Magnetism as a Residual Projection Field
The preprint is available on SSRN: Magnetism as a Residual Projection Field (September 13, 2026). http://dx.doi.org/10.2139/ssrn.7456978
Magnetism as a Residual Projection Field
Juliet Zhong
Independent
Researcher | London, United Kingdom | September 2026
ORCID:
0009-0006-5099-3671
Abstract
Natural lodestone is rare not because iron-bearing minerals
are rare but because the transition from magnetically ordered mineral to stable
permanent magnet demands an external event of specific character. Geological
magnetisation leaves ordinary magnetite with a remanence-to-saturation ratio
typically below 0.05; lodestone specimens reach 0.10–0.26 and
lightning-associated fulgurite samples 0.45–0.92, indicating exposure to fields
orders of magnitude stronger than the geomagnetic background, with lightning-induced
magnetisation the established pathway studied here. That a microsecond
discharge can write a stable macroscopic directional state across an enormous
number of microscopic sites is not answered by noting that domain walls move;
the prior question is why the quantities being arranged possess a common directional
character at all. This paper proposes that they are not independent because
each is the local resolution of a single generating relation between a
six-dimensional source structure and the projected material domain, and that
what a lightning strike writes is an arrangement, not a coincidence. Magnetic
order on this account is the residual surviving how that arrangement handles
cancellation: concordant readouts produce ferromagnetism, site-alternating
readouts produce antiferromagnetism, sublattice-inequivalent readouts produce
ferrimagnetism, and symmetry-patterned readouts that cancel globally while
preserving local directional structure produce altermagnetism—the last being
the cleanest empirical evidence that zero net moment does not entail absence of
order. Superconductivity is proposed as the limiting case of complete
relational locking, from which composite pairing and magnetic-field expulsion
follow as distinct interpretive consequences rather than coupled corollaries of
one condensate property. Four falsification and discriminating conditions are
stated.
Keywords: magnetism; projection ontology; binary
readout; altermagnetism; superconductivity; Meissner effect; lodestone;
remanent magnetisation; finite-momentum pairing; foundations of condensed
matter
1. Introduction: The Direction That Remains
Magnetism is among the best-described phenomena in physics
and among the least interrogated. Maxwell's equations govern how magnetic
fields propagate, couple to charge and current, and transform between frames
(Maxwell 1865). Heisenberg exchange and its band-theoretic descendants explain
why neighbouring atomic moments prefer particular relative orientations, and
why some materials order ferromagnetically while others do not (Heisenberg
1928; Stoner 1938). Domain theory explains why a ferromagnet with full local
order can show no net moment until a field is applied, and coercivity theory
explains why some materials retain an imposed arrangement and others do not
(Weiss 1907; Stoner and Wohlfarth 1948). The two-valued measurement spectrum of
a spin-1/2 degree of freedom along any chosen axis is fixed by SU(2)
representation theory and confirmed by every relevant experiment; the outcomes
are the eigenvalues ±ℏ/2 of the spin operator, and the choice of axis is free
while the spectrum is not. These accounts are correct, and nothing in what
follows disputes any of them.
But they answer questions of a particular type—how does the
observable behave, under what conditions, with what magnitude—and they do not
answer a question of a different type, one that is rarely posed because the
vocabulary of the field makes it hard to formulate. The two outcomes of a spin
measurement are outcomes of something. The quantisation axis is chosen by the
experimenter; the two-valuedness is not created by that choice. A magnetic
field has a direction, and that direction is expressed in the coordinate system
of the laboratory or the crystal, but the coordinate system does not
manufacture the directional content it expresses. What, then, is the physical
relation of which the two-valued readout is the local resolution, and what is
the magnetic field the macroscopic appearance of?
The clearest empirical entry point into this question is not
an exotic laboratory phenomenon but a stone. Ordinary magnetite, Fe₃O₄, is
ferrimagnetic: Fe³⁺ and Fe²⁺ sit on inequivalent crystallographic sites, the
sublattice moments are antiparallel and unequal, and a spontaneous residual
survives the compensation (Néel 1948). It is common, widely distributed, and
magnetically ordered in every specimen. Yet a piece of ordinary magnetite
picked up from a riverbed does not attract iron filings. Lodestone—the same
mineral, with the same spinel structure—does. Remanence-to-saturation ratios
for ordinary magnetite and titanomagnetite are typically below 0.05; lodestone
specimens reach 0.10–0.26; lightning-associated fulgurite samples reach
0.45–0.92 (Wasilewski and Kletetschka 1999, 2023). The geomagnetic field, at
roughly 0.5 gauss, cannot account for the high values: thermoremanent and
chemical remanence acquired under geological conditions deliver the low ones,
and the high values require a field orders of magnitude stronger. The mechanism
is lightning. A stroke discharges currents of tens to hundreds of kiloamperes
across microseconds, producing transient fields of thousands of gauss. Where
the mineral already has the right microstructure—magnetically hardened
precursor domains with high coercivity—that transient suffices to write a
stable macroscopic arrangement that the ambient geomagnetic field subsequently
cannot erase (Wasilewski and Kletetschka 2023). Laboratory replication with
pulsed currents confirms it: controlled high-current impulse experiments on
mafic and ultramafic igneous rocks, with peak currents up to approximately 80
kA, raise remanent magnetisation by factors reaching about 490 and drive REM
ratios above 0.1, into the range characteristic of lodestone and fulgurite (de
Andrade Nunes et al. 2026).
The question lodestone poses is not why lightning can
produce such strong fields—that is understood—but why the same discharge that
leaves one mineral permanently magnetised leaves another unchanged. A piece of
quartz struck by lightning is not a magnet. A soft iron needle struck by
lightning may acquire a weak moment that it subsequently loses. A magnetically
hardened magnetite specimen struck by lightning can become a lodestone it was
not before. The difference is not in the field that acts; it is in the material
it acts on. More precisely, it is in what the material already holds. To write
a persistent macroscopic arrangement, the material must already possess the
degrees of freedom that the arrangement will organise, and those degrees of
freedom must already be susceptible to collective expression by a single
directional event. Domain physics explains how a coherent arrangement is
written and why coercivity retains it. A prior question remains: why does the
quantity being arranged possess a common directional character at all?
