This working paper reframes the previous speculative Riemann Hypothesis brainstorming note into a smaller, testable research program. The empirical question is: Do ordered prime gaps contain measurable compression structure beyond their marginal distribution?
Using the first 20,000 and 100,000 primes, we compared LZ76 complexity of real prime-gap order against shuffled controls preserving the same multiset. At 100,000 primes, the raw gap encoding produced:
This does not prove RH, but establishes a first compression-detectable signal. The first E2 overnight run then tested stronger controls: block shuffle, wheel-aware shuffle, Markov-preserving surrogates, Cramér-like synthetic gaps, and 500k scaling checks.
E2 result: the broad E1 signal is partly explained by local/arithmetic structure, but the raw gap signal survives wheel-aware and Markov-preserving controls. At 500k primes, the Markov-preserving raw gap test gave delta −65.18, z −5.42, p 0.004975.
The original brainstorming paper mixed three levels: empirical observations, reasoned interpretations, and metaphysical speculation. This version separates them. The empirical core is now simple: prime gaps are treated as a sequence, the real order is compared against null models, and MDL/LZ76 decides whether additional structure exists.
Prime gaps \(g_n = p_{n+1} - p_n\) are not independent. Their distribution changes with scale, and primes obey arithmetic constraints. The sharper question is: After preserving basic statistics, does the ordered sequence still compress better than appropriate controls? That is exactly the kind of question the 8Z/MDL/DCC program is built to ask.
DCC balances between seizure (excessive order) and noise (excessive disorder). The original speculative analogy placed Re(s)=1/2 as the edge-of-chaos line. This remains a conceptual analogy, not a proof. The empirical test only asks whether prime gaps show measurable structure under compression tools.
This paper makes no claim to prove the Riemann Hypothesis. It only asserts that ordered prime gaps are more compressible than shuffled controls preserving the same gap distribution. Any bridge to RH remains speculative future motivation.
We generated the first N primes, computed gaps, and applied several encodings (gap, delta, abs_delta, mod6, bucket_log2). Each real ordered window was compared against shuffled controls. The main statistic is \(\Delta_{LZ} = LZ76(real) - mean(LZ76(control))\); a negative value indicates the real sequence is more compressible.
Configuration: 20,000 primes, 30 trials, window 5,000. Summary:
| Encoding | Real LZ76 | Shuffled LZ76 | Delta | Z-score | Interpretation |
|---|---|---|---|---|---|
| gap | 1486.75 | 1579.31 | −92.56 | −22.85 | strong |
| gap_div2 | 1486.75 | 1579.14 | −92.39 | −23.86 | strong |
| delta | 1753.00 | 1820.93 | −67.93 | −16.72 | strong, but needs special control |
| abs_delta | 1520.00 | 1537.62 | −17.62 | −4.44 | moderate/strong |
| mod6 | 714.75 | 882.10 | −167.35 | −88.43 | sanity check; partly expected |
| bucket_log2 | 1045.25 | 1054.22 | −8.97 | −3.92 | weak but present |
Configuration: 100,000 primes, 50 trials, window 10,000. Summary:
| Encoding | Real LZ76 | Shuffled LZ76 | Delta | Z-score | Interpretation |
|---|---|---|---|---|---|
| gap | 2898.80 | 3066.76 | −167.96 | −32.38 | very strong |
| gap_div2 | 2898.80 | 3067.18 | −168.38 | −32.83 | very strong |
| delta | 3395.10 | 3532.89 | −137.79 | −23.90 | very strong, needs control |
| abs_delta | 2959.50 | 3002.05 | −42.55 | −8.25 | strong |
| mod6 | 1281.40 | 1603.32 | −321.92 | −127.43 | sanity check |
| bucket_log2 | 1974.80 | 1984.15 | −9.35 | −3.01 | weak but present |
Raw prime gaps are about 5.5% more compressible in real order than in shuffled order.
Supported: The order of prime gaps carries structure beyond marginal distribution. Not yet shown: that it survives Markov-preserving controls, that it connects to zeta zeros, or that AC/Zero Framework is correct. mod6 is a sanity check; delta needs caution due to lag-1 artifacts.
Minimal success: raw gap outperforms full shuffle, block shuffle, and wheel-aware. Strong success: outperforms Markov surrogates. Very strong: survives all controls and shows scaling growth.
The first E2 overnight run tested whether the E1 shuffle signal survives harder null models. The result is positive, but narrower than the first E1 impression. The large shuffle signal is partly local/arithmetic, yet raw prime-gap order still survives the key wheel-aware and Markov-preserving controls.
