# Proof: 2026 is both a "happy number" and mathematically "perfect"

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

---

## Key Findings

- **SC1 (happy number): PROVED.** 2026 is a happy number. The digit-square-sum sequence is: 2026 → 44 → 32 → 13 → 10 → 1. Confirmed by two independent algorithms (A1, A2).
- **SC2 (perfect number): DISPROVED.** 2026 is not a perfect number. Its prime factorisation is 2 × 1013 (1013 is prime). The sum of proper divisors is 1 + 2 + 1013 = **1016**, not 2026. The deficit is 1010. Confirmed by two independent methods (A3, A4).
- **SC3 ("cosmically special"): NOT EVALUABLE.** This is a rhetorical expression with no mathematical definition. It is excluded from the verdict.
- **Compound claim fails** because SC2 is false. The claim "BOTH a happy number AND perfect" does not hold.

---

## Claim Interpretation

**Natural language claim:** 2026 is both a "happy number" and mathematically "perfect," proving the year is cosmically special.

This is a compound claim with three parts:

**SC1 — Happy number:** A positive integer n is *happy* if repeatedly replacing n with the sum of squares of its decimal digits eventually reaches 1. All other integers are *unhappy* (they cycle through a fixed loop containing 4). This is the standard mathematical definition (OEIS A007770). The claim is interpreted as a Boolean: is 2026 happy under this definition?

**SC2 — Perfect number:** A positive integer n is *perfect* if the sum of its proper divisors (all positive divisors excluding n itself) equals n exactly. This is the standard mathematical definition (Euclid, *Elements* Book IX, Proposition 36; OEIS A000396). The known perfect numbers are 6, 28, 496, 8128, 33550336, ... — only 51 are known as of 2024. The claim "mathematically perfect" is interpreted as this strict definition, not a colloquial usage, because the claim invokes mathematical precision.

**SC3 — "Cosmically special":** This is a rhetorical or metaphorical expression. No mathematical definition of "cosmically special number" exists in the literature. This sub-claim is excluded from formal evaluation.

**Compound operator:** The claim holds only if SC1 **AND** SC2 are both true.

---

## Evidence Summary

| ID | Fact | Verified |
|----|------|----------|
| A1 | SC1: Is 2026 a happy number? (Floyd cycle detection on digit-square-sum sequence) | Computed: True — sequence [2026, 44, 32, 13, 10, 1] terminates at 1 |
| A2 | SC1 cross-check: Happy-number verification via unhappy-cycle membership | Computed: True — agrees with A1 |
| A3 | SC2: Is 2026 a perfect number? (direct enumeration of proper divisors) | Computed: 1016 (need 2026 for perfect; deficit = 1010) |
| A4 | SC2 cross-check: Perfect-number via multiplicative σ formula | Computed: 1016 — agrees with A3 |
| A5 | SC2 factorisation: prime factorisation of 2026 | Computed: {2: 1, 1013: 1} (2026 = 2 × 1013) |

---

## Proof Logic

### SC1: Is 2026 a happy number?

Starting with 2026, repeatedly sum the squares of decimal digits (A1):

| Step | Number | Computation |
|------|--------|-------------|
| 0 | 2026 | start |
| 1 | 44 | 2²+0²+2²+6² = 4+0+4+36 = 44 |
| 2 | 32 | 4²+4² = 16+16 = 32 |
| 3 | 13 | 3²+2² = 9+4 = 13 |
| 4 | 10 | 1²+3² = 1+9 = 10 |
| 5 | **1** | 1²+0² = 1+0 = **1** |

The sequence reaches 1 in 5 steps. By the standard definition, **2026 is a happy number** (A1, A2 — two independent algorithms agree).

### SC2: Is 2026 a perfect number?

The prime factorisation of 2026 is established first (A5): 2026 ÷ 2 = 1013, and 1013 is prime (not divisible by any prime ≤ √1013 ≈ 31.8). So 2026 = 2¹ × 1013¹.

The proper divisors of 2026 are: {1, 2, 1013}. Their sum is 1 + 2 + 1013 = **1016** (A3, A4 — two independent methods agree).

For 2026 to be perfect, this sum must equal 2026. It does not: 1016 ≠ 2026. The deficit is 1010.

For comparison, the smallest perfect numbers are 6 (1+2+3=6) and 28 (1+2+4+7+14=28). A perfect number of roughly the size of 2026 would require σ(n) = 4052 — far more divisors than 2026's sparse factorisation allows.

**2026 is not a perfect number. It is deficient** (σ(n) = 3042 < 4052 = 2n).

### Compound claim

Because SC2 is false, the compound "BOTH happy AND perfect" claim fails. SC1 is proved; SC2 is disproved.

---

## Counter-Evidence Search

**Alternative definitions of "happy number":** The standard definition operates in base 10. Base-2 happy numbers exist but are a different concept; the unqualified term always refers to base 10 in mathematical literature. No alternative definition changes the result for 2026.

**Alternative definitions of "perfect number":** Several related concepts were checked: quasi-perfect (σ(n) = 2n+1), almost perfect (σ(n) = 2n−1), semiperfect (sum of a subset of proper divisors = n), abundant (σ(n) > 2n), deficient (σ(n) < 2n). With σ(2026) = 3042 and 2n = 4052, 2026 is *deficient* — it does not qualify under any of these variants either.

**Factorisation error check:** 2026 / 2 = 1013 exactly. Primality of 1013 was verified by trial division through all primes ≤ 31 (= ⌊√1013⌋). Two independent algorithms for the sum of proper divisors both return 1016.

**"Cosmically special" as a defined term:** No mathematical definition found. The phrase is rhetorical.

---

## Conclusion

**Verdict: PARTIALLY VERIFIED**

- **SC1 (happy number): PROVED.** 2026 is confirmed to be a happy number. The digit-square-sum sequence [2026, 44, 32, 13, 10, 1] reaches 1 in 5 steps. Two independent algorithms agree.
- **SC2 (perfect number): DISPROVED.** 2026 = 2 × 1013. The sum of proper divisors is 1016, not 2026 (deficit: 1010). This was confirmed by two independent methods. No variant of "perfect number" is satisfied by 2026.
- **SC3 ("cosmically special"): NOT EVALUABLE.** A rhetorical claim without mathematical content.

The compound claim — that 2026 is *both* a happy number *and* a perfect number — **does not hold**. 2026 earns the "happy" designation but falls well short of mathematical perfection. The nearest perfect number is 8128, and the next above it is 33,550,336.

This proof has no citations (pure mathematics). All results are Type A facts verified entirely by computation and fully reproducible by running `python proof.py`.

---

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