Natural Ratios as the Invariant Foundation of Physical
Description:
J. Rogers, SE Ohio
Abstract
Physical reality is unified. The plurality of physical laws — E =
mc², E = hf, λ = h/p, and a dozen others — arises not from independent
facts about the universe but from the fragmentation of a single unified
substrate S_u into multiple conceptual measurement axes. Once the
invariant dimensionless ratios of that substrate are identified, the
laws of physics reduce to a single tautology: X = X. This paper argues
that any description of the universe — whether expressed in the language
of information theory, string theory, loop quantum gravity, geometric
algebra, or any other coherent formalism — is equally valid provided it
preserves these natural ratios. The criterion for a valid physical model
is not ontological priority but coherence: does the description preserve
the invariant structure of S_u under projection? No model is the
territory. Every model is a viewport. The natural ratios are the only
invariant that all valid viewports share.
1. The Foundational Separation
1.1 What Is and How We Describe It
The central error in the foundations of physics has been the
conflation of physical reality with our description of physical reality.
Reality — the unified substrate S_u — exists prior to any measurement,
any axis, any unit, any model. It is a single coherent thing, presenting
dimensionless relations.
Everything above S_u is constructed by us. The conceptual axes we
call Mass, Length, Time, Temperature, Frequency — these are not
independent substances in nature. They are coordinate directions imposed
by observers to organize experience. The unit systems (SI, CGS, Planck)
are further choices layered on top of those axes. The physical constants
c, h, G, k_B are the Jacobian factors forced into existence by the
misalignment between our independently-chosen axes.
Once this separation is made cleanly, a profound consequence follows:
the description is ours, but the invariant ratios underneath it are not.
The natural ratios — the dimensionless X values that survive all unit
transformations — are the only part of the description that points
directly at S_u.
1.2 The Natural Ratios
The equivalence chain expresses this directly. For any physical
state, every measurement axis returns the same dimensionless value X
when the unit scaling is removed:
T/T_P = f·t_P = m/m_P = l_P/λ = E/E_P = p/p_P =
X
X is not a ratio involving Planck units in any fundamental sense —
the Planck units are themselves just the SI labels placed on the natural
crossing point where all reciprocal axes simultaneously equal unity. X
is the physical reality. The measurements are projections of X through
our chosen axes, each scaled by its Jacobian component.
The laws of physics are not independent discoveries. They are the
fifteen pairwise projections of X = X through six axes, taken two at a
time. Each law carries Jacobian factors — the physical constants — whose
sole function is to convert between the arbitrary scaling of one human
axis and another. Remove the Jacobians and every law collapses back to X
= X.
2. Measurement as Projection
2.1 What Measurement Actually Is
Measurement is not a neutral transparent operation. It is a
projection. When we measure a physical quantity, we are projecting X —
the invariant physical reality — onto a particular conceptual axis,
scaled by that axis's Jacobian component. The number we obtain is not
the physics. It is X multiplied by a Planck unit.
measurement = X · (Jacobian for that axis)
The Jacobian is not discovered — it is chosen, when we choose our
unit system. The physics, X, was already there. We added the scaling.
When we write a formula relating two measurements and include a
constant, we are not encoding new physical information. We are undoing,
on one side, what we added on both sides when we chose our units.
2.2 Constants as Jacobian Components
The constants are defined by the Jacobians, not the reverse. Each
constant is the ratio of Planck units — the SI expansion of the scaling
factor for a pair of axes:
c = l_P / t_P (Space-Time axis ratio)
h = m_P · l_P² / t_P (Energy-Frequency axis ratio)
k_B = m_P · l_P² / (t_P² · T_P) (Energy-Temperature axis ratio)
G = l_P³ / (t_P² · m_P) (Geometry-Mass axis ratio)
When c was fixed in the SI system, the meter and second were fixed
relative to each other. When h was fixed, the kilogram was defined — m_P
was determined given l_P and t_P already set. When k_B was fixed, the
Kelvin was defined. G follows as a consequence of all prior fixings: its
numerical value is not measured from nature but is determined by the
accumulated Jacobian choices already made. The 2019 SI redefinition was,
formally, a specification of the Jacobian components. The constants do
not have their values because the universe chose them. They have their
values because we chose our axes.
3. The Criterion for Valid Physical Models
3.1 No Model Is the Territory
S_u is not directly accessible. It has no units, no scale, no axes,
no internal structure imposed from outside itself. Any description of it
— any physical model — is a projection: a coherent mapping from the
natural ratios of S_u into some representational system chosen by the
observer.
