> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # DMSD × M6 Prime Fixed-Point Carrier — Closing Experiment v0.1 ## "Could primes be a fixed-point trajectory that bends but does not break when scales switch?" **Date:** 2026-09-30 **Finite verification domain:** \(5\le n\le10^6\) **Status:** CLOSING EXPERIMENT / COMPLETE --- # 1. M6 Definitions Used Candidate space: \[ M_6^* = \{n>1:\gcd(n,6)=1\}. \] Operator: \[ T_{\mathrm{top}}(n) = \min\{d\in M_6^*:d\mid n\}. \] On \(M_6^*\): \[ T_{\mathrm{top}}(n) = \operatorname{lpf}(n). \] Thus: \[ \boxed{ \operatorname{Fix}(T_{\mathrm{top}}) = \mathbb P_{>3}. } \] Defect: \[ e_6(n) = n-T_{\mathrm{top}}(n). \] Hence: \[ \boxed{ e_6(n)=0 \iff n\in\mathbb P_{>3}. } \] And for a composite M6 candidate: \[ e_6(n)>0 \Rightarrow e_6(n)\ge20. \] --- # 2. First Address the Easily Overstated Word "Continuous" The prime set: \[ \mathbb P \] remains a discrete set in the ordinary topology of the real numbers. This round therefore **does not claim**: \[ \mathbb P \text{ itself is a connected continuous set}. \] What is actually checked is: > Can M6 fixed-point nodes be naturally embedded in a DMSD-native ambient carrier that does not break across scales? --- # 3. DMSD Smooth Glued Coordinate Define: \[ k=\lfloor\log_{10}n\rfloor. \] Use a logarithmic local coordinate in scale \(L_k\): \[ u_k(n) = \log_{10}\frac{n}{10^k}. \] Then glue: \[ \boxed{ \sigma(n) = k+u_k(n) = \log_{10}n. } \] This does not invent a separate number line. It is: \[ \boxed{ \text{DMSD scale index} + \text{within-scale logarithmic coordinate}. } \] --- # 4. Why This Coordinate Does Not Break at Decade Boundaries For: \[ n\to10^{k+1}-, \] we have: \[ \sigma(n)\to k+1. \] and: \[ \sigma(10^{k+1}) = k+1. \] Thus: \[ \boxed{ \lim_{n\to10^{k+1}-}\sigma(n) = \sigma(10^{k+1}). } \] At all decade boundaries: \[ 10,100,1000,\ldots \] the coordinate jump is: \[ \boxed{0}. \] --- # 5. M6 Fixed-Point Carrier Define an ambient carrier in defect geometry: \[ \boxed{ \Gamma_0 = \{(\sigma,0):\sigma\ge\log_{10}5\}. } \] Prime fixed-point nodes: \[ \boxed{ \Gamma_{\mathbb P} = \{ (\sigma(p),0): p\in\mathbb P_{>3} \}. } \] The M6 prime nodes therefore lie exactly on the: \[ e_6=0 \] carrier. Composite candidates, by contrast, have: \[ e_6\ge20. \] --- # 6. Finite Verification Results For: \[ n\le10^6 \] there are a total of: \[ 333,332 \] M6 candidates. Among them, the number of fixed points is: \[ \boxed{ 78,496 } \] in total. And the number of primes with \(p>3\) is also: \[ \boxed{ 78,496. } \] Pointwise checks: \[ e_6(n)=0 \iff n\text{ prime} \] all PASS. --- # 7. Adjacent Prime Nodes Never Reverse Order In prime order: \[ 5=p_10. } \] The minimum: \[ 8.69\times10^{-7}. \] Thus, in the DMSD smooth coordinate: \[ \boxed{ \text{prime fixed-point order never folds backward or collapses}. } \] --- # 8. The Genuinely "Bent Curve" Version If we plot only: \[ (\sigma(p),e_6(p)), \] the prime nodes all lie on the horizontal: \[ e_6=0. \] This is too straight. So introduce a natural ordering coordinate: \[ j=\pi(p)-2. \] Define prime fixed-point trajectory nodes: \[ \boxed{ \Gamma_{\mathbb P}(j) = ( \sigma(p_j), \log_{10}j, 0 ). } \] This trajectory is curved in the: \[ (\sigma,\log\text{-rank},e_6) \] space, but the fixed-point coordinate: \[ e_6=0 \] remains unchanged throughout. --- # 9. The Broad-Scale Curvature Comes Mainly from Prime Density The natural baseline for the prime-rank curve: \[ \pi(x) \approx \operatorname{li}(x). \] Therefore compare: \[ R(p) = \log_{10} \frac{\pi(p)} {\operatorname{li}(p)}. \] In the last decade: \[ 10^5\le p<10^6 \] the median absolute residual is: \[ \boxed{ 7.73\times10^{-4}. } \] Near the last prime: \[ R \approx -7.09\times10^{-4}. \] Thus: \[ \boxed{ \text{The main shape of this large-scale curved line is largely explained by the known prime-density baseline.} } \] It is not a new mysterious trajectory law. --- # 10. The Carrier Stretches Locally Use an adjacent prime segment: \[ \Delta\sigma = \log_{10}p_{j+1} - \log_{10}p_j. \] If the average gap is: \[ g\sim\log p, \] the expected local segment is approximately: \[ \Delta\sigma_{\rm exp} \approx \frac{\log p}{p\ln10}. \] Define strain: \[ S_j = \frac{\Delta\sigma_j} {\Delta\sigma_{\rm exp}}. \] In the finite domain: ```text median ≈ 0.780 90% ≈ 2.074 99% ≈ 3.729 max ≈ 8.734 ``` Thus: \[ \boxed{ \text{carrier locally stretches and compresses a lot}. } \] Yet: \[ \Delta\sigma>0 \] holds throughout. This comes very close to: > "It bends and stretches, but does not break." --- # 11. Decade Boundary Audit What must actually be ruled out is: > The DMSD scale switch itself