The source result and access to the technical passage

The result is stated in Section 3.86 of the recorded October 23 universe-and-matter manuscript, Conditional No Go Theorem for a Constant Cosmological Term in Measurement Quantization. Its four-cell readout appears in Section 3.48, and Section 3.64 gives the explicit cyclic shift. The technical manuscript version was supplied for review and can be requested from the Institute. The related public preprint page inspected for this Brief does not include this passage. This Brief therefore supplies the assumptions, proof and worked arithmetic rather than implying that the linked public page contains the full argument. [1] [2] [3]

Let $\mathrm{𝑀}$ denote the Frames mapping. In MQ, the Internal Frame is the discrete count-configuration domain and the System Frame is the realized relational encoding. A diagnostic mode readout from counts is not automatically the full physical mapping. The present theorem concerns one specified readout and update class; deriving their physical selection under $\mathrm{𝑀}$ is a separate requirement. [1] [4]

The four-cell readout and what is being preserved

Use one oriented block with cells ordered zero, one, two and three. Its complex mode has real part equal to the count in cell zero minus that in cell two, and imaginary part equal to the count in cell one minus that in cell three. Thus a single unit in each successive cell contributes the phase values one, the imaginary unit, minus one and minus the imaginary unit. Equivalently, their real/imaginary coordinate pairs are (1, 0), (0, 1), (−1, 0) and (0, −1). [1]

The count total is the sum of all four allocations. The mode norm is the magnitude obtained from the two opposite-cell differences. These are distinct quantities. A count-conserving rule need not preserve that mode norm. Equal mode norm also does not determine the full count allocation, because the readout discards some count information. [1]

A unit phase is a complex multiplier of magnitude one. Multiplying by one fixed unit phase rotates every nonzero mode by the same angle and preserves its norm. The theorem demands that the same count rule induce that same multiplication for every admissible input, not merely give a norm-preserving output for a particular preparation. [1]

The convex proof

Table 1 identifies the simultaneous assumptions. Orient the count column so that the update multiplies it from the left. Conservation for every input makes each column of the update matrix sum to one. Nonnegativity makes that column a set of convex weights. Apply the rule to a unit count in one cell. Its output mode is therefore a convex combination of the four cell-phase values. [1]

These combinations fill the diamond joining the four unit-axis points in Figure 1. The diamond lies inside the unit circle and touches that circle only at its four vertices. To see why, a convex average of unit vectors has unit magnitude only when all positive-weight contributions point in the same direction. The four cell phases have distinct directions, so a unit-magnitude output from this column must select a vertex. [1]

For a unit input in cell zero, the mode initially equals the real unit point. The demanded common multiplier must therefore itself be one of the four vertex phases. The same multiplier maps each of the other three input phases to a vertex as well. Each column then has its weight on the corresponding single output cell. This yields the appropriate cyclic permutation; it cannot produce a non-quarter-turn universal multiplier under the stated hypotheses. [1]

The single-cell test is also justified if the universal relation is initially phrased for every strictly positive integer configuration. Subtract the linear readout relation for a common positive background from the relation for that background with one extra count in a selected cell. This gives the same basis-column relation. A genuinely restricted family without those variations would change the universality premise. [1]

An exact non-quarter-turn target

Choose the unit target with real part three fifths and imaginary part four fifths. Equation (1a) checks its squared modulus exactly. These numbered equations are arithmetic illustrations local to this Brief, not additional MQ physical laws. The target’s sum of absolute coordinates is seven fifths, exceeding the diamond’s bound of one.

A convex mixture with weights three sevenths on the real unit vertex and four sevenths on the positive imaginary vertex instead has coordinate pair (3/7, 4/7). Equation (2b) checks its squared modulus. Its norm is five sevenths. It points along the target direction but contracts the encoded amplitude; calling it a unit-phase update would discard the norm-preservation demand.

Quarter turns are realized by count permutations

The converse is constructive. The source’s forward shift sends the four ordered allocations to the sequence consisting of the old cell three, zero, one and two counts. The real part becomes minus the old imaginary part and the imaginary part becomes the old real part. That is a positive quarter turn. Repeated application gives the other three permitted phases and returns the original allocation after four shifts. [1]

Table 2 applies this convention to a strictly positive seed. Every row has total count fifteen and mode norm five. The numbers are an exact illustrative choice, not an experimentally selected or physically calibrated MQ state.

The norm being preserved is the encoded complex magnitude, not the Euclidean norm of the raw four-count vector. Positive counts, conservation and a supplied readout do not by themselves identify quantum probabilities, a physical clock or an action that selects a particular update. [1]

A selected non-quarter-turn endpoint pair is still possible

Figure 2 uses the same input (7, 3, 2, 3) but a different output (6, 5, 3, 1). Both have total fifteen and all entries are positive. Their modes are respectively (5, 0) and (3, 4), each with norm five. This realizes the chosen non-quarter-turn phase change on that endpoint pair. An integer transfer witness sends one count from cell zero to one, one from cell two to one, and two from cell three to two; all moves use neighboring cells of the cycle.

The endpoint witness does not contradict the convex argument. It has more total inventory than its encoded magnitude, so it is not the extremal single-cell test that forces the universal multiplier to a vertex. It only establishes reachability for this preparation. A universal rule must also meet all other input tests, including the extremal relation obtained by linearity. [1]

Norm preservation alone is a different proposition

Swapping cells one and three while leaving zero and two fixed changes the sign of the imaginary part and preserves the mode norm for every count vector. This is complex conjugation, a reflection. It is not multiplication by one fixed unit phase for all modes. It leaves the real unit input unchanged but reverses the imaginary unit input, so the required multiplier would differ between those two tests.

The zero mode also deserves care. A phase multiplier cannot be inferred from a mode whose real and imaginary parts both vanish. Examples confined to that invisible mode cannot establish an arbitrary universal phase law. The theorem’s input family supplies nonzero modes that distinguish the alternatives. [1]

What remains open in the physical program

Table 3 separates the exclusion from its possible extensions. Section 3.86 lists restricted state domains, nonlinear count dependence, coherent extra data and larger realizations as ways to leave the tested class. That list is a statement of logical possibilities, not proof that each option supplies a successful physical completion. A partial collective construction is not a universal positive linear count automorphism merely because it reproduces selected phase updates. [1]

A fixed finite collection of phase vertices has the same convex limitation on unit-modulus mixtures at its vertices. Increasing a finite phase set can enlarge a discretized menu, but a richer construction must still explain how its intended continuous evolution and normalization arise. This Brief does not promote the four-cell result into an unrestricted impossibility theorem for all encodings.

For MQ, the physical question includes which count histories and readouts are admitted, what selects their update, how physical time and action normalization are fixed, and whether the same structure supplies the required observables and records. The source’s larger action program retains those joint requirements. A hand-chosen endpoint, a coherent interpolation or an externally supplied quantum generator cannot be presented as their primitive derivation. [1] [4] [6]

Chronology and reporting guidance

October 23, 2026 identifies the recorded manuscript version. It does not establish the first discovery day, original announcement or journal publication date. This release and Brief were prepared November 2, 2026. The conditional convex proof and the displayed exact examples were checked, while no independent experiment or full audit of the larger theory is claimed. [1]

For interviews, request the precise readout, rule matrix, count domain and common multiplier condition. Establish which hypothesis changes in a proposed escape. Ask whether the result is universal or selected-state, whether norm means encoded mode magnitude or raw inventory, and what derives the physical update rather than merely specifying it. The supported conclusion is a restricted obstruction with explicit constructive limits.