The Speed of Light and Domain Boundary Velocities

The revised cosmic image is an AI-edited conceptual illustration, not measured data. It shows the relational configuration domain, a rip representing a persistent mapping defect, a narrow quantum epoch, and the expansionary illustration. A mapping obstruction is not a tear in pre-existing physical space. The quantum epoch precedes the big bang in MQ structural ordering, not elapsed time on an external clock. The quantum epoch is a pre-referenceable structural relaxation regime.

Domain allocation

Omega denotes a normalized fixed geometric volume fraction. Subscripts vis, obs, uobs, dk and tot identify visible, observable, unobserved, dark and total. The visible portion is presently referenceable. The unobserved remainder belongs to the same observable domain; it is not an additional domain outside it. The dark complement lies outside the observable allocation. MQ domain fractions are not independently fitted cosmological density parameters.

Ω_{uobs} = Ω_{obs} - Ω_{vis}

Unobserved equals observable minus visible.

Ω_{obs} = Ω_{vis} + Ω_{uobs}

Observable equals visible plus unobserved.

Ω_{dk} = 1 - Ω_{obs}

Dark equals one minus observable.

Calculated fractions for the supplied coefficient are visible 4.8488348909 percent, unobserved 26.7887489867 percent, observable 31.6375838775 percent, and dark 68.3624161225 percent. The bar uses these volume fractions. They are not radii, and the artwork is not calibrated to them.

Local step and conditional rates

Theta with italic subscript si is the dimensionless invariant coefficient. It is supplied as 3.26239030504, not independently derived here. Fundamental length l with subscript f and fundamental time t with subscript f are realized measures. The symbol c is the speed of light in vacuum, exactly 299792458 metres per second in SI. R with subscript obs denotes the observable reference radius; delta denotes its change during one common elapsed-time increment t. The local axiom advances this boundary by one fundamental length during one fundamental time step, excluding subsequent metric expansion.

Δ R_{obs} = l_{f}, Δ t = t_{f}

The observable-radius change is one fundamental length; the elapsed-time increment is one fundamental time.

v_{vis} = c \sqrt[3]{\frac{1}{2 θ_{s i}}}

The visible reference rate is c times the cube root of one divided by twice theta. The calculated value is 160,434,857.091 metres per second, rounded.

v_{obs} = \frac{l_{f}}{t_{f}} = c

The observable reference rate is fundamental length divided by fundamental time, equal to c. The calculated value is 299,792,458.000 metres per second, rounded.

v_{tot} = c \sqrt[3]{\frac{θ_{s i}^{2} + 2}{4}}

The total reference rate is c times the cube root of theta squared plus two divided by four. The calculated value is 439,966,489.984 metres per second, rounded.

The letter v identifies each incremental reference-radius rate. Constant fractions represented by concentric Euclidean spheres imply radius ratios equal to cube roots of the corresponding volume ratios. Requiring the same ratios at both ends of a common step gives proportional radial increments. This extra construction transfers the adopted observable normalization to the other radii. It does not identify observers’ distinct proper times, determine a global proper horizon speed or independently measure expansion. Digits are computational, not experimental error bounds. Total reference extent may have no physical outer boundary, and its rate above c is not a local signal velocity.

Shell thickness changes are differences of bounding-radius rates. Unobserved gives observable minus visible, 139,357,600.909 metres per second. Dark gives total minus observable, 140,174,031.984 metres per second. A cube root of an isolated shell-volume fraction would describe an equivalent-volume ball, not either shell boundary.

Sources

[1] Jody A. Geiger, Measurement Quantization: A First Principles Origin of the Universe, Informativity Institute public preprint, 2026, Section 4.0.4 and Appendix E. Public source https://www.informativity.org/origin-of-the-universe-preprint . [2] Jody A. Geiger, A First Principles Derivation of the Galactic Radial Acceleration Relation from Measurement Quantization, Informativity Institute public preprint, 2026, Section 3.9, equations 79–81. Public source https://www.informativity.org/dark-matter-galactic-rotation-curves-from-measurement-quantization-preprint . The coefficient is supplied as input. No independent empirical velocity comparison is asserted.