E2 weakens the broad version of the E1 claim but strengthens the precise version: prime-gap order contains compression-detectable sequential structure not fully explained by marginal gap distribution, mod-6 wheel structure, or first-order Markov transitions.
gap summary#| Test | Control | Delta | Relative | Z-score | p | Reading |
|---|---|---|---|---|---|---|
| 100k shuffle, 1000 trials | C0 shuffle | −168.24 | −5.49% | −31.71 | 0.000999 | E1 confirmed |
| 100k block10 | C1 block | −51.10 | −1.73% | −8.75 | 0.001996 | local signal survives |
| 100k block50 | C1 block | −14.21 | −0.49% | −2.62 | 0.014970 | weak but alive |
| 100k block100 | C1 block | −8.60 | −0.30% | −1.64 | 0.078842 | marginal |
| 100k wheel6 | C2 wheel | −19.97 | −0.68% | −3.84 | 0.002994 | survives wheel |
| 100k Markov | C3 Markov | −32.42 | −1.11% | −3.61 | 0.001996 | survives first-order transitions |
| 100k Cramér | C4 Cramér | −207.98 | −6.69% | −21.56 | 0.006623 | supportive, less decisive |
| 500k shuffle | C0 shuffle | −312.62 | −5.26% | −44.38 | 0.004975 | scales |
| 500k wheel6 | C2 wheel | −33.40 | −0.59% | −4.68 | 0.004975 | survives wheel at depth |
| 500k Markov | C3 Markov | −65.18 | −1.14% | −5.42 | 0.004975 | strongest E2 result |
Shuffle is confirmed. With 1000 trials at 100k primes, the raw gap signal remains around −5.5% relative compression advantage over shuffled controls.
Block shuffle narrows the claim. The signal is strong at block10, weak at block50, and marginal at block100. This means much of the signal is local-to-medium scale.
Wheel-aware controls are passed. After preserving simple mod-6 wheel structure, raw gap still remains more compressible than controls. The signal is not only the trivial \(6k \pm 1\) residue pattern.
Markov-preserving controls are the key win. Raw gap survives first-order transition controls at both 100k and 500k primes. This supports the stronger E2 claim: the real sequence contains compression-detectable structure beyond marginal distribution and beyond first-order transition statistics.
Cramér-like controls are supportive but not decisive. Real prime gaps strongly outperform Cramér-like synthetic gaps, but this control does not preserve enough arithmetic structure to be the hardest null model. Wheel-aware and Markov-preserving controls matter more.
The delta and abs_delta encodings looked strong under full shuffle, but weaken or flip under harder controls. They should be treated as diagnostic sensors, not primary evidence. The primary evidence is now raw gap.
Prime-gap order contains a surviving compression signal that is not fully explained by gap distribution, simple wheel residue structure, or first-order Markov transition statistics. The effect is smaller than the initial shuffle result but more meaningful.
If E2 confirms, number theory becomes a credible ninth domain for the 8Z/DCC approach. The strongest future claim would be: 8Z/DCC compression tools detect nontrivial order-sensitive structure in prime gaps.
The critical line as edge-of-chaos remains speculative motivation only. Verified: E1 signal. Reasoned: compression advantage suggests order-sensitive structure. Speculative: AC/Zero Framework explanations are not evidence.
The distinction between numeric zero \(0\) and ontological nothingness \(\emptyset\) is philosophically interesting but not yet mathematically formalized for RH.
Zeta zeros are points of exact cancellation (destructive interference), not annihilation. The stronger language is reserved for a future formal annihilation operator, if defined.
Ordered prime gaps contain measurable compression structure beyond their marginal distribution. After E2, the surviving raw-gap signal also exceeds simple wheel-aware and first-order Markov-preserving controls, though much of the broader signal is local/arithmetic.
The strongest current thesis is narrower than the original speculation but stronger than the initial shuffle-only result.
Recommended sequence: higher-trial C0, block shuffle, wheel-aware, Markov surrogate, Cramér, depth comparison. See the full paper for exact command lines.
E1 delivered a clear shuffle signal. E2 then narrowed and strengthened the result. The broad shuffle advantage is partly local/arithmetic, but the raw gap signal survives both wheel-aware and Markov-preserving controls, including at 500k primes.
The current conclusion is therefore:
The prime-gap compression signal is real, smaller under harder controls, and still alive where it matters most: raw gap order survives beyond marginal distribution, simple wheel structure, and first-order transition statistics.
This is not RH evidence yet, but it is a legitimate 8Z/DCC number-theory signal candidate.
The current compression experiments cannot prove the Riemann Hypothesis. They are empirical signal tests. A proof would require a formal mathematical invariant or theorem that applies to all relevant cases, not only to tested prime ranges.
The useful question is therefore not:
Do prime gaps compress better, therefore RH is true?
That would be invalid. The better question is:
Can compression/DCC analysis help discover a formal invariant that forbids off-critical-line zeros?
