P vs NP – The Hidden Bit: A Proof Architecture Based on Information Flow

[theauditor]

For 50 years, P vs NP has resisted combinatorics, diagonalization, algebraization, natural proofs, circuit bounds, and every other direct assault.

What if the mistake is treating it as a purely combinatorial problem?

I’ve been working on an alternative framing that reduces P vs NP to a single invariant:

Polynomial-time computation cannot causally acquire one bit of globally dispersed information.

The core idea is simple:

• SAT’s truth value is a global property — dispersed across many constraints.

• Any polynomial-time algorithm interacts with only a vanishing fraction of those constraints.

• By the chain rule of mutual information, finite interaction bounds information gain.

• Therefore:

  I(\mathrm{SAT}(\phi); S_A(\phi)) \to 0

• No information ⇒ no decision ⇒ SAT ∉ P ⇒ P ≠ NP.

The architecture rests on:

1. Standard information theory (chain rule, data processing inequality).

2. Known proof-complexity lower bounds (global coordination required).

3. An explicit modeling of computation as bounded interaction.

4. A distilled “Hidden Bit” game that makes the invariant visible.

I’ve written up the full architecture here:

👉 https://​​vakofmaya.github.io/​​pnp.html

I’m not claiming this is the final word.
I am claiming that if this architecture is wrong, the failure point should be identifiable — and if it’s right, it reframes the problem at a much more fundamental level.

I’d love rigorous critique from people here who are comfortable with:

• Information theory

• Complexity theory

• Proof complexity

• Or foundational assumptions about computation

If there’s a flaw, let’s find it.
If not, maybe P vs NP was never combinatorial — maybe it was informational.

• AI used to correct and draft text