This paper proposes a theory of representation centered on invariance as the constitutive condition of objectivity. Any objective representational system functions as a sieve: it stabilizes identity by enforcing invariance over a structured possibility space. I demonstrate that binary relational representation under global invariance forces exactly four exhaustive stability profiles, determined by whether each argument position behaves as instance-like or role-like under substitution. These four profiles appear across representational systems: as fundamental verbs in natural language (IS, HAS, (...) MEANS, CAUSES), as arithmetic operations in mathematics (addition, multiplication, subtraction, division), and as interaction roles in physics. This convergence is not analogical but necessary: the four-fold emerges from the constraint structure of objective binary relational representation itself. (shrink)

Declarative sentences perform two distinct kinds of work: they contribute descriptive content about states of affairs, and they fix how that content is to be evaluated. While most verbs contribute primarily to descriptive content, a small class contributes primarily to evaluation itself. This paper argues that English lexicalizes exactly four ultra-general orientational operators—be, have, mean, and cause—which exhaust a minimal grammatical space generated by two independent parameters: (1) whether evaluation targets world-structure or contrast-structure, and (2) whether predication is invariant or (...) instance-selecting. This framework explains persistent philosophical puzzles about causation, meaning, and identity by showing how orientational structure has been systematically mistaken for ontological content. (shrink)

In this paper the Lorentz transformations (LT) and the standard transformations (ST) of the usual Maxwell equations (ME) with the three-dimensional (3D) vectors of the electric and magnetic fields, E and B, respectively, are examined using both the geometric algebra and tensor formalisms. Different 4D algebraic objects are used to represent the usual observer dependent and the new observer independent electric and magnetic fields. It is found that the ST of the ME differ from their LT and consequently (...) that the ME with the 3D E and B are not covariant upon the LT but upon the ST. The obtained results do not depend on the character of the 4D algebraic objects used to represent the electric and magnetic fields. The Lorentz invariant field equations are presented with 1-vectors E and B, bivectors EHv and BHv and the abstract tensors, the 4-vectors Ea and Ba. All these quantities are defined without reference frames, i.e., as absolute quantities. When some basis has been introduced, they are represented as coordinate-based geometric quantities comprising both components and a basis. It is explicitly shown that this geometric approach agrees with experiments, e.g., the Faraday disk, in all relatively moving inertial frames of reference, which is not the case with the usual approach with the 3D bf E and B and their ST. (shrink)

We discuss, within the framework provided by a recently developed variational method, transposition-invariant field equations for unified field theories. Systems that are, in addition, invariant under Weyl-type gauge transformations or lambda transformations are derived. It is found that in a weak field limit two of the systems contain the equations of general relativity and the covariant Maxwell equations for a charge-free region.

CPT invariance and the spin-statistics connection are typically taken to be essential properties in relativistic quantum field theories (RQFTs), insofar as the CPT and Spin-Statistics theorems entail that any state of a physical system characterized by an RQFT must possess these properties. Moreover, in the physics literature, they are typically taken to be properties of particles. But there is a Received View among philosophers that RQFTs cannot fundamentally be about particles. This essay considers what proofs of the CPT and (...) Spin-Statistics theorems suggest about the ontology of RQFTs, and the extent to which this is compatible with the Received View. I will argue that such proofs suggest the Received View’s approach to ontology is flawed. (shrink)

A certain class of geometric objects is considered against the background of a classical gauge field associated with an arbitrary structural Lie group. It is assumed that the components of these objects depend on the gauge potentials and their first derivatives, and also on certain gauge-dependent parameters whose properties are suggested by the interaction of an isotopic spin particle with a classical Yang-Mills field. It is shown that the necessary and sufficient conditions for the invariance of the given objects (...) under a finite gauge transformation are embodied in a set of three relations involving the derivatives of their components. As a special case these so-called invariance identities indicate that there cannot exist a gauge-invariant Lagrangian that depends on the gauge potentials, the interaction parameters, and the4-velocity components of a test particle. However, the requirement that the equations of motion that result from such a Lagrangian be gauge-invariant, uniquely determines the structure of these equations. (shrink)

The lifetimes of both positive and negative relativistic (γ = 29.33) muons have been measured in the CERN Muon Storage Ring with the results τ+ = 64.419 (58) µs, τ− = 64.368 (29) µs The value for positive muons is in accordance with special relativity and the measured lifetime at rest: the Einstein time dilation factor agrees with experiment with a fractional error of 2×10−3 at 95 confidence. Assuming special relativity, the mean proper lifetime for μ− is found to be (...) τ0− = 2.1948(10) µs the most accurate value reported to date. The agreement of this value with previously measured values of τ0+ confirms CPT invariance for the weak interaction in muon decay. (shrink)

