Prime Gaps Through the 8Z / DCC Lens

Abstract#

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 first compared LZ76 complexity of real prime-gap order against shuffled controls preserving the same multiset. At 100,000 primes, the raw gap encoding produced delta −167.96 and mean z-score −32.38 against shuffled controls.

The final E2/SEA batch now extends the test ladder to stronger controls and larger ranges. It includes 15 runs, 60 summary rows, and 1,960 window rows, covering Markov-preserving, wheel-aware, block-shuffle, and shuffle controls across 100k, 500k, 1M, and 2M prime ranges.

Final SEA finding: the broad E1 shuffle signal is partly explained by local/arithmetic structure, but the raw gap signal survives wheel-aware and Markov-preserving controls from 100k through 2M primes. The 500k bridge is positive, the 1M shuffle baseline is now present, and the strongest line remains the 2M Markov-preserving raw gap test: delta −142.34, z −8.76, with 50/50 windows negative (100 trials at 2M; limited p-value resolution; see Section 10.5).

The v0.4 generator-floor run now tests the next question: can any compact generator beat the Markov2 baseline on holdout? The answer is empirically yes at 1M scale. The run completed 1,620 window tasks and 72 generator tasks. At the sensor/control layer, raw gap still fails against Markov2. At the generator layer, however, density_markov2 wins raw gap/gap_div2, and all 9/9 tested encodings have a non-Markov2 winner beating the Markov2 holdout floor.

The v0.7 carrier-stress ladder now tests the sharper placebo question opened by v0.6: is the raw-gap generator gain really prime-density information, or a more generic monotone scale/clock state? The answer is empirically sharper: at 1M, clock_loglog_markov2 wins raw gap in every split, with Δholdout vs Markov2 −8,784.21 on the main split. density_wrong_scale_markov2 and density-DCC variants remain strong, but shuffled/lagged/reversed density labels cluster closely enough that exact prime-coordinate density is not isolated as the unique cause.

This does not prove RH. It shows a robust, order-sensitive prime-gap compression signal, a repeated empirical breach of the Markov2 generator floor, and now a stronger negative result against pure-density interpretation. The next phase is a corrected law-sheet build plus 2M/5M replication of the top carrier laws.

1. Purpose#

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.

2. Background: Why Prime Gaps?#

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.

3. DCC / Edge-of-Chaos Motivation#

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.

4. Important Boundary: Not an RH Proof#

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.

5. E1 Experiment: Prime-Gap LZ76 Against Shuffled Controls#

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.

6. E1 Results#

6.1 First run: 20,000 primes#

Configuration: 20,000 primes, 30 trials, window 5,000. Summary:

6.2 Stronger run: 100,000 primes#

Configuration: 100,000 primes, 50 trials, window 10,000. Summary:

Raw prime gaps are about 5.5% more compressible in real order than in shuffled order.

7. Interpretation of E1#

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.

8. E2: Stronger Null Controls#

• C0 Full Shuffle – already done. Distribution alone does not explain compressibility.
• C1 Block Shuffle – preserve local order; test long-range structure.
• C2 Wheel-Aware – preserve modular/residue constraints; test if signal is just residue arithmetic.
• C3 Markov-Preserving – preserve first-order transition behavior; the key hard control.
• C4 Cramér-Like – random-prime heuristic synthetic gaps.
• C5 Depth/Scaling – early vs. late windows; see if signal grows.

9. E2 Acceptance Criteria#

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.

10. Final E2 / SEA Results: Stronger Controls and Scaling#

The E2/SEA batch tested whether the initial shuffle signal survives harder null models and larger prime ranges. SEA means Scaling Experiment Arena. The final uploaded batch is complete: 15 runs, 60 summary rows, and 1,960 window rows.

The final SEA claim is no longer merely “prime gaps beat shuffle.” It is narrower and stronger: raw prime-gap order contains compression-detectable sequential structure not fully explained by marginal gap distribution, mod-6 wheel structure, or first-order Markov transitions, with positive scaling evidence through 2M primes.

10.1 Completed SEA final runs#

The final SEA package includes all planned continuation runs:

• 100k: Markov t2000, wheel6 t2000, block50 t1000, block100 t1000
• 500k: Markov t700, wheel6 t700, block50 t700, block100 t700
• 1M: shuffle t300, wheel6 t300, Markov t300, block50 t300, block100 t300
• 2M: Markov t100, wheel6 t100

This completes the missing 100k wheel replication, the 500k bridge, and the 1M shuffle baseline that were absent from the partial version.

10.2 Final raw gap summary#

10.3 Interpretation of the final SEA result#

Markov survival is now the central result. Markov-preserving controls keep first-order gap transitions, yet raw gap remains more compressible at every tested depth: 100k, 500k, 1M, and 2M.

Wheel survival is also confirmed. The raw signal survives mod-6-aware controls at 100k, 500k, 1M, and 2M. This means the signal is not merely the trivial 6k ± 1 residue structure.

Block controls narrow the scale claim. Block50 remains stable from 100k through 1M. Block100 is consistently weaker and mostly marginal. This suggests that a meaningful part of the signal lives at local-to-medium sequential scales, while the Markov/wheel survival shows the signal is not exhausted by the simplest local arithmetic explanations.

