The result and the technical source

The recorded October 23 manuscript, Conditional No Go Theorem for a Constant Cosmological Term in Measurement Quantization, states a restricted inverse-square theorem in Section 3.1 and tests the additional cosmological identification in Section 3.4. Sections 3.5–3.7 qualify local gravitational, matter and variational claims. This technical version was supplied for review and can be requested from the Institute. The related public curvature preprint does not reproduce the complete current theorem and audit. The essential construction and checks are therefore given here. [1] [2]

MQ’s Internal Frame is the discrete count-configuration domain. The System Frame is the realized relational description in which dimensional observables arise through the Frames mapping. Those definitions set the intended representation. They do not establish that a chosen comparison frequency is the metric Hubble rate or that its algebraic scalar must be the coefficient in a gravitational field equation. [1] [7]

Define the comparison before interpreting it physically

Let $\mathrm{\Lambda }$ denote the constructed comparison scalar in m⁻² throughout Equations (1a)–(5e). Let ${\mathrm{𝛺}}_{\mathrm{\Lambda }}$ be the dimensionless density fraction used in the comparison, ${\mathrm{𝐻}}_{\mathrm{sys}}$ the chosen System Frame comparison frequency in s⁻¹ and $\mathrm{𝑐}$ vacuum light speed in m s⁻¹. Equation (1a) rearranges the conventional density-parameter identity using the same frequency for its critical-density normalization. At this stage it is an adopted bridge to the scalar, not a derivation of a metric solution. [1]

Let ${\mathrm{𝐴}}_{\mathrm{𝑈}}$ be the adopted positive realized age in s and ${\mathrm{𝜃}}_{\mathrm{𝑠}\mathrm{𝑖}}$ the fixed dimensionless MQ coefficient. Let ${\mathrm{𝐻}}_{\mathrm{int}}$ be the Internal Frame-resolved comparison frequency, also a physically realized frequency in s⁻¹. These names distinguish the two source prescriptions; the second is not a primitive clock inside the pre-realization configuration domain. Equation (2b) states their adopted inverse-age forms. [1]

Let ${\mathrm{\Omega }}_{\mathrm{𝑑}\mathrm{𝑘}}$ be the fixed MQ dark referenceability fraction. Equation (3c) combines its geometric expression with the additional identification as the density fraction. The upright geometric fraction and the conventional density parameter have different definitions even when equated by this model hypothesis. Similar numerical percentages do not establish that identification independently. [1]

The exact inverse-square proof

With the hypotheses in Table 1, substitution of Equations (2b) and (3c) into Equation (1a) gives Equation (4d). The coefficient multiplying inverse age squared is positive and independent of age on the stated branch. The circle constant is dimensionless; the denominator carries square metres, as required. [1]

Differentiating with respect to the realized age gives Equation (5e). The derivative has units m⁻² s⁻¹ and is strictly negative for a positive scalar and positive age. Along an evolution parameter it is strictly negative wherever the age derivative is positive. A merely increasing smooth age can have stationary derivative points; that does not make the inverse-square function constant on an increasing interval. [1]

The same age scaling holds if the second comparison frequency is used consistently, since it also has an age-independent numerator. Figure 1 and Table 2 show the normalized System Frame construction. The result is exact under these hypotheses. Inverse-area dimensions alone would not force this dependence or rule out another constant contribution. [1]

The boundary and alternative cases matter. At the square-root-of-two coefficient the fraction vanishes, outside the theorem’s positive branch. A variable fraction, another rate prescription or a separately added constant changes the construction. Such possibilities are not excluded by proving the selected positive scalar cannot remain constant. [1]

Why expansion alone does not prohibit a constant term

The manuscript gives an empty, spatially flat de Sitter solution as a direct counterexample to the unrestricted inference. It has positive constant Hubble frequency, exponential expansion and a positive constant cosmological coefficient. It can retain fixed reference standards. It does not satisfy the extra inverse-age prescription, so it is consistent with the conditional theorem while refuting the broader claim that expansion itself forbids a constant term. [1] [4]

