---
name: STEP 784 — E16_JOINT 4-tuple Joint K-Complexity (Three-Sided Negative)
description: tier2_axiom joint K-complexity attack via 4-feature tensor product — negative result confirms STEP 783 direction, plus peak=9232 shared by 23/25 atomic cores discovery
type: project
originSessionId: a08149e8-f1de-4fbc-b4eb-fb2cec8fad21
---
# STEP 784 — E16_JOINT Lens + Three-Sided Negative Result (2026-04-14)

**Why:** STEP 783 showed simple across-orbit 1D labelings (MOD_96_READY, FIBER_57_TREE, CHANG_REI_INTERSECT) are all AIT-random. Chat-Claude's world OSS survey (PyBDM 1D/2D only, ComplexityMeasures.jl n-dim but Julia, koolmogorov experimental) concluded no finished n-dim K-complexity tool exists. Adopted "Method A tensor product + custom E16_JOINT lens" as shortest path.

**How to apply:** This is the 3rd AIT measurement axis after within-orbit (780b) and across-orbit-simple (783). Three-sided negative means tier2_axiom structure does NOT live in PyBDM-measurable K-complexity channels at tuple widths ≤ 16 bits. Escalate to transfer-operator spectra, higher moduli (mod 2^k, k ≫ 8), or Lyapunov-theoretic invariants before reopening K-complexity attacks.

## Encoding (16-bit per atomic core, 25 × 16 = 400-bit joint)

```
[mod96    : 7 bits]   residue n mod 96
[t1       : 3 bits]   trailing ones in binary(n), clamped 0..7
[peak9232 : 1 bit ]   1 iff orbit peak = 9232
[Kbucket  : 5 bits]   (orbit length) / 16, clamped 0..31
```

## Result

| Measurement | K/L | Class |
|-------------|------|-------|
| JOINT_4TUPLE (400 bits) | **2.589** | RANDOM |
| MOD96_ONLY (175 bits) | 2.571 | RANDOM |
| T1_ONLY (75 bits) | 2.517 | RANDOM |
| PEAK9232_ONLY (25 bits) | 2.221 | RANDOM |
| KBUCKET_ONLY (125 bits) | 2.659 | RANDOM |
| PSEUDO_RANDOM (400 bits) | 2.648 | baseline |
| Single-feature mean | 2.492 | — |

- Joint K/L is 2.2% below pseudo-random but ABOVE single-feature mean → tensor product does NOT expose hidden cross-feature compression.
- **Verdict: HYPOTHESIS_REJECTED** — three-sided negative (780b / 783 / 784).

## Side-Discovery: peak = 9232 near-universal

STEP 696 memo flagged 6 cores sharing peak 9232 (27, 55, 91, 95, 143, 221). Full scan over 25 atomic cores reveals peak=9232 is shared by **23/25** (92%). Only exceptions:

- n = 247 → peak = 1672
- n = 255 → peak = 13120

This is a stronger structural claim than the STEP 696 record. **The 23/25 shared-peak fact is now a candidate tier2_axiom component**: any atomic core reaching peak 9232 has an orbit that passes through the fixed shared suffix (9232 → 4616 → 2308 → 1154 → 577 → ... → 1). The two exceptions deserve algebraic isolation (analogous to how STEP 694 isolated n=27 at mod 96 level).

## Files

- `scripts/pybdm-joint-4tuple.py` — Python analysis producing `data/pybdm-joint-4tuple.json`
- `src/axiom-os/collatz-joint-complexity-engine.ts` — TS bridge + `analyzeJoint4Tuple`, `summarize`, `encodeCore16Bit`
- `test/step784-joint-complexity-test.ts` — 77/77 pass
- `src/axiom-os/mandala-supreme-engine.ts` — E16_JOINT cluster + v9 header bump (34 lenses)

## Next-Attack Candidates (prioritized)

1. **Shared-peak substructure** — Why do 23/25 atomic cores funnel through 9232? Algebraic origin of n=247, n=255 exceptions. Likely cheapest next move.
2. **Higher moduli** mod 2^k for k ∈ {10, 12, 14, 16} — push past the STEP 695 saturation at 25 irreducibles.
3. **Transfer-operator spectra** — spectral gap of Syracuse map on L²(Z_2) (Tao-style ergodic).
4. **Lyapunov-theoretic invariants** — drift-based, avoids K-complexity entirely (STEP 726 E11 candidate).