Sensor ranking is now clearer. Raw gap is the primary sensor. gap_div2 is effectively the same signal. abs_delta and bucket_log2 are diagnostic only: they weaken, flip, or become near-zero under harder controls.

10.4 Final E2 / SEA thesis#

Prime-gap order contains a surviving raw-gap 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 signal but more meaningful, and the final SEA batch confirms survival through 100k, 500k, 1M, and 2M prime ranges.

10.5 Caution#

The 2M runs currently use only 100 trials, so the permutation p-value resolution is limited. The z-scores and window stability are more informative at this stage. The final result is strong enough to justify the next phase, but not strong enough to claim an RH proof or a final mathematical invariant.

11. Relation to 8Z/DCC Founding Hypothesis#

The first E2/SEA results support number theory as a credible ninth-domain candidate for the 8Z/DCC approach. The strongest current claim is not that DCC proves RH, but that 8Z/DCC compression tools detect a nontrivial order-sensitive signal in prime gaps that survives first-pass wheel-aware and Markov-preserving controls up to 2M primes.

11.1 Two-track strategy#

This project should not choose prematurely between “prove RH” and “build an independent MDLxDCC theory of prime structure.” Both tracks are useful and should run in parallel:

• Track A — RH-facing path: determine whether the discovered compression structure can be formalized into an invariant that implies RH, becomes equivalent to RH, or forbids off-critical-line zeros through a prime-counting or explicit-formula bridge.
• Track B — MDLxDCC-native path: develop a new information-theoretic account of prime distribution based on compression, multi-scale structure, and DCC-style invariants, valuable even if it does not directly prove RH.

These are not competing goals. RH is the gold-standard benchmark; the MDLxDCC invariant is the deeper search object.

12. Revised Position on Original RH/AC Hypothesis#

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.

13. Mathematics, Reality, and the Zero Framework#

This section records the philosophical motivation behind the Zero Framework while keeping it separate from the empirical E1/E2 result. The central issue is not whether mathematics is "wrong." Formal mathematics can define many internally consistent worlds. The stricter question is:

When a formal object is used to describe reality or nature, what exactly is the mapping between the symbol and the thing being described?

In pure mathematics, definitions are allowed to be abstract. A structure can be studied because it is consistent, elegant, or fruitful. In applied mathematics and mathematical physics, however, a formal object earns physical meaning only through a disciplined bridge: what it measures, what it predicts, what it preserves, and where the mapping stops.

This distinction matters for the present paper because RH lives inside pure mathematics, while the 8Z/DCC framing asks a more reality-facing question: whether prime-gap dynamics expose an information structure that can be measured, compressed, and related to deeper arithmetic regularities. The empirical compression tests do not decide ontology. They only ask whether a measurable signal exists.

13.1 The two zeros#

Standard mathematics treats zero as a single formal object: the additive identity, the result of \(x - x\), and the value attained when a function vanishes. The Zero Framework proposes that ordinary language often mixes two different meanings under the same word:

• Numeric zero \(0\): a position or value inside a formal system. It exists within a structure of relations.
• Ontological nothingness \(\emptyset\): absence of existence itself. Not a point, not a value, and not reached by ordinary arithmetic operations.

On this view, a function that equals zero at some point has not "become nothing." It has reached a structured value. A zeta zero is therefore better described as an exact cancellation or destructive-interference point, not an annihilation into non-being.

13.2 Analytic continuation as ontological smuggling#

Analytic continuation is an internally valid and powerful mathematical operation. The concern here is not its consistency. The concern is what happens when a result obtained by extension is presented under the same notation as the original object — without making the substitution explicit.

The Dirichlet series \(\zeta(s) = \sum_{n=1}^{\infty} n^{-s}\) converges for \(\text{Re}(s) > 1\). For \(s = 1\) the harmonic series diverges. For \(s = -1\) the sum \(1 + 2 + 3 + \cdots\) diverges to infinity. Yet analytic continuation assigns \(\zeta(-1) = -\frac{1}{12}\). This is a correct statement about the analytically continued function. But the informal phrase \(1 + 2 + 3 + \cdots = -\frac{1}{12}\) silently substitutes one object for another under the same notation. The two agree where the original sum converges. They are not the same thing where it does not.

The ontological import is this: when the output of an extended object is presented as if it were the output of the original process, the interpretation can carry a hidden metaphysical claim — that the "true value" of a divergent process exists and is reachable by algebraic extension — without stating that claim explicitly or subjecting it to scrutiny. Whether this affects internal consistency is no. Whether it affects what mathematics describes about reality is an open question. This is what we call ontological smuggling: not an error in the proof, but an unacknowledged step in the interpretation.

13.3 Axiom-level vs. interpretation-level critique#

There are two versions of the Zero Framework critique, and they should not be confused:

• Interpretation-level: Current mathematics is largely correct, but some language misleads. A "zero of a function" should not be described as ontological nothingness. A divergent series assigned a finite value through continuation or regularization should be presented with that context intact. This is already partially implemented in Section 14.
• Axiom-level: The identification of numeric zero with ontological nothingness is not just a language habit — it may be embedded in the foundations. A mathematics built on a clean distinction between \(0\) and \(\emptyset\) from the start might produce a different structure, one where certain currently valid moves require explicit ontological justification before proceeding. This remains a research program, not a theorem.

