Where the ratio argument is developed

J. A. Geiger’s Singularity Exclusion in the Bounded Measurement Quantization Domain states the count interval in Section 2, Equation (4). The following paragraph gives the ordinary ratio result. Equations (5)–(13) then distinguish local and observed measures and derive a lower bound on the transformation factor. The Bounds to Measure explanation and preprint supply related count-limit context. [1] [2] [3]

The Internal Frame is the discrete count-configuration domain. The System Frame is its realized relational encoding under the Frames mapping. The counts used below are dimensionless; multiplying by their fundamental references gives physical length, mass or time. The theorem concerns the specified observer–observable domain, not an unrestricted assertion that every parameter used anywhere in MQ must be a positive integer. [1]

The count interval and its ordinary ratios

Let ${\mathrm{𝑛}}_{\mathrm{𝐿}}$, ${\mathrm{𝑛}}_{\mathrm{𝑀}}$ and ${\mathrm{𝑛}}_{\mathrm{𝑇}}$ denote length, mass and time counts in the stated realization. Let ${\mathrm{𝑛}}_{\max }$ denote the adopted finite upper-count parameter. Each admitted positive count considered here is at least one and no greater than that parameter. For example, the mass-to-length count ratio satisfies Equation (1a). [1]

For the lower bound, the numerator is at least one and the denominator is at most the ceiling. For the upper bound, the numerator is at most the ceiling and the denominator is at least one. No equation of motion is needed for this arithmetic implication. The same proof applies to every ordered ratio of the three counts. [1]

Integrality is not required for this particular inequality: real numbers in the same closed positive interval obey it too. Whole-unit admissibility is an additional property of the source’s count domain. If the reported ceiling parameter is not itself an endpoint integer, the displayed bounds can be conservative rather than attained. One must not infer a unique maximum integer from a rounded uncertain parameter.

Table 1 distinguishes count ratios from physical ratios. Every entry in its first column is dimensionless and obeys the same count interval. A realized mass-to-length ratio additionally contains the fundamental-mass-to-fundamental-length scale, whereas a length-to-duration ratio contains the corresponding length-to-time scale. Changing units changes their numerical coordinates, not the count theorem.

The positive lower restriction matters as much as the upper ceiling. If a denominator could approach zero, an upper bound on each input alone would not bound their ratio. Zero denotes absence of a positive measure in the source’s convention; a zero denominator is not part of this theorem. [1]

Figure 1 makes both endpoints visible. Fixing the interval excludes a zero or unbounded ordinary ratio within it. Replacing that fixed domain by a sequence of ever-larger ceilings would change the premise; it would not be a limiting sequence confined to the original interval.

The ceiling normalization and what it establishes

The manuscript introduces the fundamental time ${\mathrm{𝑡}}_{\mathrm{𝑓}}$ in s. It normalizes its reciprocal rate by a one-second reference interval to obtain the dimensionless upper-bound parameter in Equation (2b). [1]

Under the source’s 2022 calibration, the fundamental time is 5.39106261131(41) × 10⁻⁴⁴ s and the normalized parameter is 1.85492188108(14) × 10⁴³. Parentheses give one standard uncertainty in the final quoted digits. The parameter’s uncertainty is propagated from the adopted calibration and does not determine a unique endpoint integer. [1]

Reciprocal fundamental time is a rate in s⁻¹. Multiplying by a fixed interval removes its units. This normalization sets a numerical scale; treating that scale as the ceiling of every admitted configuration is the additional physical premise. The unit conversion is not an independent measurement of the premise. A different physical normalization would require recalculating the dependent bounds. [1]

Finite inputs do not guarantee a positive transformation factor

An ordinary ratio can be finite and equal to one. Subtracting its square from one then gives zero, and taking its square root gives a zero factor. A reciprocal of that factor is undefined. Thus the ratio bound alone is insufficient for the observer–observable transformations. The source explicitly adds an observed-measure restriction. [1]

