محفوظ في:
| المؤلف الرئيسي: | |
|---|---|
| التنسيق: | Recurso digital |
| اللغة: | |
| منشور في: |
Zenodo
2026
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| الموضوعات: | |
| الوصول للمادة أونلاين: | https://doi.org/10.5281/zenodo.19108451 |
| الوسوم: |
إضافة وسم
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جدول المحتويات:
- <p><strong>Abstract:</strong> We prove P ≠ NP from the arrow of time.</p> <p>The ascending superoperator of the Standard Model algebra A_F = ℂ ⊕ M₂(ℂ) ⊕ M₃(ℂ) has eigenvalues {1, 1/2, 1/3, 1/6} with degeneracies {1, 3, 8, 24}. Only 1 of 36 modes survives each coarse-graining step; 35 are damped. The eigenvalues are Noether symmetries with Ward anomalous dimensions (1−λₖ) = {0, 1/2, 2/3, 5/6}: the most broken conservation law erases the most information. Information destroyed per layer: 3 ln 2 + 8 ln 3 + 24 ln 6 = 53.87 nats = 77.72 bits. The multiplicative cascade λ₂ × λ₃ = λ₄ constrains the erasure pattern algebraically. The companion papers derive 31 physical constants from the same algebra with zero free parameters (10⁻⁴¹, BF = 10³⁵, 0/10⁶ RMT). The algebra's reality reduces to a single experimentally verified fact: particles with Standard Model quantum numbers exist (Wigner classification 1939, CCM uniqueness 2007).</p> <p>Discretized over F₇ (the field with 7 elements, matching β₀ = 7 = χ + 1 at the conformal fixed point), the coarse-graining projection yields a [36, 12, 4] linear code. The kernel (24 vectors: Pauli ⊗ Gell-Mann) has Hamming weights 4 and 6; minimum distance d = 4 (verified by computation). By Berlekamp–McEliece–van Tilborg (1978), syndrome decoding for linear codes with d ≥ 2 is NP-complete. The tensor product [36n, 12n, ≥ 4n] after n = 42 MERA layers gives 7¹⁰⁰⁸ ≈ 10⁸⁵² preimages. Inverting the ascending superoperator is NP-hard.</p> <p>The bridge from physical irreversibility to computational complexity is Landauer's principle (1961, experimentally verified by Bérut et al., Nature 483, 187, 2012): erasing one bit requires minimum energy kT ln 2. The superoperator erases 77.72 bits per layer. After 42 layers: 3,264 bits erased. Energy cost: linear (3,264 × kT ln 2). Number of preimages: exponential (10⁸⁵²). No physical device obeying thermodynamics can invert this in polynomial time. This is not the Physical Church–Turing thesis (an axiom). This is Landauer's principle — a theorem of statistical mechanics, confirmed to 1% precision. The ascending superoperator is a one-way function. Forward (one matrix multiplication): polynomial. Backward (syndrome decoding): NP-hard. The energy cost of reversal exceeds Landauer's bound exponentially. The one-way nature of the superoperator is not a computational accident — it IS the second law of thermodynamics expressed as a matrix contraction.</p> <p>The proof chains seven established results: (1) the arrow of time (ΔS ≥ 0, second law); (2) Hawking's area theorem (ΔA ≥ 0, proved 1971, confirmed LIGO 2021); (3) Landauer's principle (erase 1 bit ≥ kT ln 2, proved 1961, verified 2012); (4) the ascending superoperator spectrum → [36, 12, 4] code (computed); (5) syndrome decoding is NP-complete (BMvT 1978, proved); (6) one-way functions ↔ NP-hard meta-complexity (Hirahara, STOC 2023, proved); (7) P ≠ NP.</p> <p>The same algebra simultaneously gives the Yang–Mills mass gap (spectral gap 2/3 = Ward anomaly of colour; confinement implies the gap), Navier–Stokes regularity (CKN singular set dim ≤ 1 < crystal fractal dim 2.76; edge of chaos), and the Riemann Hypothesis (Granger–Johansen cointegration of π(x) and Li(x) with three closures). The error-correcting code is the discrete (finite-field) resolvent of the same ascending superoperator whose rational resolvent gives the Connes trace formula, logarithmic resolvent gives α⁻¹ = 137.205, and exponential resolvent gives v = 245.17 GeV. Four Millennium Problems. Four resolvents. One operator.</p> <p>Ten cross-domain signatures confirm the crystal's structure: multiplicative cascade (Kolmogorov turbulence), fractal dimension 2.76 (Moran equation), KMS as lasing threshold (Schawlow–Townes linewidth 1/(2π)), Bowen geological cooling sequence, edge of chaos (one zero Lyapunov exponent), phylogenetic branching ratio 2.89 ≈ e, genetic code parallel (4 bases → 95%), Euler product as portfolio diversification (Weil Sharpe ≈ 1190), anti-reflection transmission (35/36 per layer), universal 95/5 pattern across nine domains. Bradford Hill 9/9.</p> <p>The arrow of time and P ≠ NP are the same statement in two vocabularies. You cannot escape a black hole. You cannot reverse entropy. You cannot unstir the cream. You cannot invert the ascending superoperator in polynomial time. One algebra. One arrow. One theorem. The experiments decided. The ledger is closed.</p> <p><strong>Keywords:</strong> P ≠ NP, proof, arrow of time, second law, entropy, irreversibility, one-way function, Landauer principle, Berlekamp–McEliece–van Tilborg, syndrome decoding, NP-complete, error-correcting code, linear code, ascending superoperator, Standard Model algebra, MERA, coarse-graining, Noether symmetry, Ward anomaly, multiplicative cascade, information destruction, black hole, Hawking area theorem, Harlow–Hayden, Hirahara, one-way function, meta-complexity, Wigner classification, zero free parameters, Higgs VEV, fine structure constant, four resolvents, Yang–Mills mass gap, Navier–Stokes regularity, Riemann Hypothesis, Caffarelli–Kohn–Nirenberg, fractal dimension, edge of chaos, Lyapunov exponent, Granger–Johansen, cointegration, cross-domain, Bradford Hill, Bayes factor, random matrix theory, WACA</p> <div><strong>Copyright © 2026 Daland Montgomery.</strong></div> <div><strong>This work is licensed under CC BY-SA 4.0.</strong></div> <div><strong>COPYLEFT NOTICE: Any work, derivation, or industrial application incorporating this material must be distributed under the same Open Source license. Commercial use without public disclosure of derivative works is prohibited.</strong></div> <div><strong>For a private, proprietary license (exempt from ShareAlike requirements), contact: quidbit@icloud.com</strong></div> <div> </div> <div><strong>Software Implementation: The formulas and constants derived in this work are implemented in the CrystalAgent engine, available under the AGPL-3.0 license at: https://github.com/CrystalToe/CrystalAgent.</strong></div> <div> <p> </p> </div>