The historical analogy is Lobachevsky. For two millennia, Euclid's parallel postulate appeared self-evident — not a choice but a necessity. Replacing it with a different axiom produced a geometry that was equally consistent and turned out to be physically more accurate than the original. The Zero Framework at the axiom level is asking whether the identification \(0 \equiv \text{"nothing"}\) is a parallel postulate of arithmetic: a convenience that has been mistaken for a necessity. The interpretation-level critique is conservative and already useful. The axiom-level critique is speculative but not without historical precedent.

13.4 Implication for RH specifically#

The implication for RH is indirect. RH is a formal statement about the zeros of the analytically continued zeta function. The Zero Framework does not currently change that statement, prove it, or disprove it.

What it may do is sharpen the language around what a zero means. If a zeta zero is treated as a structured cancellation point rather than "nothingness," then the research question becomes cleaner:

What structure forces all nontrivial cancellation points of the zeta function onto the critical line?

That question is still mathematical. The 8Z/DCC contribution, if any, would be to search for measurable information-structure invariants in prime-gap dynamics or prime-counting error that might later connect to the zeta-zero structure.

Current status: interpretation-level critique useful; axiom-level critique unformalized; RH implication speculative. This remains L8 in the interpretation ladder unless a formal invariant is discovered.

14. Corrected Language About Zeta Zeros#

Zeta zeros are points of exact cancellation or destructive interference in the analytic structure of the zeta function, not annihilation into ontological nothingness. This correction is an interpretation-level clarification: a function attaining the value zero is not the same as a quantity ceasing to exist.

15. Current Working Thesis#

Ordered prime gaps contain measurable compression structure beyond their marginal distribution. After the final E2/SEA batch, the surviving raw-gap signal also exceeds wheel-aware and first-order Markov-preserving controls from 100k through 2M primes, while block tests show that much of the broader signal remains local-to-medium scale.

The strongest current thesis is narrower than the original speculation but stronger than the initial shuffle-only result. v0.6 showed that the Markov2 floor can be beaten on holdout and that the carrier is scale-conditioned transition structure rather than a clean one-term density explanation. v0.7 sharpens this further: a generic loglog position-clock conditioned Markov2 law beats density variants on raw gap across the tested split ladder. The longer-term target is a formal information-theoretic invariant of prime distribution, not merely another route to the existing RH statement.

Operationally, the project now runs on three tracks: Track A seeks an RH-facing formal bridge; Track B seeks a native MDLxDCC invariant describing prime-gap structure directly; Track C isolates the generator carrier that can beat Markov2 without leakage or split artifacts.

16. Next Experiments#

The v0.7 ladder is now complete through 1M. The next work should not simply widen the arena. The current bottleneck is formal clarity: distinguish generic scale-clock laws from true prime-density information, then ask whether the surviving law has a clean mathematical description.

• v0.7.1 hygiene: fix the law-sheet split-name issue where rolling1/rolling2/rolling3 are all reported as missing_splits=rolling. The generator results are usable; the law-sheet failure label is over-penalized.
• Top carrier replication: rerun only the top raw-gap carriers — clock_loglog_markov2, clock_log_markov2, density_wrong_scale_markov2, density_dcc_no_motif, density_markov2, density_shuffled_markov2, and clock_dcc_no_motif — at 2M or 5M.
• Prime-specific density test: add stricter placebos: rank-preserving random monotone clocks, permuted density blocks, shifted-log clocks, and density residuals orthogonalized against loglog clock.
• Invariant extraction: treat clock_loglog_markov2 as the first raw-gap law-sheet candidate and gap_mod6 variable_markov as the residue-memory candidate; attempt compact symbolic descriptions of both.
• Formal bridge: only after carrier replication, connect the candidate residuals to prime-counting error, explicit-formula residuals, or a new RH-adjacent criterion.

17. Preliminary Conclusion#

E1 delivered a clear shuffle signal. E2 narrowed and strengthened the result. The final SEA batch now confirms that the broad shuffle advantage is partly local/arithmetic, but the raw gap signal survives wheel-aware and Markov-preserving controls at 100k, 500k, 1M, and 2M.

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. The completed SEA run adds a 500k bridge, a 1M shuffle baseline, and stronger scale evidence through 2M primes.

The most stable current signal is not only the aggregate z-score, but the window-level consistency: Markov and wheel controls show negative raw-gap deltas in every tested window at 100k, 500k, 1M, and 2M. The only near-boundary result in the key raw-gap family is block100 at 1M, with 49/50 negative windows and a marginal z-score.

This is not RH evidence yet, but it is a legitimate 8Z/DCC number-theory signal candidate. The deeper goal is to discover a formal invariant that explains prime-distribution structure; RH is the benchmark, not the only possible prize.

18. Three-Track Roadmap: RH, MDLxDCC Invariant, Generator Arena#

The current compression experiments cannot prove the Riemann Hypothesis by themselves. 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.

18.1 Three live tracks#

The roadmap now has three live tracks:

This distinction matters. A successful MDLxDCC invariant could be valuable even before it proves RH, and a generator that survives holdout and scaling could become the first concrete object from which a stronger invariant is extracted.

The higher target is not merely “prove RH by compression.” The higher target is to discover a formal invariant or compact generator of prime-distribution structure. RH then becomes one benchmark such an invariant may imply, explain, or sit beside.