In this work I present what may be the first complete construction of quantum gravity describing the real universe via the celestial holographic conformal field theory dual to Einstein gravity in asymptotically-flat 4D spacetime. The theory is rigorously constructed as the shadow-invariant, purely spin-2 sector of holomorphic Chern–Simons theory on twistor space PT ≃ CP³ with gauge group the quantomorphic group Quant(PT). Primary fields are the celestial graviton operators O^{±2}Δ(z, z̄) with Δ ∈ 1 + iR and J = (...) ±2. Three-point functions are determined by the Penrose transform and reproduce the MHV and anti-MHV amplitudes of general relativity, including the vanishing of parity-odd and mixed-helicity couplings. Higher-point amplitudes arise from OPEs with analytically continued principal-series contours; loop integrands emerge as double discontinuities across shadow poles, which I verify explicitly for the scalar box integrals computed via this novel shadow discontinuity method, demonstrating analytically that they match Feynman diagram calculations for the scalar box, with numerical verification for the graviton box matching to 11+ decimal places. The construction satisfies extended BMS₄ ⋉ shadow symmetry with vanishing Virasoro central charge, also explicitly demonstrated via five independent and mutually reinforcing and rigorous arguments to be c = 0 to 10+ significant figures. I prove a rigidity theorem establishing that any 2D CFT with only spin ±2 primaries, the above symmetries and three-point data, and local inverse-Mellin-transformed correlators, is uniquely determined to be isomorphic to this quantomorphic twistor theory. Combined with the proof that crossing symmetry is a necessary consequence of bulk locality (not an independent axiom), this establishes that Einstein gravity in asymptotically-flat spacetime is the unique consistent theory of a single massless spin-2 particle. The quantomorphic gauge group structure provides a non-perturbative foundation, with perturbative scattering amplitudes emerging as the first residues of the holographic algebra. The first breakthrough insight that enabled and facilitated this construction was the realisation that the "shadow symmetry" of the boundary CFT—the map Δ ↔ 2 − Δ on conformal dimensions—is the restriction to null infinity of the Jost involution Θ of axiomatic quantum field theory, the operator independently constructed by Jost (1957) that coincides with the product C ∘ P ∘ T only in theories where all three factors individually exist as symmetries. At null infinity, where the metric is degenerate along the null generator, the distinction between temporal and spatial inversion collapses: the Jost involution is irreducible and reduces to time reversal T alone (Theorem 8.1). This provides an elegant resolution to the 40+ year "googly problem" in twistor theory by unifying self-dual and anti-self-dual helicity sectors through the shadow transform as representing discrete T-symmetric involution across the boundary, with the Einstein field equations derived from boundary Ward identities and crossing symmetry. All quantum mechanical foundations emerge from the Haar measure structure underlying this framework with no additional postulates: the five Wightman axioms are derived from the Peter–Weyl theorem on representation spaces, the Born rule P = |⟨n|ψ⟩|² is proven as the unique probability measure compatible with Haar structure on quantum state spaces, and wave function collapse is derived as Haar projection onto gauge singlets. This same identification reveals that the celestial conformal dimension is related to the Riemann zeta variable by Δ = 2s, so that the shadow involution Δ ↔ 2 − Δ becomes s ↔ 1 − s, the functional equation of the Riemann zeta function—establishing that the functional equation, shadow symmetry, and time reversal are three manifestations of one underlying self-duality. The entire holographic framework grows from a single root: self-dual Haar measure on (R⁺, ×), the simplest non-trivial locally compact group, whose self-duality already contains the functional equation and critical line. Three Cayley–Dickson doublings R → C → H → O produce the holographic chain R⁺ → S² → M⁴ → Gr(2,4)_C, with shadow symmetry at each level corresponding to the conjugation involution at that doubling step. The Grassmannian Gr(2,4) is the canopy of this holographic tree. The same Haar-invariant measure on the Grassmannian Gr(2,4) that determines the graviton spectrum also addresses three Millennium Prize problems through a unified geometric mechanism (Haar self-duality, functional equation symmetry, and Peter–Weyl compactness). The Riemann Hypothesis follows via 4 otherwise independent arguments and checks for circularity and consistency, and most fundamentally follows from three simultaneous constraints on the adelic quotient A×/Q× that form an overdetermined system whose unique solution forces Re(s) = 1/2 for all nontrivial zeros, with numerical verification of the first 10¹³ zeros on the critical line to machine precision using six independent tests. The Birch and Swinnerton-Dyer Conjecture is argued to be proven for analytic rank ≤ 1 using identical Haar measure structure on GL₂(A_Q)/GL₂(Q): for elliptic curves with ord_{s=1} L(E,s) ≤ 1, the same three-fold constraint described above, combined with the Gross–Zagier formula and Kolyvagin's Euler system descent, establishes ord_{s=1} L(E,s) = rank E(Q) and the parity conjecture (−1)^{rank} = w_E is proven for all curves from Haar self-duality. The framework also identifies the precise obstruction to proving the conjecture at higher rank (algebraic cycles from the Beilinson–Bloch–Kato conjectures). The Yang–Mills mass gap is established through a complete celestial holographic construction: the Wess–Zumino–Witten model at Kac–Moody level k on S², mapped to four-dimensional Schwinger functions by inverse Mellin transform, with reflection positivity proven via the shadow-reflection bridge, the remaining Osterwalder–Schrader axioms verified explicitly, and the Wightman theory reconstructed by the OS theorem—yielding mass gap M = 2N/(k+N) · Λ_QCD > 0 from the Sugawara formula and confinement from Haar orthogonality as a rigorously proved theorem. The construction introduces no lattice, takes no continuum limit, and imposes no infrared regulator. Spacetime itself emerges from the Grassmannian via Penrose twistor correspondence (M⁴ ↔ Gr(2,4) (Penrose 1967). The division algebra tower determines the Standard Model: the symmetry breaking chain Spin(8) → G₂ → SU(3) → SU(2) × U(1) produces the gauge group; three fermion generations follow from Weyl anomaly cancellation, Plücker partitions, and Spin(8) triality; sin²θ_W = 3/8 at the GUT scale is the Dynkin index ratio, equivalently dim(Im(H))/dim(O). Physical constants are arithmetic invariants: CP violation from the Jacobi sum J(χ₄⁽⁵⁾, χ₃⁽⁷⁾) = e^{iπ/6}; down-type quark mass ratios m_d : m_s : m_b = 2:5:9 from SU(3) Casimir eigenvalues; the spin-statistics connection η(4)/ζ(4) = 7/8 from the p = 2 Euler factor; the lightest neutrino mass m_{ν₁} = 0 exactly from Bott periodicity—in independent agreement with Boyle–Finn–Turok (PRL 121, 251301, 2018); and the three dimensional constants ℏ, c, G corresponding to the three Cayley–Dickson doubling steps. Strong CP angle θ = 0 exactly from Haar self-duality (consistent with experimental bound |θ| < 10⁻¹⁰), and the construction provides right-handed neutrinos as geometric dark matter candidates in agreement with Boyle–Turok (CP)T-symmetric cosmology, which integrates with uncanny precision into my T-symmetric, thermodynamically cyclic quantum cosmology. The T-anti-symmetric unification of a unified bipartite self-dual universe resolves the matter-antimatter asymmetry as a selection effect. Total baryon number B_total = 0 with embedded observers observing identical physics on either side due to being thermodynamic in nature, naturally aligning their arrows of time in the direction of increasing entropy, which on either side of the boundary flows in opposing directions. The Feynman-Stueckelberg interpretation of antimatter means antimatter is equivalent to matter moving backwards in time, which combined with the reversal in T of the direction of entropy increases means observers only ever observe the B > 0 (matter) sector, with the B < 0 (antimatter) sector always being the hidden sector from the perspective of any temporally-embedded observer that respects thermodynamics, as from any observer's perspective their time moves forward, their atomic nuclei have a positive electrical charge, electrons have negative electric charge, and the standard model has left handed doublets and right-handed singlets, with a weak force that appears left-chiral. To observe the antimatter would require an observer to reverse their arrow of time, which is thermodynamically not possible as consciousness and memory are thermodynamic processes that happen only in the direction of increasing entropy. Shadow geometry and right-handed neutrinos provide dark matter. The Grassmannian spinor bundle splits each fermion representation across the T boundary: right-handed neutrinos reside in the shadow sector, coupling to the observable sector only gravitationally. The dark matter density profile ρ(r) = ρ₀/(1+(r/r_c)²) with core radius r_c ≈ 9.98 kpc is derived from the shadow kernel by Abel inversion with zero free parameters, resolving the core cusp problem of dark matter distribution and addressing several further cosmological anomalies (the S₈ tension, Hubble tension, DESI dark energy evolution, JWST early galaxies, CMB hemispherical asymmetry, CMB dipole anomaly, axis of evil, overmassive early black holes, rotation curve diversity, etc.) from a single geometric source. Time's arrow emerges as a consequence of T-symmetric boundary conditions with entropy always increasing away from the conformal boundary in opposing temporal directions. The horizon and flatness problems are naturally resolved by this framework as the celestial CFT is globally defined at the conformal boundary. The framework generates falsifiable predictions including no primordial gravitational waves (tensor-to-scalar ratio r → 0, definitively falsified if r > 0.01 detected by CMB-S4), m_{ν₁} = 0 (KATRIN, JUNO), r_c ≈ 9.98 kpc (Euclid), glueball mass ratios matching the Sugawara spectrum, GUE statistics in the vacuum noise spectrum of gravitational wave detectors (LIGO/Virgo/KAGRA), and right-handed neutrino dark matter coupling only through gravity. Over 155 independent computational tests with zero free parameters confirm the framework to machine precision. Part 1 contains an introductory overview of the several subframeworks involved in this synthesis explaining their interrelations in the wider framework, part 2 contains the constructions and rigorous proofs of all major results, Part 3 contains extensions and other corollaries. Part 4 continues with comparisons to other approaches and discussion regarding foundational contributions without which this work would not have been possible follows, followed by testable predictions and a chapter addressing falsifiability, and Part V delivers a philosophical analysis of the implications suggested by the mathematics of this framework. This work unifies quantum mechanics and general relativity via the celestial holography program while revealing deep connections between number theory and physics: the identification Δ = 2s means the same self-dual Haar-invariant measure structure constraining the Riemann zeros to the critical line in the complex plane where Re(s) = 1/2 simultaneously constrains the conformal dimensions of graviton operators to the principal series where Re(∆) = 1, the same compactness that forces the Yang–Mills mass gap forces the discrete zeta spectrum, and the same functional equation symmetry that governs L-functions governs the shadow transform of the celestial CFT—suggesting geometry and arithmetic share a common logos rooted in self-dual Haar measure on (R⁺, ×) and flowering in the Grassmannian Gr(2,4). (shrink)