The answer proposed here is that the sites are never
independent in the relevant sense, because each is the local resolution of the
same generating relation between a source structure and the projected material
domain. A transient directional event does not orchestrate a coincidence among
independent quantities; it supplies the external condition under which a
relation already common to all sites becomes macroscopically expressed. What
the high-coercivity microstructure retains is the expression, not anything
deposited from outside. Nothing is added to the mineral by the lightning
strike. What changes is how what was already there is distributed.
This is an ontological claim about the nature of magnetic
order, not a replacement of any part of the standard treatment. The account
proposed in this paper identifies what the state variables of standard magnetic
theory are variables of, rather than deriving new values for those variables.
Three prior results from the author's earlier work are imported as premises and
not re-derived here. The radiation domain, in which proper time is identically
zero along every worldline, and the matter domain, in which proper time is
strictly positive, are ontologically distinct—the separation Lorentz invariant
and derivable from Newton's two categories of time combined with special
relativity's two proper-time classes (Zhong 2026c; 2026d). The observable
material domain is a structural projection from a six-dimensional source
structure, a specification shown by explicit calculation of the worldsheet
conformal invariance conditions of string theory, with the timelike direction
removed and six taken as the total, to be not merely permissible but determined
(Zhong 2026a, 2026e). A projected relation leaves observable residuals in S³D;
this has been established for gravitation, where the perturbation of an ambient
radiation field by condensed matter recovers the inverse-square law from flux
geometry (Zhong 2026b). Two labels from this framework are used below and no
others: S⁶D for the six-dimensional source structure and S³D for the observable
material domain. Observer structure and the measurement ground are established
elsewhere and not revisited (Zhong 2026f, 2026g).
2. Two Classes of Direction and the Ontological Distinction
Two kinds of direction must be distinguished, and the
distinction is categorical rather than a matter of scale or origin.
A Class A direction is the direction of the
generating relation between S⁶D and a projected material point. It is not
produced by the material, not a quantity the material can alter, and—this is
the restriction that most requires emphasis—it is not a vector in the material
domain's own coordinate system. Asking for its components in the laboratory
frame is asking for the components of something that does not live in that
frame. The consequences are accepted throughout: no Class A quantity appears in
any expression in this paper, no experiment claims to measure one, and no
prediction requiring access to a Class A value is offered.
A Class B direction is any direction the material
domain generates for itself. Crystallographic axes are Class B. The Néel axis
is Class B. The easy axis, the quantisation axis chosen by the experimenter,
the orientation of an applied field, the laboratory frame—all Class B.
Everything that can be given components, rotated by turning the sample, or
shifted by choosing a different reference frame is Class B. Nothing in the
standard treatment of these quantities is disputed; they behave exactly as the
standard theory says, and their calculation is untouched.
The two are not two members of one set. Class B directions
are internal to a coordinate system the material carries with it. A Class A
direction is the relation by virtue of which there is a material point to carry
a coordinate system at all. The distinction is not a matter of scale or origin
but of logical category: Class B presupposes the existence of a projected
domain to provide coordinates; Class A is what generates the projection. To ask
which Class B direction corresponds to the Class A relation is a category error
of the same shape as asking for the spatial location of a logical proposition.
Class B geometry determines how a binary degree of freedom
is resolved in the material coordinate system; it does not generate the binary
character of the readout itself. Quantum mechanics establishes that the spin
measurement spectrum is ±ℏ/2 along any chosen axis, with the choice of axis
free and the spectrum not. The present account agrees entirely and adds one
question: given that the two-valuedness is not a product of the chosen axis,
what is it a product of? The axis is Class B. The two-valuedness is not. The
proposal is that the two outcomes correspond to the two complementary registrations
available to a two-ended generating relation—one read source-first, one read
point-first. This is an ontological identification made after the
two-dimensional Hilbert-space structure has been established empirically and
mathematically; it is not a derivation of that structure. A relation with two endpoints
does not mathematically entail a two-dimensional state space; two-ended systems
routinely carry continuous parameters, complex phases, and higher-multiplet
structure. What is proposed is that, given the empirically confirmed binary
spectrum, the two outcomes are interpretable as the two registrations of a
two-ended generating relation. The registrations are not spatial orientations:
"source-first" and "point-first" name orderings, not
directions in S³D space.
If a Class A direction cannot be measured in the material
domain, what content does it have? Class A is not a hidden variable in the
sense Bell's analysis constrains: nothing here assigns pre-existing values to
measurement outcomes, and the two registrations are not two states of an
electron that pre-exist measurement. Class A supplies an identification—of what
the readout is a readout of—and identifications of this kind are tested through
their structural consequences, all of which are material-domain measurements.
The condition that would defeat the distinction most directly is that the
two-valued readout axis be shown exhaustively determined by crystallographic
and laboratory geometry, with no residue. That condition is stated formally in
Section 5. Class A has content only if removing it leaves something unaccounted
for; Section 5 specifies what that something is.
3. From Readout to Magnetic Appearance: The Four-Stage Structure
An unmeasurable relation becomes a measurable field through
four stages.
Stage one is the fixed Class A generating relation at
each projected point—not measurable in S³D, not expressible as a vector within
it. It is the precondition for the existence of a material point, not a
property that point possesses the way it possesses a crystal structure.
Stage two is the local binary registration of that
relation at each site—the quantity standard treatment calls intrinsic spin.
Everything the standard treatment says about how spin is measured, how it
couples to orbital degrees of freedom, and how it enters the exchange
Hamiltonian is retained without modification. What the present account adds is
the identification of what this quantity is a registration of.
