The Conceptual Tension between the Equation with a Cosmological Term and the Equation without One —An Epistemological Diagnosis and a Response in Terms of a Modified Hypothesis

How the equation with a cosmological term and the equation without one areto be reconciled conceptually has no conceptually self-consistent account in thecommunity to date. The common reconciliation—treating the equation with acosmological term as the more fundamental equation, and interpreting the equationwithout one as its limit as the cosmological term tends to zero—is mathematicallysound and fits observation well phenomenologically, but it performs an unarguedsubstitution at the epistemological level: it replaces the analysis of the physicalidentity of the cosmological term with a description in terms of “different effectivevalues at different scales,” re-describing “the physical identity of the cosmologicalterm is unclear” as “the cosmological term behaves differently at different scales.”This paper provides a systematic diagnosis of this operation.The diagnosis starts from a premise: general relativity is itself a hypotheticalsystem, not a theorem derived from more fundamental principles; it is preciselyverified in the weak-field and classical regimes, but verification is not proof, andits incompatibility with quantum field theory, its failure at singularities, and theunclear identity of the cosmological term are all signals that this hypothetical systemrequires modification. Under this premise, the object of criticism in this paper isnot “the equation with a cosmological term is wrong,” but the community’s treatinga hypothesis that requires modification as an unmodifiable benchmark, and using“two limits” to reconcile it with the equation without a cosmological term—thismistakes the status of a hypothesis for the status of a principle.The diagnosis proceeds in three layers. The first layer: the equation with acosmological term is an equation with an undetermined parameter; treating it asa first-principles equation misplaces its status at the level of fitting capacity as itsstatus at the level of principle. The second layer: the reconciliation by “two limits”presupposes that the effective-field-theory framework can be applied universally, butthis presupposition itself is not argued; and the community lacks consensus on “whatthe problem is.” The third layer: the enormous scale difference between solar-systemand cosmological constraints on the cosmological term is a magnitude difference, anda magnitude difference does not constitute a logical entailment between equations; “two limits” is a phenomenological description, not a structure derived from firstprinciples.After the diagnosis, this paper positions the equal-footing equation frameworkas a modified hypothesis for the general-relativistic hypothetical system: it doesnot take the equation with a cosmological term as its starting point, but the equation without one; the effective cosmological term emerges from the behaviour ofthe matter gravitational-field energy-momentum tensor on large scales, rather thanbeing input as a free parameter. This response modifies the source-term structureof the equation, and offers a modified hypothesis that allows the equation withouta cosmological term to become the conceptual starting point again.