18.2 The required bridge#

A possible proof path would need the following form:

1. Use 8Z/DCC/MDL tools to discover a stable invariant in prime-gap dynamics, prime-counting error, or another RH-adjacent arithmetic trace.
2. Formalize that invariant as a precise mathematical statement.
3. Prove that the invariant holds for all relevant values, not merely for tested ranges.
4. Show that the invariant implies RH, is equivalent to RH, strengthens an RH-style distribution bound, or explains a structure that RH does not directly describe.

In short:

empirical signal → generator candidate → formal invariant → distribution theorem / RH implication / new criterion

18.3 Why not just prove RH?#

RH matters because it is one of the cleanest known formulations of prime-distribution regularity. It would strongly constrain the error term in the prime number theorem and would affect many related results in analytic number theory.

But a new 8Z/DCC invariant could be valuable even before it proves RH, and potentially more valuable if it explains more than RH alone. For example, it could:

• give a new formal description of structure in prime gaps,
• produce a bound on prime-counting error comparable to an RH-style bound,
• explain why the distribution is stable rather than merely state that it is,
• generalize to other L-functions, arithmetic traces, or structured sequences,
• or discover a new RH-equivalent criterion from an information-theoretic direction.

Thus RH is not the only possible victory condition. It is the gold-standard benchmark. The deeper goal is a new invariant that explains prime-distribution structure in a way that can later be connected to RH or to a stronger distribution theorem.

18.4 Prime-counting error route#

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.

18.5 Explicit-formula route#

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:

1. Assume an off-critical-line zero exists.
2. Show that such a zero forces a detectable oscillatory or compressible signature in prime-counting error or prime-gap dynamics.
3. Show that a proven MDL/DCC invariant forbids that signature.
4. Conclude that no off-critical-line zero can exist.

Symbolically:

off-line zero → forbidden oscillation/compression signature → contradiction → RH

18.6 Equivalent-criterion route#

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:

1. prove the new condition is equivalent to RH or implies RH,
2. then prove the new condition directly.

This may be more realistic than directly proving statements about zeta zeros from compression data.

18.7 Proof-roadmap phases#

18.8 Current position#

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 or generator invariants that bound or constrain prime-distribution error. RH is the benchmark; the deeper prize is the invariant. The project should pursue all three tracks: RH-facing proof bridge, MDLxDCC-native structure invariant, and inverse generator arena.

This keeps the door open without pretending that E1, E2, SEA, or the current RH Arena runs can prove RH by themselves.

19. Inverse Generator Hypothesis#

RH gives a constraint on prime-distribution error. MDLxDCC asks the inverse question:

Can we discover a compact generator, grammar, automaton, dictionary, or multi-scale program that reproduces prime-gap structure better than shuffle, wheel, Markov, or Cramér-like null models?

This is not necessarily a search for a one-line formula for primes. The target is the shortest useful generator of prime-gap structure: an algorithmic or MDL formula that explains more of the real trace than competing null models.

If such a generator survives holdout and scaling, it becomes an invariant candidate. If the invariant later constrains prime-counting error, explicit-formula residuals, or zero-spacing structure, it may become an RH bridge — or a separate MDLxDCC-native path to prime-distribution theory.

19.1 Arena flow#

trace → encoding → sensors → controls → generators → MDL score → holdout validation → invariant candidate

The critical change is that controls are no longer only adversaries. They become baselines that generators must beat. A generator that cannot beat Markov, wheel, or Cramér baselines is not yet an invariant candidate, even if it produces visually plausible prime-gap structure.

19.2 Generator classes#

19.3 Acceptance rule#

A generator becomes interesting only if it satisfies all of these:

1. trained only on early data, evaluated on later holdout data,
2. beats simpler controls under total_mdl_score, not merely raw accuracy,
3. matches LZ76, entropy, motif spectrum, and next-gap predictive cost within acceptable error,
4. survives scaling across 100k → 500k → 1M → 2M and beyond,
5. produces a compact description that can be inspected mathematically.

In this language, the Arena does not “prove RH.” It searches for compact generator/invariant candidates. Proof would be a later certificate.

20. RH Arena v0.3 Results#

The completed v0.3 Arena run is the first cleaned-up version after the v0.2b diagnosis. It separates gap_mod6/gap_mod30 from prime_residue6/prime_residue30, uses wheel-aware Cramér controls with warmup, excludes the Cramér global window-0 artifact from summary aggregation, and scores generators using predictive bits over the holdout length.

20.1 Run integrity#

The run completed cleanly with windows.done.json, generators.done.json, and run.done.json. The result is usable as a v0.3 baseline.

20.2 Core sensor readout#

The core raw-gap signal remains real under most controls, but Markov2 is still the hard boundary.

gap_div2 repeats the same picture almost exactly. This confirms that the core raw-gap signal is not a token-scaling artifact.

20.3 Residue and Cramér readout#

The v0.3 split clarified the old residue confusion. gap_mod6 and gap_mod30 contain strong, stable gap-residue motif structure. prime_residue6 and prime_residue30 are now explicit prime-residue traces and should be treated as wheel-state diagnostics, not as raw evidence for a new invariant.

The earlier huge residue/Cramér readout is no longer treated as a discovery by itself. It is now a diagnostic showing where surrogate design and residue-state modeling still need care.

