CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Counterexample: a bounded one-point record of the cosmological redshift slot does not determine the emitter scale factor

Physical scope

Conceptual counterexample inside the classical gravitational framework: it shows that a declared one-point record with positive declared uncertainty for the cosmological redshift readout of the worked interface instance is consistent with two distinct parameters of the same operational model, so such a record does not determine the parameter uniquely. The numbers are declared illustrative values of the worked instance; the item asserts no measurement, no apparatus, no sampling model, no statistical inference and no empirical conclusion about any model or postulate.

Statement refuted

In the operational model $M_{o p}$ of Worked instance of the interface vocabulary on the cosmological redshift slot: an explicit de Sitter ray family, an operational model and the exact window of an illustrative bounded record clause (b), with the declared comparison rule of clause (c) of that example and its declared one-point record with positive declared uncertainty, the parameter is determined uniquely: if two parameters $a_{E}, a_{E}^{'} \in (0, a_{R})$ of $M_{o p}$ are consistent with the record, then $a_{E} = a_{E}^{'}$ .

Facts & Assumptions

Given: The de Sitter FLRW configuration of Worked instance of the interface vocabulary on the cosmological redshift slot: an explicit de Sitter ray family, an operational model and the exact window of an illustrative bounded record with the declared flat vacuum parameter $Λ = 3 H^{2} / c^{2}$ , the operational model $M_{o p} = (S, Θ, θ, D, F)$ with $S = Θ = (0, a_{R})$ , $θ = i d$ and redshift readout $F (a_{E}) = a_{R} / a_{E} - 1$ , the declared one-point record $R$ with closed box $[1 / 2, 3 / 4]$ on $z$ , and the declared comparison rule that $a_{E}$ is consistent with $R$ exactly when $F (a_{E}) = z (a_{E}) \in [1 / 2, 3 / 4]$ . All of these are declared illustrative structures of the worked instance; none is a measurement.

[F1]

The worked instance: its vacuum FLRW datum has $Λ = 3 H^{2} / c^{2} > 0$ and satisfies the adopted field law by clause (b) and fact [A2] of Worked instance of the interface vocabulary on the cosmological redshift slot: an explicit de Sitter ray family, an operational model and the exact window of an illustrative bounded record. The operational model of that example's clause (b) has parameter set $(0, a_{R})$ and prediction map $F (a_{E}) = a_{R} / a_{E} - 1 = z (a_{E})$ ; its declared record has $N = 1$ , $d = 1$ , $z_{1} = 5 / 8$ , $ε_{1} = 1 / 8$ , so that the declared closed box is $[1 / 2, 3 / 4]$ ; its declared comparison rule makes $a_{E}$ consistent exactly when $F (a_{E}) = z (a_{E})$ lies in that box; and it records the consistency set $C (R, M_{o p}) = [4 a_{R} / 7, 2 a_{R} / 3]$ in clauses (b)-(c).

[L1]

Reciprocal ratio windows: for $a_{*} > 0$ and $0 < u_{1} \leq u_{2}$ the preimage under $a \mapsto a_{*} / a$ of $[u_{1}, u_{2}]$ is $[a_{*} / u_{2}, a_{*} / u_{1}]$ , and that map is strictly decreasing and injective (The reciprocal ratio map $a \mapsto a_{*} / a$ : strict decrease, inverse, and the exact preimages of points and closed intervals claims 2-4).

Counterexample

technique · direct

1.1F1L1algebra

(Two consistent parameters.) Let $a_{1} : = 4 a_{R} / 7$ and $a_{2} : = 2 a_{R} / 3$ , both defined since $a_{R} > 0$ . They lie in the parameter set $(0, a_{R})$ because $0 < 4 / 7 < 2 / 3 < 1$ . Their predictions are $F (a_{1}) = a_{R} / (4 a_{R} / 7) - 1 = 7 / 4 - 1 = 3 / 4$ and $F (a_{2}) = a_{R} / (2 a_{R} / 3) - 1 = 3 / 2 - 1 = 1 / 2$ by field arithmetic; both values lie in the declared closed box $[1 / 2, 3 / 4]$ . Hence $a_{1}$ and $a_{2}$ are both consistent with the record under the declared comparison rule.

1.2F1L1algebra

(Distinctness.) The two parameters differ: $a_{2} - a_{1} = a_{R} (2 / 3 - 4 / 7) = a_{R} (14 - 12) / 21 = 2 a_{R} / 21 > 0$ , so $a_{1} \neq a_{2}$ . In fact no uniqueness holds even after fixing the box: by [L1] with $a_{*} = a_{R}$ , $u_{1} = 3 / 2$ and $u_{2} = 7 / 4$ , the consistency set is the whole closed interval $[4 a_{R} / 7, 2 a_{R} / 3]$ , in agreement with the consistency set recorded in [F1], and every one of its elements is consistent with the record.

2.1F1L1step 1.1step 1.2∎

(Conclusion.) The record $R$ is a one-point data record of the interface vocabulary with positive declared uncertainty $ε_{1} = 1 / 8 > 0$ , and steps 1.1 and 1.2 exhibit two distinct parameters of the same operational model that are both consistent with it; the uniqueness assertion of the Statement refuted is therefore false. This is an exact algebraic consistency computation on declared illustrative values: it asserts no measurement, no calibration, no probability model, no statistical inference and no empirical conclusion about the model or about any postulate.

Remarks

Positive uncertainty is what fails to pin the parameter. With $ε_{1} = 0$ the box would be the singleton ${5 / 8}$ , whose preimage under the strictly monotone prediction map is the single parameter $8 a_{R} / 13$ ; the failure of uniqueness in the counterexample is a property of the closed box of positive width together with the comparison rule, not of the prediction map's invertibility.
Scope. The two witnesses are illustrative parameters of a hypothetical configuration; identifying any of them with a physical source would require the apparatus, calibration and analysis layer that the research queue records as unsupplied.

Counterexample: a bounded one-point record of the cosmological redshift slot does not determine the emitter scale factor

Statement refuted

Facts & Assumptions

Counterexample

Remarks

Depends on

Dependency tree · 2 levels

Sources