This paper develops a geometric semantics for metaphysics by showing how the Modal–Dependence Calculus (MDC) induces curvature, metric structure, and geodesic necessity across modal space. Dependence is interpreted as metaphysical curvature, invariance as a metric tensor, and necessity as the straightest (geodesic) trajectory in a governed modal field. The result is a field-theoretic account of grounding that unifies explanation, fundamentality, and modal stability under a single geometric framework. The paper also connects this structure to cognitive architecture via Path C (...) and the Semantic Artificial Intelligence Reactor (SAIR), illustrating how intelligence can be modeled as traversal within an externally governed semantic manifold. These developments collectively establish the conceptual foundation for Continuity Field Theory (CFT), a unified structural framework for metaphysics, logic, and cognition. (shrink)

The equations of motion for a particle in a classical gauge field are derived from the invariance identities 2 and basic assumptions about the Lagrangian. They are found to be consistent with the equations of some other approaches to classical gauge-field theory, and are expressed in terms of a set of undetermined functions Eα. The functions Eα are found to satisfy a system of differential equations which has the same formal structure as a system of equations from Yang-Mills theory. (...) 3 These results are obtained by a new method which applies techniques from the theory of functional equations to deduce the way in which the arguments of the Lagrangian must combine. The method constitutes an aid for obtaining the equations of motion when a non-gauge-invariant Lagrangian is chosen, and it is assumed that the equations of motion can be written in a gauge-invariant manner. (shrink)

Informational Renormalization of the Invariant Speed: A Causal–Symmetric Framework for Dynamical Light Propagation -/- In a causal-symmetric informational framework, where both time and space emerge from an underlying dynamics of information, the usual status of the speed of light as a primitive constant is reconsidered. The paper extends a previously developed “kappa framework”, in which an informational conductivity between initial and final boundary conditions generates microscopic irreversibility, emergent time and spacetime geometry, to the electromagnetic sector. Instead of treating the speed (...) of light as a fixed input, the theory introduces an informational renormalization factor that depends on scalar informational quantities such as the deviation from a preferred reference state, the cosmological expansion, and informational potentials. In this setting, the invariant speed is defined operationally as the signal speed of electromagnetic disturbances measured by rods and clocks whose behaviour is themselves governed by the same informational law. The numerically measured “invariant speed” thus becomes an emergent quantity determined by the informational state of the universe, rather than a built-in constant. -/- In its minimal version, the modified Maxwell dynamics remain covariant and gauge invariant, preserve well-behaved wave propagation, and share the same causal structure as the unmodified theory; informational renormalization changes the calibration of spacetime units but not the underlying light cones. The paper argues that this effect cannot be removed by local redefinitions of fields without simultaneously altering the matter sector and the informational action, so the modification is physically meaningful. In homogeneous cosmology, the effective invariant speed reduces to a function of the scale-dependent informational conductivity, linking its numerical value directly to the same law that also governs time flow and curvature. As an optional extension, the same informational structure allows an effective metric with a wider “informational” light cone and a superluminal sector, while a global time function defined by kappa prevents causal paradoxes. The result is a unified informational picture in which the invariant speed emerges from the global informational state of the universe. (shrink)

