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

- **Generated:** 2026-03-28
- **Reader summary:** [proof.md](proof.md)
- **Proof script:** [proof.py](proof.py)

---

## Claim Specification

*Source: proof.py JSON summary `claim_formal`*

| Field | Value |
|-------|-------|
| Subject | 2026 |
| Compound operator | AND (SC1 AND SC2; SC3 excluded as non-evaluable) |
| SC1 property | happy number |
| SC1 operator | == True |
| SC1 operator note | Standard definition: sum of squares of decimal digits repeatedly, reaches 1. OEIS A007770. |
| SC2 property | perfect number |
| SC2 operator | == True |
| SC2 operator note | Standard definition: sum of proper divisors = n. OEIS A000396. "Mathematically perfect" interpreted strictly. |
| SC3 property | cosmically special |
| SC3 verdict | NOT_EVALUABLE — rhetorical/metaphorical, no mathematical definition exists. |

---

## Fact Registry

*Source: proof.py JSON summary `fact_registry`*

| ID | Type | Label |
|----|------|-------|
| A1 | Computed | SC1: Is 2026 a happy number? (iterative digit-square-sum algorithm) |
| A2 | Computed | SC1 cross-check: Happy-number verification via cycle membership (OEIS A007770 structure) |
| A3 | Computed | SC2: Is 2026 a perfect number? (compute sum of proper divisors) |
| A4 | Computed | SC2 cross-check: Perfect-number verification via multiplicative sigma formula |
| A5 | Computed | SC2 factorisation: prime factorisation of 2026 |

---

## Full Evidence Table

*Source: proof.py JSON summary `fact_registry`*

### Type A (Computed) Facts

| ID | Fact | Method | Result |
|----|------|--------|--------|
| A1 | SC1: Is 2026 a happy number? | `is_happy_iterative(2026)`: Floyd cycle detection on digit-square-sum sequence | True — sequence [2026, 44, 32, 13, 10, 1] terminates at 1 |
| A2 | SC1 cross-check: happy-number via cycle membership | `is_happy_cycle_check(2026)`: halt on UNHAPPY_CYCLE membership | True — agrees with A1 |
| A3 | SC2: Is 2026 a perfect number? | `sum_of_proper_divisors_direct(2026)`: O(√n) enumeration | 1016 (need 2026 for perfect; deficit = 1010) |
| A4 | SC2 cross-check: perfect via multiplicative σ | `sigma_multiplicative(2026)`: σ(n)=∏(p^(a+1)−1)/(p−1) minus n | 1016 — agrees with A3 |
| A5 | SC2 factorisation | `prime_factorisation(2026)`: trial division | {2: 1, 1013: 1} |

### Type B (Empirical) Facts

None. This is a pure-math proof.

---

## Computation Traces

*Source: proof.py inline output (reproduced verbatim)*

```
--- SC1: Happy Number ---
  digit-square-sum sequence for 2026: [2026, 44, 32, 13, 10, 1]
  Sequence terminates at: 1
  Primary (Floyd cycle detection): is_happy = True
  Cross-check (unhappy-cycle membership): is_happy = True
  SC1: 2026 is a happy number: True == True = True

--- SC2: Perfect Number ---
  Prime factorisation of 2026: {2: 1, 1013: 1}
  Primary (direct enumeration): sum of proper divisors = 1016
  Cross-check (multiplicative σ formula): sum of proper divisors = 1016
  SC2: (sum of proper divisors) - 2026 [must be 0 for perfect]: s_direct - n = 1016 - 2026 = -1010
  SC2: sum_of_proper_divisors(2026) == 2026: 1016 == 2026 = False
  Compound: SC1 AND SC2 (both must hold for year to be cosmically special): False == True = False
```

---

## Cross-Checks (Rule 6)

*Source: proof.py JSON summary `cross_checks`*

### SC1: Two independent happy-number algorithms agree

- **Values compared:** True (Floyd detection) vs True (unhappy-cycle membership)
- **Agreement:** Yes (exact boolean)
- **Method independence:** Primary uses Floyd cycle detection — stops when `current` has been seen before in the iteration. Cross-check stops when `current` is a member of the precomputed UNHAPPY_CYCLE = {4, 16, 37, 58, 89, 145, 42, 20}. These use different stopping criteria: a bug in the Floyd detection's seen-set logic would not affect the cycle-membership check, and vice versa.

### SC2: Two independent proper-divisor-sum algorithms agree

- **Values compared:** 1016 (direct enumeration) vs 1016 (multiplicative σ formula)
- **Agreement:** Yes (exact integer)
- **Method independence:** Primary iterates candidate divisors from 2 to √n (arithmetic). Cross-check uses the multiplicative formula σ(n) = ∏(p^(a+1)−1)/(p−1) derived from number theory (algebraic identity for multiplicative functions). A computational error in the loop structure of the primary method would not affect the algebraic formula, and vice versa.

---

## Adversarial Checks (Rule 5)

*Source: proof.py JSON summary `adversarial_checks`*

| # | Question | What Was Done | Finding | Breaks Proof? |
|---|----------|---------------|---------|---------------|
| 1 | Alternative definition of "happy number"? | Reviewed OEIS A007770. Checked base-10 vs base-2 definitions; standard unqualified usage is always base-10 (Grundman & Teeple 2001). | No alternative definition changes the result. 2026 is happy under any standard base-10 algorithm. | No |
| 2 | Alternative definition of "perfect number" under which 2026 qualifies? | Checked quasi-perfect (σ=2n+1), almost perfect (σ=2n−1), semiperfect, abundant, deficient. σ(2026)=3042; 2n=4052. 2026 is deficient. | 2026 does not qualify under any variant of "perfect number." It is strictly deficient. | No (SC2 was already disproved) |
| 3 | Factorisation error could change SC2? | Verified 2026/2=1013 exactly; primality of 1013 by trial division through all primes ≤ 31 = ⌊√1013⌋; two independent sum algorithms agree. | Factorisation confirmed. Sum of proper divisors is unambiguously 1016. | No |
| 4 | "Cosmically special" as a falsifiable claim? | Searched number theory, combinatorics, mathematical physics literature. | No mathematical definition found. SC3 is rhetorical and excluded. | No |

---

## Hardening Rules Checklist

*Source: validate_proof.py output*

| Rule | Status | Notes |
|------|--------|-------|
| Rule 1: No hand-typed extracted values | PASS (auto) | Pure-math proof — no value extraction |
| Rule 2: Citations verified by fetching | PASS (auto) | No empirical facts — not needed |
| Rule 3: System time anchor | PASS (auto) | No time-dependent computations |
| Rule 4: CLAIM_FORMAL with operator_note | PASS | Present for all sub-claims |
| Rule 5: Adversarial checks | PASS | 4 structurally independent checks |
| Rule 6: Independent cross-checks | PASS (auto) | Pure-math: two mathematically distinct methods per sub-claim |
| Rule 7: No hard-coded constants | PASS | No magic constants; `compare()` and `explain_calc()` used |

*Validator result: 15/16 checks passed, 0 issues, 1 warning. Warning resolved: `compound_holds` rewritten to use `compare()` for auditable output.*

---

## Sub-Claim Verdicts

*Source: proof.py JSON summary `sub_claim_verdicts`*

| Sub-claim | Verdict |
|-----------|---------|
| SC1: happy number | **PROVED** |
| SC2: perfect number | **DISPROVED** |
| SC3: cosmically special | NOT EVALUABLE (opinion/metaphor) |
| **Compound (SC1 AND SC2)** | **PARTIALLY VERIFIED** |

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