The Speed of Light and the Observable Universe

A local advance of one fundamental length during one fundamental time increment fixes a propagation scale. When the MQ domain fractions are additionally represented by concentric Euclidean spheres whose radii grow proportionally, that scale determines conditional rates for the visible radius and the total reference radius. The derivation below separates the local boundary rule from this geometric construction and from the subsequent expansion of space. A further conditional construction examines how directional boundary observations could constrain an observer’s offset within a specified spherical domain.

The local boundary rule

Let $l_{f}$ denote the realized fundamental length, $t_{f}$ the realized fundamental time, and $c$ the speed of light in vacuum. Let $R_{obs}$ denote the observable reference radius. The symbol $Δ$ denotes the change over one step, and $t$ denotes the common elapsed-time parameter used for that step. The proposed axiom states that the observable boundary advances by one fundamental length relative to its immediately preceding local position during one fundamental time increment. The displacement excludes the additional change caused by subsequent metric expansion.

Δ R_{obs} = l_{f}, Δ t = t_{f}

Define $v_{obs}$ as this incremental displacement divided by the step duration. The MQ fundamental-measure relation $t_{f} = \frac{l_{f}}{c}$ then gives

v_{obs} = \frac{Δ R_{obs}}{Δ t} = \frac{l_{f}}{t_{f}} = c

This specifies an advance relative to a previous local position. It does not assign a nonzero velocity to a boundary relative to itself at the same event, nor require a physical observer to ride on a lightlike boundary. The axiom selects which boundary increment is normalized by the fundamental-measure relation; that selection is an additional assumption.

For numerical evaluation, $c = 299792458 m/s$ is exact in SI.^[1]

The MQ fractions and their spherical realization

In MQ usage, the visible domain is presently referenceable, the observable domain includes the visible domain and what can become referenceable with further elapsed time, and the total domain includes their dark complement. These meanings should not be silently exchanged with the conventional cosmological particle horizon. The symbols $Ω_{vis}$ , $Ω_{obs}$ and $Ω_{tot}$ denote the corresponding normalized domain fractions. The dimensionless MQ invariant coefficient $θ_{s i}$ is supplied here as an input with value $3.26239030504$ . It is not independently derived by this calculation.

The adopted fraction relations appear in Section 3.9, equations (79)–(80), of Jody A. Geiger’s preprint “A First Principles Derivation of the Galactic Radial Acceleration Relation from Measurement Quantization.” The normalized total is unity.^[2]

Ω_{obs} = \frac{4}{θ_{s i}^{2} + 2}

Ω_{vis} = \frac{2}{θ_{s i} (θ_{s i}^{2} + 2)}

$Ω_{tot} = 1$ . These are MQ referenceability fractions. They are not being identified with fitted matter or dark-energy density parameters.

To turn fractions into radius ratios, impose the additional geometry used on this page. Represent the visible, observable and total reference domains as concentric Euclidean balls with constant volume fractions relative to one finite total reference volume. Let $R_{vis}$ and $R_{tot}$ denote the visible and total reference radii. Because a Euclidean ball’s volume scales with the cube of its radius, the radius ratios are

\frac{R_{vis}}{R_{obs}} = \sqrt[3]{\frac{Ω_{vis}}{Ω_{obs}}} = \sqrt[3]{\frac{1}{2 θ_{s i}}}

\frac{R_{tot}}{R_{obs}} = \sqrt[3]{\frac{Ω_{tot}}{Ω_{obs}}} = \sqrt[3]{\frac{θ_{s i}^{2} + 2}{4}}

The Euclidean volume law is part of this construction. Spherical symmetry alone would not justify it for proper volumes in a spatially curved geometry. Neither the fractions alone nor the local light-speed rule establishes that actual cosmological boundaries realize these concentric balls.

