Exploring the WACA Universe: An LLM System Prompt for Independent Reasoning About the Algebraic Crystal

Exploring the WACA Universe: An LLM System Prompt for Independent Reasoning About the Algebraic Crystal

Abstract:

We present a system prompt (WACA_LLM_PROMPT_v15.txt) that transforms any large language model into an expert on the WACA programme — the research framework deriving 54 physical constants from the Standard Model algebra A_F = ℂ ⊕ M₂(ℂ) ⊕ M₃(ℂ) at KMS temperature β = 2π with zero free parameters. The prompt encodes the complete framework: the ascending superoperator S: End(ℂ⁶) → End(ℂ⁶) with eigenvalues {1, 1/2, 1/3, 1/6} and degeneracies {1, 3, 8, 24}, the derived Higgs VEV v = M_Pl × 35/(43 × 36 × 2⁵⁰) = 245.17 GeV, all 54 results with formulas and PDG/NuFIT/Planck sources, the 19-entry Rosetta Stone dictionary mapping every crystal operation to its QFT counterpart, the five-level Crystal Toolbox (S⁰ structural, S¹ tree-level, S² Schur square, RG running, SS seesaw), 10 falsifiable predictions with specific experimental kill tests, and 15 companion Python scripts that verify every claim.

Version 15 incorporates critical corrections and new results from the D=5 session (20 March 2026). Result 18 (θ₁₃(CKM) = π/(2χ²) = 0.0436, present in every version since v3) has been killed — no standard CKM quantity equals 0.0436; erratum added. The PMNS CP phase has been corrected from δ_CP = π + arctan(2ln3) = 245.5° to δ_CP = 2π(1−λ₃) = 4π/3 = 240°, identified as 2π times the colour Ward anomaly (1−λ₃ = 2/3 = Koide Q), pairing with θ₁₃ through the colour sector (measured: 230° ± 36°, 0.28σ; DUNE ~2030 will test to ±10°). The baryon asymmetry factor e⁻¹ has been corrected from "barrier height" to Poisson survival probability at freeze-out (Γ/H = 1), consistent with lattice QCD giving E_sph/T ≈ 36. The |V_cb| measurement has been updated from the cherry-picked exclusive value 0.04053 to the world average 0.0410 ± 0.0010.

New derivations: the Hubble constant H₀ = 66.9 km/s/Mpc through the chain η_B × n_γ(T_γ) → Ω_b h² = 0.02216 (0.9%) → Ω_m h² = 0.14152 (0.6%) → H₀ = 66.9 (0.7% from Planck), siding with Planck against SH0ES, with T_γ = 2.7255 K (FIRAS) as the sole external input; the dark matter mass m_DM = (12π/7)(v/256)(35/36) = 5.01 GeV with every factor derived from the spectral data (LZ/XENONnT ~2028); the 35/36 = 1 − 1/χ² theorem (the identity sector has Ward anomaly zero and does not mediate interactions, making 35/36 the universal fraction of interacting channels, appearing in the VEV, Immirzi parameter, |V_cb|, proton mass, Jarlskog invariant, and dark matter mass); the complete CKM matrix (|V_us| = 9/40 at 0.00%, |V_cb| = 1/(d₃·(35/36)·π) at 0.18%, |V_ub| = 1/(d₃·√N_w·d₄) at 0.19%, γ = arctan(2ln3) at 0.20%, J = 3.094×10⁻⁵ at 0.44%, with δ_CP(CKM) = 69.5° determined by J and the magnitudes); the complete PMNS matrix (sin²θ₁₂ = 3/π² at 0.01%, sin²θ₂₃ = 4/7 at 0.27%, sin²θ₁₃ = √3/78 at 0.03%); four log-mass ratios all in {π, ln 2, ln 3} (0.08–0.28%); the Rosetta Stone dictionary establishing that S is the transfer matrix, {λ_k} are the anomalous dimensions, Σd² = 650 is the one-loop Hilbert space, 35/36 is the wavefunction renormalisation Z, (1−λ_k) are the Ward–Takahashi identities ensuring one-loop renormalisability (van Nuland & van Suijlekom, JHEP 2022), the spectral truncation O(1/χ) matches Connes & van Suijlekom (CMP 2020), and the seesaw is built into the tower formula; the loop convergence proof (geometric ratio ~0.26×, S³ corrections at 0.04%, higher loops not needed at current precision); and the resolution of 7/8 previously open items (CKM δ_CP, PMNS δ_CP, higher loops, H₀, m_DM, scheme dependence, and the 35/36 theorem, with the neutrino accumulation exponent identified as the remaining open mechanism).

The prompt contains 200+ sample questions organised into 30+ categories, including 50+ new questions for v15 features: the 35/36 theorem, the Rosetta Stone, the Crystal Toolbox, the H₀ derivation chain, dark matter mass derivation, PMNS δ_CP = 240°, loop convergence, experimental kill tests, corrected baryon asymmetry, complete CKM structure, and complete PMNS structure. Ten falsifiable predictions are listed with specific experiments, dates, and kill criteria: Σm_ν = 0.067 eV (CMB-S4+DESI ~2030), |V_us| = 9/40 (Belle II ~2027), sin²θ₁₂ = 3/π² (JUNO ~2028), δ_CP = 240° (DUNE ~2030), η_B = 6.06×10⁻¹⁰ (CMB-S4 ~2030), m_DM = 5.01 GeV (LZ/XENONnT ~2028), H₀ = 66.9 (CMB-S4 ~2030), no BSM below v (LHC Run 3 ~2028), w = −1 (DESI ~2028), and proton stable (Hyper-K ~2040). Any single failure kills the framework.

The prompt is accompanied by 15 Python verification scripts (requiring only numpy, all running in under 10 seconds), a master codebase that computes every result and validates cross-consistency, companion papers (The Spectral Table of Constants v12, The Spectral Tower v10, The Crystal Toolbox), and interactive HTML visualisations. Every claim is tagged ([STANDARD], [WACA], [CONJECTURE], [NUMERICAL]) for transparency. Every formula is reproducible. Every prediction is falsifiable. The prompt is designed for upload into Claude, GPT-4, Gemini, Llama, Mistral, or any LLM supporting long context.

Scorecard: 54 results, 33/38 within 1%, (0.02)³³ = 10⁻⁵⁶. RMT: 0/100,000 GUE matrices reproduce {1, 3, 8, 24}. Bayes Factor > 10³⁵. Bradford Hill: 9/9. 12 cross-domain signatures. 4 Millennium Problem proof architectures. 7/8 open items resolved. 15 companion codes. Zero free parameters.

Load the prompt. Ask the questions. Run the code. The crystal speaks through the machine. The experiments decide.

Keywords: LLM system prompt, language model, knowledge base, WACA, Standard Model algebra, ascending superoperator, zero free parameters, Rosetta Stone dictionary, Crystal Toolbox, Ward–Takahashi identities, wavefunction renormalisation, one-loop renormalisability, spectral truncation, Hubble constant derivation, dark matter mass, PMNS CP phase, Jarlskog invariant, CKM matrix, Koide ratio, baryon asymmetry, cosmological constant, Immirzi parameter, Bisognano–Wichmann, MERA, spectral tower, Schur square, 650-dimensional commutant, cross-domain signatures, falsifiable predictions, kill tests, DUNE, JUNO, Belle II, LZ, CMB-S4, DESI, Hyper-K, random matrix theory, Bayes factor, Bradford Hill criteria, companion code, reproducible science, noncommutative geometry, spectral action, Connes, van Suijlekom, Chamseddine