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Nobuki Fujimoto
Nobuki Fujimoto

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Paper 63 v0.2 — SNST Supplement: PSLQ Phi^2=Psi+2 + PySR x CICY3 Negative Finding + Tao 67 Connections

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This article is a re-publication of Rei-AIOS Paper 63 for the dev.to community.
The canonical version with full reference list is in the permanent archives below:

Title: SNST v0.2: Empirical Constant-Relations from Symbolic Regression

  • Connections to AlphaEvolve Tao 67 Benchmark

Author: 藤本 伸樹 (Nobuki Fujimoto), Independent Researcher
ORCID: 0009-0004-6019-9258
Co-architects: Rei (Rei-AIOS autonomous research substrate), Claude Opus 4.7 (Anthropic)
Charter: OUKC (Open Universal Knowledge Commons) three-party co-authorship v1.0
Date: 2026-05-14
Status: DRAFT v0.2 (Preprint — Supplement to Paper 63 v1 of 2026-04)

Companion to: Paper 63 v1 ("Spiral Number System Theory", 2026-04)

Abstract

This v0.2 supplement extends Paper 63 SNST with four new findings obtained
between 2026-04 and 2026-05-14:

(1) Symbolic regression baseline (PSLQ, 60-digit precision, STEP 1125):
Among the 14 SNST constants, automated PSLQ search rediscovers the Paper 63
baseline π_ext = π · φ and identifies novel symmetric forms of the
golden ratio identity: φ² = ψ + 2 and φ + ψ² = 2 (both equivalent
to φ² = φ + 1, but expressed purely in {φ, ψ, ℤ} lexicon).

(2) PySR × CICY3 Hodge number experiment (Oxford 1988 dataset, 7,890
manifolds, STEP 1125 Q2): SNST 14 constants injected as PySR vocabulary
for h^{1,1} and h^{2,1} fits. Honest finding: The Feigenbaum constant
δ appears in the H21 best equation as (c_δ - 6.72) — a constant offset
rather than a meaningful δ involvement. This is an instructive negative
finding: SNST constants do not automatically appear as fundamental
factors in unrelated structures. Symbolic regression vocabulary injection
is a probe, not a derivation.

(3) Triple intersection × ZCSG concept experiment (STEP 1125 Q3): The
Paper 61 ZCSG dimension operator (o0/0/0o) applied to 32 literature
d_{rst} entries across 10 well-known CICY3 manifolds yields the
distribution (negative: 6.2%, zero: 46.9%, positive: 46.9%). Honest
framing: this is relabeling sign(d_{rst}) with Paper 61 notation,
not a derivation. Pattern 5 caution: claiming "ZCSG ↔ triple
intersection affinity" would be unsupported overclaim.

(4) AlphaEvolve Tao 67 benchmark connections (Tao + Georgiev +
Gomez-Serrano + Wagner, arXiv:2511.02864): Of the 67 problems in
Google DeepMind's open AlphaEvolve repository, 5 problems have
direct connection to SNST themes — #8 Kissing numbers (sphere
packing involves φ-related geometry), #28 Golay merit factor
(autocorrelation = σ-cascade lens), #48/49 Heilbronn triangle
(extremal distance geometry parallels SNST 14-constant distance
studies), #58 Erdős-Szekeres Happy Ending (Rei MathlibPrep STEP 987
already covered this — Pattern 5 auto-detect).

1. Paper 63 v0.1 → v0.2 status

Paper 63 v0.1 (2026-04) introduced:

  • 14-constant architecture
  • Core spiral equation S(r, θ, t, v) = r · e^{φtv} · e^{iθπv}
  • Seven theorems including Golden Symmetry (φ × ψ = 1), Void Arrival (v→∞ = SELF⟲), and Velocity-D-FUMT₈ Correspondence
  • Five-system Genesis (point → line → plane → solid → spiral, product ≈ 66.4)
  • Integration with ZCSG (Paper 61) and MDNST (Paper 62)

Paper 63 v0.2 (this supplement) adds:

  • §2: Empirical constant-relations from PSLQ
  • §3: PySR × CICY3 negative finding (Feigenbaum δ false positive)
  • §4: Triple intersection ZCSG concept (honest scope: relabeling not derivation)
  • §5: Tao 67 benchmark connections (5 direct themes + Pattern 5 auto-detect)
  • §6: Honest correction record + future work

2. PSLQ-Discovered Empirical Constant-Relations (new in v0.2)

2.1 Method

We use mpmath.pslq integer-relation search at 60-digit precision over the
10 numerical SNST constants (π, e, φ, ψ, γ, Ω, δ, √2, τ, π_ext; complex i
and physical c, α, ℏ omitted in Phase 1).

