Palantir v2.0 "Bounty Hunter": Riemann ζ-zeros CF pipeline + Beal counterexample scanner

[gHashTag]

Prize Target: $2,000,000 (Clay + AMS)

Context

Palantir v1.1 CF pipeline is complete (13 blind spots tested, φ²/π² proven ARITHMETICALLY GENERIC).
Now extending to attack prize problems using our CF + PSLQ toolkit.

Target #1: Riemann Hypothesis ($1M Clay)

Nobody has applied systematic CF analysis to ζ-zeros.

New modules needed:

1. src/sacred/zeta_zeros.zig — Odlyzko zeros database

• Import first 10,000 non-trivial zeros of ζ(s) from Odlyzko's tables
• Store as f128 array of imaginary parts t_n where ζ(½ + it_n) = 0
• First 100 zeros hardcoded, rest loaded from data file
• Reference: https://www.lmfdb.org/zeros/zeta/

2. src/sacred/cfrac_zeta.zig — CF expansion of zero spacings

• Compute Δt_n = t_{n+1} - t_n (consecutive zero gaps)
• Compute CF expansion of each Δt_n (up to 50 terms)
• Apply full 10-test battery from Palantir v1.1:
  • Irrationality measure μ
  • Khinchin constant K
  • Gauss-Kuzmin χ² test
  • Lévy constant convergence
  • Autocorrelation of partial quotients
  • Digit normality (base 10 + base 2)
• CLI: trinity cfrac-zeta --zeros 1000 --terms 50

3. src/sacred/pslq_spacing.zig — PSLQ on zero spacings

• Run PSLQ integer relation detection on normalized spacings
• Basis: {Δt_n, π, ln(2), φ, e, √2, √3, ζ(3)}
• Search for relations: a₁Δt + a₂π + a₃ln2 + ... = 0 with |a_i| ≤ 10
• If FOUND → potential breakthrough (arithmetic structure in zeros)
• If GENERIC → confirms GUE random matrix universality
• CLI: trinity pslq-zeta --zeros 100 --complexity 10

Target #2: Beal Conjecture ($1M AMS)

A^x + B^y = C^z, x,y,z > 2, gcd(A,B,C) = 1 → contradiction?

4. src/sacred/beal_scan.zig — SIMD brute force scanner

• Search for counterexamples: A^x + B^y = C^z with gcd(A,B,C) = 1
• SIMD vectorized inner loop (8× parallel on AVX2)
• Search space: A,B ≤ 10000, C ≤ 20000, x,y,z ∈ {3,4,5,...,20}
• Output near-misses: |A^x + B^y - C^z| < C^(z/2)
• CLI: trinity beal-scan --max-base 10000 --max-exp 20

5. src/sacred/beal_near.zig — Near-miss analyzer

• Collect near-misses from beal-scan
• Statistical analysis: do near-misses thin out (conjecture true) or cluster (counterexample nearby)?
• PSLQ on near-miss residuals
• CLI: trinity beal-near --input near_misses.dat

Pipeline Architecture

CURRENT (v1.1):
  cfrac-scan → cfrac-stats → cfrac-approx → cfrac-detect → cfrac-verdict

NEW (v2.0 "BOUNTY HUNTER"):
  zeta-zeros ──→ cfrac-zeta ──→ pslq-spacing ──→ verdict-prize
  beal-scan  ──→ beal-near  ──→ pslq-beal    ──→ verdict-prize

CLI Commands to Add

trinity cfrac-zeta    # CF analysis of ζ-zero spacings
trinity pslq-zeta     # PSLQ integer relations in zero spacings  
trinity beal-scan     # Brute force Beal counterexample search
trinity beal-near     # Near-miss statistical analysis

Acceptance Criteria

• zeta_zeros.zig compiles and stores first 100 zeros
• cfrac_zeta.zig computes CF of Δt_n and runs 10-test battery
• pslq_spacing.zig runs PSLQ on normalized spacings
• beal_scan.zig scans A,B ≤ 1000 (small test) with SIMD
• beal_near.zig collects and analyzes near-misses
• All modules integrate with existing build.zig
• zig build test passes for all new modules

References

• Odlyzko zero tables: https://www-users.cse.umn.edu/~odlyzko/zeta_tables/
• Beal conjecture: https://en.wikipedia.org/wiki/Beal_conjecture
• GUE hypothesis: Montgomery (1973), Odlyzko (1987)
• PSLQ algorithm: Ferguson-Bailey-Arwade (1999)

[juzigu40-ui]

Hi team, Juzi Valley Research Team is interested in this issue. We are reviewing the architecture and interface contracts now, and we will align the implementation with your contribution standards and test expectations. Could you please assign this to us so we can deliver a reviewable PR within 48 hours?

[gHashTag]

Status Update: Palantir v2.0 "Bounty Hunter"

Most of this pipeline is already implemented. Here's the breakdown:

Target #1: Riemann ζ-zeros — ~90% done

Module from issue	Implemented as	LOC	Status
zeta_zeros.zig — Odlyzko zeros DB	src/sacred/zeta_import.zig	269	✅ Complete
cfrac_zeta.zig — CF expansion of spacings	src/sacred/zeta_cf.zig	444	⚠️ Structure done, TODO in expandCF() integration
pslq_spacing.zig — PSLQ on spacings	src/sacred/zeta_pslq.zig	325	✅ Complete (7 constants: π, ln2, φ, e, √2, √3, ζ(3))
Palantir v1.1 pipeline	src/sacred/cfrac_palantir.zig + cfrac_palantir_v11.zig	1,037	✅ Complete (6 stages + 4 extensions)
CLI commands	src/sacred/zeta_commands.zig	237	✅ 5 subcommands: tri math zeta {import,spacing,cf,pslq,verdict}

Target #2: Beal Scanner — ~85% done

Module from issue	Implemented as	LOC	Status
beal_scan.zig — SIMD scanner	src/beal/ (7 files)	2,482	✅ Complete: GCD filter, modular arithmetic (3 primes), SIMD NEON, BigInt verify, multi-threaded search
beal_near.zig — Near-miss analyzer	Logic in src/beal/search.zig	—	⚠️ No standalone CLI yet
FPGA acceleration	fpga/openxc7-synth/beal_*.v	600+	✅ Synthesized bitstream for Artix-7 (125x speedup vs CPU)
CLI	zig build beal	—	✅ Working, 27 tests passing

Remaining work (3 items)

1. zeta_cf.zig integration — Wire expandCF() to actual cfrac_palantir.expandCF() instead of the current TODO panic stub
2. beal-near CLI — Extract near-miss statistical analysis from search.zig into a standalone tri beal-near command
3. Hardcoded first 100 zeros — Currently requires external Odlyzko data file; issue spec asks for first 100 zeros hardcoded in source

Acceptance Criteria Status

• Zeta zeros module compiles and loads zeros
• zeta_cf.zig fully integrated with Palantir pipeline (TODO stub remains)
• PSLQ runs on normalized spacings
• Beal scanner works with SIMD for A,B ≤ 1000
• beal_near standalone CLI
• All modules integrate with build.zig
• zig build test passes

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@juzigu40-ui — The remaining items are relatively contained integration tasks. Please review the existing code in src/sacred/ and src/beal/ before starting to avoid duplicating what's already built. The main gap is wiring zeta_cf.zig to the real Palantir CF expansion engine.