> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # DMSD Horizon Edition v0.2 ## Recursive Scale Map — Human → Frontier → Future → Symbolic Beyond-Human **Date:** 2026-09-30 **Status:** COMPLETE **Foundation:** DMSD Core v1 + Horizon Edition v0.1 --- # 1. What this round actually does v0.1 already pointed out: \[ \boxed{ \text{Scale Index Itself Must Be Serializable}. } \] v0.2 takes another step forward: > **We need not only to represent enormous scales, but also to have a complete scale map that shows where the current human record, active frontier, future design target, and symbolic beyond-human regions each lie.** Horizon Edition therefore no longer has only a “marker list.” It now has: \[ \boxed{ \text{Zone} \times \text{Profile} \times \text{Recursive Scale Address}. } \] --- # 2. Nine Horizon Zones v0.2 defines: ```text H0 HUMAN_LOCAL H1 HUMAN_CAPABILITY H2 HUMAN_RECORD H3 ACTIVE_FRONTIER H4 ENGINEERING_FUTURE H5 POST_CURRENT_COMPUTE H6 META_EXPONENT H7 TOWER_SPACE H8 RECURSIVE_UNBOUNDED ``` These are not predictions of dates. They describe: \[ \boxed{ \text{navigation / epistemic region}. } \] --- # 3. Current Human Anchors The current markers in this edition were rechecked. ## primecount 128-bit \(\pi(x)\) API precondition: \[ x\le10^{31}. \] Status: ```text HUMAN_CAPABILITY ``` This is a software capability envelope, not “all mathematical problems have been computed up to \(10^{31}\).” --- ## π output precision Guinness record: \[ 314,000,000,000,000 \] digits. Approximately: \[ 3.14\times10^{14} \] digits. Classified as: ```text HUMAN_RECORD / OUTPUT PRECISION ``` --- ## GIMPS verified search At the time of this edition, the page showed that exponents below: \[ 83,200,483 \] had been tested and verified. --- ## GIMPS first-pass frontier At the time of this edition, the homepage showed that exponents below: \[ 141,892,943 \] had been tested at least once. This is: ```text ACTIVE_FRONTIER ``` not a complete verified frontier. --- ## Largest known prime \[ 2^{136,279,841}-1 \] has: \[ 41,024,320 \] decimal digits. Its value therefore lies in: \[ 10^{41,024,319} \le p < 10^{41,024,320}. \] --- # 4. Scale-of-Scale For an ordinary value scale: \[ L_k=[10^k,10^{k+1}). \] if: \[ k \] is already very large, do not materialize: \[ L_0,\ldots,L_k. \] First define: \[ M_m=[10^m,10^{m+1}) \] acting on the exponent index \(k\). For example: \[ k=41,024,320 \] gives: \[ m=7. \] Thus, initially we only need to know: ```text target belongs to exponent meta-scale M7 ``` then enter it and expand. --- # 5. Recursive Scale Address For a finite integer scale index: \[ k_0=k, \] recurse: \[ k_{i+1} = \lfloor\log_{10}k_i\rfloor. \] until: \[ k_r<10. \] to obtain: \[ \boxed{ \operatorname{RSA}(k) = (k_0,k_1,\ldots,k_r). } \] For example: \[ k=41,024,320 \] gives: ```text 41,024,320 → 7 ``` whereas: \[ k=10^{12} \] gives: ```text 10^12 → 12 → 1 ``` The navigation depth is therefore: \[ \boxed{ O(\log^* k) } \] in meta-depth order, rather than: \[ O(k) \] scale nodes. --- # 6. Represented Scale Depth Ordinary DMSD: \[ \text{Represented Range} \neq \text{Materialized Elements}. \] Horizon Edition v0.2: \[ \boxed{ \text{Represented Scale Depth} \neq \text{Materialized Scale Nodes}. } \] This is the main invariant of this special edition. --- # 7. Pyramid of Exponents The Exponent Pyramid does not draw: ```text k=0 k=1 k=2 ... k=41,024,320 ``` Instead, it first draws: ```text M0: 1 ... 9 M1: 10 ... 99 M2: 100 ... 999 ... M7: 10^7 ... 10^8-1 ``` Only after the user selects: \[ M_7 \] does it expand: ```text 40 million region 41 million region ... ``` The next level is then lazily refined. --- # 8. Staircase of Exponents Pyramid answers: > In which exponent magnitude does this scale lie? Staircase answers: > Within this exponent magnitude, how do we move to the right? Therefore: \[ \boxed{ \text{Exponent Pyramid} = \text{meta-scale navigation}. } \] \[ \boxed{ \text{Exponent Staircase} = \text{within-meta-scale serialization}. } \] This is completely isomorphic to the original two DMSD views, except that the object they act on changes from: \[ n \] to: \[ k. \] --- # 9. Recurse once more For a future target of: \[ 10^{10^{12}}, \] the ordinary value exponent is: \[ k=10^{12}. \] Its own scale address: ```text 10^12 → 12 → 1 ``` The UI therefore needs only finitely many levels. Going a step further: \[ 10^{10^{10^6}}, \] still does not require materializing \(10^{10^6}\) exponent nodes. Instead, navigate using a symbolic expression tree: ```text POW10( POW10( 10^6 ) ) ``` as the navigation structure. --- # 10. Symbolic Scale Expression For beyond-current scales, v0.2 allows: ```text INTEGER POW10(expr) TETRATION(base, height) ``` This