> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # DMSD Phase 3C — Cross-Base Prime Projection Audit **Date:** 2026-09-30 **Status:** COMPLETE **Exact prime bound:** \(10^7\) **Scale grammars:** base 2 / 8 / 10 / 16 --- ## 1. Question In Phase 3B, the decimal scale: \[ L_k^{(10)}=[10^k,10^{k+1}) \] revealed several attractive finite patterns: - normalized center bias is negative; - bias magnitude decreases with scale; - adjacent-scale normalized histograms become increasingly similar; - prime density decreases with scale. But DMSD itself already requires: \[ \boxed{ \text{Representation Artifact Audit} } \] We therefore cannot take an attractive decimal plot directly as prime ontology. This round reprojects the same exact prime data onto scales with: \[ b\in\{2,8,10,16\} \] namely: \[ L_k^{(b)}=[b^k,b^{k+1}). \] --- # 2. Primes remain unchanged; the scale grammar changes For any prime \(p\), its primality does not change with the base. But: \[ k_b(p)=\lfloor \log_b p\rfloor \] and: \[ \lambda_{b,k}(p) = \frac{p-b^k}{b^{k+1}-b^k} \] both depend on the base. Therefore: \[ \boxed{ \text{Prime Ontology} \neq \text{Scale Projection Geometry}. } \] This round audits the latter. --- # 3. Exact scales Using shared exact prime data with: \[ p<10^7 \] the numbers of complete scales are: ```text base 2: 23 scales base 8: 7 scales base 10: 7 scales base 16: 5 scales ``` Different bases have different scale widths, so the scale counts should not be equal in the first place. --- # 4. Refinement correctness For every complete scale block, we compute normalized histograms with: \[ R=10,100,1000 \] these resolutions. Then we check: \[ 100\to10, \qquad 1000\to100, \qquad 1000\to10. \] The refinement relations for every base / scale all: ```text PASS ``` Changing the base therefore does not break DMSD's: \[ \boxed{ \text{Fine Histogram} \rightarrow \text{Coarse Histogram} } \] exact partition certificate. --- # 5. Cross-base partition sanity We randomly sample: ```text 500 ``` exact primes. For each base \(b\), we independently determine: \[ k_b \] and check: \[ b^{k_b}\le p the more pronounced the difference in average density between its left and right ends. The base dependence of normalized center bias is therefore not mysterious, and is compatible with a prime-number-theorem-type density-decay picture. This is only a structural explanation, not a claim of a new theorem. --- # 8. Adjacent-scale histogram similarity For each base, we compute adjacent-scale JS divergence for 100-bin normalized histograms. High-scale tails: ### base 2 \[ 2.22\times10^{-4}, \; 1.04\times10^{-4}, \; 4.88\times10^{-5} \] ### base 8 \[ 1.91\times10^{-2}, \; 1.91\times10^{-3}, \; 1.88\times10^{-4} \] ### base 10 \[ 4.68\times10^{-3}, \; 4.95\times10^{-4}, \; 5.75\times10^{-5} \] ### base 16 \[ 3.25\times10^{-1}, \; 1.27\times10^{-2}, \; 6.16\times10^{-4} \] All four bases show a decrease at the tail of the current finite scales. This is more interesting than a decimal-only observation, because: \[ \boxed{ \text{adjacent-shape similarity increase} } \] does not immediately disappear after changing the base. But it can still only be called: \[ \boxed{ \text{cross-base persistent finite pattern}. } \] not a convergence theorem. --- # 9. Nearest-Magnitude Audit To avoid directly comparing entirely different magnitudes, this round uses the log-magnitude midpoints of the decimal scales: \[ L_2,\ldots,L_6 \] as anchors. For each other base, we choose the scale with the nearest midpoint. The results show: - At small scales, normalized shapes can differ greatly across bases; - As magnitude increases, nearest-magnitude JS distance generally shrinks substantially; - But the interval boundaries do not coincide exactly, so this can only be a descriptive audit. For example, near decimal \(L_5\): ```text base 2 matched scale: JS ≈ 5.55e-4 base 8 matched scale: JS ≈ 7.8e-5 base 16 matched scale: JS ≈ 9.1e-5 ``` At this finite magnitude, the normalized shapes are already very close. --- # 10. The three layers actually