A scalar’s geometric role likewise does not determine its time dependence. In an Einstein-form equation a constant coefficient can multiply a time-dependent metric. Nor does the construction cancel arbitrary constant matter-vacuum contributions. A source or action derivation would have to show how any such contribution enters the full effective coefficient. [1] [4]

The conservation test

With fixed gravitational coupling and light speed, the contracted Bianchi identity and metric compatibility link a varying field-equation coefficient to the covariant divergence of its source. If the complete source is separately conserved, that coefficient must be spacetime constant. Identifying the constructed decreasing scalar with it is then incompatible with those simultaneous assumptions. [1]

An evolving coefficient could instead require source exchange or modified field dynamics. Moving a term from one side of the equation to the other only changes its bookkeeping; it does not remove total conservation. The source does not derive an extra interaction merely by writing the necessary balance condition. The isolated conserved timelike matter branch retains this restriction. [1]

A flat-dust background check even when exchange is allowed

Section 3.4 next tests a spatially flat homogeneous metric with fixed coupling, pressureless matter and a vacuum-form scalar contribution. The adopted age is identified with elapsed cosmic proper time from a fixed origin, and the chosen comparison frequency is identified with logarithmic metric expansion on an interval. Total conservation is retained while exchange between matter and the scalar contribution is allowed. [1]

Differentiating the flat Friedmann relation and using the total balance eliminates the exchange term. Combining that acceleration requirement with the prescribed inverse-age frequency gives the necessary dimensionless rate–age condition in Equation (6f), source Eq. (44f). It is a requirement of this restricted extension, not another MQ calibration choice. [1]

A 90-digit calculation using the shared fixed-branch inverse fine-structure calibration reproduces the manuscript’s coefficients. The adopted fraction is 0.683624161224… and Equation (6f) requires a rate–age product of 2.107198417068…. The System Frame prescription instead gives 0.962972654970…, while the Internal Frame-resolved prescription gives 1.037027345030…. The same derivation applies to the second choice if it is used consistently as the metric rate. [1]

Table 3’s final column would equal two for a consistent solution of this restricted system. All entries depend on the same coefficient; they are not independent observational estimates. Their mismatch is not assigned a sigma value. Permitting exchange alone cannot repair it while the other assumptions remain fixed. [1]

Changing pressure, the fraction, the metric-rate prescription or the field equations would define a different extension requiring a derivation and tests. Retuning the coefficient to force this check would also conflict with the independent calibration held fixed here. The theorem does not select such a repair. [1]

Local recovery does not complete global cosmology

Section 3.5 derives the stationary high-acceleration Newtonian limit of the galactic law. Section 3.6 obtains restricted local equivalence of an isolated timelike action to irrotational pressureless matter on a common smooth metric. These support specific comparisons, not equality of every gravitational or cosmological prediction. [1] [8]

The metric investigation separately distinguishes smooth representability from count-admissible variations. Its trace-free variation test retains an undetermined constant cosmological integration contribution. Thus conservation and a restricted variational structure do not themselves delete that constant. Full gravitational equivalence still requires an appropriate action, sufficiently general admissible variations, matter coupling and compatible boundary and initial conditions. [1]

A physical cosmology must also establish the metric meaning of the adopted age and diameter, solve the background consistently and calculate perturbations and observed distances. A diameter proportional to age is not automatically a comoving scale factor or a causal horizon. Matching selected present-epoch scalars does not supply those missing steps. [1] [6]

Chronology, source access and reporting

October 23, 2026 is the recorded technical source version. It does not establish the first discovery, original announcement or journal publication date. This release and Brief were prepared November 2, 2026 using the supplied conditional theorem and dynamical qualifications. The related public preprint is useful context, while the full current theorem and audit can be requested from the Institute. Journal publication of that version is not established. [1] [2]

For interviews, request the rate and fraction hypotheses, the metric-rate identification, the age convention and the action/source law. Report the inverse-square theorem, the de Sitter counterexample and the failed flat-dust identification as distinct results. The supported milestone is a precisely bounded theoretical restriction and a diagnostic of its simplest cosmological extension, not a universal solution of the cosmological constant problem.