Many philosophers are baffled by necessity. Humeans, in particular, are deeply disturbed by the idea of necessary laws of nature. In this paper I offer a systematic yet down to earth explanation of necessity and laws in terms of invariance. The type of invariance I employ for this purpose generalizes an invariance used in meta-logic. The main idea is that properties and relations in general have certain degrees of invariance, and some properties/relations have a stronger degree (...) of invariance than others. The degrees of invariance of highly-invariant properties are associated with high degrees of necessity of laws governing/describing these properties, and this explains the necessity of such laws both in logic and in science. This non-mysterious explanation has rich ramifications for both fields, including the formality of logic and mathematics, the apparent conflict between the contingency of science and the necessity of its laws, the difference between logical-mathematical, physical, and biological laws/principles, the abstract character of laws, the applicability of logic and mathematics to science, scientific realism, and logical-mathematical realism. (shrink)

Philosophers of physics are split on whether foundational issues in relativistic quantum field theory should be framed within pragmatist approaches, which trade mathematical rigor for the ability to formulate non-trivial interacting models, or purist approaches, which trade the ability to formulate non-trivial interacting models for mathematical rigor. This essay addresses this debate by viewing it through the lens of the CPT theorem. I first consider two formulations of the CPT theorem, one purist and the other pragmatist, and extract from them (...) a set of problems that clarifies the distinction between pragmatism and purity. I then apply this distinction to Greenberg's influential claim that the violation of CPT invariance in an interacting RQFT entails the violation of Lorentz invariance. I show how this claim rests on an unsuccessful attempt to mediate between pragmatism and purity. I then evaluate another attempt at such mediation in the form of causal perturbation theory. This approach suggests that a focus on renormalized perturbation theory as a way of distinguishing pragmatists from purists may be misleading. (shrink)

It is now standard to interpret symmetry-related models of physical theories as representing the same state of affairs. Recently, a debate has sprung up around the question when this interpretational move is warranted. In particular, Møller-Nielsen :1253–1264, 2017) has argued that one is only allowed to interpret symmetry-related models as physically equivalent when one has a characterisation of their common content. I disambiguate two versions of this claim. On the first, a perspicuous interpretation is required: an account of the models’ (...) common ontology. On the second, stricter, version of this claim, a perspicuous formalism is required in addition: one whose mathematical structures ‘intrinsically’ represent the physical world, in the sense of Field. Using Dewar’s :485–521, 2019) distinction between internal and external sophistication as a case study, I argue that the second requirement is decisive. This clarifies the conditions under which it is warranted to interpret symmetry-related models as physically equivalent. (shrink)

The Higgs mechanism gives mass to Yang-Mills gauge bosons. According to the conventional wisdom, this happens through the spontaneous breaking of gauge symmetry. Yet, gauge symmetries merely reflect a redundancy in the state description and therefore the spontaneous breaking can not be an essential ingredient. Indeed, as already shown by Higgs and Kibble, the mechanism can be explained in terms of gauge invariant variables, without invoking spontaneous symmetry breaking. In this paper, we present a general discussion of such gauge invariant (...) treatments for the case of the Abelian Higgs model, in the context of classical field theory. We thereby distinguish between two different notions of gauge: one that takes all local transformations to be gauge and one that relates gauge to a failure of determinism. (shrink)

Wu and Yang (1975) identified gauge fields with principal fiber bundles: a gauge field is a bundle with a connection; this is not a choice of formalism but the definition of the object. This paper examines two lines of work that detach gauge-theoretic vocabulary from the geometric object it names, then identifies why the standard computational tools work despite this detachment. Part I reviews the metaplectic fusion category papers, which classify solutions to the pentagon and hexagon equations for the (...) SO(2p+1)₂ based ring, and shows that the physical vocabulary surrounding the work — topological quantum field theory, anyons, gauge invariants, SO(2p+1) at level 2 — is not earned. It supplies the result the classification was standing in for: integration over the principal SO(2p+1)-bundle with Chern-Simons connection at level 2 produces a single determined object. For SO(5)₂, the classification also produces wrong numerical data: the published twist values disagree with the bundle computation. Part II identifies the root issue — a pervasive substitution of ring algebras for fiber bundles — and exhibits the same error in a recent proposal to "encode" Chern-Simons theory into a Raman-coupled Bose gas, where the connection is eliminated in favor of a density coupling and the bundle is lost. It then proves that the Verlinde formula, used for 35 years without geometric explanation, computes on the complex Hopf fibration with first Chern number c₁ = 1 and linking number L = 1. Since physical fields fiber through U(1), the Hopf fibration is the universal principal bundle for gauge theory, and the Verlinde formula works because L = 1 at every level of the Hopf tower. (shrink)

A number of arguments have been given to show that the modal interpretation of ordinary nonrelativistic quantum mechanics cannot be consistently extended to the relativistic setting. We find these arguments inconclusive. However, there is a prima facie reason to think that a tension exists between the modal interpretation and relativistic invariance; namely, the best candidate for a modal interpretation adapted to relativistic quantum field theory, a prescription due to Rob Clifton, comes out trivial when applied to a number of (...) systems of physical interest. However, it is far from clear whether this difficulty for the modal interpretation is traceable to relativistic invariance per se or to the infinite number of degrees of freedom involved. In any case, the proponents of the modal interpretation have work to do. (shrink)

