Subinvariance Validator

Problem Scope

The Subinvariance Validator targets a narrow number-theoretic automation task: given a downward Markov chain and a weight function, check whether the chain satisfies subinvariance and then derive an upper bound for the Erdős sum of a primitive set. These proofs often require repeated manual checks of chain legality, weight recursion, and comparison targets; this package turns those steps into repeatable Python calls.

How It Works

The core module is organized around four entry points:
- validate_chain(chain, test_range): checks that the chain is valid over a specified range before any subinvariance test is run.
- test_subinvariance(chain, weight, name, test_range): executes a subinvariance test for a named weight, making it practical to turn paper examples or section conditions into runnable assertions.
- estimate_f_A_bound(chain, weight, x, max_N): after subinvariance has been validated, estimates an upper bound for f(A), giving a numerical boundary for Erdős-sum-style conclusions.
- compare_chains(): compares built-in chains, helping choose the von Mangoldt, Mertens, prime-power-modified, or odd-prime construction closest to the target problem.
Built-in objects include chain_von_mangoldt, chain_mertens, chain_prime_power_mod, and chain_section6_odd_primes, along with weight_nu0 and weight_nu2. The usual workflow is: choose a chain, choose a weight, validate the chain, run subinvariance, then estimate the bound.

Boundaries and Caveats

This tool is best used for localized automation of the method described in arXiv:2605.00301, not as a general proof generator. Its output depends on the chosen chain and weight fitting the paper's assumptions; for a custom chain with different boundary conditions, recursion structure, or test range, the result is a numerical check rather than a proof. When connecting the output to the Erdős primitive set conjecture, the Erdős–Sárközy–Szemerédi large-number conjecture, or the revised Banks–Martin conjecture, theorem preconditions still need to be verified.