# Proof: An object moving at exactly 0.95c experiences a Lorentz factor greater than 3.2

- **Generated:** 2026-03-28
- **Verdict:** PROVED
- **Audit trail:** [proof_audit.md](proof_audit.md) | [proof.py](proof.py)

## Key Findings

- The Lorentz factor for v = 0.95c is **γ ≈ 3.2026**, which is strictly greater than 3.2.
- Three independent computation methods (IEEE 754 float, 50-digit Decimal, exact rational arithmetic) all agree to >10 decimal places.
- Exact rational arithmetic confirms γ² = 400/39 > 256/25 = (3.2)² with **zero floating-point involvement**, eliminating any doubt about rounding.
- The margin above 3.2 is ~0.0026 — small but unambiguous.

## Claim Interpretation

**Natural language:** An object moving at exactly 0.95c relative to a stationary observer experiences a Lorentz factor γ greater than 3.2.

**Formal interpretation:** The Lorentz factor γ = 1/√(1 - v²/c²), evaluated at v/c = 0.95 (exact), must be strictly greater than 3.2. "Greater than" is interpreted as the strict inequality (>). If γ were exactly 3.2, the claim would be FALSE. This is the more conservative interpretation — using ≥ would make the claim easier to prove.

The Lorentz factor is a standard, universally accepted definition from special relativity with no competing formulations. With v/c given as an exact value, this is a pure mathematical computation with no empirical ambiguity.

## Evidence Summary

| ID | Fact | Verified |
|----|------|----------|
| A1 | Primary computation of γ via direct formula | Computed: γ ≈ 3.2025630761 |
| A2 | Cross-check via high-precision decimal arithmetic | Computed: γ ≈ 3.2025630761 (agrees with A1) |
| A3 | Cross-check via algebraic simplification γ² = 1/(1-β²) | Computed: γ² = 400/39 > 256/25 = (3.2)² (exact rational proof) |

*Source: proof.py JSON summary*

## Proof Logic

The Lorentz factor is defined as:

γ = 1 / √(1 - β²), where β = v/c

**Step 1 — Compute β²:** With β = 0.95, β² = 0.9025 (A1).

**Step 2 — Compute 1 - β²:** 1 - 0.9025 = 0.0975 (A1).

**Step 3 — Compute γ:** γ = 1/√(0.0975) ≈ 3.2025630761 (A1).

**Step 4 — Compare:** 3.2026 > 3.2 = True.

**Cross-verification:** The same result was confirmed via 50-digit Decimal arithmetic (A2), yielding γ = 3.2025630761017426696650733953537... — agreement to all displayed digits. Additionally, exact rational arithmetic (A3) shows β = 19/20, so β² = 361/400, 1 - β² = 39/400, and γ² = 400/39. Since 400/39 > 256/25 (equivalently 10000/975 > 9984/975), we have γ² > (3.2)² and therefore γ > 3.2, proved without any floating-point operations.

*Source: author analysis*

## Counter-Evidence Search

1. **Alternative definitions of γ?** — The Lorentz factor γ = 1/√(1-v²/c²) is the only definition used in special relativity. The reciprocal 1/γ is sometimes referenced but is clearly a different quantity. No alternative would change the result.

2. **Floating-point error risk?** — Three independent computation methods (float, Decimal, exact Fraction) all agree. The exact rational computation uses no floating-point at all, confirming the result is independent of representation.

3. **Margin too small for confidence?** — γ ≈ 3.2026 vs threshold 3.2 gives a margin of ~0.0026. While small, this is confirmed by exact arithmetic (γ² = 400/39 ≈ 10.2564 vs 3.2² = 10.24, margin ~0.016 in squared domain) and is many orders of magnitude above floating-point epsilon.

*Source: proof.py JSON summary*

## Conclusion

**PROVED.** The Lorentz factor for an object moving at v = 0.95c is γ ≈ 3.2026, which is strictly greater than 3.2. This result is confirmed by three mathematically independent computation methods — IEEE 754 floating-point, 50-digit Decimal, and exact rational arithmetic — all in perfect agreement. The exact rational proof (γ² = 400/39 > 256/25 = 3.2²) establishes the result with no floating-point involvement whatsoever.

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Generated by [proof-engine](https://github.com/yaniv-golan/proof-engine) v0.10.0 on 2026-03-28.