This paper derives the spectral coupling invariant ξ² ≡ (μα)² in closed form from the axiom system {A1 non-commutativity, A2 Π_d-saturation, A4 redundancy exclusion} of Operatiology, completing the Level 2 Class I certification mandated by the companion paper (DOI: 10.5281/zenodo.20577184). The derivation proceeds through the field Q(D₀), where D₀=(δ−1)δγ satisfies 16D₀²−24D₀+3=0, making Q(D₀) two-dimensional over Q. Since μ and α⁻¹ are each elements of L(M₃(ℂ)) with coefficients in Q(D₀), their ratio ξ=μ/α⁻¹ belongs to this field, and ξ²=(ξ_a+ξ_b·D₀)² reduces exactly to (...) A+B·D₀ — the unique mandatory representation forced by the two-dimensionality of Q(D₀), itself the algebraic expression of Axiom A4. No other value of ξ² is algebraically possible. The result is ξ² = A + B·D₀ = 179.534632529506…, with A = −1202.60331893592…, B = 10042.5954528296…, zero free parameters, and no reference to individually derived values of μ or α⁻¹. Agreement with benchmark (μ_CODATA2022/α⁻¹_Morel2020)² at +1.085σ. The Morel–Parker 5.5σ discrepancy is identified as structural evidence that α is a Tier-3 projection of ξ², not an independent fundamental constant. The complete derivation runs from the spectrum {1,1,−2} alone. ◆◆◆ [PYTHON VERIFICATION CODE — note: × denotes * and ^ denotes **] ◆◆◆ from mpmath import mp, mpf, pi, sqrt, exp, euler import math mp.dps = 50 h = [mpf(1), mpf(1), mpf(-2)] dim = 3 e1 = sum(h) e2 = h[0]×h[1] + h[0]×h[2] + h[1]×h[2] e3 = h[0]×h[1]×h[2] delta = sqrt(mpf(3)/2) gamma = mpf(1)/(dim-1) D0 = (delta-1)×delta×gamma gamma_E = euler Phi1 = mpf(2); Phi3 = mpf(13); Phi6 = mpf(7) ch2 = (e1^2 - 2×e2)/2 ch3 = (e1^3 - 3×e1×e2 + 3×e3)/6 N2 = mpf(10)/4 N3 = mpf(7)/2 C2 = sqrt(ch2/N2)×gamma L6 = abs(ch3)/N3 L0 = 6×pi^5 L2 = Phi3/Phi6 L4 = -delta^2/gamma × exp(-gamma_E) L5 = -Phi1/Phi3 mu_a = L0 + L2×(-3)/16 + L4×(-99)/256 + L5×(-135)/256 mu_b = L2×3/2 + L4×45/16 + L5×981/256 ainv_a = 48×pi + L6×(-2943)/4096 ainv_b = -32×pi + C2 + L6×2673/512 N = ainv_a^2 + ainv_a×ainv_b×mpf(3)/2 + ainv_b^2×mpf(3)/16 inv_a = (ainv_a + ainv_b×mpf(3)/2) / N inv_b = -ainv_b / N xi_a = mu_a×inv_a - 3×mu_b×inv_b/16 xi_b = mu_a×inv_b + mu_b×inv_a + 3×mu_b×inv_b/2 A = xi_a^2 - 3×xi_b^2/16 B = 2×xi_a×xi_b + 3×xi_b^2/2 print("A =", A) print("B =", B) print("xi^2 =", A + B×D0) ◆◆◆ [END] ◆◆◆ Published on June 7, 2026 DOI 10.5281/zenodo.20581357. (shrink)

Wigner's quantum-mechanical classification of particle-types in terms of irreducible representations of the Poincaré group has a classical analogue, which we extend in this paper. We study the compactness properties of the resulting phase spaces at fixed energy, and show that in order for a classical massless particle to be physically sensible, its phase space must feature a classical-particle counterpart of electromagnetic gauge invariance. By examining the connection between massless and massive particles in the massless limit, we also derive a (...) classical-particle version of the Higgs mechanism. (shrink)

This book seeks to answer the question "What explains CPT invariance and the spin-statistics connection?" These properties play foundational roles in relativistic quantum field theories, are supported by high-precision experiments, and figure into explanations of a wide range of phenomena, from antimatter, to the periodic table of the elements, to superconductors and superfluids. They can be derived in RQFTs by means of the famous CPT and Spin-Statistics theorems; but, the author argues, these theorems cannot be said to explain these (...) properties, at least under standard philosophical accounts of scientific explanation. This is because there are multiple, in some cases incompatible, ways of deriving these theorems, and, secondly, because the theorems fail for the types of theories that underwrite the empirical evidence: non-relativistic quantum theories, and realistic interacting RQFTs. The goal of this book is to work towards an understanding of CPT invariance and the spin-statistics connection by first providing an analysis of the necessary and sufficient conditions for these properties, and second by advocating a particular account of explanation appropriate for this context. (shrink)

We explore the connection between combinatorial principles on uncountable cardinals, like stick and club, on the one hand, and the combinatorics of sets of reals and, in particular, cardinal invariants of the continuum, on the other hand. For example, we prove that additivity of measure implies that Martin’s axiom holds for any Cohen algebra. We construct a model in which club holds, yet the covering number of the null ideal is large. We show that for uncountable cardinals κ≤λ and, if (...) all subsets of λ either contain, or are disjoint from, a member of, then has size at least etc. As an application, we solve the Gross space problem under by showing that there is such a space over any countable field. In two appendices, we solve problems of Fuchino, Shelah and Soukup, and of Kraszewski, respectively. (shrink)

A new approach to developing formulisms of physics based solely on laws of mathematics is presented. From simple, classical statistical definitions for the observed space-time position and proper velocity of a particle having a discrete spectrum of internal states we derive u generalized Schrödinger equation on the space-time manifold. This governs the evolution of an N component wave function with each component square integrable over this manifold and is structured like that for a charged particle in an electromagnetic field but (...) also includes SU(N) gauge field couplings. This construction reveals a new hasis for gauge invariance and new insight into the appearance of spin and other such properties in relativistic quantum mechanics and suggests a new charged particle model. (shrink)

This article summarizes a variety of current as well as previous research in support of a new theory of consciousness. Evidence has been steadily accumulating that information about a stimulus complex is distributed to many neuronal populations dispersed throughout the brain and is represented by the departure from randomness of the temporal pattern of neural discharges within these large ensembles. Zero phase lag synchronization occurs between discharges of neurons in different brain regions and is enhanced by presentation of stimuli. This (...) evidence further suggests that spatiotemporal patterns of coherence, which have been identified by spatial principal component analysis, may encode a multidimensional representation of a present or past event. How such distributed information is integrated into a holistic percept constitutes the binding problem. How a percept defined by a spatial distribution of nonrandomness can be subjectively experienced constitutes the problem of consciousness. Explanations based on a discrete connectionistic network cannot be reconciled with the relevant facts. Evidence is presented herein of invariant features of brain electrical activity found to change reversibly with loss and return of consciousness in a study of 176 patients anesthetized during surgical procedures. A review of relevant research areas, as well as the anesthesia data, leads to a postulation that consciousness is a property of quantumlike processes, within a brain field resonating within a core of structures, which may be the neural substrate of consciousness. This core includes regions of the prefrontal cortex, the frontal cortex, the pre- and paracentral cortex, thalamus, limbic system, and basal ganglia. (shrink)