From radius ratios to incremental rates

Now require the radius ratios to remain fixed across the same step, using the same radial convention and elapsed-time parameter for all three reference radii. Subtracting each fixed-ratio relation at the beginning of the step from the relation at its end gives the identical ratio for the radial increments. Define $v_{vis} = \frac{Δ R_{vis}}{t_{f}}$ and $v_{tot} = \frac{Δ R_{tot}}{t_{f}}$ as the corresponding reference-radius rates. Inserting the observable increment then yields

v_{vis} = c \sqrt[3]{\frac{Ω_{vis}}{Ω_{obs}}} = c \sqrt[3]{\frac{1}{2 θ_{s i}}}

v_{tot} = c \sqrt[3]{\frac{Ω_{tot}}{Ω_{obs}}} = c \sqrt[3]{\frac{θ_{s i}^{2} + 2}{4}}

This is a finite-step derivation; no replacement of the axiom by a global time derivative is needed. The common parameter transfers the observable normalization to the other radii. It does not equate distinct observers’ proper times. Nor does it require each reference-radius increment to be an integer number of fundamental lengths. Only the observable boundary is assigned the one-length step in the axiom.

Table 1 gives the numerical evaluation. Volume percentages refer to the same total reference volume, while the rate ratios are relative to the observable increment.

Table 1. Conditional reference-radius rates for the supplied invariant coefficient.
Reference domain	Volume (%)	Rate / c	Rate (m/s)
Visible	4.8488348909	0.535153079438	160,434,857.091
Observable	31.6375838775	1.000000000000	299,792,458.000
Total reference	100.0000000000	1.467570241491	439,966,489.984

The displayed digits are rounded computational results for the supplied decimal input, not measured velocities or experimental error bounds. No uncertainty for that input or for the spherical realization is assigned here. The observable rate inherits its value from the axiom; the other two rates inherit it from the additional geometry.

Unobserved and dark regions are shells

Let $Ω_{uobs}$ denote the unobserved fraction and $Ω_{dk}$ the dark fraction. The MQ partition relations are

Ω_{uobs} = Ω_{obs} - Ω_{vis}, Ω_{dk} = Ω_{tot} - Ω_{obs}

Within the concentric representation, the unobserved shell lies between the visible and observable radii. It is the remainder inside the observable domain, not an additional domain outside it. The dark shell lies between the observable and total reference radii. Their volume fractions evaluate to 26.7887489867% and 68.3624161225%, respectively. “Dark” here names the MQ referenceability complement and does not identify this shell with a conventional dark-matter fluid.

Each shell has an inner and an outer bounding radius. Its thickness is the difference of those radii, so its thickness-growth rate is the difference of their rates. For the unobserved shell this gives

\frac{Δ (R_{obs} - R_{vis})}{t_{f}} = v_{obs} - v_{vis}

The result is 139,357,600.909 m/s, or 0.464846920562 times c, after rounding. Its inner boundary has the visible rate and its outer boundary has the observable rate in Table 1.

For the dark shell the corresponding result is

\frac{Δ (R_{tot} - R_{obs})}{t_{f}} = v_{tot} - v_{obs}

This evaluates to 140,174,031.984 m/s, or 0.467570241491 times c, after rounding. Its inner boundary has the observable rate and its outer boundary has the total reference rate. These differences describe shell-thickness changes, not a unique velocity shared by all points of a shell.

Taking a cube root of an isolated shell-volume fraction would instead construct the radius of a filled ball with the same volume. That ball’s radius is neither the shell thickness nor either of its bounding radii. It therefore supplies no additional physical boundary speed.

Local propagation and global horizons

The local step excludes subsequent metric expansion. A cosmological horizon’s proper-radius growth also depends on the intervening geometry and requires an expansion history and a horizon relation. Particle horizons, event horizons and Hubble spheres differ. See Davis and Lineweaver’s “Expanding Confusion: Common Misconceptions of Cosmological Horizons and the Superluminal Expansion of the Universe,” Section 2 and Appendix A.^[3]

Can lookback time locate an observer?

The reference-radius construction suggests a further question. Could directional differences in how far back an observer sees determine that observer’s position within a larger domain? A conditional position estimator can be derived if an observer-independent spherical boundary is specified and can be identified observationally. The reference radii developed above do not themselves establish such a boundary or a global center.