This is not a limitation of current physics. It is a structural
necessity. Kant identified S_u as the noumenon — the thing-in-itself,
forever behind the phenomenon. The Grothendieck fibration framework
formalizes this: S_u is the terminal object of the base category of
conceptual types, and every measurement is a Cartesian lifting from that
terminal object through a chosen fiber. The observer never touches S_u
directly. The observer always touches a projection.
3.2 Coherence as the Validity Criterion
Given that no model is the territory, the question of which model is
the 'correct' or 'true' description of reality is malformed. The
meaningful question is: does the model preserve the natural ratios? Is
it a coherent projection of S_u?
A model is a valid description of the universe if and only if:
1. It correctly encodes the invariant dimensionless ratios X.
2. Its internal structure is coherent — it does not contradict
itself.
3. It reduces back to X = X under the removal of its representational
scaffolding.
These criteria do not privilege any particular representational
system. They do not demand spacetime, or particles, or fields, or
information bits, or strings. They demand only that the representational
system faithfully carry the invariant ratios of S_u into whatever
language it speaks.
4. Projections Into Different Models
4.1 The Standard Model and Quantum Field Theory
The Standard Model projects S_u into the language of quantum fields,
gauge symmetries, and particle excitations. It is a highly successful
projection — it encodes the natural ratios with extraordinary precision.
The constants appearing in its Lagrangian are Jacobian components in
disguise. The dimensionless coupling constants — the fine structure
constant α ≈ 1/137, the mass ratios of particles — are genuine invariant
content, genuine X values. The dimensionful constants are unit scaling
artifacts. The model is valid not because it is the true description but
because it coherently preserves the invariant ratios it claims to
describe.
4.2 General Relativity
General Relativity projects S_u into the language of spacetime
geometry. The metric tensor encodes how the time gradient field I = m/r
varies across the universe. Geodesics are the natural flows of that
field. Felt force is deviation from those flows. The constants G and c
in Einstein's field equations are Jacobian components — G encodes the
Mass-to-geometry axis ratio, c encodes the Space-Time axis ratio. In
natural ratios, the field equations simplify to a statement about how
the I field curves the description space. The model is valid because it
coherently carries the invariant structure of how mass distorts time
ratios.
4.3 Information-Theoretic Models
An information-theoretic model projects S_u into the language of
bits, qubits, entropy, and computation. This is equally valid provided
the model encodes the correct natural ratios. The Bekenstein-Hawking
entropy formula, for instance, is in natural ratios a statement about
the dimensionless ratio of a black hole's area to the Planck area — a
pure X value. The appearance of constants in the formula is unit
scaling. Strip the scaling and you have a dimensionless ratio that any
coherent information-theoretic model must reproduce if it is to be a
valid description of that physical state.
4.4 String Theory and Loop Quantum Gravity
String theory projects S_u into the language of one-dimensional
extended objects vibrating in higher-dimensional spaces. Loop quantum
gravity projects it into discrete networks of spin-foam. Both are
coherent representational systems to the extent that they preserve the
natural ratios. The apparent conflict between these frameworks is not a
conflict about S_u — it is a conflict between projections. S_u does not
prefer strings over loops. The question is purely whether each
framework, in its own language, correctly encodes X.
Critically: neither framework should introduce new constants whose
values are not determined by the natural ratios. Any free parameter with
a dimensionful value is a signal of unit scaling not yet eliminated. The
presence of such parameters is diagnostic — it means the model has not
yet fully reduced to the natural ratios it claims to describe.
4.5 The I Field as a Natural-Ratio Model
Working directly in natural ratios, a unified picture emerges from
the dimensionless quantity I = m/r — the local time gradient set by mass
m at distance r, expressed as a pure ratio. In this language:
Gravitational interaction: F_nat = I₁ · I₂
Velocity-gravity unification: β² = 2I
Lorentz factor: γ = 1/√(1 - 2I)
GPS correction: M / (1/r_earth + 3/(2r_sat))
No constants appear because no unit scaling has been introduced. The
same quantity I that describes gravitational potential also describes
velocity, time dilation, and orbital dynamics — because these were
always descriptions of the same thing. The I field is not a new theory.
It is what becomes visible when the Jacobian epicycles are removed.
5. What Cannot Vary and What Can
5.1 Constants Cannot Vary
A universe with a different value of c is not a different universe.
It is the same observer with a different ruler. c = l_P/t_P is
determined entirely by the choice of length and time units. Transport
any coherent unit system to any coherent universe and c will take the
value fixed by those unit choices. The fine-tuning literature — which
asks what would happen if c, G, or h were different — is asking what
would happen if we redefined our axes. The answer is: the equations
would look different, but the physics, X, would be unchanged.