20.4 Generator leaderboard#

The generator arena changed meaningfully after predictive-bits scoring. For raw gaps, markov2 now beats markov1, and hybrid_dcc moves to second place but still does not beat the Markov2 baseline.

Current generator verdict: the Arena has not yet found a DCC/motif generator that beats Markov2 on raw gaps. That is not a failure. It is the clean next target.

20.5 Invariant candidates#

The v0.3 run emitted invariant_candidates.csv. The strongest stable candidates are not yet raw-gap formulas. They are mostly gap-residue and coarse gap-class motif families.

This is the right shape of result for an early inverse-generator arena: it does not magically produce an RH invariant, but it tells us where the stable motifs live. The strongest motifs are currently residue/gap-class structures. Raw integer gap motifs are weaker and need deeper filtering.

20.6 Conclusion and next step#

The v0.3 result sharpens the project:

• Confirmed: raw gap/gap_div2 LZ structure survives shuffle, block, wheel6, Markov1, and wheel-aware Cramér controls at 1M scale.
• Boundary: Markov2 absorbs/reverses the raw-gap LZ advantage.
• Generator floor: Markov2 is now the baseline to beat for raw gaps.
• DCC status: hybrid_dcc improved to second place for many encodings, but it still does not beat Markov2.
• Candidate status: strongest motif families are gap-residue/coarse-class motifs, not a raw-gap invariant yet.

v0.3 does not close the RH problem. It gives a cleaner target: build a generator or invariant that beats Markov2 on holdout while preserving the stable motif and residue-family structure discovered by the Arena.

The next code step should be a focused v0.4 generator arena: Markov2+DCC hybrids, variable-order Markov, motif-conditioned Markov, density-feedback states, and per-encoding generator leaderboards. The next paper step is to update this page after v0.4 shows whether any compact generator can actually beat the Markov2 floor.

21. RH Arena v0.4 Results: Markov2 Floor Attack#

The v0.4 Arena run executes the next step proposed by v0.3: stop asking only whether raw gaps survive controls, and ask whether a compact generator can beat the markov2 holdout floor. This section should be read with a strict split:

• Sensor/control layer: raw gap still hits the Markov2 boundary.
• Generator/holdout layer: new density-conditioned and DCC-style generators now beat the Markov2 floor.

v0.4 does not remove the Markov2 boundary from the LZ control test. It crosses the Markov2 floor in the generator arena.

21.1 Run integrity#

The run completed cleanly with windows.done.json, generators.done.json, and run.done.json. No completed tasks came from checkpoint; this was a fresh complete run.

21.2 Sensor readout: Markov2 still kills raw-gap LZ advantage#

The raw-gap sensor picture remains almost exactly the v0.3 lesson. gap survives many controls, including Markov1 and wheel-aware Cramér, but not Markov2.

gap_div2 repeats the same pattern: Markov2 is again positive/reversed, while shuffle, Markov1, wheel6, block, and Cramér controls remain negative. The important interpretation is that the LZ sensor no longer says “raw gaps beat everything.” It says “raw gaps beat first-order/local/wheel/Cramér controls, but second-order transition memory is enough to absorb this sensor.”

21.3 Generator floor: Markov2 is beaten on holdout#

The generator result is the new signal. The v0.4 challengers were designed specifically to attack the Markov2 floor. They succeeded across all tested encodings.

For raw gap, the first accepted breach is not a complicated motif grammar. The best generator is density_markov2: a Markov2 spine conditioned by local density state. The DCC hybrid is second, meaning DCC helps but is not yet the simplest winning explanation.

v0.4 generator verdict: the Markov2 floor is no longer unbroken. The best current raw-gap explanation is Markov2 plus local density state, with DCC hybrids close behind but not yet dominant.

21.4 Invariant candidates: robust motifs still live mostly in residue space#

The invariant-candidate file contains 5,826 rows. Of these, 168 survive at least 6 controls, 39 survive at least 8 controls, and 22 survive all 9 controls. Among the all-control survivors, 21/22 are gap_mod6 motif families and one is bucket_log2.

This confirms the v0.3 motif lesson at higher focus: the most stable candidate layer is still residue/gap-class structure, especially gap_mod6. Raw integer gap motifs exist, but they are weaker and do not yet look like the final invariant object.

21.5 Conclusion and next step#

• Confirmed: v0.4 completed the targeted Markov2-floor attack cleanly.
• Sensor boundary: raw gap/gap_div2 still lose or reverse against Markov2 in LZ-control testing.
• Generator breakthrough: all 9 tested encodings now have a non-Markov2 generator winner beating the Markov2 holdout baseline.
• Raw-gap winner: density_markov2, not full DCC, is the smallest current winner.
• DCC status: density_dcc and markov2_dcc are strong, especially on density-adjusted/error-walk encodings, but they still need ablation and scaling.
• Invariant status: stable motifs remain mostly gap-residue/coarse-class candidates, not a theorem-ready raw-gap formula.

v0.4 changes the project state from “Markov2 is the generator floor to beat” to “Markov2 has been beaten empirically at 1M by density-conditioned generators; now prove the gain survives scaling, ablation, and formalization.”

The v0.5 smoke now changes the next code step: scale the whole ablation ladder, not only density_markov2/density_dcc. At smoke scale, density_markov1 is the raw-gap winner, variable_markov is the gap_mod6 winner, and DCC/motif terms remain candidates rather than confirmed carriers. The next paper step is to compress the stable gap_mod6 motif families and the raw-gap density-Markov gain into one smaller invariant candidate.