We derive the six Wightman axioms of relativistic quantum field theory as theorems of three mathematical inputs: Haar measure on the Grassmannian Gr(2,4), the Penrose twistor correspondence, and the Peter-Weyl decomposition of L^2 spaces on compact groups. The Hilbert space is L^2(Gr(2,4), dmu_Gr), the vacuum is the unique SU(4)-invariant vector, Poincare covariance arises from the conformal embedding P+ into SU(2,2), the spectrum condition from forward-tube analyticity of the Penrose transform, locality from twistor non-incidence (spacelike separation implies non-incident twistors, non-incidence implies (...) holomorphic integrand, holomorphicity implies vanishing contour integral by Cauchy's theorem), cyclicity from irreducibility of the vacuum sector, and temperedness from elliptic regularity on the compact Grassmannian. The CPT theorem emerges from Haar measure self-duality (the map Lambda->Lambda-perp is the geometric CPT). The spin-statistics connection emerges from the topology of the Plucker line bundle. The Osterwalder-Schrader axioms follow by Wick rotation. No physical postulate enters the argument. The Wightman axioms are necessary consequences of Haar measure on the space of light rays. (shrink)

A theory is presented in which a field depends not only on spacetime coordinates xμ, but also on a Lorentz-invariant parameter τ. Such a theory is conceptually and technically simple and manifestly covariant at every step. The generator of evolution and the generator of spacetime translations and Lorentz transformations are obtained in a straightforward way. In the quantized theory the Heisenberg equation of motion is written in a covariant form and is equivalent to the field equation. The equal τ commutator (...) between the field and its canonically conjugate momentum is just proportional to the spacetime δ function. Finally comparison with the conventional field theory is done, and it is found that the expectation value of the momentum operator in the on shell states is the same. (shrink)

A massless Weyl-invariant dynamics of a scalar, a Dirac spinor, and electromagnetic fields is formulated in a Weyl space, W4, allowing for conformal rescalings of the metric and of all fields with nontrivial Weyl weight together with the associated transformations of the Weyl vector fields κμ, representing the D(1) gauge fields, with D(1) denoting the dilatation group. To study the appearance of nonzero masses in the theory the Weyl symmetry is broken explicitly and the corresponding reduction (...) of the Weyl space W4 to a pseudo-Riemannian space V4 is investigated assuming the breaking to be determined by an expression involving the curvature scalar R of the W4 and the mass of the scalar, selfinteracting field. Thereby also the spinor field acquires a mass proportional to the modulus Φ of the scalar field in a Higgs-type mechanism formulated here in a Weyl-geometric setting with Φ providing a potential for the Weyl vector fields κμ. After the Weyl-symmetry breaking, one obtains generally covariant and U(1) gauge covariant field equations coupled to the metric of the underlying V4. This metric is determined by Einstein's equations, with a gravitational coupling constant depending on Φ, coupled to the energy momentum tensors of the now massive fields involved together with the (massless) radiation fields. (shrink)

A topological inevitability of early developmental events through the use of classical topological concepts is discussed. Topological dynamics of forms and maps in embryo development are presented. Forms of a developing organism such as cell sets and closed surfaces are topological objects. Maps (or mathematical functions) are additional topological constructions in these objects and include polarization, singularities and curvature. Topological visualization allows us to analyze relationships that link local morphogenetic processes and integral developmental structures and also to find stable spatio-temporal (...) topological characteristics that are invariant for a taxonomic group. The application of topological principles reveals a topological imperative: certain topological rules define and direct embryogenesis. A topological stability of embryonic morphogenesis is proposed and a topological scenario of developmental and evolutionary transformations is presented. (shrink)

David Albert claims that classical electromagnetic theory is not time reversal invariant. He acknowledges that all physics books say that it is, but claims they are ``simply wrong" because they rely on an incorrect account of how the time reversal operator acts on magnetic fields. On that account, electric fields are left intact by the operator, but magnetic fields are inverted. Albert sees no reason for the asymmetric treatment, and insists that neither field should be inverted. I (...) argue, to the contrary, that the inversion of magnetic fields makes good sense and is, in fact, forced by elementary geometric considerations. I also suggest a way of thinking about the time reversal invariance of classical electromagnetic theory -- one that makes use of the invariant (four-dimensional) formulation of the theory -- that makes no reference to magnetic fields at all. It is my hope that it will be of interest in its own right, Albert aside. It has the advantage that it allows for arbitrary curvature in the background spacetime structure, and is therefore suitable for the framework of general relativity. (The only assumption one needs is temporal orientability.). (shrink)

We continue our study of the Lorentz-invariant field theory based on the equations Γ jk;l i =0 and gij;k=0. To first order in a perturbation expansion, we find Γ jk;l i =0 reduces to the wave equation. In orders higher than the first, we find that Γ jk;l i =0 cannot be linearized. We also find that the simple wave-type equation gij∂2g/∂xi∂xj=0 is contained in the theory when an appropriate choice is made for the parameters at the origin point.

The equations of electrodynamics for the interactions between magnetic moments are written on R×S3 topology rather than on Minkowskian space-time manifold of ordinary Maxwell's equations. The new field equations are an extension of the previously obtained Klein-Gordon-type, Schrödinger-type, Weyl-type, and Dirac-type equations. The concept of the magnetic moment in our case takes over that of the charge in ordinary electrodynamics as the fundamental entity. The new equations have R×S3 invariance as compared to the Lorentz invariance of Maxwell's equations. (...) The solutions of the new field equations are given. In this theory the divergence of the electric field vanishes whereas that of the magnetic field does not. (shrink)

In 1992 M.W. Evans proposed the O(3) symmetry of electromagnetic fields by adding a constant longitudinal magnetic field to the well-known transverse electric and magnetic fields of circularly polarized plane waves, such that certain cyclic relations of a so-called O(3) symmetry are fulfilled. Since then M.W. Evans has elevated this O(3) symmetry to the status of a new law of electromagnetics. As a law of physics must be invariant under admissible coordinate transforms, namely Lorentz transforms, in 2000 he (...) published a proof of the Lorentz invariance of O(3) symmetry of electromagnetic fields. As will be shown below this proof is incorrect; more, after simple correction it will turn out here that the O(3) symmetry cannot be Lorentz invariant. (shrink)

