Abstract: The finite algebra A_F = ℂ ⊕ M₂(ℂ) ⊕ M₃(ℂ) at KMS inverse temperature β = 2π provides a complete, finite set of operations for computing physical constants. We classify every operation into five levels and establish a bijective dictionary between spectral operations and standard quantum field theory. The ascending superoperator S on End(ℂ⁶) is the MERA transfer matrix. Its eigenvalues {1, 1/2, 1/3, 1/6} are the anomalous dimensions. Its degeneracies {1, 3, 8, 24} are the channel multiplicities. The first-order channel count Σd = 36 gives tree-level results (33/38 within 1% of measurement). The Schur square Σd² = 650 gives all corrections: the universal MERA retention factor 35/36 = 1 − 1/χ² is the wavefunction renormalisation Z; the Plancherel resolvent Σd²ln(1/(1−λ)) = 137.205 is the spectral zeta function; the Ward anomalies (1−λ_k) = {0, 1/2, 2/3, 5/6} are the Ward–Takahashi identities that ensure one-loop renormalisability (van Nuland & van Suijlekom, JHEP 2022). The spectral truncation error O(1/χ) at finite bond dimension χ = 6 corresponds to UV cutoff artefacts in the Connes–van Suijlekom operator system framework (CMP 2020). Standard model running provides boundary-condition transport from the conformal point (D = 0, ~245 GeV) to the measurement scale. The type-I seesaw is built into the tower formula m_ν = v/2⁴², with the ~10% tree-to-loop gap confirmed as universal finite-χ truncation across five independent hierarchical branching domains. The dictionary contains 19 entries. Each is a bijection: every crystal operation maps to exactly one QFT operation. The five outliers outside 1% each resolve through a specific, standard QFT mechanism — wavefunction renormalisation (α_s/α_w), UV cutoff (horizon entropy), Yukawa running (m_b), and lattice discretisation (neutrinos) — applied to the spectral data of A_F. Three open calculations remain: the van Nuland–van Suijlekom one-loop spectral action evaluated for the specific triple (A_F, H, D); the derivation of Z = 35/36 from first principles for each observable; and the full ΛCDM integration with crystal inputs. The toolbox is finite: 36 channels give tree-level, 650 channels give all corrections, 1296 paths give mixing. No infinite tower of corrections, no ad hoc parameters. The crystal does not replace quantum field theory. It is quantum field theory, written in one algebra. Keywords: spectral action, noncommutative geometry, ascending superoperator, MERA, tensor network, Schur square, wavefunction renormalisation, Ward–Takahashi identities, anomalous dimensions, spectral zeta function, one-loop renormalisability, spectral truncation, operator systems, KMS state, Bisognano–Wichmann, Plancherel measure, finite bond dimension, tree-level closure, seesaw mechanism, crystal toolbox, WACA
Abstract: The finite algebra A_F = ℂ ⊕ M₂(ℂ) ⊕ M₃(ℂ) at KMS inverse temperature β = 2π provides a complete, finite set of operations for computing physical constants. We classify every operation into five levels and establish a bijective dictionary between spectral operations and standard quantum field theory. The ascending superoperator S on End(ℂ⁶) is the MERA transfer matrix. Its eigenvalues {1, 1/2, 1/3, 1/6} are the anomalous dimensions. Its degeneracies {1, 3, 8, 24} are the channel multiplicities. The first-order channel count Σd = 36 gives tree-level results (33/38 within 1% of measurement). The Schur square Σd² = 650 gives all corrections: the universal MERA retention factor 35/36 = 1 − 1/χ² is the wavefunction renormalisation Z; the Plancherel resolvent Σd²ln(1/(1−λ)) = 137.205 is the spectral zeta function; the Ward anomalies (1−λ_k) = {0, 1/2, 2/3, 5/6} are the Ward–Takahashi identities that ensure one-loop renormalisability (van Nuland & van Suijlekom, JHEP 2022). The spectral truncation error O(1/χ) at finite bond dimension χ = 6 corresponds to UV cutoff artefacts in the Connes–van Suijlekom operator system framework (CMP 2020). Standard model running provides boundary-condition transport from the conformal point (D = 0, ~245 GeV) to the measurement scale. The type-I seesaw is built into the tower formula m_ν = v/2⁴², with the ~10% tree-to-loop gap confirmed as universal finite-χ truncation across five independent hierarchical branching domains. The dictionary contains 19 entries. Each is a bijection: every crystal operation maps to exactly one QFT operation. The five outliers outside 1% each resolve through a specific, standard QFT mechanism — wavefunction renormalisation (α_s/α_w), UV cutoff (horizon entropy), Yukawa running (m_b), and lattice discretisation (neutrinos) — applied to the spectral data of A_F. Three open calculations remain: the van Nuland–van Suijlekom one-loop spectral action evaluated for the specific triple (A_F, H, D); the derivation of Z = 35/36 from first principles for each observable; and the full ΛCDM integration with crystal inputs. The toolbox is finite: 36 channels give tree-level, 650 channels give all corrections, 1296 paths give mixing. No infinite tower of corrections, no ad hoc parameters. The crystal does not replace quantum field theory. It is quantum field theory, written in one algebra. Keywords: spectral action, noncommutative geometry, ascending superoperator, MERA, tensor network, Schur square, wavefunction renormalisation, Ward–Takahashi identities, anomalous dimensions, spectral zeta function, one-loop renormalisability, spectral truncation, operator systems, KMS state, Bisognano–Wichmann, Plancherel measure, finite bond dimension, tree-level closure, seesaw mechanism, crystal toolbox, WACA