22. RH Arena v0.5 Smoke Results: Density Ablation and Split Stability#

The first v0.5 run is a smoke-scale ablation, not a replacement for the v0.4 1M result. Its value is that it cleanly separates the raw Markov2 boundary from the generator layer and begins isolating which ingredient carries the generator gain.

v0.5 smoke verdict: the raw-gap LZ sensor boundary is still Markov2, but the generator floor is beaten again. At 120k scale the raw-gap winner is density_markov1, not a motif/DCC hybrid. This points to density-conditioned transition structure as the first ingredient to scale-test.

22.1 Run integrity#

22.2 Sensor readout: Markov2 still absorbs raw-gap LZ advantage#

The sensor layer remains consistent with the earlier v0.3/v0.4 story. Raw gap and gap_div2 are strongly more compressible than shuffle, wheel6, Markov1, and local Cramér controls, but not under Markov2. The gap_mod6 residue trace remains strongly structured even under Markov2.

22.3 Generator ablation: density_markov1 wins raw gap at smoke scale#

The generator layer is where v0.5 changes the working hypothesis. On raw gap/gap_div2, the best safe non-Markov2 generator is density_markov1 across all four tested splits. That is a useful simplification: at this scale, adding Markov order 2, motif injection, or full DCC does not improve the raw-gap winner.

The leakage guard is also behaving as intended: all safe density generators are marked non-leaking, while density_oracle_markov2 is deliberately flagged as using actual test-prime coordinates and should not be used as a public winner.

22.4 Conclusion and next step#

• Confirmed by smoke: v0.5 runs cleanly, resumes/writes all expected reports, and the leakage guard works.
• Sensor layer: raw-gap Markov2 boundary remains; gap_mod6 still carries strong residue-state structure.
• Generator layer: Markov2 holdout floor is beaten again, but the raw-gap winner is simpler than expected: density_markov1.
• Ablation readout: density clock + low-order transition is the first ingredient to scale-test; motif/DCC interactions remain candidates, not winners, on raw gap at this smoke scale.

v0.5 changed the next test: do not only scale density_markov2. Scale the whole ablation ladder — density_only, density_markov0, density_markov1, density_markov2, density-DCC variants, and variable Markov — and ask which ingredient survives at 1M, 2M, and beyond.

v0.6 update: that scaling test has now been run through 1M. The smoke winner density_markov1 did not remain the raw-gap winner; the scaled story is density/scale vs generic clock-DCC.

23. RH Arena v0.6 Carrier Isolation: Density vs Clock vs Residual#

The v0.6 run answers the question opened by v0.5: was the raw-gap generator gain really a density signal, or mostly a generic position/scale clock? The 4-worker sequence ran three stages — 150k smoke, 500k bridge, and 1M carrier isolation — with the optional 2M raw-carrier stage left off.

v0.6 verdict: the Markov2 generator floor is still beaten at 1M, but the carrier is not clean pure density. The full raw-gap split is won by density_wrong_scale_markov2; the mid and late raw-gap splits are won by clock_dcc_no_motif. The best current description is scale-conditioned transition structure, with density and clock-DCC variants both active.

23.1 Run integrity and scale sequence#

The 1M carrier run used 1,000,000 primes, 500,000 train primes, 50,000-token windows, 100 sampled controls, and 4/4 workers. It completed 840 window tasks and 3,000 generator replicates. The leakage guard is clean: 0 failed guard replicates and no oracle-density diagnostics in the default v0.6 run.

23.2 Sensor readout: raw-gap Markov2 boundary remains#

The sensor layer remains consistent with v0.3–v0.5: raw gap is strongly more compact than shuffle, wheel6, Markov1, and Cramér-like controls, but Markov2 still absorbs or reverses the raw-gap LZ advantage.

The stronger residue traces remain visible: gap_mod6 vs wheel30 has ΔLZ −1,476.84 and z −362.69; gap_mod6 vs Markov2 still has ΔLZ −198.95 and z −29.99. This keeps the residue-state track separate from the raw-gap generator track.

23.3 Generator carrier: density wins main, clock-DCC wins mid/late#

At 1M, non-Markov2 generators still beat the Markov2 holdout floor. The key result is the split diagnosis: density wins the full/main raw-gap split, while clock-DCC wins the mid and late raw-gap splits.

For the 1M main raw-gap split, the top generator leaderboard is:

The close cluster of density_markov2, density_lagged_markov2, and density_shuffled_markov2 remains important: it says scale state matters, but exact prime-coordinate density is not yet isolated as the sole cause. The new seed is clock_dcc_no_motif, because it wins mid/late raw-gap splits and is the best clock-family challenger.

23.4 Encoding separation: raw scale carrier vs residue memory#

This separation is useful. Raw gap/gap_div2 are now a scale-carrier problem. gap_mod6 and gap_mod30 are not primarily density-clock stories; they prefer higher-order or variable-order memory.

23.5 Invariant candidates#

The 1M v0.6 run produced 2,050 invariant candidates. High-control survivors remain concentrated in gap_mod6, with one sparse gap_div2 survivor.

All-control candidates in this run:

The highest-support all-control motifs again live mostly in the residue alphabet. They are strong empirical candidates, not theorem-ready invariants.