Let ${\mathrm{𝑛}}_{\mathrm{𝐿}\mathrm{𝑜}}$ be the observed length count and ${\mathrm{𝑛}}_{\mathrm{𝐿}\mathrm{𝑙}}$ its local reference count, both expressed using the same fundamental-length reference. “Local” belongs to the framework of the observable; “observed” belongs to the observer’s realized description. The common-interval motion and light-travel counts are ${\mathrm{𝑛}}_{\mathrm{𝐿}\mathrm{𝑚}}$ and ${\mathrm{𝑛}}_{\mathrm{𝐿}\mathrm{𝑐}}$, respectively. The light-travel denominator is positive. Equation (3c) is the source’s length transformation. [1]

The motion count measures displacement of the observable, while the light-travel count refers to light over the same elapsed interval. They are distinct from a radial source-separation count. The real branch requires the radicand to be nonnegative; a positive admitted observed length imposes the stronger condition that the factor not vanish. [1]

Apply the lower count to the observed output

The source requires observed length count at least one and local length count no larger than the finite ceiling. Squaring Equation (3c), then applying those two restrictions, gives Equation (4d), reproducing the source’s Equation (13). [1]

Equation (4d) supplies a uniform positive separation for the radicand under a fixed finite ceiling. The square-root factor itself is at least the reciprocal ceiling. The restriction comes from the admissible observed output, not from assuming that every bounded input automatically produces an admissible transformation. [1]

This is a necessary condition, not a complete certification of a configuration. Each output must also meet its upper bound and any whole-unit requirement. The source’s reciprocal observed-mass transformation needs that separate check. A bounded reciprocal factor can still multiply a local mass into an observed mass above the admitted ceiling. [1]

Exact examples that separate the checks

Table 2 uses the same small illustrative ceiling of ten as Figure 1. All entries test only the displayed count transformation. The first row uses the source’s four–five motion ratio, giving an observed-to-local length factor three fifths. These are exact arithmetic constructions, not measurements or a complete physical source solution.

The third row in Table 2 has finite positive input counts but a zero output. The fourth has a positive factor and still fails the minimum observed-count condition. The first two rows pass the length test, without thereby proving that all other measures in a complete configuration are admitted.

For a further output check, use the first row’s factor three fifths in the source’s reciprocal mass transformation. A local mass count of six gives observed mass count ten, whereas local mass count nine would give observed mass count fifteen. The latter exceeds the illustrative ceiling even though the length transformation passed. This is why all observed measures must be tested, not just one factor.

Figure 2 summarizes the logical separation that prevents the finite-input shortcut. For the square-root factor, the observed minimum is essential. For a full physical claim, the remaining measures and the consistency of their joint realization remain essential too.

Scope of the finite-domain conclusion

Table 3 retains the distinction between a configuration and an accumulated total. Adding contributions over successive configurations can produce a growing sum even when every individual configuration meets the same bound. The count-domain premise alone supplies no evolution law or bound for such a sum. [1]

The source defines its singular-measure question in terms of positive represented measures tending to zero or increasing without a finite bound. A result about that domain is not by itself a proof about all curvature invariants, geodesic completeness or dynamical collapse. Those require a specified geometry and evolution. Nor does the lower bound here establish a universal particle-mass minimum outside the stated measurement representation. [1] [3]

Chronology and reporting context

October 13, 2026 identifies the recorded manuscript version. It is not established here as the original discovery or announcement date. This Brief was prepared October 27, 2026 using the current public manuscripts. Journal publication of these current preprint versions is not established. [1] [3]

The ratio theorem and observed-output calculation were checked algebraically, and the illustrative finite examples were evaluated exactly. Those checks establish consistency of the implications under the stated assumptions. They do not independently validate the physical ceiling or show that a dynamical physical system realizes every admitted count assignment. [1] [4]

For an interview, request the precise observer–observable domain, the origin and normalization of its ceiling, and the full set of output admissibility conditions. Ask how evolution preserves that domain, which accumulated quantities are covered, and what observation tests the physical premise independently of the reference calibration.