prover proves that the product is valid.
• The Verifier proves they have the true ability to evaluate the product correctly.

Without this mutual validation, even if the Prover presents a valid NP-complete problem, an incapable Verifier will fail to recognize its value, leading to wasted resources and potential failure.

2-3. P vs. NP-Complete Problems in Product Development

An NP-complete problem is one where verification is easy, but finding the solution is computationally difficult. This is distinct from P (Polynomial Time) problems, which can be solved procedurally. For example, manufacturing a standard automobile is a P problem since it follows well-defined steps.

For a startup launching a new product, market acceptance resembles an NP-complete problem because:

• If a successful product exists, an experienced Verifier can recognize it easily.
• However, proving the product’s success requires massive trial-and-error efforts.

2-4. Startups Must Work with NP-Complete Problems

A startup must tackle NP-complete problems rather than P problems. Products that have already been proven and commercialized belong to the P category and are widely available in the market. Startups must solve unknown problems, where the solution is correct but difficult to prove.

This leads back to the original assertion: A startup (Prover) that possesses the ability to succeed (a product) may not be immediately recognized as a future success by the majority. However, to a buyer (Verifier) with strong evaluation capabilities, the elements of success are evident and easily verifiable. An Accredited Verifier can confirm the Prover holds the truth essential for success without requiring any information disclosure regarding the proof problem (Zero Knowledge Proof).

3. Proving Verifier’s Ability to Evaluate

Beyond having a valid product, a startup must ensure that capable Verifiers exist. If Verifiers lack proper evaluation ability, even optimal products will fail due to misclassification or delayed adoption.

3-1. Risks When Verifiers Lack Evaluation Capabilities

If corporate buyers or VCs fail to assess startups properly, the following scenarios occur:

1. NP-hard but unsatisfiable – The startup presents a truly innovative product, but Verifiers fail to recognize its value, stalling its adoption.
2. PSPACE-hard but unsatisfiable – The startup faces computational intractability, requiring excessive resources for validation.
3. Verifier misclassification – Investors or customers misjudge the startup, causing incorrect funding or adoption decisions.

4. Using ZKP to Classify Startup Success and Failure

Mutual ZKP is essential for hyper-growth. Many failed startups were cases where either the Prover lacked a valid solution or the Verifier misclassified a promising product.

4-1. Success When Mutual ZKP Holds

If a startup correctly solves an NP-complete problem and finds Verifiers within a limited timeframe, the problem transitions to NP-easy (solved within the market), leading to hyper-growth.

4-2. Failure Due to Lack of Mutual ZKP

• Failure Case 1: The startup holds a valid solution but fails to find a capable Verifier within the required timeframe.
• Failure Case 2: A startup misclassifies a known P problem as an NP-complete opportunity, leading to lack of differentiation.
• Failure Case 3: A startup falsely believes it holds an NP-complete solution, resulting in wasted resources.

5. Post-Success Failure Cases

Even after successful hyper-growth, startups can fail if they lose Mutual ZKP:

• Case 1: The Prover (startup) loses its original truth, as seen in Kodak’s downfall.
• Case 2: The Prover transitions from solving NP-complete problems to merely iterating on NP-easy solutions, leading to competitive displacement (e.g., Sony Walkman → iPhone).

6. Mutual ZKP Framework for Hyper-Growth Strategy

To ensure sustainable success, startups must:

• Prove their product’s truth via ZKP.
• Find Accredited Verifiers capable of validating their claims.
• Transition from NP-complete to NP-easy within a defined timeframe (e.g., 10 years).
• Continuously present new NP-complete solutions after solving prior ones.

6-2. Mutual ZKP Enables Product-Led Organic Growth

Startups can minimize Customer Acquisition Cost (CAC) by leveraging Mutual ZKP. This approach shifts from evidence-based decision-making to an efficient trust model, accelerating adoption and scaling.

TANAAKK claims that Mutual ZKP is a template for product-led hyper-growth, driving superior capital returns and operating leverage.