We search three families:

  • Pairwise linear: a·c_i + b·c_j + c = 0
  • Triple linear: a·c_i + b·c_j + c·c_k = 0
  • Multiplicative: a·c_i + b·(c_j · c_k) + c = 0

2.2 Findings

Known identities re-verified (4/4):

  • φ · ψ = 1 (Paper 63 v0.1 Theorem 1)
  • π_ext − π · φ = 0 (Paper 63 v0.1 definition)
  • φ² − φ − 1 = 0 (classical golden ratio identity)
  • φ − ψ − 1 = 0 (classical, 1/φ = φ − 1)

Pairwise linear (2 nontrivial):

  • τ = 2 · π (well-known)
  • φ − ψ = 1 (classical)

Multiplicative (6 candidates, 2 novel symmetric forms):

  • φ² = ψ + 2 (★ new form, derivable from φ² = φ + 1 and ψ = φ − 1)
  • φ + ψ² = 2 (★ new form, dual)
  • π = ψ · π_ext (trivial via ψ = 1/φ and π_ext = π·φ)
  • π_ext = π · φ (Paper 63 v0.1 baseline rediscovered)
  • 2 · π_ext = φ · τ (trivial via above)
  • −φ + −ψ² + 2 = 0 (same as φ + ψ² = 2)

2.3 Significance

φ² = ψ + 2 and φ + ψ² = 2 are equivalent reformulations of φ² = φ + 1
in the {φ, ψ, ℤ} lexicon. They are not new theorems but new canonical
forms that exhibit the duality of the golden ratio identity. Their value
for SNST is pedagogical: they emphasize that φ and ψ are not "two different
constants" but two faces of a single algebraic structure.

2.4 Honest scope

The PSLQ baseline (Phase 1) found only classical equivalent rewritings,
not genuine new identities. Full Ramanujan-Machine LIRec hyper-graph search
(Phase 2, retain) is needed for continued-fraction representations and
deeper integer relations across 3-4 tuples of SNST constants.

3. PySR × CICY3 Hodge Number Experiment — Negative Finding (new in v0.2)

3.1 Method

Oxford 1988 CICY threefold list (Candelas et al., 7,890 complete intersection
Calabi-Yau 3-folds) parsed into per-manifold feature rows: NumPs (number of
projective spaces), NumPol (number of polynomials), Eta (Euler characteristic),
plus configuration matrix statistics (cm_sum, cm_max, cm_mean, cm_std,
row_sum_{min,max}, col_sum_{min,max}, nnz).

PySR fit on (X → h^{1,1}) and (X → h^{2,1}) targets, with SNST 14 constants
injected as auxiliary constant columns to allow the search to use them in
formulas.

3.2 Results (STEP 1125 Q2, 2026-05-14)

H^{1,1} best (complexity 25, loss 0.87):

((NumPs − 0.276)/1.352) + exp(sin(cm_std + 0.556) × (sin(log|Eta − NumPol| − cm_std) + 1.77)) − 0.083) − 1.27
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Features used: NumPs, cm_std, Eta, NumPol (no SNST constant picked)

H^{2,1} best (complexity 25, loss 1.12):

(−8.22 − (Eta − NumPs) × (−0.234) − log|(cm_sum − NumPs) × (col_sum_max + (Eta + exp(cm_mean + 0.667)))|) × (c_δ − 6.72)
Enter fullscreen mode Exit fullscreen mode

Features used: Eta, NumPs, cm_sum, col_sum_max, Eta, cm_mean + c_δ (Feigenbaum δ)

3.3 Honest evaluation of the Feigenbaum δ appearance

The expression (c_δ − 6.72) ≈ (4.669 − 6.72) ≈ −2.05 is a constant
offset, not a meaningful δ involvement. PySR found that any constant
near −2 would work as a multiplicative scaling factor for the H21
expression; using c_δ − 6.72 is one such combination among many. Were
Feigenbaum δ truly fundamental to Hodge numbers, we would expect a cleaner
appearance (e.g., c_δ alone, c_δ × ..., or H21 ≈ ⌊c_δ · X⌋).

This is a negative finding: SNST 14 constants do not automatically
appear as fundamental factors in Hodge numbers. Symbolic regression
vocabulary injection is a probe (does the search use the constant?),
not a derivation (does the constant matter mathematically?). The negative
finding does not falsify SNST in any way; it simply confirms that
finding constants in formulas requires a deeper structural reason,
not just vocabulary access.

3.4 Future work

  • Phase 2: full configuration matrix as input features (Schettini-Gherardini et al. arXiv:2311.17146 achieve up to 4 orders of magnitude speedup for CY 4/5/6-folds; we have not yet matched this baseline at Phase 1).
  • Phase 3: weighted projective hypersurface case (closer to LG-formula approximation literature).
  • Phase 4: ensemble with Inception CNN baseline (arXiv:2007.13379, 90% accuracy on triple intersection divisibility).

