> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # DMSD Phase 3A — Prime Layer Baseline **Date:** 2026-09-30 **Status:** COMPLETE **Positioning:** The first DMSD property layer to allow `UNKNOWN` as a formal mathematical state --- ## 1. Why the Prime Layer differs from the preceding layers The three preceding property classes: ### Parity \[ S(B) \] can be computed exactly directly from the interval endpoints. ### Residue \[ V_m(B) \] can be computed exactly directly from the quotient/remainder. ### Square \[ C_\square(a,b) = \lfloor\sqrt{b-1}\rfloor-\lfloor\sqrt{a-1}\rfloor \] can be computed exactly from the integer-square-root boundary. The prime baseline does not assume that the same kind of compact exact block operator exists. For the first time, DMSD must therefore explicitly distinguish: \[ \boxed{ \text{Represented Block} \neq \text{Known Exact Property Summary} } \] --- ## 2. PrimeSummary State This round defines: ```text EXACT_COMPUTED EXACT_MERGED UNKNOWN ``` ### EXACT_COMPUTED The entire interval has been computed exactly using the registered method. ### EXACT_MERGED All of the parent's children are exact, so the parent summary can be reconstructed exactly with the merge operator. ### UNKNOWN DMSD knows that this block exists, but currently has no admitted exact prime summary for it. This is neither false nor zero. It is simply: \[ \boxed{ \text{UNKNOWN}. } \] --- ## 3. Exact computation boundary for this round The local reference sieve covers: \[ 0\le n\le10^7. \] Thus, complete computation of: \[ L_k=[10^k,10^{k+1}) \] for: \[ 0\le k\le6 \] provides exact summaries. | \(k\) | interval | prime count | |---:|---|---:| | 0 | \([1,10)\) | 4 | | 1 | \([10,100)\) | 21 | | 2 | \([100,1000)\) | 143 | | 3 | \([10^3,10^4)\) | 1,061 | | 4 | \([10^4,10^5)\) | 8,363 | | 5 | \([10^5,10^6)\) | 68,906 | | 6 | \([10^6,10^7)\) | 586,081 | For: \[ k\ge7 \] this round's baseline leaves the count unfilled: ```text UNKNOWN ``` Importantly, this is only the **current runtime / validation boundary**, not a claim of any “mathematical impossibility of computation.” --- ## 4. Validation results This round completed: ```text Exact scale layers: 7 Hierarchy merge checks: 7 Nested exact refinement trials: 250 Independent trial-division oracles: 300 ``` All: ```text PASS ``` The independent oracle performs trial division element by element in small intervals, then compares the results with the sieve summary. --- ## 5. PrimeSummary The minimal summary for this round: \[ S_{\mathbb P}(B) = ( \sigma, C, p_{\min}, p_{\max}, M, \Gamma ) \] where: - \(\sigma\): knowledge state; - \(C\): prime count; - \(p_{\min}\): first prime; - \(p_{\max}\): last prime; - \(M\): method; - \(\Gamma\): coverage / provenance. If: \[ \sigma=\texttt{UNKNOWN}, \] then \(C,p_{\min},p_{\max}\) must remain unfilled. --- ## 6. Prime Merge Algebra If: \[ B=\bigsqcup_iB_i \] and **all** children are exact, then: \[ C(B)=\sum_i C(B_i). \] The first prime is the first prime of the first non-empty child; the last prime is the last prime of the last non-empty child. Therefore: \[ \boxed{ S_{\mathbb P}(B) = \bigoplus_i S_{\mathbb P}(B_i) } \] But this upgrade is allowed only when: \[ \forall i,\quad \sigma(B_i)\in \{\texttt{EXACT\_COMPUTED},\texttt{EXACT\_MERGED}\} \] holds. If any child has: \[ \sigma(B_i)=\texttt{UNKNOWN}, \] the children cannot establish that the parent is exact. --- ## 7. Knowledge-State Monotonicity DMSD can now distinguish numerical-domain refinement from knowledge refinement. Spatially: \[ B\rightarrow B_1,\ldots,B_m. \] In terms of knowledge, the progression may be: ```text UNKNOWN parent → UNKNOWN children → compute selected children → EXACT_COMPUTED children → all children closed → EXACT_MERGED parent ``` DMSD thus has not only a resolution tree, but is also beginning to have: \[ \boxed{ \text{Knowledge-State Tree}. } \] --- ## 8. The most important change in this round For the preceding properties, a global block could already have an exact summary before refinement. The prime baseline instead allows a represented scale on the order of: \[ 10^{100} \] to exist while: ```text prime_count = UNKNOWN ``` For the first time, this lets DMSD truly separate: \[ \boxed{ \text{Representation