Stage three is the distribution of the local
registrations over Class B geometry, together with the persistence of that
distribution in the measured domain. This is the stage at which the
overwhelming majority of standard condensed matter physics operates, and it is
the stage that does the most work in this account. Crystal structure determines
the symmetry constraints on the arrangement: which readouts can be concordant,
which must alternate, and which are related by symmetry operations that
generate momentum-space rather than real-space patterns. Sublattice
inequivalence determines whether compensation, where it occurs, is exact or
structural. Exchange coupling determines the energetically preferred alignment
of neighbouring registrations, driving the arrangement toward the
configurations that minimise free energy and fixing the ordering temperature at
which long-range arrangement becomes stable. Domain formation partitions the
arrangement into regions of different Class B orientation, each internally ordered
but collectively cancelling at the macroscopic scale; domain walls are the
boundaries at which the arrangement switches its Class B orientation, and their
mobility under applied fields is what hysteresis and coercivity measure. Thermal
fluctuations compete with exchange coupling and disrupt long-range arrangement
above the ordering temperature, so that the macroscopic stage-four residual
averages to zero in the absence of an applied field. The principal calculations
of standard band magnetism, exchange theory, spin-group symmetry analysis, and
micromagnetics operate at stage three. The present account does not replace any
part of this; it identifies what the arranged quantity is.
Stage four is the macroscopic residual of the
stage-three arrangement, expressed through the magnetisation M and, via the
standard Maxwell relations between magnetisation, bound current and field,
through the electromagnetic field B that accompanies it. The magnetic field is
identified here as the observable electromagnetic expression of that residual,
not as a quantity that bypasses the magnetisation layer: what stage three
yields directly is a distribution of local moments, and the passage from that
distribution to B is the standard one, unaltered. The residual's macroscopic
magnitude reflects the degree to which the stage-three arrangement fails to
cancel; its observable direction is expressed through Class B geometry; its
existence is attributed to the underlying Class A relation.
The mapping from cancellation completeness to field
magnitude is supplied by standard magnetic theory and is not duplicated here. The
relation between local moments and macroscopic magnetisation is already
supplied by standard magnetic theory—exchange models, band magnetism,
micromagnetics—and the electromagnetic constitutive relations connect the
resulting magnetisation to the observable field. What those theories compute is
the stage-three distribution and its stage-four residual. The present account
does not add a competing function; it identifies what the computed quantities
are quantities of. The identification operates at a different level from a
field equation and does not compete with one.
What the four-stage structure provides is an account of why
stage two and stage four can come apart so completely. In a paramagnet the
local stage-two degrees of freedom remain available: the relevant 3d or 4f
states support effective moments, which can be inferred from the Curie-law
susceptibility χ = C/T (Curie 1895; Langevin 1905). Yet no spontaneous
macroscopic residual appears at stage four. On any view in which the magnetic
field simply expressed the presence of local moments, this would require special
explanation. On the present account it requires none: the stage-three
arrangement has no persistent long-range form, thermal energy disrupts any
coherent distribution before it can produce a surviving residual, and stage
four reflects stage three, not stage two. The local readout exists; the global
arrangement does not; the appearance is null.
The same point comes from the other direction in an
unmagnetised ferromagnet below the Curie temperature. Here the stage-three
arrangement is locally complete and highly ordered within each domain: exchange
coupling has aligned the readouts with near-perfect concordance within every
domain volume. But the arrangement at the macroscopic scale is partitioned into
domains of different Class B orientation, and the contributions of differently
oriented domains substantially cancel. The macroscopic stage-four residual
again vanishes, not because stage two is absent or because stage three is
disordered, but because the domain partition produces a global cancellation.
Apply a field, and the macroscopic response is produced primarily through
changes in domain populations and orientations, with local rotation and other
reversal mechanisms also contributing—the Class B partition shifts, and the
stage-four residual appears. Remove the field in a soft magnet, and the domain
configuration relaxes and the residual largely disappears again. In a hard
magnet with high coercivity, the domain partition is pinned and the residual
persists. In every case, the appearance is a function of stage three—of what
the arrangement does with cancellation—not a direct function of stage two.
These four arrangements are the structurally distinct
limiting cases the present argument requires, and they are not a conventional
magnetic phase classification. Canted antiferromagnetism, weak ferromagnetism,
non-collinear and spiral order, metamagnetic transitions, spin glasses and spin
liquids are all arrangements at stage three and none is excluded. The four
cases below are chosen because each isolates a different relation between a
distribution of binary readouts and its own cancellation.
Ferromagnetism is reinforcement: the readouts are
predominantly concordant across the arrangement, so the residual from one
region does not cancel the residual from another, and a macroscopic stage-four
field survives. The exchange interaction that drives the concordance is
standard and untouched; what the present account identifies is what the
concordant quantity is. The domain structure is a Class B partitioning of the
arrangement, and the stage-four behaviour under applied and removed fields—appearing
when domains align, vanishing when they randomise—is the clearest
material-domain demonstration that stage four depends on stage three, not on
stage two.
Antiferromagnetism is strict compensation: the
readouts alternate between sublattices related by translation combined with
time reversal, so their bulk contributions cancel and the macroscopic net
moment vanishes despite full local order at stage three. Local order is
complete and macroscopic appearance is absent—the same decoupling that
paramagnetism demonstrates from the thermal side is here demonstrated from the
structural side.
Ferrimagnetism is unequal compensation: the two
sublattices are crystallographically and chemically inequivalent, carrying
different effective moments, and the compensation is incomplete by
construction. Magnetite, Fe₃O₄, is the canonical case—Fe³⁺ on tetrahedral A
sites carrying moments anti-parallel to the combined Fe³⁺ and Fe²⁺ on
octahedral B sites, with the A and B sublattice moments unequal (Néel 1948). A
stage-four residual survives not because the readouts are concordant but
because the compensation is structurally prevented from being exact. What
lightning writes into magnetite when it creates lodestone is therefore a
macroscopic expression of this ferrimagnetic unequal compensation, not a
ferromagnetic concordance the mineral does not possess.