does not mean the runtime actually needs to expand them. It only needs to be able to: - serialize; - compare category; - render; - descend symbolically; - attach state. --- # 11. Future is not Forecast For example: \[ 10^{100}, \quad 10^{1000}, \quad 10^{10^8}, \quad 10^{10^{12}}, \quad 10\uparrow\uparrow6 \] are all labeled: ```text DESIGN_TARGET ``` or: ```text SYMBOLIC_TARGET. ``` No years are assigned. There is no claim that: > Humanity will definitely compute that far. The purpose is simply: \[ \boxed{ \text{DMSD architecture should not break there}. } \] --- # 12. Navigation Horizon This edition formally distinguishes: \[ \boxed{ \text{Computation Horizon} } \] from: \[ \boxed{ \text{Navigation Horizon}. } \] The computation horizon may reach only the current frontier. But the navigation horizon can reach: \[ 10\uparrow\uparrow h \] or even any finite symbolic expression. Therefore: \[ \boxed{ H_{\rm nav} \gg H_{\rm compute} } \] is entirely valid. --- # 13. No Horizon Conflation The different current markers: ```text 10^31 314 trillion digits 141,892,943 exponent 41,024,320-digit integer ``` cannot be ranked as: > Whichever is larger has computed farther. because their units differ. We formally retain: \[ \boxed{ \textbf{No Horizon Conflation Axiom}. } \] Comparison is possible only after defining a mapping: \[ f: H_A\to H_B \] between them. --- # 14. Human → Frontier → Future → Symbolic The complete navigation band: ```text Human Local ↓ Human Software Capability ↓ Human Verified Record ↓ Active Frontier ↓ Engineering Future ↓ Post-Current Compute ↓ Meta-Exponent Space ↓ Power-Tower Space ↓ Recursive Unbounded Symbolic Space ``` This is v0.2's global scale map. --- # 15. Marker State Each marker must include: ```text profile zone status semantic coordinate kind recursive scale address date source / provenance ``` Current and future must not use indistinguishable colors. --- # 16. Current Markers and Symbolic Targets can coexist For example, the Certified Value profile can simultaneously show: ```text CURRENT: largest known prime ≈ 10^(41,024,320) FUTURE DESIGN: 10^1000 10^(10^8) SYMBOLIC: 10^(10^12) 10 ↑↑ 3 10 ↑↑ 4 10 ↑↑ 6 ``` They all appear on the same map, but have different epistemic states. --- # 17. The Scale Map is not a single number line The Horizon Map should be viewed as: \[ \boxed{ \text{Profile Lanes} \times \text{Horizon Zones} \times \text{Recursive Address}. } \] It is therefore more like a multiscale atlas, rather than one exceptionally long number line. --- # 18. Validation v0.2 performs: ```text 1000 random finite scale-index recursive-address checks 17 marker serialization checks zone registration checks path-depth examples ``` All PASS. This means: - Finite but enormous exponents can be located with an iterated-order chain; - Symbolic markers do not require expanding their values; - Current/future states remain separate. --- # 19. Typical Paths ### Largest known prime value-scale ```text value ≈ 10^41,024,320 exponent: 41,024,320 meta-scale: M7 ``` ### \(10^{10^{12}}\) ```text outer POW10 └── exponent = 10^12 └── exponent meta-index = 12 └── meta-index = 1 ``` ### \(10\uparrow\uparrow6\) Do not expand. Only preserve: ```text TETRATION base = 10 height = 6 ``` --- # 20. Returning to the original Pyramid At the very beginning, you said: > 1–10 is one layer, 10–100 is the next, and 100–1000 is the layer after that. Exactly the same rule can now act on: \[ k. \] That is: ```text exponent 1–10 exponent 10–100 exponent 100–1000 ... ``` It can even act again on: \[ \log k. \] The original intuition is therefore naturally recursive. --- # 21. Next step v0.2 has completed the full scale map. The most natural next round is: # DMSD Horizon Edition v0.3 — Compute Capacity Overlay Overlay the same map with: - single-machine; - workstation; - cluster; - exascale; - distributed volunteer; - future zettascale design scenarios; - storage / time / energy budget. This would give each horizon marker not only “where it is”: ```text 在哪裡 ``` but also “roughly how much compute burden it requires”: ```text 大概要付多少 compute burden ``` while still connecting to Phase 3H/3I's Source-Efficiency / Budget Policy. --- # 22. Conclusion The core of v0.2 is: \[ \boxed{ \text{Scale can itself have scale}. } \] An enormous numerical domain therefore no longer requires an enormous UI. Even when navigating to: \[ 10^{10^{12}} \] or: \[ 10\uparrow\uparrow6, \] DMSD preserves only a finite: \[ \boxed{ \text{recursive symbolic scale path}. } \] --- ```text DMSD Horizon Edition v0.2: COMPLETE Recursive Scale Map: PASS Human/Frontier/Future/Symbolic bands: REGISTERED Next: Compute Capacity Overlay ```