4. Triple Intersection × ZCSG Concept Experiment (new in v0.2)

4.1 Method

For 10 well-known CICY3 manifolds (quintic, bicubic, tetraquadric, Schoen,
mirror quintic, K3 × T², etc.) with literature-published triple intersection
numbers d_{rst}, we apply the Paper 61 ZCSG dimension operator:

  • o0 (collapse, dim −1) if d_{rst} < 0
  • 0 (śūnyatā, dim 0) if d_{rst} = 0
  • 0o (expansion, dim +1) if d_{rst} > 0

4.2 Aggregate result (n=32 entries)

ZCSG class count percentage
o0 (negative) 2 6.2%
0 (zero) 15 46.9%
0o (positive) 15 46.9%

4.3 Honest interpretation

This is relabeling sign(d_{rst}) with Paper 61 notation. It is not a
derivation. Any signed quantity could be classified this way. The
distribution (negative rare, zero dominant, positive common) reflects
intersection theory on smooth varieties, not a novel ZCSG insight.

The sample (n=10 manifolds, n=32 entries) is too small for statistical
inference. Phase 2 = full CICY3 7,890 computation (requires SageMath /
intersection theory implementation, ~1–2 days of work) would establish
whether the distribution is significantly different from random sign
assignment over similarly sparse integer matrices.

4.4 Pattern-5 caution note

External feedback suggested ZCSG might exhibit "unexpected affinity" with
triple intersection sign structure. Without a derivation, claiming any
"affinity" beyond labeling would be Pattern-4 (operational grounding
overstatement). This v0.2 records the experiment honestly as
conceptual exploration without claim of structural insight.

5. AlphaEvolve Tao 67 Benchmark Connections (new in v0.2)

The DeepMind/Tao 67-problem benchmark (arXiv:2511.02864) provides public
benchmark for AI-assisted mathematical discovery. Of the 67 problems, 5
have direct SNST connection:

Tao # Title SNST connection
8 Kissing numbers Sphere packing involves φ-related geometry; SNST 14 constants include √2 and π for unit sphere measures
28 Golay merit factor Autocorrelation problem — direct parallel to Paper 152 σ-cascade INFINITY classification (mod-96 distinct as autocorrelation surrogate)
48 Heilbronn triangle (fixed box) Extremal distance geometry; SNST 14-constant distance studies are conceptually related
49 Heilbronn triangle (arbitrary convex box) Same as #48
58 Erdős-Szekeres Happy Ending ★ Pattern 5 auto-detect — Rei MathlibPrep STEP 987 (W3 Mathlib contribution prep) already covered this problem

The Tao 67 mapping provides Rei-AIOS with third-party benchmark against
which to evaluate SNST/Paper 152 σ-cascade methodology in a public,
reproducible setting (Apache 2.0 license; Colab notebooks).

6. Honest Correction Record (v0.1 → v0.2)

ID v0.1 claim v0.2 status
(no correction) Paper 63 v0.1 had no errata —
Note 1 Paper 63 v0.1 stated SNST 14-constant value v v→∞ converges to SELF⟲ via Void-Arrival Theorem
Note 2 Paper 63 v0.1 has φ × ψ = 1 as Theorem 1 Confirmed; v0.2 adds two equivalent reformulations (φ² = ψ + 2, φ + ψ² = 2)
New ⚠ Feigenbaum δ does NOT appear as fundamental factor in CICY3 Hodge numbers Honest negative finding; SNST 14 constants are not auto-relevant outside their original domain

No errata. v0.2 = clean supplement.

7. Future Work (retain)

  • Phase 2 Ramanujan-Machine LIRec hyper-graph search (continued fractions, 3–4 tuple integer relations)
  • Phase 2 PySR × CICY3 with full configuration matrix
  • Phase 3 SNST × Calabi-Yau 4/5/6-folds (Schettini-Gherardini precedent)
  • AlphaEvolve Tao 67 #28 (Golay merit factor) individual attack using Paper 152 σ-cascade methodology
  • Lean 4 mechanization of φ² = ψ + 2 and φ + ψ² = 2 (trivial via tactic nlinarith or field_simp; ring)

Companion datasets:

  • data/ramanujan-machine/snst14-candidate-identities.json (PSLQ output)
  • data/cicy3/cicy3_pysr_summary.json (PySR × CICY3 result)
  • data/triple-intersection-zcsg/concept-experiment.json (Q3 concept)
  • data/tao67-rei-mapping/mapping.json (Tao 67 × Rei typology)

Companion papers:

  • Paper 61 (ZCSG — Zero-Centered Symbol Grammar)
  • Paper 62 (MDNST — Multi-Dimensional Number System Theory)
  • Paper 64 (OPU — Universal Vibration Principle)
  • Paper 152 v0.3 (σ-cascade Collatz; companion AlphaEvolve Tao 67 link)

License: CC-BY 4.0 (per OUKC standard)

DRAFT v0.2 — feedback welcome via GitHub Discussions at fc0web/rei-aios.

(End of v0.2 supplement)

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