Altermagnetism is patterned compensation: the global
moment vanishes exactly by crystal symmetry, while a non-trivial local
directional pattern survives in the electronic structure. The defining feature
is a momentum-dependent spin splitting of the electronic bands that arises from
the crystal rotation operations relating the two magnetic sublattices, not from
spin-orbit coupling or net magnetisation (Šmejkal, Sinova and Jungwirth 2022a,
2022b). In a d-wave altermagnet the splitting takes the form ΔAM(k) = α(kx² −
ky²), transforming under the crystal point group as ΔAM(C₄k) = −ΔAM(k): it
changes sign under ninety-degree rotation and vanishes along nodal lines kx =
±ky, resulting in momentum-space spin splitting that is large, anisotropic, and
exactly zero in the global moment. Room-temperature realisation in the metallic
d-wave altermagnet KV₂Se₂O is established (Jiang et al. 2025), and
altermagnetic spin splitting of g-wave symmetry is observed by photoemission in
MnTe (Krempaský et al. 2024) and CrSb (Ding et al. 2024), where the
higher-order nodal pattern is fixed by the spin group in the same way.
Each of these four arrangements has a standard
order-parameter description, and the mapping is exact. Ferromagnetism is
characterised by a non-vanishing magnetisation M; antiferromagnetism by a
non-vanishing staggered magnetisation, or Néel vector, with M vanishing;
ferrimagnetism by inequivalent sublattice magnetisations M_A and M_B whose sum
does not vanish; altermagnetism by a vanishing M together with a
momentum-dependent spin polarisation whose symmetry is fixed by the spin group.
Every one of these is a stage-three quantity and its calculation is untouched
here. What the present account states is that they are order parameters of the
same underlying readout, differing in how that readout is distributed and
therefore in what survives. This is why the four are grouped by cancellation
structure rather than by ordering temperature, moment magnitude, or itinerancy,
and the grouping cuts across the usual ones: ferromagnetism and ferrimagnetism
both leave a residual, but through concordance in one case and sublattice
inequivalence in the other; antiferromagnetism and altermagnetism both cancel
globally, but through strict alternation in one case and symmetry-enforced
patterning in the other. Cancellation structure is what relates stage three to
stage four, so it is the structure by which the arrangements must be
classified.
The standard spin-group symmetry analysis defines and
explains altermagnetism entirely within the material domain, and nothing in the
present account displaces it. What altermagnetism provides for the present
account is an empirical demonstration of the principle that zero net moment
does not entail absence of order. This principle is the logical core of the
whole paper. Global cancellation is a geometric property of the stage-three
arrangement, not evidence about the stage-two readouts. Antiferromagnetism demonstrates
it in real space; altermagnetism demonstrates it in momentum space, and
particularly cleanly, because the patterned compensation is symmetry-enforced
rather than approximate.
4. Relational Locking, the Superconducting Limit, and Field Expulsion
In ferromagnetism, antiferromagnetism, ferrimagnetism, and
altermagnetism, the stage-three distribution arranges readouts relative to one
another—some concordant, some alternating, some patterned—and the stage-four
field is what survives that mutual arrangement. There is a limiting arrangement
in which the readouts coincide with their own Class A relations rather than
with each other: in which the arrangement ceases to be a distribution of readouts
relative to neighbours and becomes instead a coincidence of each readout with
the generating relation of its own site. This is the limiting case of complete
relational locking.
If magnetic directionality traces to one relation, what is
to be made of the manifest multiplicity of pairing wavevectors Q observed in
altermagnetic superconductors? Does a plurality of Q values not imply a
plurality of generating relations?
A prior symmetry analysis of pairing in altermagnetic
systems, carried out without any ontological premises whatever, supplies the
anchor here (Zhong 2026h). In an altermagnet, the momentum-dependent breaking
of time reversal makes the conventional pairing partners energy-non-degenerate:
ε₊(k) ≠ ε₋(−k), with mismatch δε(k) = 2ΔAM(k) = 2α(kx² − ky²). Pairing in the
opposite-spin channel compensates by acquiring a finite centre-of-mass
momentum. The optimal pairing vector has magnitude |Q*| ~ ΔAM(kF)/vF, and the
crystal point group leaves the states with Q* along the two antinodal
directions degenerate, so the pairing vector is symmetry-generated and locked
to the crystal axes rather than externally imposed. Independent results from
different methods confirm the finite-momentum instability and its altermagnetic
origin (Chakraborty and Black-Schaffer 2024; Zhang, Hu and Neupert 2024;
Brekke, Brataas and Sudbø 2023; Mazin 2025). Symmetry already explains this
degeneracy, and the present account adds its interpretive point: the doublet is
two resolutions, in the material domain's own coordinate system, of one pairing
problem posed by one crystal. The multiplicity is generated at stage three; it
is a property of how a Class B geometry accommodates a constraint. Distinct
material-domain resolutions do not imply distinct generating relations, just as
patterned compensation shows local directional order surviving global
cancellation.
The pair as the minimal complete relational carrier.
Superconductivity is proposed as the material realisation of the limiting
arrangement. In BCS theory, pairing is an instability consequence: an
arbitrarily weak attractive interaction destabilises the Fermi sea against the
formation of bound pairs, and the pair is the carrier because the interaction
makes it so (Cooper 1956; Bardeen, Cooper and Schrieffer 1957). That account is
complete and is not being competed with. The question the present account addresses
is a different one: why, from the relational standpoint, is a composite the
minimal carrier capable of entering the limiting arrangement?
A single carrier supplies a single local readout. Whatever
the registration at that site, it is one end of a two-ended relation resolved
in one way. It cannot, by itself, represent the complete relation—it represents
a registration, not a coincidence. A paired composite, by contrast, can
coherently represent the two complementary registrations of one relation as a
single collective object. This is the sense in which the pair is the minimal
complete relational carrier. The claim is not a one-to-one assignment of
registrations to electrons within the pair: the internal structure of a
superconducting order parameter—singlet and triplet channels, anisotropic gap
functions, and in the altermagnetic case a finite centre-of-mass momentum—is
considerably richer than any such simple assignment, and the present account
does not pretend otherwise (Sigrist and Ueda 1991). The claim is at the level
of relational structure: complete coincidence with the generating relation
requires representing both ends, and a single registration represents only one.