A stationary spherical-boundary construction

For this auxiliary geometry, let $R_{b}$ denote the radius of a hypothetical spherical domain at the observation epoch and $r_{o}$ the observer’s distance from its center at that epoch, with $R_{b} > 0$ and $0 \leq r_{o} \leq R_{b}$ . These quantities are not identifications with the earlier MQ reference radii. Assume first that the sphere is stationary, its interior is transparent and Euclidean, and the same identifiable boundary has emitted light for long enough to be observed in opposite radial directions.

Define $t_{L, near}$ and $t_{L, far}$ as the lookback times to the nearest and farthest boundary points, respectively. The subscript $L$ labels lookback time. The corresponding path lengths are the sphere’s radius minus and plus the observer’s offset, giving

t_{L, near} = \frac{R_{b} - r_{o}}{c}, t_{L, far} = \frac{R_{b} + r_{o}}{c}

At the center, both lookback times equal $\frac{R_{b}}{c}$ . At the edge, the far-side travel time is $\frac{2 R_{b}}{c}$ and the near-side time is zero. If $X$ denotes the center-to-boundary travel time, so that $X = \frac{R_{b}}{c}$ , the far-side relation can also be written $t_{L, far} = X (1 + \frac{r_{o}}{R_{b}})$ . This is the precise stationary geometry behind the proposed change from one center-to-boundary travel time to twice that value.

Adding and subtracting the two travel-time expressions recovers the radius as $R_{b} = \frac{c (t_{L, far} + t_{L, near})}{2}$ and the offset as $r_{o} = \frac{c (t_{L, far} - t_{L, near})}{2}$ . Their ratio gives the dimensionless position estimator

\frac{r_{o}}{R_{b}} = \frac{t_{L, far} - t_{L, near}}{t_{L, far} + t_{L, near}}

The directional difference supplies the positional information. A single maximum lookback time cannot determine both the radius and the offset unless one is independently constrained. The oldest detected galaxies are also not automatically measurements of this hypothetical boundary. Their observed ages and visibility depend on when sources formed, emitted and became detectable.

Accounting for expansion

To extend this construction to an expanding geometry, the common time parameter $t$ must additionally be identified with cosmic time. Let $t_{0}$ denote its present value and $a (t)$ the positive dimensionless scale function, normalized by $a (t_{0}) = 1$ . Assume a prescribed spatially flat cosmological geometry in which the observer and the hypothetical spherical boundary remain at fixed comoving positions. Under this normalization, $R_{b}$ and $r_{o}$ also give their comoving radius and offset. Their present-distance meanings are retained.

Let $t_{near}$ and $t_{far}$ denote the emission times of the near-side and far-side boundary signals. The previously defined lookback times are then $t_{L, near} = t_{0} - t_{near}$ and $t_{L, far} = t_{0} - t_{far}$ . Define $χ_{near}$ and $χ_{far}$ as the corresponding comoving light-path distances. Null propagation gives

χ_{near} = \int_{t_{near}}^{t_{0}} \frac{c}{a (t)} d t

χ_{far} = \int_{t_{far}}^{t_{0}} \frac{c}{a (t)} d t

Comoving distance accounts for expansion along the photon’s path, whereas lookback time is the elapsed cosmic time since emission. These quantities are distinguished in David W. Hogg’s “Distance Measures in Cosmology,” Section 4, equation (14), and Section 10, equation (29).^[4] The integral construction requires an independently specified expansion history; the local fundamental-length step does not supply that history.

For the assumed comoving sphere, the geometric path lengths satisfy $χ_{near} = R_{b} - r_{o}$ and $χ_{far} = R_{b} + r_{o}$ . The expanding-geometry estimator is consequently

\frac{r_{o}}{R_{b}} = \frac{χ_{far} - χ_{near}}{χ_{far} + χ_{near}}

If the scale function remains unity throughout the photon journeys, each comoving path distance reduces to $c$ multiplied by its lookback time and the stationary estimator is recovered. Otherwise, doubling a present distance does not generally double its lookback time. Both boundary signals must also lie within the available emission and propagation history. An observer’s offset cannot make light arrive from an epoch before the relevant sources or boundary emitted it.

What would make the estimator observationally useful?

