---
paper_id: PAPER_1193
title: "P vs NP as Conjecture-Grade UQFF Anchor: Informational Buoyancy Argument and Audit Closure"
session: 301
date: 2026-05-17
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: ["P-vs-NP", "complexity", "conjecture-grade", "UQFF", "informational-buoyancy"]
crosslinks: [PAPER_1186]
sm_anchor: "CVW v2.0.0 -- G6 SM Anchor Gate compliant"
arxiv_class: "cs.CC"
---

# PAPER_1193: P vs NP as Conjecture-Grade UQFF Anchor --- Informational Buoyancy Argument and Audit Closure

## Abstract

We close audit gap \#11 of the UQFF calibration program by explicitly registering the P vs NP problem as a CONJECTURE-tier entry with zero observational anchors. The UQFF informational-buoyancy stance asserts that exhaustive search has positive vacuum (SCm) buoyancy cost, $C_{\rm search}(n)=B_{\rm inf}\cdot 2^n$, which diverges from any polynomial alternative $C_{\rm solve}(n)=B_{\rm inf}\cdot n^k$ at $n\to\infty$, formally suggesting $\text{P}\neq\text{NP}$. We make this conjecture explicit and recordable in the calculator registry, but emphasize that no falsifiable observational anchor exists. Calculator registered as `cp4_id = 445`; 20/20 smoke tests pass.

## 1. Motivation

Audit entry \#11 noted that the UQFF framework had no formal handling of conjecture-grade mathematical claims with no observational anchor, breaking the G6 SM Anchor Gate audit trail. The fix is structural: explicitly mark such entries as CONJECTURE with `anchor = None`.

## 2. Informational Buoyancy Cost Functions

For an NP-complete decision problem of size $n$:

$$C_{\rm search}(n) \;=\; B_{\rm inf}\cdot 2^n, \quad C_{\rm solve}(n) \;=\; B_{\rm inf}\cdot n^k, \quad k\in\mathbb{N}. $$

UQFF buoyancy non-negativity requires

$$\boxed{\;\Delta_C(n) \;\equiv\; C_{\rm search}(n) - C_{\rm solve}(n) \;=\; B_{\rm inf}\bigl(2^n - n^k\bigr) \;\ge\; 0 \;\forall n\ge n_*. \;}$$

By contrapositive (informal): if $\text{P}=\text{NP}$ existed then $\Delta_C\to 0$ for all $n$, in tension with the asserted positivity of $C_{\rm search}$. Hence the UQFF conjectural stance is $\text{P}\neq\text{NP}$.

\textbf{This argument is conjectural and has no observational anchor.}

## 3. Statement Registry

Four formal statements are exposed in `STATEMENTS`:

| ID | Name | Grade | Anchor | UQFF Position |
|---|---|---|---|---|
| S1 | $P \ne NP$ | conjecture | None | $\Delta_C > 0\;\forall n \Rightarrow P\ne NP$ |
| S2 | $NP \subseteq AM$ | conjecture | None | orthogonal (no claim) |
| S3 | $NP \ne \text{co-}NP$ | conjecture | None | proof/disproof informational asymmetry |
| S4 | NP-intermediate exists (Ladner 1975) | conjecture | None | consistent, not asserted |

## 4. UQFF Aether Modulation

Even though the P vs NP problem is purely mathematical, the calculator returns a nominal aether factor $f_A=1+\beta_i(\rho_{\rm SCm}/\rho_{\rm amb})\cos(\pi t_n)$ for registry-consistency; it does not affect the conjectural verdict.

## 5. Anchor Validation (Tier = CONJECTURE)

| Anchor | Type | Verdict |
|---|---|---|
| --- | none | n\_anchors = 0; n\_statements = 4; all unanchored |

This is the correct outcome for a conjecture-tier entry. The calculator's `query_result` exposes `tier = "CONJECTURE"` so downstream audit tools can filter conjectural entries from observational anchor counts.

## 6. Code Registration

Module `_session301_pnp_conjecture_anchor.py`; class `PvsNPConjectureAnchor`; `cp4_id = 445`. Registered in `CondensedPhysics3.py` under `SESSION_301_CALCULATORS`. Smoke-test result: 20 / 20.

## 7. Falsifiable Predictions

(i) Any future proof of $P=NP$ would falsify the UQFF informational-buoyancy stance.
(ii) The conjecture-tier flag must propagate unchanged in all downstream UQFF audit reports.

## 8. Status

\textbf{[CLOSED]} \quad audit gap \#11 \quad cp4\_id = 445 \quad session = 301 \quad tier = CONJECTURE.

## 9. References

Cook 1971; Karp 1972; Ladner 1975. Clay Mathematics Institute Millennium Prize Problems. UQFF\_CALIBRATION\_AUDIT.md.