The minimal-carrier statement is a structural interpretation and not a theorem
following from the two-ended character alone; no mathematical necessity carries
a two-ended relation into a two-particle composite. It acquires physical
content because conventional electronic superconductivity is, experimentally
and theoretically, represented by a composite paired condensate, so the
interpretation has something determinate to be true or false of.
BCS asks: what interaction drives carriers to pair? The
present account asks: if the target state is complete relational coincidence,
what is the minimum carrier structure that can reach it? The answers are
complementary, and the second does not displace the first.
BCS establishes that a weak attractive interaction, mediated
by phonons in conventional superconductors, creates a pairing instability: for
any net attractive interaction, however weak, the normal Fermi sea is unstable
against pair formation below some temperature. The pair is therefore the
carrier because the instability dictates it, and the theory is a complete
account of the mechanism. What BCS does not address is the question of
relational structure: given that pairs form, what does a pair hold that a single
carrier does not? The present account's answer is that a single carrier holds
one registration of a two-ended relation—one end resolved in one way—while a
composite that coherently represents both registrations holds something that
can, in principle, coincide with the complete relation. This is not a claim
about the microscopic wavefunction of individual Cooper pairs; the BCS
wavefunction is a many-body state and its description in terms of individual
pair wavefunctions is a convenience. It is a claim about the minimum structural
unit capable of relational coincidence, at the level of collective objects
rather than individual particles.
Complete relational locking and global phase coherence.
When the carriers coincide with their own Class A relations rather than with
each other, the stage-three arrangement ceases to be a distribution of readouts
relative to neighbours. Global phase coherence—the macroscopic quantum
coherence of the condensate, its rigidity against perturbation, the
quantisation of flux—is read, on this account, as the S³D expression of many
pairs standing in coincidence with the same relation. It is not proposed as an
additional quantity requiring separate explanation; it is what many local
coincidences with one relation look like when described in the material
domain's own terms. The macroscopic quantum coherence of the condensate and its
electromagnetic response are established results treated as such.
The critical temperature does not perform the locking.
Reaching it opens a condition latent in the material, allowing carriers to
return to coincidence with the relation from which they were projected. This is
the material's original condition, not an achieved one, and no external agent
drives the transition. The mechanism of the temperature gate is the subject of
separate work not addressed here; temperature is not the alignment, and the
critical temperature is not the source of the ordered state.
Dissipationless transport. Standard theory accounts
fully for zero resistance in the superconducting state: the condensate, the
energy gap, macroscopic phase coherence, and London electrodynamics together
explain why current flows without dissipation and why the resistance is exactly
zero rather than merely small (London and London 1935; Bardeen, Cooper and
Schrieffer 1957). Nothing in the present account characterises that explanation
as mere scattering suppression, and nothing disputes it. The resistance of a
normal metal is not simply a matter of magnetic readout discordance between
neighbours—electron-phonon scattering, impurity and defect scattering,
electron-electron interaction, and boundary scattering all contribute through
entirely standard mechanisms. The present account addresses a different
question: what does the locked state lack that makes dissipative rearrangement
unavailable as a channel at all? Within the proposed interpretation,
dissipation is associated with the availability of carrier configurations that
can undergo irreversible rearrangement through coupling to the surrounding
material degrees of freedom. Complete relational locking removes that
independently available channel: where the carriers stand in coincidence with
their own relations rather than in a mutually arranged state, the
configurational freedom on which the ordinary dissipative mechanisms operate is
no longer present in the same form. The claim is not that relational locking
suppresses electron-phonon or impurity scattering as such, but that the locked
state does not offer the independently rearrangeable carrier configurations
through which those mechanisms produce dissipation in a normal metal.
Field expulsion. A superconductor expels magnetic
flux from its interior, doing so actively rather than as a consequence of
perfect conductivity alone: a sample cooled through its transition temperature
in an applied field expels the flux already present (Meissner and Ochsenfeld
1933). The standard account grounds this in the rigidity of the condensate
wavefunction and the London electrodynamics (London and London 1935). One point
must be affirmed before any reinterpretation: the superconducting condensate couples
to the electromagnetic field. It must. The London equations relate current
density to vector potential; flux through a superconducting ring is quantised
in units of h/2e; the Josephson effect makes the phase difference across a
junction directly electromagnetic (Deaver and Fairbank 1961; Doll and Näbauer
1961; Josephson 1962). Any account claiming the condensate presents nothing for
the field to couple to would be refuted by all three. The present account does
not say this.
In a normal conductor the applied field acts on electronic
and collective degrees of freedom that remain available for independent
rearrangement, and the resulting magnetic response is governed by the ordinary
electrodynamic and material mechanisms. In the condensate, those degrees of
freedom are locked into coincidence with the shared relation and are no longer
independently available for rearrangement. The present interpretation
attributes flux expulsion not to the mere absence of dissipation but to the loss
of independently available configurational response under equilibrium
electromagnetic conditions. Standard superconductivity describes
dissipationless transport and field expulsion within the same condensate
electrodynamics; the present account assigns them distinct interpretive roles.
The condensate remains fully electromagnetically coupled; what changes is the
set of configurations through which that coupling can produce a bulk
rearrangement. This distinction is what Section 5 tests.
The present account does not claim to have derived Hc, Hc1,
or Hc2. The vortex state, flux penetration in quantised units, pinning, and the
field dependence of the condensation energy belong entirely to standard
treatment, applied here without modification (Abrikosov 1957). The interpretive
statement is qualitative: an applied Class B field of sufficient magnitude
presents a configuration the locked state cannot accommodate while remaining
locked, and the state gives way in the manner standard theory describes.