A possible proof path would need the following form:
In short:
empirical signal → formal invariant → equivalence/implication → proof
One possible route is through prime-counting error. RH is deeply connected to how tightly prime-counting functions stay near their expected main terms. A compression-based approach would need to discover a bound or regularity in the prime-gap trace that implies an appropriate bound on prime-counting error.
A rough proof-seed form would be:
If every prime-gap window has bounded normalized MDL excess, then prime-counting error remains within an RH-compatible bound.
This is not yet a theorem. It is a target shape for future formalization.
The most RH-adjacent route would connect compression structure in prime gaps to the oscillatory terms generated by zeta zeros in explicit formulas. In that direction, the desired contradiction would look like this:
Symbolically:
off-line zero → forbidden oscillation/compression signature → contradiction → RH
Another route is discovery of a new RH-equivalent condition. 8Z/DCC could search for a new boundedness, monotonicity, spectral, or compression condition that appears empirically stable. The mathematical task would then be:
This may be more realistic than directly proving statements about zeta zeros from compression data.
| Phase | Goal | Status |
|---|---|---|
| 1 | Detect empirical signal in prime-gap order | started; E1 positive |
| 2 | Test signal against stronger null controls | active; E2 running |
| 3 | Build multi-scale MDL/compression spectrum | planned |
| 4 | Relate spectrum to prime-counting error or explicit-formula terms | planned |
| 5 | Extract candidate invariant | future |
| 6 | Prove invariant and RH implication/equivalence | future |
| 7 | Formal verification where possible | future |
The current strongest honest formulation is:
8Z/DCC may help discover a new RH-adjacent regularity by treating prime gaps as a multi-scale information trace and searching for stable compression invariants that bound or constrain prime-distribution error.
This keeps the door open without pretending that E1 or E2 can prove RH by themselves.
Tables are shown in Section 6 above.
| Level | Claim | Status |
|---|---|---|
| L1 | Prime-gap order compresses better than shuffled gaps | ✅ supported by E1 |
| L2 | Signal is not only marginal distribution | ✅ supported by E1 |
| L3 | Signal survives wheel-aware controls | ⏳ pending E2 |
| L4 | Signal survives Markov-preserving controls | ⏳ pending E2 |
| L5 | Signal scales with prime depth | ⏳ pending E2/E3 |
| L6 | Signal relates to zeta-zero statistics | 🔮 speculative |
| L7 | DCC edge-of-chaos explains critical line | 🔮 speculative |
| L8 | AC/Zero Framework explains RH truth/proof difficulty | 🔮 highly speculative |
Use: “compression-detectable structure”, “order-sensitive signal”, “preliminary empirical result”, “null controls”, “not an RH proof”. Avoid: “RH confirmed”, “proof”, “annihilation of zeta zeros”, “AC requires RH”.
For each analysis window independently, the exact multiset of gap tokens inside that window is shuffled. This preserves the local gap distribution of that window and destroys only the ordering inside the window.
This matters because the E1 claim is not that prime gaps have a special distribution. The claim is narrower: the real order of those same gaps is more compressible than shuffled order.
These are non-overlapping main windows in the reported E1 runs. Later E2/E3 runs may add overlapping or depth-specific windows, but those should be reported separately.
Each encoding maps the prime-gap sequence into a token sequence before LZ76 is computed:
gap: raw integer prime-gap values.gap_div2: normalized even gap values after the initial special case.delta: first difference, \(g_{n+1} - g_n\).abs_delta: absolute first difference, \(|g_{n+1} - g_n|\).mod6: gap values modulo 6.bucket_log2: logarithmic bucket encoding of gap size.The primary statistic is:
Delta_LZ = LZ76(real) - mean(LZ76(control))
A negative value means the real ordered sequence is more compressible than the control sequences.
The permutation p-value is computed as:
p = (1 + count(control_LZ <= real_LZ)) / (trials + 1)
With 50 trials, the minimum possible p-value is \(1/51 \approx 0.0196\). Therefore, the E1 p-values should be read as “the real sequence beat all sampled controls,” not as a final high-resolution significance estimate.
The E1 compression advantage may partly reflect known arithmetic constraints and scale effects, including:
This is why E2 adds block, wheel-aware, Markov-preserving, Cramér-like, and depth/scaling controls. The purpose is to separate obvious arithmetic structure from deeper sequential compression structure.
The current E1 result defeats full-shuffle controls only. It does not yet defeat block-shuffle, wheel-aware, Markov-preserving, or Cramér-like null models. Those are the active E2 tests.
Next: Add full per-control CSV/JSON audit tables, window-level depth plots, and a sensor leaderboard. If the 500k/1M tests remain stable, the paper advances to v0.5 with a dedicated multi-scale signal-arena section.
Version 0.4 · May 2, 2026 · Working HTML page · E2 night results added · Next: v0.5 after deeper scaling/sensor-arena tests.