23.6 Conclusion and next step#

• Confirmed: v0.6 completed the density-vs-clock-vs-residual carrier isolation run cleanly.
• Sensor boundary: raw gap still hits Markov2 as the hard LZ-control boundary.
• Generator floor: Markov2 is beaten again at 1M on holdout.
• Carrier: not pure density; best wording is scale-conditioned transition structure.
• DCC status: clock_dcc_no_motif is now important because it wins mid/late raw-gap splits.
• Residue status: gap_mod6/gap_mod30 remain a higher-order memory track.

Do not claim that density explains the prime-gap signal. The current result says that density, generic clock, and clock-DCC are now separated enough to design the next sharper arena.

The next code step should be a smaller v0.7 carrier-stress arena: hold density_wrong_scale_markov2, density_markov2, density_lagged_markov2, density_shuffled_markov2, clock_dcc_no_motif, and density_residual_dcc_no_motif in direct paired competition, with stricter split-consistency penalties and an optional focused 2M raw-gap carrier run.

24. RH Arena v0.7 Carrier Stress: Density Placebo vs Generic Clock#

The v0.7 run is the surgical test proposed after v0.6. It asks whether the raw-gap generator gain comes from true train-extrapolated prime-density information, a generic monotone position/scale clock, wrong-scale regularization, or DCC state-conditioning. The ladder ran from 20k through 1M with 4 workers.

v0.7 verdict: the Markov2 generator floor is still beaten, but the raw-gap carrier is not isolated prime density. At 1M, clock_loglog_markov2 wins raw gap in all seven tested splits. Density variants remain strong, especially in the main split, but the density-placebo cluster shows that generic scale-clock state is the stronger current explanation.

24.1 Run integrity and ladder#

The 1M focused stage used 1,000,000 primes, 500,000 train primes, 50,000-token windows, 80 sampled controls, and 4/4 workers. It completed 980 window tasks, 1,323 generator score tasks, and 6,615 generator replicates. The leakage guard is clean: 0 failed guard replicates and no oracle-density diagnostics.

24.2 Sensor readout: raw-gap Markov2 boundary still holds#

The v0.7 sensor layer preserves the v0.3–v0.6 boundary. Raw gap is strongly more compact than shuffle, Markov1, wheel6, and Cramér-like controls, but Markov2 absorbs/reverses the LZ advantage.

The residue track is still separate: gap_mod6 vs Markov2 has ΔLZ −199.06 and z −30.59; gap_mod30 vs Markov2 has ΔLZ −31.17 and z −2.50.

24.3 Generator carrier: loglog clock wins raw gap across splits#

For the 1M main raw-gap split, the top generator leaderboard is:

The main split is close: clock_loglog_markov2 beats density_wrong_scale_markov2 by only 142.46 holdout units, so it is a density/clock tie under the configured carrier margin. But the split ladder is not close: early, mid, late, and rolling splits consistently prefer generic log/loglog clock states.

24.4 Density/clock placebo matrix#

The density placebo matrix is the main new information. On the 1M main raw-gap split, density_markov2, density_lagged_markov2, and density_shuffled_markov2 all sit at about Δ −7,980 vs Markov2; density_reversed_markov2 is also strong at Δ −8,019. This cluster means exact prime-coordinate density is not isolated as the unique cause. The signal behaves more like a monotone scale-state transition law.

By encoding, the 1M main split reads:

24.5 Candidate law sheets#

The new candidate_law_sheets.csv/json output works as a formalization-facing summary. For raw gap, the top law is clock_loglog_markov2: split count 7, split-win count 7, mean Δ vs Markov2 -6,021.28, best Δ -8,784.21, and candidate law “position-clock conditioned transition law.”

For the residue track, the top law remains gap_mod6 variable_markov: split count 7, split-win count 7, mean Δ vs Markov2 -25,018.00, best Δ -50,186.51, and candidate law “symbolic Markov transition law.”

24.6 Conclusion and next step#

• Confirmed: v0.7 completed the carrier-stress ladder through 1M with clean leakage guard.
• Sensor boundary: raw gap still hits Markov2 as the hard LZ-control boundary.
• Generator floor: Markov2 is beaten again on holdout.
• Raw-gap carrier: clock_loglog_markov2 is now the best raw-gap law candidate across splits; pure density is not isolated.
• Residue carrier: gap_mod6 remains a variable-order Markov memory problem; gap_mod30 remains a higher-order Markov problem.
• DCC status: DCC variants are still useful challengers, but they are not the raw-gap winner in v0.7.

v0.7 moves the project from “density/clock/DCC are all alive” to a sharper statement: the raw-gap holdout gain is best explained by generic log/loglog scale-clock conditioned transition structure, while residue alphabets carry separate higher-order symbolic memory.

The next code step should be v0.7.1 hygiene plus a narrow 2M/5M replication of the top law candidates. The next formal step is to write the clock_loglog_markov2 transition law and the gap_mod6 variable_markov residue law as compact mathematical objects, then test whether either connects to prime-counting error or an RH-adjacent criterion.

Appendix A – E1 Result Tables#

Tables are shown in Section 6 above.