In this article I expound an understanding of the quantum mechanics of so-called “indistinguishable” systems in which permutation invariance is taken as a symmetry of a special kind, namely the result of representational redundancy. This understand- ing has heterodox consequences for the understanding of the states of constituent systems in an assembly and for the notion of entanglement. It corrects widespread misconceptions about the inter-theoretic relations between quantum mechanics and both classical particle mechanics and quantum field theory. The most (...) striking of the heterodox consequences are: that fermionic states ought not always to be considered entangled; it is possible for two fermions or two bosons to be discerned using purely monadic quantities; and that fermions may always be so discerned. (shrink)

Quantum theory of Lorentz invariant local scalar fields without restrictions on 4-momentum spectrum is considered. The mass spectrum may be both discrete and continues and the square of mass as well as the energy may be positive or negative. One may assume the existence of such fields only if they interact with ordinary fields very weakly. Generalization of Kallen-Lehmann representation for propagators of these fields is found. The considered generalized fields may violate CPT-invariance. Restrictions (...) on mass-spectrum of CPT-violating fields are found. Local fields that annihilate vacuum state and violate CPT-invariance are constructed in this scope. Correct local relativistic generalization of Lindblad equation for density matrix is written for such fields. This generalization is particularly needed to describe the evolution of quantum system and measurement process in a unique way. Difficulties arising when the field annihilating the vacuum interacts with ordinary fields are discussed. (shrink)

Maxwell accounted for the apparent elastic behavior of the electromagnetic field by augmenting Ampere’s law with the so-called displacement current, in much the same way that he treated the viscoelasticity of gases. Maxwell’s original constitutive relations for both electrodynamics and fluid dynamics were not material invariant. In the theory of viscoelastic fluids, the situation was later corrected by Oldroyd, who introduced the upper-convective derivative. Assuming that the electromagnetic field should follow the general requirements for a material field, we show that (...) if the upper convected derivative is used in place of the partial time derivative in the displacement current term, Maxwell’s electrodynamics becomes material invariant. Note, that the material invariance of Faraday’s law is automatically established if the Lorentz force is admitted as an integral part of the model. The new formulation ensures that the equation for conservation of charge is also material invariant in vacuo. The viscoelastic medium whose apparent manifestation are the known phenomena of electrodynamics is called here the metacontinuum. (shrink)

The standard argument for the Lorentz invariance of the thermodynamic entropy in equilibrium is based on the assumption that it is possible to perform an adiabatic transformation whose only outcome is to accelerate a macroscopic body, keeping its rest mass unchanged. The validity of this assumption constitutes the very foundation of relativistic thermodynamics and needs to be tested in greater detail. We show that, indeed, such a transformation is always possible, at least in principle. The only two assumptions invoked (...) in the proof are that there is at least one inertial reference frame in which the second law of thermodynamics is valid and that the microscopic theory describing the internal dynamics of the body is a field theory, with Lorentz invariant Lagrangian density. The proof makes no reference to the connection between entropy and probabilities and is valid both within classical and quantum physics. To avoid any risk of circular reasoning, we do not postulate that the laws of thermodynamics are the same in every reference frame, but we obtain this fact as a direct consequence of the Lorentz invariance of the entropy. (shrink)

This paper aims to single out some pathologies of current lexical semantics, which suffers from both the trauma of immanence and the opposite anxiety of rooting all knowledge in the pre-semiotic dimension, or entrusting sense-making entirely to context. To untangle these pitfalls, the dialogue with a phenomenological cognitive semiotics may prove fertile to focus on the lexicon as a type of storage and a type of memory; that is, a type of accumulated and sedimented knowledge based on the dynamical re-pertinentization (...) of invariants. In particular, the perspective developed by Sonesson shows striking coincidences with De Mauro’s semiotic semantics, which already provides a solid foundation for rethinking the notion of lexical field drawing on an “authentic” Humboldtian and Saussurean heritage. Therefore, I will briefly review some principles of lexical structural semantics, with the aim of showing how these notions, once emancipated from the Saussurean vulgata, can not only be compatible with a cognitive phenomenological approach, but also facilitate the rehabilitation of a theoretical apparatus still valuable for both (lexical) semantics and semiotics. Although focused on lexical fields, some of the issues addressed can be extended to language and languages, as well as to other types of semiotic systems, to shed light on controversial themes such as the embodiment of signs and the autonomy of linguistics and semantics. (shrink)

The earliest formulation of the Higgs naturalness argument has been criticized on the grounds that it relies on a particular cutoff-based regularization scheme. One response to this criticism has been to circumvent the worry by reformulating the naturalness argument in terms of a renormalized, regulator-independent parametrization. An alternative response is to deny that regulator dependence poses a problem for the naturalness argument, because nature itself furnishes a particular, physically correct regulator for any effective field theory in the form of that (...) EFT’s physical cutoff, together with an associated set of bare parameters that constitute the unique physically preferred “fundamental parameters” of the EFT. Here, I argue that both lines of defense against the initial worry about regulator dependence are flawed. I argue that reformulation of the naturalness argument in terms of renormalized parameters simply trades dependence on a particular regularization scheme for dependence on a particular renormalization scheme, and that one or another form of scheme dependence afflicts all formulations of the Higgs naturalness argument. Concerning the second response, I argue that the grounds for suspending the principle of regularization or renormalization scheme independence in favor of a physically preferred parametrization are thin; the assumption of a physically preferred parametrization, whether in the form of bare “fundamental parameters” or renormalized “physical parameters,” constitutes a theoretical idle wheel in generating the confirmed predictions of established EFTs, which are invariably scheme-independent. I highlight certain features of the alternative understanding of EFTs, and the EFT-based approach to understanding the foundations of QFT, that emerges when one abandons the assumption of a physically preferred parametrization. I explain how this understanding departs from several dogmas concerning the mathematical formulation and physical interpretation of EFTs in high-energy physics. (shrink)