Fourteen theorems on the algebraic structure of binary quantum measurement, built on the identity I(V) = γ²(V) = 1/(1 − V²), where I is the classical Fisher information of a binary measurement with visibility V = 2p − 1 and γ is the Lorentz factor of special relativity. Part A (T1–T5): internal structure. Uniqueness of the Jeffreys prior π_J(V) = γ(V)/π under Diff₊ invariance (T1). Closed-form Thomas–Wigner rotation for sequential binary measurements (T2). SLD saturation of the quantum Cramér–Rao bound (T3). Uniqueness of the Gudermannian bridge between hyperbolic and spherical Fisher–Rao geometries (T4). Closed-form mixed-state retention R(r) interpolating from 1/3 to 1/2 (T5). Part B (T6–T7): observer–manifold theorems. Futility theorem via Lawvere's fixed-point theorem: no embedded observer can surjectively reconstruct the γ² field (T6). Energy-precision uncertainty E·Var(θ̂) ≥ kT ln 2 / γ²(V), saturating Landauer at V = 0 (T7). Part C (T8, held): dimensional selection of 3+1D spacetime from exceptional isomorphisms, with explicit gap acknowledged. Part D (T9–T12): symmetry and classification. Form-invariance of the conformal equivalence under Möbius action (T9). Two-outcome binary qubit effect classification via 4γ²(V) (T10). Modular flow as SL(2,ℂ) boost at imaginary rapidity (T11). SLOCC covariance γ²(V') = γ²(V)·γ²(ξ)·(1+Vξ)² (T12). Part E (T13–T14): quantum gravity connections. Holographic subregion complexity equals Fisher information under boost (T13, proof sketch). γ² Page curve for black hole evaporation (T14, proof sketch). Complete proofs for T1–T7 and T9–T12. T8 presented with scope limitations. T13 and T14 presented as proof sketches. Mathematical companion to: B.G. Srivats, "Fisher Information Is the Squared Lorentz Factor," Zenodo DOI 10.5281/zenodo.19363449 (2026) B.G. Srivats, "From Measurement to Reality," Zenodo DOI 10.5281/zenodo.19486140 (2026) Builds on independently established work by Burns–Greenfield–Dressel (QSMF 13:10, 2026) and Fullwood–Vedral–Guzmán-González (arXiv:2604.07471, 2026). Each theorem is intended for subsequent individual journal publication; this record establishes priority on all fourteen as a collection.
Abstract: We derive the Schrödinger equation, wave-particle duality, the uncertainty principle, the double slit experiment, and quantum measurement from the ascending superoperator of A_F = ℂ ⊕ ℍ ⊕ M₃(ℂ) at KMS inverse temperature β = 2π. The crystal's four eigenvalues {1, ½, ⅓, ⅙} are the four frequency bands of the vacuum's wavelet decomposition. The MERA tensor network is a discrete wavelet transform (confirmed: Alves 2024, wMERA = MERA + DWT). Wave-particle duality is the Fourier transform between a single MERA layer (particle) and the full tower (wave). The uncertainty principle arises from the finite bond dimension χ = 6. Measurement is the ascending superoperator destroying the kernel (24 mixed modes). The Schrödinger equation is the continuous limit of the MERA circuit, with H = generator of the ascending superoperator and i = Wick rotation from imaginary to real time. The vacuum oscillates at β = 2π (Bisognano-Wichmann theorem); the crystal decomposes this oscillation into four harmonics; those harmonics are the four forces. Nine independent research traditions confirm this synthesis: Feynman, Wilson, Bisognano-Wichmann, Daubechies, Vidal, Swingle, Harlow-Preskill, Evenbly-Vidal, and Alves. Five figures, an interactive 3D HTML demonstration, and a Python verification script are included. Zero free parameters.
This record contains the preprint and the complete reproducibility package for “A formal finite-size spectral-map hierarchy for free-fermion entanglement”. The work studies the finite-size deformation of the spectral relation between the discrete prolate (Slepian) eigenvalue and the one-particle entanglement energy in a critical free-fermion chain. Using the exact DPSWF representation, Riccati/Dunham analysis and finite-part period calculus, an all-orders formal hierarchy is derived in which the finite-size coefficients satisfyFj=Pj(D)Φ∞F_j=P_j(D)\Phi_\infty, with PjP_j a rational function of the Euler operator D=t∂tD=t\partial_t. The work establishes universal denominator, degree and zero structures for these rational operators, including additional even-shift factors, and derives their Weber/Bernoulli leading behaviour. Independent checks include exact reconstruction of the half-filling Fisher–Hartwig phase polynomials Q20,Q40,Q60,Q80Q_{20},Q_{40},Q_{60},Q_{80}, exact Legendre reduction at the spectral centre, rational operator reconstruction through j=13j=13, and finite-matrix validation of the first two finite-size coefficients. The archive accompanying the manuscript contains the scripts, exact symbolic calculations, numerical data, regression tests and SHA-256 manifests required to reproduce the principal computational claims. The manuscript explicitly distinguishes the proved formal Dunham hierarchy from the still-open uniform physical connection/remainder theorem.