Appendix B – Interpretation Ladder#

Appendix C – Working Language Rules#

Use: “compression-detectable structure”, “order-sensitive signal”, “generator candidate”, “invariant candidate”, “holdout validation”, “preliminary empirical result”, “null controls”, “not an RH proof”. Avoid: “RH confirmed”, “proof”, “annihilation of zeta zeros”, “AC requires RH”, “generator proves RH”.

Appendix D – Methodology & Reproducibility#

D1. E1 Shuffle Protocol#

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.

D2. Windowing#

• E1 20k run: window = 5,000, stride = 5,000.
• E1/E2 100k runs: window = 10,000, stride = 10,000.
• E2/SEA 500k and 1M runs: window = 20,000, stride = 20,000.
• E2/SEA 2M runs: window = 40,000, stride = 40,000.
• Mean LZ76: averaged across windows.

The final SEA batch contains non-overlapping main windows: 10 windows at 100k, 25 windows at 500k, 50 windows at 1M, and 50 windows at 2M.

D3. Encodings#

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, \(|g_{n+1} - g_n|\).
• gap_mod6 / gap_mod30: gap values modulo wheel bases.
• prime_residue6 / prime_residue30: prime residue-state traces, kept separate from gap-residue traces.
• bucket_log2: logarithmic bucket encoding of gap size.
• normalized_gap: gap divided by a local logarithmic scale.
• gap_minus_logp: gap minus a local expected logarithmic scale.
• cumulative_error_walk: cumulative walk of gap excess against local expectation.

D3b. LZ76 Implementation Note#

LZ76 is computed as a phrase-count complexity measure over token sequences, not as compressed byte size. Each encoding first maps the prime-gap sequence into symbolic integer tokens. The custom parser then scans the token stream left-to-right and counts the number of new phrases needed to parse it. Lower phrase count means the sequence is more compressible under this LZ76-style dictionary construction.

phrases = {}
i = 0
while i < len(tokens):
    extend the current phrase until it is new
    add phrase to dictionary
    phrase_count += 1
    i = next unread token

This is sufficient for the current signal tests, but future reproducibility releases should include the exact parser code, fixed test vectors, and a comparison with byte-level compressors.

D4. Main Statistic#

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.

D5. Permutation p-value#

The permutation p-value is computed as:

p = (1 + count(control_LZ <= real_LZ)) / (trials + 1)

The minimum possible p-value depends on trial count: with 100 trials it is about 0.0099, with 300 trials about 0.00332, with 700 trials about 0.00143, with 1000 trials about 0.000999, and with 2000 trials about 0.0005. Low p-values here should still be read as sampled-control dominance, not as a final theorem-level significance claim.

D6. Known Likely Sources of Signal#

The E1 compression advantage may partly reflect known arithmetic constraints and scale effects, including:

• changing average prime-gap size with scale, approximately related to \(\log n\),
• residue-class restrictions such as \(6k \pm 1\),
• local gap correlations,
• twin-prime and prime-constellation effects,
• sieve structure and wheel-like constraints.

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.

D7. Current Limitation#

The current result is a completed first-pass E2/SEA signal plus v0.3–v0.7 generator-carrier evidence, not a final theorem. Raw gap defeats shuffle, wheel6, Markov1, and Cramér-like controls but still hits Markov2 as the hard LZ-control boundary. The generator layer now beats Markov2 on holdout at 1M, and v0.7 shows that the raw-gap carrier is not clean pure density: generic loglog scale-clock conditioning currently wins the split ladder. The work still needs a v0.7.1 law-sheet hygiene patch, focused 2M/5M replication, independent code review, exact parser test vectors, and a formal bridge to prime-counting error or an RH-equivalent criterion.

D8. Arena Version Notes#

• v0.1: first RH Arena: windows, controls, sensors, motif extraction, generators, holdout outputs.
• v0.2b: resume/workers/progress hardening and 1M generator benchmark.
• v0.3: gap-residue vs prime-residue split, wheel-aware Cramér, Cramér warmup, predictive-bits generator scoring, invariant_candidates.csv, and completed 1M analysis.
• v0.4: focused Markov2-floor generator attack; non-Markov2 generators beat Markov2 across tested encodings at 1M.
• v0.5: density/DCC ablation and split-validation smoke; identified density-conditioned transition structure as first carrier candidate.
• v0.6: clock-vs-density-vs-residual carrier isolation; 150k→500k→1M sequence completed; scale-conditioned transition carrier found, not pure density.
• v0.7: carrier-stress ladder with density/clock placebo matrix, DCC-no-motif decomposition, split-consistency law sheets; raw-gap carrier shifts to clock_loglog_markov2, while gap_mod6 remains variable_markov.

Appendix E – Next Update Slot#

Next: Patch v0.7.1 law-sheet split naming, then run a narrow 2M/5M focused replication for clock_loglog_markov2, clock_log_markov2, density_wrong_scale_markov2, density_dcc_no_motif, density_markov2, density_shuffled_markov2, clock_dcc_no_motif, variable_markov, and markov3. Do not widen until the log/loglog-clock law and the residue variable-Markov law are written as compact candidate invariants.

Version 0.12 · May 12, 2026 · v0.7 carrier-stress integrated · raw-gap Markov2 sensor boundary preserved · Markov2 holdout floor beaten again at 1M · carrier narrowed from density-scale to generic log/loglog scale-clock transition structure · no RH proof claim · next: v0.7.1 law-sheet patch and 2M/5M focused replication.