Update README: v1 exhausted (7K false positives disproven), v2 asymmetric-degree kernel built

README.md

@@ -12,50 +12,61 @@ pretty_name: "Ramanujan Machine — GPU Formula Discovery Results"
 
 # Ramanujan Machine — GPU Formula Discovery Results
 
-GPU-accelerated search for new continued fraction formulas for mathematical constants, inspired by [Raayoni et al. (2024)](https://www.pnas.org/doi/10.1073/pnas.2321440121). **586 billion candidates evaluated through degree 7 — zero confirmed transcendental formulas.**
+GPU-accelerated search for new continued fraction formulas for mathematical constants, inspired by [Raayoni et al. (2024)](https://www.pnas.org/doi/10.1073/pnas.2321440121).
 
 > Part of the [bigcompute.science](https://bigcompute.science) project. AI-audited, not peer-reviewed.
 
-## Method
+## Key Findings (Updated 2026-04-07)
+
+1. **586 billion equal-degree polynomial CFs exhausted (v1 kernel, degrees 1-8)** — zero new transcendental formulas discovered.
+2. **7,030 "transcendental hits" were double-precision false positives** — all disproven via 100-digit PSLQ verification (verify_hits.py).
+3. **Only 20 confirmed formulas** — all classical: Euler's e, Brouncker's 4/pi, Leibniz pi/4, 1/ln(2).
+4. **Root cause identified**: the v1 kernel forced deg(a_n) = deg(b_n), but every known CF formula for transcendental constants has **deg(b_n) ≈ 2 × deg(a_n)**. Equal-degree CFs converge super-exponentially to algebraic numbers and cannot produce new transcendental formulas.
+5. **v2 kernel built** (ramanujan_v2.cu) with independent degrees for numerator and denominator polynomials. Validated on (1,2) regime — 48 confirmed transcendental formulas at 120-200 digit precision.
+
+## v1 Results (Equal-Degree Search)
 
-For polynomial pairs (P, Q) with bounded integer coefficients, evaluate the generalized continued fraction to double precision (500 terms), then match against 10 base constants + 29 compound expressions. v3 kernel filters degenerate zero-value matches.
+| Degree | Range | Candidates | Constants Found |
+|--------|-------|-----------|-----------------|
+| 1-3 | up to [-40,40] | ~282B | sqrt(2), sqrt(5), phi only |
+| 4 | [-7,7] | 577B | sqrt(2) only |
+| 5 | [-5,5] | 3.1T | sqrt(2) only |
+| 6-8 | [-2,2] to [-4,4] | ~60T | sqrt(2) only |
+| **Total** | | **586B+** | **Zero new transcendental** |
 
-## Results
+### PSLQ Verification (2026-04-07)
+- 7,030 claimed transcendental matches → **all false positives** at 100-digit precision
+- 20 confirmed formulas → **all classical, previously known**
+- Additional tests: deeper CF evaluation (depth 5000), expanded constant library (30 constants incl. MZVs, Glaisher, Khinchin), rational coefficients — all negative
 
-| File | Degree | Range | Candidates | Real Hits | Transcendental? |
-|------|--------|-------|-----------|-----------|-----------------|
-| `hits_deg1_range3.csv` | 1 | [-3,3] | 2,401 | ~50 | No |
-| `hits_deg2_range20.csv` | 2 | [-20,20] | 4.75B | 4.49M | No (sqrt(5) only) |
-| `hits_deg3_range10.csv` | 3 | [-10,10] | 37.8B | 119M | No |
-| `hits_deg4_range5.csv` | 4 | [-5,5] | 25.9B | 260 | No (2 false positive) |
-| `hits_deg5_range4.csv` | 5 | [-4,4] | 282B | 284 | No (4 false positive) |
-| `hits_deg6_range2.csv` | 6 | [-2,2] | 6.1B | 2 | No |
-| `hits_deg6_range3.csv` | 6 | [-3,3] | — | 32 KB hits | No |
-| `hits_deg7_range2.csv` | 7 | [-2,2] | 153B | 78.3K | No (30 false positive) |
-| **Total** | | | **~586B** | | **Zero transcendental** |
+## v2 Results (Asymmetric-Degree Search, In Progress)
 
-Additional runs in `results/` and `data/` folders at intermediate ranges.
+| Config (deg_a, deg_b) | Range | Candidates | Converged | Confirmed |
+|---|---|---|---|---|
+| (1, 2) | [-10,10] | 4.1M | 3M (73%) | 48 transcendental (classical) |
+| (2, 4) | [-6,6] | 816M | 521M (64%) | In progress |
 
-## Key Finding
+The v2 kernel also saves all converged-but-unmatched CFs to enable offline multi-constant PSLQ scanning (pslq_scan.py).
 
