# Open Questions

**Last updated:** 2026-07-03

## Q1: Does 3-SAT have a spectral signature?
The first NP-complete test case for the octagon. If YES → P=NP route
via spectral methods. If NO → natural P≠NP witness.
[Link: OCTAGON_PRINCIPLE.md §P vs NP]

## Q2: Can eigenvalue products replace SLOS for linear optical circuits?
SLOS is linear → U^(⊗m) determined by U's spectrum → eigenvalue products
should determine the output. Measured K=1 failed, but FULL spectrum
(all products) is untested.
[Link: MERGED_O1_TRANSFORM.md §SLOS linearity]

## Q3: Is the cmix weight matrix low-rank?
If the 23 singular values are dominated by top-5, the compression
shape lives in 5D. The search for better compressors becomes 5D,
not 23×461.
[Link: cmix_epigenetic_analysis.md]

## Q4: Does every DNA computation factor into reaction-primes?
The algebraic version of "is there a fundamental theorem of
arithmetic for computation?" If yes → conservation law = FTA.
If no → the framework needs a different algebraic structure.
[Link: REACTION_PRIMES.md]

## Q5: Can the three O(1) transforms merge into one physical step?
Adleman (search) + CRT lift (reconstruct) + CRT gradient (verify)
as one DNA hybridization. The wall is O(n) readout. The decision-problem
shortcut (1-bit answer) is the most promising escape.
[Link: MERGED_O1_TRANSFORM.md]

## Q6: What is the Φ-metric?
The "geometry of observability" — what defines what counts as "same
object" before physics processes it. Is it a kernel? A distance?
An operator-valued metric? This is the formalization gap between the
vision and a computable system.
[Link: INVARIANT_COMPUTATION_GEOMETRY.md]

## Q7: At what n does the DNA hybridization energy gap fail?
The SNR cliff question. For the merged O(1) transform: the energy
gap between correct and near-correct hybridization is constant
(~1 mismatch), but the number of near-correct candidates grows with n.
When does the gap become too small for thermodynamic selection?
Same cliff as superposition (k=16: lossless, k=48: lost).
[Link: MERGED_O1_TRANSFORM.md §The Energy Gap Problem]

## Q8: Does the Sidon spectral signature generalize to other NP properties?
Sidon = pairwise-sum matrix → eigenvalue degeneracy. Does the
unit-distance property have a spectral signature in the distance
matrix? Does graph coloring (beyond Hoffman bound)? Does clique
number (beyond Lovász theta)?
[Link: OCTAGON_PRINCIPLE.md §Fast-Forward]