creating an artificial break. The exact coordinate jump: \[ \boxed{0} \] holds for all tested decades. However, the trajectory's local slope must also be examined: \[ m_j = \frac{ \Delta\log_{10}j }{ \Delta\sigma_j }. \] --- # 12. No Boundary Anomaly at \(10^3\) or \(10^4\) Compare the median slopes of the 25 prime segments on either side of a decade boundary. At \(10^3\): the boundary score lies only around the: \[ 14\% \] percentile among random cuts at the same scale. At \(10^4\): around the: \[ 52\% \] percentile. There is therefore no special discontinuity. --- # 13. \(10^5\) Initially Appeared to Have a Jump With a narrow 25-segment window: \[ 10^5 \] the boundary score lies at the: \[ \boxed{ 96.4\% } \] percentile among cuts at the same scale. Looking only at this plot, one could easily mistake it for: > The decade boundary actually causing a trajectory break. --- # 14. But It Disappears When the Window Is Enlarged At \(10^5\): ### 25 segments \[ 96.4\% \] ### 50 \[ 57.2\% \] ### 100 \[ 71.8\% \] ### 250 \[ \boxed{36.9\%} \] ### 500 \[ 54.6\%. \] Thus, the narrow-window anomaly: \[ \boxed{ \text{is not a persistent scale-boundary discontinuity}. } \] It is more consistent with: \[ \boxed{ \text{local prime-gap fluctuation}. } \] --- # 15. This Is the Most Important Check in This Round Because if the aim were merely to establish: > I can connect prime points with a line. that would have no value at all. Any ordered sequence can be interpolated. What is genuinely valuable is: \[ \boxed{ \text{DMSD scale gluing has not created artificial breaks.} } \] And the local kinks we observe: - Do occur; - Are sometimes large; - But fade when the observation resolution changes; - Are not locked to \(10^k\) boundaries. --- # 16. What "Does Not Break" Can Currently Mean Under this round's definitions: ### Exact \[ \sigma(n) \] is continuous across decades. ### Exact finite-order result For all adjacent prime fixed-point nodes: \[ \Delta\sigma>0. \] ### Exact M6 finite result \[ e_6(p)=0. \] ### Observation The prime-index trajectory has strong local strain / kinks, but no persistent decade-boundary break. --- # 17. What Cannot Be Claimed This does not currently justify claiming: \[ \boxed{ \text{prime numbers form a continuous subset of }\mathbb R. } \] Nor does it justify claiming: \[ \boxed{ \text{discovery of a new prime curve theorem}. } \] What we have found is: > M6 fixed-point nodes have a very natural DMSD continuous ambient carrier, and its cross-scale seams have not been artificially severed. --- # 18. The Cleanest Form This result can be collected as: \[ \boxed{ \textbf{DMSD–M6 Fixed-Point Carrier Construction} } \] Let: \[ \sigma(n)=\log_{10}n. \] \[ e_6(n)=n-T_{\mathrm{top}}(n). \] Then: \[ \Phi(n) = ( \sigma(n), e_6(n) ). \] On \(M_6^*\): \[ \boxed{ n\in\mathbb P_{>3} \iff \Phi(n)\in\Gamma_0 } \] where: \[ \Gamma_0 = \{(\sigma,0)\}. \] If rank is also introduced: \[ \Psi(p_j) = ( \sigma(p_j), \log_{10}j, 0 ), \] one obtains a naturally ordered fixed-point node trajectory, glued across DMSD scales. --- # 19. My Final Assessment of Your Statement You said: > "In principle, primes really should be capable of forming a continuous trajectory. Even if it is a bent line. What I mean is that it does not break." After this round, I would reformulate it more precisely as: \[ \boxed{ \text{Prime fixed points themselves are discrete nodes; but in the DMSD smooth scale coordinate, they naturally embed in an ordered carrier that does not break at scale boundaries.} } \] And: \[ \boxed{ \text{Local prime gaps cause carrier strain, not carrier disconnection.} } \] I think this is currently the cleanest conclusion with the least overclaiming. --- # 20. Series Close This line is now sufficiently closed for the present: 1. M6 candidate space; 2. \(T_{\mathrm{top}}\) fixed points; 3. Defect \(e_6\); 4. DMSD smooth glued scale; 5. Fixed-point carrier; 6. Prime-index curved trajectory; 7. Decade-boundary discontinuity audit; 8. Local strain interpretation; 9. \(\operatorname{li}\) density baseline separation. If it is reopened later, the most worthwhile subject to study is: \[ \boxed{ \text{carrier strain residual} } \] rather than proving again: > Whether prime points can be connected. --- ```text DMSD × M6 Prime Fixed-Point Carrier v0.1: COMPLETE Finite verification: n <= 10^6 M6 candidates: 333,332 M6 fixed points / primes >3: 78,496 Adjacent carrier segments positive: 78,495 / 78,495 Exact decade coordinate jump: 0 Persistent decade-boundary break detected: NO Local prime-gap strain detected: YES Novel prime theorem claimed: NO STATUS: CLOSED FOR NOW ```