# Current Research Targets

**Last updated:** 2026-07-03
**Changes daily.** Old versions in git history.

## Active Targets

### 1. Invariant Computation Geometry [OPEN]
**Goal:** Determine which NP properties have polynomial-size spectral
invariant embeddings (the octagon question at O(n) dimension).

**Status:** One measured success (Sidon 4/4), one known failure
(cospectral graphs on Hamiltonicity), one proven existence at O(n²)
(Etesami-Haemers 2019). The O(n) question is open.

**Links:**
- [Octagon Principle](../research/OCTAGON_PRINCIPLE.md)
- [Invariant Computation Geometry](../research/INVARIANT_COMPUTATION_GEOMETRY.md)

### 2. Reaction Prime Decomposition [SPECULATIVE]
**Goal:** Define irreducible DNA computational operations (reaction
primes) and determine if every computation factors uniquely.

**Status:** Framework proposed (reaction algebra, information primes,
category theory). No measurements yet. Connects to the conservation
law (prime total ≥ K(data)) and the pipeline (each stage = one
prime decomposition step).

**Links:**
- [Reaction Primes](../research/REACTION_PRIMES.md)
- [O(1) Transforms](../research/O1_TRANSFORMS.md)

### 3. Merged O(1) Transform [SPECULATIVE]
**Goal:** Determine if DNA hybridization can simultaneously search,
reconstruct (CRT), and verify (energy) in one physical step.

**Status:** Framework documented. The wall is O(n) readout (conservation
law). The decision-problem shortcut (1-bit answer = O(1) readout) is
the most promising path. Needs: design CRT-coprime hachimoji pairing
rules, simulate energy landscape, measure gap vs n.

**Links:**
- [Merged O(1) Transform](../research/MERGED_O1_TRANSFORM.md)
- [Conservation Law](../weird_machine_conservation_law.md)

### 4. Formal Verification Cleanup [MEASURED]
**Goal:** Close all remaining sorries and remove all native_decide
from the active build.

**Status:** 10 active sorries (all tagged CITED/CONJECTURE, all
genuinely blocked on Mathlib API). 21 axioms (all justified).
Anti-smuggle scanner passes. 84 modules registered in lakefile.

**Links:**
- [AGENTS.md](../../AGENTS.md) — current status table

### 5. Encoder Fidelity [MEASURED]
**Goal:** Maintain 20/20 injective DNA encoding with exact arithmetic
(no floats) across all input types.

**Status:** 20/20 injective (p-adic + neg-pi). BioSight embed.py
updated to v3 (exact arithmetic). Universal pipeline 14/14 PASS.
Round-trip lossless on all text types including LaTeX/CJK/emoji.

**Links:**
- [Compression Findings](../research/COMPRESSION_HONEST_FINDINGS.md)

## Dead Targets (measured, not pursuing)

### ~~Compression via spectral methods~~ [DEAD]
**Result:** Conservation law forbids it. 8 branches measured, all
fail. Polynomial = receipt, not compressor. xz beats everything.

### ~~16D braid / golden spiral GW compression~~ [DEAD]
**Result:** 583x was zero-noise artifact. 1.5x at 30dB (ties LPC).
16D adds nothing over 50-year-old LPC.

### ~~Braille/T9 text compression~~ [DEAD]
**Result:** 4.167 b/B, worse than order-2 PPM (3.088), far behind
xz (1.989). 64-cell alphabet too small for 256 bytes.

## Candidate Targets (not yet committed)

### 6. cmix Weight Matrix SVD [OPEN]
**Goal:** Extract the 23×461 weight matrix from cmix, compute SVD,
determine if the top-k singular values capture most compression
quality. If yes, search the low-rank subspace for better configs.

### 7. 3-SAT Spectral Embedding [OPEN]
**Goal:** Build a clause-incidence matrix for 3-SAT, compute its
spectrum, check if satisfiability is spectrally detectable. This
is the P vs NP experimental program's first NP-complete test case.

### 8. Quandela SLOS Shortcut [OPEN]
**Goal:** For linear optical circuits, replace full SLOS (O(n × M_n))
with eigenvalue product computation (O(n^m)). Only works for linear
systems. Needs: verify eigenvalue products match SLOS output on
the Sidon crossing matrix.

## ~~Prime-Sidon spectral detection~~ [DEAD — 2026-07-03]

**Result:** Honest negative. 35 test cases, 0/35 significant after
Bonferroni. The prime distribution does NOT have a Sidon-detectable
spectral signature through SLOS. Adversarial review caught a tautology
in the original methodology, null was properly added, negative is
bounded. ENE session: prime-sidon-negative-001 (promoted).
