Source: https://guthlabs.ai/docs/instruments/prime-factorization-divisor-explorer/mechanism/

instrument mechanism / verified-clean

# Prime Factorization & Divisor Explorer mechanism

The inspectable method record for Prime Factorization & Divisor Explorer, kept separate from implementation and qualification.

Status: **VERIFIED_CLEAN**Lifecycle: **verified-clean**

## Model scope

The named approaches and divisor-function mathematics are standard and correct, but the specification needs exact definitions for σ, d, and the classification convention, plus an explicit BigInt/arbitrary-precision requirement and a fully specified Pollard-rho/primality implementation. The honest label is appropriate, especially for large semiprimes and complete divisor-list generation.

- The formulas field names σ(n) and d(n) but does not explicitly define them: for n=∏p_i^a_i, d(n)=∏(a_i+1) and σ(n)=∏(p_i^(a_i+1)-1)/(p_i-1), equivalently ∏(1+p_i+...+p_i^a_i).

- Perfect/abundant/deficient classification must explicitly use the proper-divisor sum σ(n)-n: perfect if σ(n)=2n, abundant if σ(n)>2n, and deficient if σ(n)<2n. The convention that n=1 is deficient should be stated.

- Pollard's rho is a correct standard factorization method, but the spec does not define the primality test, recursion/termination criteria, or deterministic handling of its randomized iteration.

## Units and conventions

- Input is an exact positive integer n≥1 with no physical units.

- Large integers must use exact arbitrary-precision arithmetic such as JavaScript BigInt; IEEE-754 Number arithmetic is unsafe above 9,007,199,254,740,991.

- Divisors are returned as positive divisors in ascending order.

- For n=1, the factorization is conventionally represented as 1, its sole divisor is 1, and it is deficient because its proper-divisor sum is 0.

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "inputs": {
    "n": 1
  },
  "expected_outputs": {
    "factorization": "1",
    "divisors": [
      "1"
    ],
    "divisor_count": "1",
    "divisor_sum": "1",
    "proper_divisor_sum": "0",
    "classification": "deficient"
  },
  "working": "1 has no prime factors. Its positive divisor list is [1]. Therefore d(1)=1 and σ(1)=1. The proper-divisor sum is σ(1)-1=1-1=0. Since 0<1, 1 is deficient."
}
````

## Failure boundary

- Reject zero, negative values, nonintegers, and non-finite inputs; n=0 has infinitely many divisors and would make the requested functions undefined.

- A divisor-count or divisor-sum implementation that uses ordinary JavaScript Number values can silently lose precision for integers above 2^53−1.

- Trial division can be impractically slow for large semiprimes; Pollard's rho can also require unbounded retries and needs a correct primality test and factor-completion logic.

- Pollard's rho is commonly randomized; an unseeded random implementation is not reproducible, so deterministic testing requires a specified seed or deterministic test properties.

- Generating the complete divisor list can require substantial memory even when factorization itself is fast.

- Classification requires computing or comparing σ(n) against 2n exactly, not using floating-point arithmetic.

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/prime-factorization-divisor-explorer/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/prime-factorization-divisor-explorer/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/prime-factorization-divisor-explorer/qualification/>)

## Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

- `VERIFIED_LEDGER.json`

- `evidence-tranche6/tranche6-results.json`