The standard treatment grounds both zero dissipation and
field expulsion in the condensate and its electrodynamics, and this grounding
is correct. The London equations describe the Meissner effect and zero
resistance together as properties of the same condensate state, and the BCS
microscopic theory provides the condensate from first principles. Nothing in
the present account disputes any of this. What the present account claims is
that, at the ontological level, zero dissipation and field expulsion arise from
the same underlying condition—relational locking—through two logically distinct
pathways: one through the removal of dissipative rearrangement channels, the
other through the removal of independently available electromagnetic
configurational degrees of freedom. The standard treatment does not require
these to be logically distinct; the present account asserts that they are. This
is the specific claim that the separability test either confirms or defeats. If
dissipationless transport and full field expulsion always appear and disappear
together in every material and every geometry, the present account's claim to
logical distinctness has no observable grip. If they can be separated even in
principle, the present account's identification has a structure that the
standard treatment does not share. The search for candidate systems is
therefore not an optional empirical footnote; it is the primary test of the
interpretive claim made in Sections 4 and 5.
Standard theory is not being replaced. Maxwell field theory,
exchange interaction and band magnetism, spin-orbit coupling, domain theory and
coercivity, the London response and Josephson effect, BCS pairing and the gap
equation, and the spin-group symmetry analysis of altermagnetism are all used
above and contradicted nowhere. No numerical field equation is replaced and no
prediction of standard electrodynamics is contradicted. The operational
relation by which a current produces a magnetic field—the Biot-Savart law—is
retained in full; the assertion is not that this relation fails, but that the
field so produced is not, at the level of what exists, an entity independently
emitted by the current-carrying body. What the account supplies is a structural
identification and the consequences that follow.
5. External Writing, Empirical Consequences, and Limits
The four-stage structure answers the lodestone question in
precise terms. Ordinary magnetite possesses stage-two readouts—the 3d states of
Fe²⁺ and Fe³⁺ supply the microscopic magnetic degrees of freedom—and a
stage-three ferrimagnetic arrangement arising from its two inequivalent
magnetic sublattices. What ordinary magnetite does not possess is a macroscopic
persistent stage-three expression: the domain arrangement has not been written
into a single preferred orientation. Lodestone is magnetite with the same two
prior stages, plus a third condition: the domain arrangement has been written
by a transient external event into a macroscopic preferred orientation, and the
high-coercivity microstructure of the magnetically hardened precursor has
retained it.
This reformulation makes the selectivity of the lightning
mechanism precise without requiring new physics. Not every iron-bearing mineral
becomes a lodestone when struck. Quartz lacks the relevant local magnetic
degrees of freedom, so there is nothing at stage two for the event to arrange.
A sufficiently soft magnetite state lacks the stage-three retention condition:
its coercivity is too low to preserve the field-written domain arrangement
after the transient event has ended. The magnetically hardened precursor has
both the stage-two degrees of freedom and the stage-three retention structure;
the transient event writes the macroscopic orientation; the material holds it. The
entire sequence—material selectivity, event-driven writing,
microstructure-dependent retention—is accounted for by the four-stage structure
without any modification of the standard coercivity and domain physics that
describes the writing and retention mechanics.
What the four-stage structure adds is an account of what is
written. Standard domain and coercivity physics describes how the domain
arrangement changes under an applied field and why the change is retained or
lost depending on the microstructure. It does not address why the quantity
arranged—the quantity that is concordant within one domain, anti-concordant
between two antiparallel domains, and globally summed to give the remanent
moment—has the directional character it has. On the present account, that quantity
is a stage-two readout of a stage-one relation, and the directional character
it has is the character of the relation it registers. The external event does
not create the directional character; it selects which macroscopic expression
of an already-present directional relation is retained.
Falsification and discriminating tests.
Four conditions follow, in decreasing order of experimental
tractability.
The sharpest discriminating test concerns the separability
of dissipationless transport and magnetic-field expulsion. Standard
superconductivity describes both within the same condensate electrodynamics.
The present account assigns them to two distinct consequences of relational
locking—one tracking the availability of dissipative rearrangement channels,
the other tracking the availability of independently rearrangeable
configurational degrees of freedom. A reproducible bulk state with
experimentally established dissipationless transport but without the
corresponding equilibrium magnetic-field expulsion would falsify the
identification of expulsion with locking; robust bulk flux expulsion coexisting
with reproducible finite bulk dissipation would falsify it from the other side.
Candidate settings for the search include strongly disordered superconductors
near the superconductor-insulator transition, granular systems where bulk
transport and bulk expulsion probe different length scales, and any regime
where a diamagnetic response persists on one side of the resistance transition
while the other property is absent. Observations of such regimes are
conventionally attributed to inhomogeneity and to fluctuation effects; the test
is whether any separation survives once those effects are controlled for.
Decidable with existing technique.
The second test concerns the binary character of the
intrinsic readout under Class B operations. A Class B transformation—point-group
operation, strain, reorientation, change of reference frame—may change the
representation or the spatial arrangement of the readout but cannot generate a
new intrinsic value of the binary spectrum, because the material domain does
not supply the values. A demonstration that a specific Class B transformation
generates a third intrinsic spin eigenvalue at a single site under a single
observable—not a rearrangement of binary readouts distributed over Class B
geometry but a genuinely new intrinsic value—directly falsifies the Class A /
Class B distinction and with it the entire account. This is a logical falsifier
rather than an active experimental programme, since the two-dimensional
structure of a spin-1/2 degree of freedom is among the most firmly established
results in physics. The corresponding experimental expectation is that in
patterned-compensation materials the symmetry operations generating the
altermagnetic directional pattern will reorganise the arrangement while leaving
the local binary spectrum unchanged, a prediction directly testable by
spin-resolved photoemission across different crystal orientations.