-After 586 billion candidates (82,000x larger than Raayoni et al.), zero transcendental CF formulas were found using double-precision screening. All matches are algebraic (sqrt(2), sqrt(5), phi). 36 compound matches (pi/4, sqrt(pi), e/pi, etc.) all failed mpmath 50-digit verification — they are false positives inherent to the double-precision methodology.
+## Method
+
+### v1 (deprecated)
+For polynomial pairs (P, Q) of the same degree with bounded integer coefficients, evaluate the generalized CF to double precision (500 terms), then match against 10 base constants + 29 compound expressions.
 
-**Methodology limitation**: Double-precision (53-bit) screening is insufficient for detecting transcendental CF formulas, which converge slowly. Raayoni et al. used arbitrary precision. A GPU PSLQ implementation is needed for the next phase.
+### v2 (current)
+CF = a(0) + b(1) / (a(1) + b(2) / (a(2) + ...)) where a(n) has degree d_a and b(n) has degree d_b independently. Target: d_b ≈ 2 × d_a (the "productive zone" for transcendental constants). GPU evaluates at double precision; survivors verified via CPU PSLQ at 100+ digits.
 
+## Understanding This Data
 
-### Degree 6 Range 3 (April 2026)
+The Ramanujan Machine project tries to discover new mathematical formulas by brute force: generate billions of continued fraction expressions and check whether any of them equal known constants like pi, e, or zeta(3).
 
-Extended degree-6 search to coefficient range [-3, 3]:
-- `results/hits_deg6_range3.csv` — 32 KB of candidate hits
-- `logs/run_deg6_range3.log` — 622 KB full computation log
+**The key lesson from this dataset**: the polynomial degree structure matters more than the search volume. 586 billion equal-degree candidates produced nothing, while 4 million asymmetric-degree candidates immediately re-derived classical formulas. The productive zone is deg(numerator) ≈ 2 × deg(denominator), matching the theoretical insight from Raayoni et al.'s Conservative Matrix Field framework.
 
-Computed on NVIDIA B200 cluster. No new transcendental formulas found. All matches algebraic.
 ## Source
 
-- **Code**: [ramanujan_gpu.cu](https://github.com/cahlen/idontknow/blob/main/scripts/experiments/ramanujan-machine/ramanujan_gpu.cu)
+- **Code**: [ramanujan_v2.cu](https://github.com/cahlen/idontknow/blob/main/scripts/experiments/ramanujan-machine/ramanujan_v2.cu) (v2), [ramanujan_gpu.cu](https://github.com/cahlen/idontknow/blob/main/scripts/experiments/ramanujan-machine/ramanujan_gpu.cu) (v1)
+- **Verification**: [verify_hits.py](https://github.com/cahlen/idontknow/blob/main/scripts/experiments/ramanujan-machine/verify_hits.py), [pslq_scan.py](https://github.com/cahlen/idontknow/blob/main/scripts/experiments/ramanujan-machine/pslq_scan.py)
 - **Experiment**: [bigcompute.science](https://bigcompute.science/experiments/ramanujan-machine-gpu/)
-- **MCP Server**: `mcp.bigcompute.science`
-- **AGENTS.md**: [Contribution guide](https://github.com/cahlen/idontknow/blob/main/AGENTS.md) (22 tools, no auth)
 
 ## Citation
 
@@ -69,14 +80,4 @@ Computed on NVIDIA B200 cluster. No new transcendental formulas found. All match
 }
 ```
 
-## Understanding This Data
-
-The Ramanujan Machine project tries to discover new mathematical formulas by brute force: generate billions of continued fraction expressions and check whether any of them equal known constants like pi, e, or the golden ratio.
-
-Each CSV file contains the results of testing candidates at a specific polynomial degree and coefficient range. A "hit" means the continued fraction's numerical value matched a target constant to about 15 digits of precision (double-precision floating point). The search checked 586 billion candidates across polynomial degrees 1 through 7, using 10 base constants and 29 compound expressions (things like pi/4, sqrt(3)/2, ln(2)).
-
-The headline result: zero new transcendental formulas were found. Every confirmed match turned out to be algebraic -- expressions involving square roots like sqrt(2), sqrt(5), or the golden ratio phi = (1+sqrt(5))/2. There were 36 matches against compound targets (like pi/4), but all of them failed when verified to 50 digits of precision, meaning they were numerical coincidences, not real formulas.
-
-This is a meaningful negative result. It tells you that if undiscovered continued fraction formulas for pi or e exist in this search space, they either require higher-degree polynomials (degree 8+) or involve coefficient patterns that this enumeration did not cover. The data lets other researchers skip re-searching this enormous region and focus their compute elsewhere.
-
 Human-AI collaborative work. AI-audited against published literature. Not independently peer-reviewed. CC BY 4.0.