Mathematical foundations / Conditional separation arguments
Conditional complexity separation: what the added axioms must do
The manuscript P ≠ NP under Davis Axioms proposes a geometric separation under additional assumptions. Its title is conditional, and that condition is the central mathematical issue to examine.
Research settingIndependent
TL;DR
In brief
The question
If two added geometric axioms are accepted, does a separation between polynomial-time and NP-complete problems follow, and could those axioms hold in standard models?
What the work does
A deposited preprint pairing an energy-landscape analysis of random 3-SAT, counting metastable basins and curvature barriers, with two stated axioms that link that geometry to computational cost.
What is still open
Whether the axioms follow from ordinary machine or circuit cost, and whether the barrier argument constrains every polynomial-time algorithm rather than one search dynamics, remains unaudited.
Available recordDeposited conditional argument; an unconditional separation is not established here.
From the research record
Inspect the work
Original figures, recorded values, and explicit model details. Each exhibit identifies its source and what the comparison means.
The assumptions carrying the final separation argument
Scroll horizontally to inspect the full table →
| Source statement | Mathematical form | Role in the proposed argument |
|---|---|---|
| Axiom 8.8: information-flow law | C = τ / K(A, B) | Connects geometric barrier cost to information flow |
| Axiom 8.9: tolerance intensivity | τ = Θ(1) as n grows | Prevents the tolerance budget from growing with input size |
| Standard-model correspondence | A separate proof obligation | Must connect these quantities to every algorithm in the ordinary computation model |
The first two rows state the final section’s assumed axioms; the third is an editorial proof obligation. Earlier sections introduce additional geometric correspondences. This table neither audits every lemma nor establishes an unconditional P versus NP separation.
The research in context
P versus NP asks whether every problem whose solutions can be checked quickly can also be solved quickly. Satisfiability, deciding whether a logical formula's variables can be set to make it true, is the canonical case: 2-SAT is solvable in polynomial time, while 3-SAT is NP-complete. A separate line of work studies random satisfiability instances as energy landscapes, counting their valleys (metastable basins) and the barriers between them. Such landscapes describe the terrain a search procedure must cross; turning that description into a statement about every possible algorithm is the step standard complexity theory requires.
The deposited preprint (January 2026) states two axioms. The first, the Davis Law, takes information flow across a barrier to be inversely proportional to its curvature, C = τ/K. The second, tolerance intensivity, holds that the tolerance budget τ stays bounded as problem size grows. The geometric part argues, through a Kac–Rice formula (a method for counting critical points of a random function) and supporting lemmas, that random 3-SAT has exponentially many metastable basins separated by curvature barriers of constant height, with sparse solution basins in the frozen regime. It also reports curvature scaling as k(k−1)/2 for k-SAT, a threefold difference between 2-SAT and 3-SAT.
The record further argues that the technique avoids the relativization, natural-proofs and algebrization barriers, and its file list includes scripts for Hessian-positivity tests and phase scans whose results the summary does not describe. The site's metadata check confirms the deposit but records that its scientific claims have not been independently reproduced. Because the separation enters through the two axioms, their status in the ordinary model of computation determines what the result means outside the framework.
From a search landscape to a complexity statement
The historical description contrasts polynomial and exponential energy landscapes through geometric barriers. To obtain a complexity theorem, the construction must specify how an input and computation are encoded and how ordinary computational cost relates to the proposed geometric cost.
A hard path for one algorithm does not rule out every polynomial-time algorithm. Nor does a continuous energy barrier imply a lower bound on discrete computation unless the correspondence is established. The extra axioms must carry that bridge explicitly.
What “under the axioms” leaves to prove
A valid implication from a stated axiom system can be worth studying without settling the standard P versus NP problem. The next question is whether the axioms follow from the ordinary model of computation, restrict it to a special class, or already encode the desired separation.
The reviewed site record identifies the manuscript but does not supply a completed independent audit of that bridge. This note therefore presents the proposed line of attack without replacing the missing argument with a numerical landscape illustration.
The next research step
The representative revision should put the axioms, encoding map, cost correspondence, and one critical lower-bound lemma on the page. Review can then target a concrete implication instead of an analogy between roughness and difficulty.
Background mathematics and definitions
Fix the input encoding, length measure, allowed operations, precision costs, and uniformity requirement. State extra geometric axioms separately from standard assumptions. A lower bound for one algorithm family and a separation between complexity classes have different logical forms.
- Input length
- Length of the encoded instance, including required precision.
- Uniformity
- How the algorithm or circuit family is specified across input sizes.
- Reduction
- A transformation with a justified cost and answer-preservation property.
Illustrative example
A search that checks every assignment of n Boolean variables considers 2ⁿ candidates. That count describes this enumeration strategy. It does not prove that every method for the same decision problem must examine each assignment.
This background example is separate from the source records and the main technical figure.
Source record
No new experiments are reported in this research note. Figure captions distinguish saved research records, source-model calculations, and explanatory diagrams.