Prime projection tomography

### PDF page 14 Page 13 of 14 Murray - Prime residue projection - submission manuscript The terms torus and information space are used operationally. The finite primorial wheel is a product of modular residue circles, and the inverse system of such wheels is naturally related to profinite integer structure. Information space here means the available residue-coordinate state space of the observer, not a physical cosmology claim. 5.4 How the result relates to Hardy-Littlewood and LOS The Hardy-Littlewood framework predicts different frequencies for admissible prime constellations through singular- series factors. Lemke Oliver and Soundararajan showed that those asymmetries produce striking consecutive-prime residue biases. The present method bundles constellations into recurrence observables and asks whether the bias is better described on a small wheel or on a larger hidden residue coordinate. The results are consistent with the analytic picture: the singular-series weights create the finite-scale gap-word structure, while non-lumpable projection explains why hidden gap-word structure appears as recurrence dependence on mod 30. 5.5 Why the residual matters The shrinking residual shows that primorial projection explains a large component of the observed dependence. The order-reversal tests show that the part left over is structured, not featureless noise: local gap words such as (2,10) and (10,2) can produce different recurrence behavior. The remaining problem is therefore sharper than before. Either larger wheels or higher-order lifted chains will absorb these ordered-word effects, or a positive finite-scale residue will remain and require a more refined operator-level account of gap-conditioned dynamics. 6. Falsifiers and next tests Large-wheel closure. If Delta_k(W) tends to zero as W increases, then the remaining residual is unresolved primorial state rather than a positive floor. Positive-floor residual. Simple decay-versus-floor fits to the current six wheel sizes do not decide whether Delta_k(W) tends to zero or to a positive floor. If larger wheels stabilize above zero despite sparse-aware out-of- sample fitting, then the residual is not exhausted by first-order primorial projection. Higher-order lifted closure. If second-order or third-order lifted chains on feasible W close the residual, the remaining effect is local chord-word memory inside the lifted torus. Asymptotic scale decay. Repeating the same experiment on much larger prime intervals will show whether the residual is finite-scale LOS/Holt behavior that decays slowly, or a stable structural feature over the measured range. Lumpability and commutator prediction. A stronger theoretical version would link an observable-specific projection-defect functional, or gap-conditioned commutator norms the norm of G_a G_b - G_b G_a, to recurrence residuals across moduli and lags rather than merely measuring both. 7. Conclusion Prime residues modulo 30 show recurrence dependence that low-order Markov controls on the observed alphabet do not reproduce, including under a chronological train-test check. Prime-residue projection tomography shows that CRT-aligned primorial lifts recover a large component of that dependence through the held-out-stable range, with W=510,510 as the main validated high-resolution wheel and W=9,699,690 retained only as a sparse diagnostic. Scrambled projections with matched fine-state counts do not reproduce the same coherent recovery trajectory. The remaining finite-scale structure is not concentrated in twin primes or in decisive two- or three-gap anomalies; it appears most clearly as scalar order-dependence in reversed gap-word contrasts. Projection converts hidden residue state into apparent recurrence dependence, and gap order reveals the local path geometry left unresolved by first-order primorial lifting. Data and code availability All computations were performed in Python using exact sieve-generated primes, NumPy, pandas, SciPy sparse matrices, and matplotlib. The submission package includes scripts and CSV outputs for the recurrence ratios, observed-state Markov controls, chronological Markov train-test checks, primorial lifts, dependence-corrected SE checks, block-length sensitivity checks, scrambled projection controls, lumpability defect, out-of-sample transition

component of the k=5 mod-30 residual through the held-out-stable wheel W=510,510 (+0.0537 at the mod-30 Markov baseline to +0.0081), while W=9,699,690 gives a smaller but sparse in-sample diagnostic (+0.0045) and is not used as the main predictive claim. Scrambled projections do not form the same coherent primorial-ordered recovery trajectory. The residual is not mainly a twin-prime effect and no two- or three-gap word remains decisive after FDR correction; the clearest residual signature is scalar order-dependence in reversed two-gap words, including (2,10) versus (10,2) at k=5. Conclusion. The results do not contradict Hardy-Littlewood or Lemke Oliver-Soundararajan theory. They provide a reproducible tomography of how known consecutive-prime biases appear under coarse observation: hidden

checks, not proof of generalization to unseen fine residues. I additionally perform a chronological 80/20 check for W=9,699,690; at this resolution a substantial fraction of held-out observations fall on fine states unseen in training, so the largest wheel is treated as a high-resolution diagnostic rather than a fully validated predictive model. 4. Results 4.1 Mod-30 recurrence and observed-state controls k real R_k ±2 SE 1 0.335012 0.001850

Table 9. Chronological 80/20 train-test check for observed-state Markov controls. The first- and second-order controls are fitted on the first 80% of prime indices and evaluated on the last 20%. Positive deltas indicate that the Markov model predicts more same-residue recurrence than is observed in held-out primes. 4.2 Primorial lifts reduce the recurrence residual W phi(W) observed states lifted R_5 Delta_5 z (real SE) 30 8 8 0.996525 +0.053727 +36.1 210 48 48 0.971079 +0.028281 +19.0

space here means the available residue-coordinate state space of the observer, not a physical cosmology claim. 5.4 How the result relates to Hardy-Littlewood and LOS The Hardy-Littlewood framework predicts different frequencies for admissible prime constellations through singular- series factors. Lemke Oliver and Soundararajan showed that those asymmetries produce striking consecutive-prime residue biases. The present method bundles constellations into recurrence observables and asks whether the bias is better described on a small wheel or on a larger hidden residue coordinate. The results are consistent with the analytic picture: the singular-series weights create the finite-scale gap-word structure, while non-lumpable projection explains larger wheels or higher-order lifted chains will absorb these ordered-word effects, or a positive finite-scale residue will remain and require a more refined operator-level account of gap-conditioned dynamics. 6. Falsifiers and next tests Large-wheel closure. If Delta_k(W) tends to zero as W increases, then the remaining residual is unresolved primorial state rather than a positive floor. Positive-floor residual. Simple decay-versus-floor fits to the current six wheel sizes do not decide whether Delta_k(W) tends to zero or to a positive floor. If larger wheels stabilize above zero despite sparse-aware out-of- residual is finite-scale LOS/Holt behavior that decays slowly, or a stable structural feature over the measured range. Lumpability and commutator prediction. A stronger theoretical version would link an observable-specific projection-defect functional, or gap-conditioned commutator norms the norm of G_a G_b - G_b G_a, to recurrence residuals across moduli and lags rather than merely measuring both. 7. Conclusion Prime residues modulo 30 show recurrence dependence that low-order Markov controls on the observed alphabet do