This can be made vivid: if c, h, and k_B were set to have the digits
of √2, 2Ï€, and e respectively — which is formally permissible given that
all three are now set by convention — the equations of physics would
become opaque with irrational transcendental mantissas. Physicists would
search for deep meaning in numbers that carry no physical information
whatsoever. The current mantissas of our constants are less dramatic but
equally arbitrary.
5.2 Natural Ratios Can Vary
What can genuinely vary between physical states, between regions of
the universe, or in principle between different universes, are the
dimensionless ratios themselves — the X values. The proton-to-electron
mass ratio (~1836), the fine structure constant (~1/137), the ratio of a
black hole's mass to its Hawking temperature, the density of the
universe relative to critical density — these are genuine invariant
physical content.
A denser universe has different dimensionless ratios between physical
states. A younger universe has a different global I field. These are
physically meaningful statements about X. They are not statements about
constants, which remain purely conventional.
6. Time, Observation, and the Eternal Now
6.1 Time Is Not a Dimension
The framework dissolves the mystery of time. Time is not a dimension
through which the universe moves. It is the rate of self-interaction of
S_u — the local I field setting how fast each part of the universe
processes the next interaction. There is no block universe of eternally
co-existing past, present, and future. There is one eternal now: S_u in
the act of updating itself.
What we call the past is a log of interactions — physical patterns
encoded in the current state of the universe, shaped by previous
updates. Memory is not access to a past that still exists. It is a
pattern in the present that was written by prior interactions. What we
call the future is pattern-matching on that log — extrapolation, not
access to something that already exists.
The arrow of time is not thermodynamic. It is structural: interaction
is irreversible. You cannot un-interact. The universe cannot unwrite a
log entry. This is not a statistical tendency — it is the definition of
what interaction is.
6.2 Observers as Log-Reading Processes
An observer is a part of the universe that maintains sufficient
internal structure to store and read the interaction log. We do not see
the universe. We model it — inside a physical system that is itself part
of S_u, using log entries written by prior interactions, running
predictions about the next update.
This connects directly to the fragmentation of S_u into axes. The
axes are not features of S_u. They are features of the model the
observer runs. The constants are not facts about the universe. They are
facts about the model's coordinate system. The observer's model imposes
fragmentation on a unified substrate, then constructs constants to
repair the damage — all while believing it is discovering facts about
nature.
7. The Plurality of Valid Models
7.1 The Model Wars Are Viewport Wars
The conflicts between competing physical frameworks — strings versus
loops, fields versus geometry, continuous versus discrete — are not
conflicts about S_u. They are conflicts about which representational
language to project S_u into. S_u does not prefer any language. The
universe has no opinion about whether we describe it with differential
geometry or spin networks.
This does not mean all models are equally useful. A model may be more
or less computationally tractable, more or less intuitive, more or less
complete in the range of X values it can encode. These are pragmatic
criteria. But ontological priority — the question of which model is
really true — is a category error. No projection is S_u. Every
projection is a viewport.
7.2 The Coherence Test
Given the framework, we can now state the coherence test for any
proposed physical model precisely. The model must:
1. Reproduce the equivalence chain: T/T_P = f·t_P = m/m_P = l_P/λ =
E/E_P = p/p_P in appropriate translation to its own language.
2. Contain no free dimensionful parameters whose values are not
determined by the natural ratios.
3. Reproduce the 15 pairwise projections — or their analogs — as
necessary consequences of its structure, not as independent
postulates.
4. Treat its own representational scaffolding — whatever corresponds
to axes and units in its language — as conventional, not
ontological.
A model that passes these tests is a valid description of the
universe. Not the true description. A valid description. The distinction
matters. Truth is reserved for X = X. Models are projections of that
truth.
8. Conclusion
The central claim of this paper is simple. Once the natural ratios of
the unified substrate S_u are correctly identified — once X is isolated
from the unit scaling and Jacobian artifacts that have obscured it — any
coherent model that faithfully projects those ratios is a valid
description of the universe.
This reframes the project of theoretical physics. The goal is not to
find the one true model — the final theory that is the universe. No
model can be the universe. The goal is to find models that are maximally
coherent, maximally complete in the X values they encode, and maximally
honest about what is representational scaffolding and what is invariant
content.
The constants are not fundamental. They are Jacobians — the price of
fragmentation. The laws are not independent. They are tautologies —
projections of X = X. Time is not a dimension. It is a rate — the
self-interaction of S_u. Observers are not external to the universe.
They are log-reading processes within it, modeling S_u from inside using
the only tools available: the patterns written by prior
interactions.
The universe is one thing. We invented many ways to describe
it. All coherent descriptions are valid. None are final. The natural
ratios are the only ground truth any description can stand
on.
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