The observable requirement is an independently identified center and boundary, together with directional observations and a justified propagation model. An observer-centered referenceability horizon cannot independently locate that observer within a larger universe. The concentric volume fractions and conditional rates on this page establish neither a physical outer edge nor its observed near-side and far-side signals.

A possible extension is to derive the sky pattern predicted for different observer offsets in a specified cosmological realization and test it against directional observations. Håvard Alnes and Morad Amarzguioui’s “CMB anisotropies seen by an off-center observer in a spherically symmetric inhomogeneous universe” provides a published example of calculating such positional effects in particular inhomogeneous cosmologies.^[5] Those models demonstrate a method of investigation; they do not validate the spherical-boundary assumptions adopted here or the MQ construction.

The boundary-position relations are therefore auxiliary geometric consequences of the stated assumptions, not an independently established MQ localization law. No numerical position for our observer follows from the supplied coefficient and local boundary axiom alone.

What the calculation determines

Given the local step, constant volumetric fractions, Euclidean concentric geometry and proportional radial increments, the three rates follow uniquely for the supplied coefficient. This is a conditional mathematical consequence. It does not by itself determine a global proper horizon speed, the Hubble expansion rate, or an observed expansion history.

The invariant coefficient remains an input throughout. Solving the visible-rate relation backward after calculating that rate from the same coefficient only recovers the input. It cannot serve as an independent determination or a new test. An independent test would require an independently established observable boundary relation and a justified connection between that observable and these reference-radius increments, including metric expansion and uncertainty.

The observer-position extension identifies a different requirement. Directional measurements of an independently identifiable boundary, interpreted through a specified geometry and expansion history, could constrain an offset. The maximum observed lookback time by itself does not supply those conditions or determine our position.

Source notes

[1] Bureau International des Poids et Mesures. SI base unit, metre (m). Official definition giving the exact SI value of the speed of light in vacuum.

[2] Jody A. Geiger. A First Principles Derivation of the Galactic Radial Acceleration Relation from Measurement Quantization. Informativity Institute, public preprint, 2026, Section 1.2, Section 3.9, equations (79)–(81), and Appendix C. Cited for the MQ domain definitions and fraction relations, not as an independent test of the boundary axiom or of the rates derived on this page. Public version consulted October 30, 2026.

[3] Tamara M. Davis and Charles H. Lineweaver. Expanding Confusion: Common Misconceptions of Cosmological Horizons and the Superluminal Expansion of the Universe. Publications of the Astronomical Society of Australia 21(1), 97–109 (2004). Section 2 and Appendix A. Public author manuscript. Supports the distinction between local light propagation and cosmological recession; it does not establish the MQ axiom.

[4] David W. Hogg. Distance Measures in Cosmology. Public technical notes, arXiv:astro-ph/9905116 (1999; revised 2000), Section 4, equation (14), and Section 10, equation (29). Public comoving-distance discussion and lookback-time discussion, hosted by NASA/IPAC Extragalactic Database. Cited for distance and lookback-time definitions, not for evidence of a physical outer boundary.

[5] Håvard Alnes and Morad Amarzguioui. CMB anisotropies seen by an off-center observer in a spherically symmetric inhomogeneous universe. Physical Review D 74, 103520 (2006). Public author manuscript. Cited as an example of model-dependent directional predictions for an off-center observer; it is not a test of the MQ domain construction.

AIP Advances
8: Resolving the Hubble tension 68.261 and 73.510 km s Mpc

International Journal of Geometric Methods in Modern Physics
7: Measurement Quantization

Journal of High Energy Physics, Gravitation and Cosmology
6: Physically Significant Units of Measure
5: Measurement Quantization Unites Classical and Quantum Physics
4: Quantum Model of Gravity Unifies Relativistic Effects, Describes Inflation/Expansion Transition, Matches CMB Data
3: Measurement Quantization Accounts for Galactic Rotational Velocities and Obviates Dark Matter
2: Measurement Quantization Describes History of Universe -
Quantum Inflation, Transition to Expansion, CMB Power Spectrum

International Journal of Theoretical and Mathematical Physics
1: Measurement Quantization Describes the Physical Constants