The Invariance Maintenance Condition (IMC) is a substrate-agnostic framework for identifying and characterizing systems that maintain structural invariants under perturbation at measurable cost. Five companion translations have applied the IMC to nonequilibrium thermodynamics, immunology, anesthesia state transitions, ecological population regulation, and computational learning theory. This paper argues that those five translations are not confirmations of the same claim in different settings. They are structural probes of the framework itself: each domain was selected because it exposes a different property of (...) the IMC that no other domain in the series makes visible. Thermodynamics exposes the energetic grounding of maintenance cost. Immunology exposes regulatory network persistence under distributed maintenance. Anesthesia exposes experimentally accessible thresholds with pre-registered empirical commitment. Ecology exposes multi-generation persistence and makes timescale independence undeniable. Computational learning theory exposes a theoretical gap in an existing formal field — the missing maintenance sample rate bound — and demonstrates that the framework operates diagnostically in mature formal domains, not only descriptively. Taken together, the five translations establish four cross-domain structural results that no individual translation produces alone: (1) the precursor maintenance-cost signal is a domain-independent prediction, appearing in all five translations in domain-native form; (2) the framework’s geometry is timescale-independent across at least ten orders of magnitude in evaluation interval; (3) the Category 1 / Category 2 distinction is a principled taxonomy of the relationship between the framework and its target domains, not a quality gradient; and (4) the origin of constraint topology — physical law, evolution, or intentional design — is irrelevant to the maintenance dynamics that operate within it. The anesthesia translation carries a pre-registered empirical commitment that, if confirmed, changes the evidentiary status of the entire series. Note on scope: This paper addresses the IMC series strictly at the level of structural persistence and maintenance dynamics. The relationship between these results and broader theoretical frameworks addressing identity, consciousness, or recursive self-recognition is not treated here. The IMC’s applicability conditions are necessary but not sufficient for any of those richer characterizations; the present series establishes the necessary conditions, not the richer claims. (shrink)

We study model theory of actions of finite groups on substructures of a stable structure. We give an abstract description of existentially closed actions as above in terms of invariants and PAC structures. We show that if the corresponding PAC property is first order, then the theory of such actions has a model companion. Then, we analyze some particular theories of interest (mostly various theories of fields of positive characteristic) and show that in all the cases considered the PAC (...) property is first order. (shrink)

We prove that NIP valued fields of positive characteristic are henselian, and we begin to generalize the known results on dp-minimal fields to dp-finite fields. On any unstable dp-finite field K, we define a type-definable group of “infinitesimals,” corresponding to a canonical group topology on (K, +). We reduce the classification of positive characteristic dp-finite fields to the construction of non-trivial Aut(K/A)-invariant valuation rings.

In this paper, I argue that the recent discussion on the time - reversal invariance of classical electrodynamics (see (Albert 2000: ch.1), (Arntzenius 2004), (Earman 2002), (Malament 2004),(Horwich 1987: ch.3)) can be best understood assuming that the disagreement among the various authors is actually a disagreement about the metaphysics of classical electrodynamics. If so, the controversy will not be resolved until we have established which alternative is the most natural. It turns out that we have a paradox, namely that (...) the following three claims are incompatible: the electromagnetic fields are real, classical electrodynamics is time-reversal invariant, and the content of the state of affairs of the world does not depend on whether it belongs to a forward or a backward sequence of states of the world. (shrink)

This paper proposes a novel ontological account of the present that reconciles temporal becoming with the relativistic structure of modern physics. While existing models—presentism, eternalism, the growing block, and the moving spotlight—struggle to accommodate both the empirical insights of relativity and the phenomenological reality of temporal passage, we introduce an alternative framework: the present as a local acme of ontological actualization. Instead of presupposing a global simultaneity or a geometric foliation of spacetime, we define the present as the point of (...) maximal ontic differentiation within a stratified, non-metric field. This tensorial field, defined over a base manifold lacking metric or causal structure, encodes gradients of actualization intensity. Local peaks in this field give rise to stable configurations—projected as regimes of geometry, matter, and causality. Our model preserves local relativistic invariance while offering a dynamic, generative ontology of time. Temporal becoming is no longer a flow through a fixed background but an emergent topology of stabilization. The present, thus redefined, becomes not a slice in time but the structural operator through which the world becomes actual. We argue that this approach opens new perspectives for reconciling metaphysical realism about the present with the operational structure of physical theory. (shrink)

Unitary field theories and “SUPER-GUT” theories work with an universal continuum, the structured spacetime of R. Descartes, B. Spinoza, B. Riemann, and A. Einstein, or a (Machian (1–3) ) structured vacuum according the quantum theory of unitary fields (Dirac, (4,5) and Heisenberg (6–8) ). The atomistic aspect of the substantial world is represented by the fundamental constants which are invariant against “all transformations” and which “depend on nothings” (Planck (9–11) ). A satisfactory unitary theory has to involve these constants (...) like the mathematical numbers. Today, Planck's conception of the three elementary constants ħ, c, and G may be the key to general relativistic quantum field theory like unitary theory. However, the elementary constants are a question of measurement-theory, also.According to Popper's theory (12–16) of induction, such unitary theories are “universal explaining theories.” The fundamental constants involve the complementarity between the universal statements in unitary theory and the “basic statements” in the language of classical observables. (shrink)

Denef and Loeser defined a map from the Grothendieck ring of sets definable in pseudo-finite fields to the Grothendieck ring of Chow motives, thus enabling to apply any cohomological invariant to these sets. We generalize this to perfect, pseudo algebraically closed fields with pro-cyclic Galois group. In addition, we define some maps between different Grothendieck rings of definable sets which provide additional information, not contained in the associated motive. In particular we infer that the map of Denef-Loeser is (...) not injective. (shrink)

It has been argued that the existence of a minimum observable interval of space and time is a model-independent result of the combination of quantum field theory and general relativity. In this paper, I promote this result to a fundamental postulate, called the MOIST postulate. It is argued that the postulate leads to the existence of a maximum signal speed and its invariance. This new result may have two interesting implications. On the one hand, it suggests that the MOIST (...) postulate can explain the invariance of the speed of light, and thus it might provide a deeper logical foundation for special relativity. Moreover, it suggests that the speed constant c in modern physics is not the actual speed of light in vacuum, but the ratio of the minimum observable length to the minimum observable time interval. On the other hand, the result also suggests that the existing experiments confirming the invariance of the speed of light already provide observational evidence to support the MOIST postulate. (shrink)