The third test concerns composite pairing in conventional
electronic superconductors. A conventional electronic superconducting
condensate—BCS in character, fermionic carriers—whose experimentally
established microscopic charge-carrying unit is demonstrably non-composite
would falsify the relational argument that the minimum carrier for complete
relational coincidence is a paired composite. The restriction to conventional
electronic superconductors is deliberate: charged-boson condensation and
excitonic mechanisms are not within the scope of the argument made in Section
4, and this condition does not claim they are. Decidable with existing
technique.
The fourth condition is constrainable but not decidable in
the strong ontological sense, and this limit is stated plainly. The claim that
the two-valued readout axis is not exhaustively determined by Class B geometry—that
there is a Class A residue—cannot be proved absent by any finite measurement
programme, only bounded progressively more tightly. Experimental evidence can
raise the constraint on any Class A contribution that manifests differently
from what Class B structure alone would predict, but it cannot close the
ontological question. What can be falsified is the account's claim to
explanatory necessity: if all observable consequences of Class A can be
reproduced within a purely Class B framework, the identification becomes idle
even if not formally refuted. The account locates its explanatory content in
the conditions above, and above all in the separability test.
A related empirical expectation concerns the writing process
itself. In transient high-field writing on a magnetically hardened precursor,
the acquired remanence direction should be set entirely by the geometry of the
external event, with no dependence on the sample's pre-existing weak remanence
beyond its coercivity. This expectation is largely shared with standard domain
physics; where the present account is more specific is in the limiting case,
expecting the prior weak remanent arrangement to become negligible once the
writing field substantially exceeds the relevant coercive scale, since on this
reading what is overwritten is an arrangement and nothing else.
The formal limitation, recorded. The account is not
yet expressed as a quantitative field theory. It supplies no equation from
which the magnitude of a magnetisation may be computed, no functional relating
cancellation completeness to observable field strength, and no governing law for
the stage-three arrangement. What the account supplies is a structural
identification of what the state variables of standard magnetic theory are
variables of, and the specific empirical consequences that follow from taking that
identification seriously.
6. Conclusion
Three descriptions, one relation.
Magnetic order describes how readouts are arranged.
Ferromagnetism, antiferromagnetism, ferrimagnetism, and altermagnetism are four
ways one binary readout can be distributed over a material geometry, and the
four phases they organise are not four independent mechanisms but four
arrangements of one relation, differing in how the distribution handles its own
cancellation—through concordance, strict alternation, incomplete sublattice
compensation, or symmetry-enforced patterned compensation. Where the arrangement
lacks persistence, as in the paramagnetic regime, the readouts remain and the
appearance does not, which is the clearest evidence that appearance is a fact
about stage-three arrangement rather than about stage-two readouts.
Superconductivity describes what happens when the readouts
coincide with the relation rather than with each other. Within the proposed
interpretation, the minimal carrier of that coincidence is a composite, because
a single registration does not constitute a complete relational coincidence;
and the independently rearrangeable configuration through which dissipative response
proceeds is absent because the carriers are no longer in a mutually arranged
state for it to act on. Magnetic-field expulsion describes what an external
field can no longer independently rearrange once the coincidence is
established: the condensate couples to the electromagnetic field, as London
electrodynamics, flux quantisation, and the Josephson effect require; what it
no longer offers is a bulk magnetic configuration that an externally applied
field can rearrange, where the transport case concerns rearrangement internal
to the carriers themselves. These are two consequences of one locking, not two
expressions of one consequence, and the separability test of Section 5 is the
paper's sharpest experimental handle on the distinction.
The lodestone with which this paper began is not a curiosity
appended for illustration. It is the ontological question the standard account
does not formulate: why is the directional quantity that coercivity retains one
rather than many? Domain physics explains the mechanics of writing and
retention. The present account proposes an answer to the prior question: the
quantity being written and retained is an arrangement of readouts of one common
relation, and the sites are collectively writable because they share that
relation. A transient event does not manufacture alignment among independent
quantities. It supplies the condition under which a relation already common to
all sites becomes macroscopically expressed. What the high-coercivity
microstructure retains is the expression.
Matter does not generate the underlying directional relation
itself. Under specific conditions matter re-coincides with the relation from
which it was projected, and what has been called the magnetic field is what
remains visible when that coincidence is incomplete.
Statements and Declarations
Funding: No funding was received for conducting this
study.
Competing interests: The author has no competing
interests to declare that are relevant to the content of this article.
Data availability: No datasets were generated or
analysed during the current study; all results are analytical and interpretive
and follow from the arguments presented in the manuscript.
Author contributions: Juliet Zhong: conceptualisation, methodology, formal analysis, investigation, writing.
Declaration of generative AI and AI-assisted technologies in
the manuscript preparation process: During the preparation of
this work, the author used Claude (Anthropic) and ChatGPT (OpenAI) for
assistance with drafting text, searching and formatting references, and
language editing. The author reviewed and edited the output as needed and takes
full responsibility for the content of the published article.
References
Abrikosov,
A. A. (1957). On the magnetic properties of superconductors of the second
group. Soviet Physics JETP, 5, 1174–1182.
de Andrade
Nunes, M., Trindade, R. I. F., Salminen, J., Albrecht, R. I., & Silva Neto,
A. (2026). Insights on lightning-induced remanent magnetization from
high-current impulse experiments. Journal of Geophysical Research: Solid Earth,
131, e2025JB033096. doi:10.1029/2025JB033096
Bardeen, J.,
Cooper, L. N., & Schrieffer, J. R. (1957). Theory of superconductivity.
Physical Review, 108, 1175–1204. doi:10.1103/PhysRev.108.1175
Brekke, B.,
Brataas, A., & Sudbø, A. (2023). Two-dimensional altermagnets:
Superconductivity in a minimal microscopic model. Physical Review B, 108,
224421. doi:10.1103/PhysRevB.108.224421
Chakraborty,
D., & Black-Schaffer, A. M. (2024). Zero-field finite-momentum and
field-induced superconductivity in altermagnets. Physical Review B, 110,
L060508. doi:10.1103/PhysRevB.110.L060508
Cooper, L.