Capacity} \neq \text{Knowledge Capacity} \neq \text{Current Compute Capacity}. } \] these three capacities. --- ## 9. Why estimates cannot be filled in Asymptotic information such as: \[ \pi(x)\sim\frac{x}{\log x} \] may exist. But this round deliberately does not put an approximation into the: ```text EXACT prime count ``` field. If an estimation layer is added in the future, it must at least introduce: ```text ESTIMATED interval / error model method scope ``` It must not rename: \[ \text{estimate} \] as: \[ \text{exact}. \] --- ## 10. Why probable primes have not been smuggled in either Likewise, this round does not automatically label a probable-prime result for an arbitrary enormous integer as: ```text EXACT CERTIFIED PRIME ``` If a probabilistic / probable-prime layer is introduced in the future, it must explicitly state: ```text PROBABLE algorithm bases error assumptions scope ``` An upgrade to an exact certificate requires a corresponding proof object / deterministic guarantee. --- ## 11. Lazy Computation MVP `DMSD_Phase3A_PrimeBaseline_MVP.html` supports: - Kmax up to 100; - Pyramid / Staircase; - upward / downward traversal; - scale-level known / unknown visual distinction; - L0–L6 cached exact summaries; - huge scales preserved as UNKNOWN; - local lazy refinement; - on-demand segmented sieve for small intervals; - exact children → EXACT_MERGED parent; - any child UNKNOWN → parent merge remains UNKNOWN. The Browser MVP's local exact envelope is provisionally: ```text b ≤ 10^7 span ≤ 250,000 ``` These are engineering safety limits, not mathematical boundaries. --- ## 12. The first DMSD principle for the Prime Layer ### No False Closure Principle \[ \boxed{ \text{No certificate / no complete computation} \Rightarrow \text{do not mark exact}. } \] This principle is particularly important for primes, but it can in fact extend to all future property layers. --- ## 13. Three different kinds of Closure We can now distinguish: ### Domain Closure How far does the current runtime represent the domain? For example: \[ K_{\max}=100. \] ### Resolution Closure Which blocks have been expanded, and to what level of detail? ### Knowledge Closure Which blocks already have a property summary sufficient to label them exact? Therefore: \[ \boxed{ \text{Domain Closure} \neq \text{Resolution Closure} \neq \text{Knowledge Closure}. } \] This may be one of DMSD's most important structural differentiations after reaching the Prime Layer. --- ## 14. Relationship to the previous line of prime research For now, we do not use the previous SDRPC source-insufficiency conjecture as a premise for the Prime Layer. DMSD does something more conservative: > If there is currently no compact exact summary operator, honestly retain UNKNOWN until computation, a certificate, or a new mathematical operator provides closure. Thus, if a new exact summary mechanism for primes is indeed found in the future, DMSD can directly register it as a new property operator. The architecture itself does not prejudge its nonexistence. --- ## 15. Next step The most worthwhile next round is: # DMSD Phase 3B — Prime Multiscale Local Structure But validation should remain finite: - exact computed scales; - normalized prime density; - prime local-position histograms; - center-relative distribution; - staircase gap summaries; - refinement convergence; - representation artifact controls. Only in that round do we begin to “look at” primes. Phase 3A's job is simply to first ensure: \[ \boxed{ \text{We will not pretend to know what we have not computed.} } \] --- ## 16. Conclusion The Prime Layer Baseline is established: \[ \boxed{ \text{Huge represented domain} + \text{partial exact knowledge} + \text{explicit unknown regions} + \text{lazy local computation} } \] can coexist. Previously, the core of DMSD was: \[ \text{Represented Range} \neq \text{Materialized Elements}. \] We now add: \[ \boxed{ \text{Represented Range} \neq \text{Known Mathematical State}. } \] This takes DMSD beyond a multiscale visualization / summary runtime, toward: \[ \boxed{ \text{Multiscale Epistemic Mathematical Runtime}. } \] --- ```text DMSD Phase 3A: COMPLETE Validation: PASS Next: DMSD Phase 3B — Prime Multiscale Local Structure ```