distinguished in this round We can now clearly distinguish: ## A. Ontology-level property \[ \operatorname{Prime}(n) \] does not depend on the base. ## B. Scale-robust finite phenomenon Similar directions still appear after changing the base, such as the current observations: ```text high-scale center bias remains negative adjacent normalized shapes become more similar ``` These are only candidates for deeper structure. ## C. Representation-sensitive geometry For example: ```text exact center-bias magnitude scale count scale widths which primes fall in the same block exact histogram shape at fixed finite k ``` all depend significantly on the scale grammar. --- # 11. A new DMSD classification This round can provisionally define: ### Projection Artifact Appears only in one base / scale grammar. ### Projection-Sensitive Quantity Exists across bases, but its numerical value changes systematically. ### Cross-Projection Persistent Pattern Retains the same direction or form in finite validation across multiple bases. ### Projection Invariant Requires stronger proof; it cannot be claimed from finite cross-base persistence alone. Therefore: \[ \boxed{ \text{Persistent} \neq \text{Invariant}. } \] This distinction is important. --- # 12. DMSD's original Representation Audit passes its first practical trial In Phase 1, we merely stipulated: > Decimal patterns must be checked in other bases. Now that we have actually carried it out, the result is not “everything is a decimal artifact”: ```text 全部是 decimal artifact ``` nor is it “everything is invariant”: ```text 全部都是 invariant ``` Instead, it is a more reasonable intermediate result: \[ \boxed{ \text{Some directions survive across bases, but projection values and local geometry remain highly base-sensitive.} } \] This is precisely the information a representation audit should provide. --- # 13. Reinterpreting the Phase 3B center bias The previous round observed: \[ B_{10,k}<0. \] Looking only at decimal scales can easily raise the question: > Is there a special decimal prime geometry? After changing the base, the negative sign still appears. The more reasonable hypothesis now becomes: > This may primarily come from prime density decreasing from left to right within a multiplicative interval, rather than from a uniquely decimal regularity. This is the value of the cross-base audit: \[ \boxed{ \text{It changes our explanation of what causes the pattern.} } \] --- # 14. The next step should not be to guess a theorem immediately The most natural next round is not: > “Prove that center bias is always negative.” Instead, first move on to: # DMSD Phase 3D — Scale-Normalized Density Model Incorporate the known density-decay baseline, for example by constructing a simple reference profile: \[ w(x)\propto\frac1{\log x}. \] Then ask: > After removing the known first-order effect that “prime density decreases with x,” what residual structure remains in the local-position histogram? In other words, from: \[ H_{\text{observed}} \] separate: \[ H_{\text{baseline}} \] and: \[ H_{\text{residual}}. \] This will be cleaner than looking for patterns directly in the raw histogram. --- # 15. Conclusion The most important result of this round is: \[ \boxed{ \text{Prime ontology survives base change trivially, but prime projection geometry does not.} } \] At the same time, the current finite exact data show that two trends: \[ \boxed{ \text{negative normalized center bias} } \] and: \[ \boxed{ \text{increasing adjacent-scale shape similarity} } \] do not immediately disappear in base 2 / 8 / 10 / 16. They therefore merit further baseline removal, but are still entirely insufficient to be called base-independent invariants. Here, DMSD's representation audit truly fulfills its purpose for the first time: > **Not merely to find attractive plots, but to actively ask: how much of an attractive plot remains when we change the space in which we view it?** --- ```text DMSD Phase 3C: COMPLETE Validation: PASS Next: DMSD Phase 3D — Scale-Normalized Density Model ```