N. (1956). Bound electron pairs in a degenerate Fermi gas. Physical Review,
104, 1189–1190. doi:10.1103/PhysRev.104.1189
Curie, P.
(1895). Propriétés magnétiques des corps à diverses températures. Annales de
Chimie et de Physique, 5, 289–405.
Deaver, B.
S., & Fairbank, W. M. (1961). Experimental evidence for quantized flux in
superconducting cylinders. Physical Review Letters, 7, 43–46.
doi:10.1103/PhysRevLett.7.43
Ding, J., et
al. (2024). Large band splitting in g-wave altermagnet CrSb. Physical Review
Letters, 133, 206401. doi:10.1103/PhysRevLett.133.206401
Doll, R.,
& Näbauer, M. (1961). Experimental proof of magnetic flux quantization in a
superconducting ring. Physical Review Letters, 7, 51–52.
doi:10.1103/PhysRevLett.7.51
Heisenberg,
W. (1928). Zur Theorie des Ferromagnetismus. Zeitschrift für Physik, 49,
619–636. doi:10.1007/BF01328601
Jiang, B.,
et al. (2025). A metallic room-temperature d-wave altermagnet. Nature Physics, 21,
754–759. doi:10.1038/s41567-025-02822-y
Josephson,
B. D. (1962). Possible new effects in superconductive tunnelling. Physics
Letters, 1, 251–253. doi:10.1016/0031-9163(62)91369-0
Krempaský,
J., et al. (2024). Altermagnetic lifting of Kramers spin degeneracy. Nature,
626, 517–522. doi:10.1038/s41586-023-06907-7
Langevin, P.
(1905). Magnétisme et théorie des électrons. Annales de Chimie et de Physique,
5, 70–127.
London, F.,
& London, H. (1935). The electromagnetic equations of the supraconductor.
Proceedings of the Royal Society A, 149, 71–88. doi:10.1098/rspa.1935.0048
Maxwell, J.
C. (1865). A dynamical theory of the electromagnetic field. Philosophical
Transactions of the Royal Society of London, 155, 459–512.
doi:10.1098/rstl.1865.0008
Mazin, I. I.
(2025). Notes on altermagnetism and superconductivity. AAPPS Bulletin, 35, 18.
doi:10.1007/s43673-025-00158-6
Meissner,
W., & Ochsenfeld, R. (1933). Ein neuer Effekt bei Eintritt der
Supraleitfähigkeit. Naturwissenschaften, 21, 787–788. doi:10.1007/BF01504252
Néel, L.
(1948). Propriétés magnétiques des ferrites: ferrimagnétisme et
antiferromagnétisme. Annales de Physique, 3, 137–198.
doi:10.1051/anphys/194812030137
Sigrist, M.,
& Ueda, K. (1991). Phenomenological theory of unconventional
superconductivity. Reviews of Modern Physics, 63, 239–311.
doi:10.1103/RevModPhys.63.239
Šmejkal, L.,
Sinova, J., & Jungwirth, T. (2022a). Beyond conventional ferromagnetism and
antiferromagnetism: A phase with nonrelativistic spin and crystal rotation
symmetry. Physical Review X, 12, 031042. doi:10.1103/PhysRevX.12.031042
Šmejkal, L.,
Sinova, J., & Jungwirth, T. (2022b). Emerging research landscape of
altermagnetism. Physical Review X, 12, 040501. doi:10.1103/PhysRevX.12.040501
Stoner, E.
C. (1938). Collective electron ferromagnetism. Proceedings of the Royal Society
A, 165, 372–414. doi:10.1098/rspa.1938.0066
Stoner, E.
C., & Wohlfarth, E. P. (1948). A mechanism of magnetic hysteresis in
heterogeneous alloys. Philosophical Transactions of the Royal Society A, 240,
599–642. doi:10.1098/rsta.1948.0007
Wasilewski,
P., & Kletetschka, G. (1999). Lodestone: Nature's only permanent magnet—what
it is and how it gets charged. Geophysical Research Letters, 26, 2275–2278.
doi:10.1029/1999GL900496
Wasilewski,
P., & Kletetschka, G. (2023). Iron ore to lodestone: With lightning assist.
Journal of Applied Geophysics, 219, 105225. doi:10.1016/j.jappgeo.2023.105225
Weiss, P.
(1907). L'hypothèse du champ moléculaire et la propriété ferromagnétique.
Journal de Physique Théorique et Appliquée, 6, 661–690.
doi:10.1051/jphystap:019070060066100
Zhang,
S.-B., Hu, L.-H., & Neupert, T. (2024). Finite-momentum Cooper pairing in
proximitized altermagnets. Nature Communications, 15, 1801.
doi:10.1038/s41467-024-45951-3
Internal references
Zhong, J.
(2026a). Ripple-Instantiation Cosmogenesis [Part I]: The six-dimensional
spherical cascade as an alternative to temporal assembly. SSRN.
doi:10.2139/ssrn.6753518
Zhong, J.
(2026b). Gravity as an emergent effect of projected radiation. SSRN.
doi:10.2139/ssrn.7251340
Zhong, J.
(2026c). The boundary between matter and radiation: An unresolved
classificatory inconsistency in modern physics. SSRN. doi:10.2139/ssrn.7186058
Zhong, J.
(2026d). The two times of the universe: Deriving the radiation–matter
dimensional separation from Newton and Einstein. SSRN. doi:10.2139/ssrn.7197481
Zhong, J.
(2026e). The nucleus that returns every signal: A six-dimensional Riemannian
string background. SSRN. doi:10.2139/ssrn.7419918
Zhong, J.
(2026f). The observer and spacetime: On the existential precondition of
physical theory. SSRN. doi:10.2139/ssrn.7316678
Zhong, J.
(2026g). The observer and consciousness: On the ontological ground of
measurement. SSRN. doi:10.2139/ssrn.7406418
Zhong, J.
(2026h). Symmetry-constrained pairing and finite-momentum superconductivity in
altermagnetic systems. SSRN. doi:10.2139/ssrn.7130138
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