The gap addressed by the revision

The source’s singular-measure question concerns positive represented measures tending to zero or growing without bound within an adopted observer–observable domain. Its ordinary positive count ratios are bounded by a minimum and a finite ceiling. That result alone does not establish that a square root of one minus a squared ratio is positive: finite equal counts still give a zero radicand. [1] [4]

The revised step is to apply the minimum count to the transformed observed length. Restricting only the local reference leaves that output uncontrolled. A mathematical continuation toward a zero factor must therefore be distinguished from an admitted count assignment satisfying all the source’s conditions. [1]

Specify the length and motion count roles

Let ${\mathrm{𝑛}}_{\mathrm{𝐿}\mathrm{𝑙}}$ be the positive local length reference count and ${\mathrm{𝑛}}_{\mathrm{𝐿}\mathrm{𝑜}}$ the observed length count, both using the same fundamental-length reference. Let ${\mathrm{𝑛}}_{\mathrm{𝐿}\mathrm{𝑚}}$ be the nonnegative motion length count and ${\mathrm{𝑛}}_{\mathrm{𝐿}\mathrm{𝑐}}$ the positive light-travel length count accumulated over the same elapsed interval. Each count is dimensionless. Equation (1a) gives the source’s nonnegative length branch. [1] [2]

The light-travel count is not a radial gravitational source count or automatically the domain’s overall ceiling. Its role is to normalize displacement over the chosen interval. Using a common interval and length reference makes the motion-to-light count ratio equal to the physical relative-speed-to-light-speed ratio. No numerical calibration of a fundamental unit is needed for the dimensionless examples below. [1] [2]

For the special-relativistic longitudinal interpretation, local length is rest length and observed length uses endpoint positions simultaneous in the observer’s inertial frame. The motion manuscript supplies Lorentz-compatible kinematics for the interior square-root dependence; the count lower bound is a further domain premise, not a consequence of the endpoint triangle alone. [2]

Put the minimum on the observed output

Write the adopted finite ceiling as ${\mathrm{𝑛}}_{\max }$. The particular premises needed in the proof are Equation (2b). The local count is positive and no larger than that ceiling; the observed count is at least one. Complete domain admission also requires the remaining counts and outputs to obey their stated restrictions. [1]

Divide Equation (1a) by the positive local count and square. Apply the observed minimum first, then the local ceiling. Equation (3c) is the source’s Eq. (13). Every inequality has an identified premise: the first uses the output minimum, the next uses the finite upper bound on the local reference. [1]

Consequently the square-root factor is at least the reciprocal local count and at least the reciprocal ceiling. Its reciprocal is finite in that fixed domain. A factor small enough to produce an observed count below one is excluded before the factor reaches zero. The proof uses both the transformed lower bound and the finite local reference; neither can be omitted.

Rearranging the local part of Equation (3c) gives Equation (4d). For a local count of five, the squared motion ratio is at most 24/25 and the factor is at least 1/5. The corresponding necessary motion-ratio limit is √24/5, about 0.979796. These are conditional count restrictions, not a measured new universal speed limit. [1]

Check the source’s exact count assignment

Table 1 reproduces the motion, length and assigned mass counts used in the source’s compatible example. The motion-to-light ratio is 4/5, so the radicand is 9/25 and its nonnegative square root is 3/5. Applying that factor to local length count five gives observed length count three. Figure 1 contrasts the length and mass directions. [1]

The reciprocal mass output needs another check

Let ${\mathrm{𝑛}}_{\mathrm{𝑀}\mathrm{𝑙}}$ and ${\mathrm{𝑛}}_{\mathrm{𝑀}\mathrm{𝑜}}$ be the local and assigned observed mass counts in the source comparison, using the same fundamental-mass reference. With a positive length factor, the reciprocal mass relation can be written as Equation (5e). In the current inertial interpretation the observed mass parameter corresponds to total energy divided by light speed squared; it is not a Lorentz transformation of invariant rest mass. [1] [2]

Multiplying Equation (5e) by the observed length count yields Equation (6f). In Table 1, both products are exactly fifteen. This is an internal algebraic consistency check on the assigned measures, not an independent experiment or a proof that every possible count tuple is realized. [1] [2]

Bounding the reciprocal factor does not automatically bound the mass output by the same ceiling. A large local mass can still be multiplied above it. For the positive factor, Equation (7g) states the additional upper-output check, obtained directly from Equation (5e). It is equivalent to a restriction on the local mass for the selected length ratio.

Examples that separate the necessary conditions

Table 2 deliberately uses a small illustrative ceiling of 25 to expose the arithmetic. It does not replace the source’s calibrated physical-domain parameter. Each listed input is a finite positive integer within this toy ceiling. The first row passes the displayed length and mass checks. Later rows show why a finite input or a positive factor is not a complete output-admission test.

The second row’s factor is exactly 5/13, so its observed length is positive but less than one. It is already outside the minimum-count premise. Its fractional outputs would also need the separate whole-unit check wherever that requirement is imposed. In the third row, the length is admitted and both mass counts are integers, yet the observed mass exceeds the toy ceiling. The last row reaches the formal zero factor, where the reciprocal mass expression is undefined.

Figure 2 shows why merely excluding equality of motion and light counts is insufficient. There is a positive-factor region that already fails the output minimum. Conversely, passing this necessary scalar inequality does not establish a whole-unit realization or a compatible mass output. At local count one, the length minimum permits only the zero-motion case in Equation (4d). Larger finite local references allow a wider necessary ratio domain.

Domain, calibration and physical scope

The source normalizes its upper-count parameter using the reciprocal fundamental-time rate multiplied by a stated reference interval. A rate in inverse seconds is not a dimensionless count until that interval is supplied. Its reported conditional calibration is not a determination of one exact endpoint integer. This Brief needs only a fixed finite ceiling for the proof and an explicit toy value for Table 2, not a new numerical calibration. [1] [4]

Table 3 keeps the scope of the conclusion explicit. The bound is uniform only while the same finite domain is retained. Sending the ceiling to infinity removes that uniform lower bound. Bounding an individual configuration also does not bound an unrestricted sum across successive configurations or supply a dynamical law preserving admission. [1]

The Internal Frame is the discrete count-configuration domain and the System Frame the realized encoding connected through the Frames mapping. Their distinction does not independently establish every finite-domain physical premise. Nor does a scalar-factor bound prove that a spacetime has finite curvature everywhere, a regular collapse evolution or extendible geodesics. Those are different mathematical and physical questions. [1] [3] [6]

Chronology and reporting context

October 13, 2026 identifies the recorded source version in this retrospective archive. It is not an established original discovery, public announcement or journal publication date. This Brief was prepared November 2, 2026 using the current public manuscripts. Journal publication of these current versions is not established. [1] [2]

The proof and diagnostic examples were checked exactly. They establish the stated implications under the adopted lower and upper restrictions, not independent validation of those restrictions. For an interview, request the full output-domain test, the reference interval defining the ceiling and a dynamical prescription preserving admission. Ask what observation would distinguish the physical premise from a count representation of already established kinematics. [1] [5]