The Channel Decomposition of a Coupling Arm Need Channels, Aggregation Operators and the Attention Budget in Intimate Dyads Where the signs of the nine cells come from, why they flip, and five ways a two-person system ends
Year: 2026
Type: preprint
Abstract: The Adaptation–Fidelity Law classifies the head–tail loop into nine cells by the signs of its two coupling arms, and the paper on merit overshadowing the sovereign applied that grid to ruler and minister. The nine cells read signs only, and they cannot answer three questions: where the sign of an arm comes from; why, in one and the same couple, an arm flips sign without any conflict event; and why two couples in the same cell can withstand completely different shocks. This paper writes each arm as an attention-weighted sum of signed need channels within one layer, b = Σₖ wₖ·jₖ, where jₖ is the signed contribution of channel k and wₖ the share of attention it currently holds. The main results are as follows. First (tier-A algebra), as long as an arm contains both positive and negative channels, a reallocation of attention shares alone can flip it, with no channel changing; the distance to the flip hyperplane w·j = 0 has a computable lower bound, and that bound is the quantity behind “same cell, different fate”; an arm whose channels all share one sign cannot be flipped by reweighting. Second, when the share of a temporary channel (the novelty of romance, child-rearing) falls below s* = −ŵ·j/(j_c − ŵ·j), the baseline mismatch it had masked appears; flips after the children leave home therefore have two distinguishable mechanisms, unmasking and withdrawal of a shared support. Third (tier-A algebra), partners read a perceived arm, not the objective one: splitting every channel into a positive and a negative part, each multiplied by a booking gain, an objectively positive arm is read as negative once the ratio of negative to positive booking gains exceeds the ratio of positive to negative totals—a “pseudo-flip” with no change in any channel or share. Fourth, whether a mismatch can be borne is decided by the aggregation operator, not by the weight: series core channels aggregate by the minimum, parallel peripheral channels nearly by the mean; this paper therefore does not write need weights as an “80/20 rule” or a power law; the candidate list of series constraints must be fixed in advance, and a switch of the binding constraint is a kink, not a phase transition. Fifth, there are at least two linearisations between partners: the two “reward → capability” arms of the resource-and-role layer, which are the nine cells, and the “deviation → deviation” coupling product K of the emotion-regulation layer. In the latter a negative product is counter-regulation and never destabilises in continuous time; but partners mostly adjust weekly or monthly, and when stability is judged on the discrete map, strong counter-regulation combined with slow adjustment diverges in oscillation (Proposition 6). Sixth, the calm steady state of long relationships is not a tenth cell but a small-amplitude mutual cell under share-conserved closure, maintained by an attachment stock; the income lever of that stock is co-present attention and regime change, and it can be counted by a shared clock. In addition, a structure can persist after participation has left: once the share of the relationship’s structural income that does not depend on the two partners’ participation passes a threshold, the system falls into hollow lock-in, and “money in place of company” brings it on sooner. Seventh, on the reading layer and the time leg, merging the two partners’ readings caps the effective number of independent watchtowers at 2/(1+ρ); attribution gating can lock one person, or both together, into a pride branch; avoidance closes into a readability–investment loop through the interval between contacts (tier B); the price of an outlay is set by the environment, and on flow budgets only the sign judgement survives. Because partners respond to each other, the identification criteria of the series leave their domain inside the dyad (Proposition 3); the most robust prescriptions are structures that do not depend on recognition, which this paper writes as periodic forced observation against a fixed standard, with the conditions under which it improves the persistence margin, its cost and its failure boundaries (Proposition 7). The paper classifies five ways a two-person system ends, gives rules for their superposition and three conditions under which the return cost of a weak wall diverges within any window that can be run (Proposition 8), and marks that only the slow-drive type is a phase transition in the sense of T12, while the shock type cannot in principle be warned of from inside. It also gives four boundaries against ruler–minister coupling, transfer conditions for other dyads and the limits of transfer to kin dyads, and uses a love video as a tier-C example. Section 13 compares the main results item by item with existing models in relationship science—nonlinear difference-equation models of marital interaction, emotional coregulation, the vulnerability–stress–adaptation model, research on the timing of divorce, marital attribution, self-expansion, risk regulation, transactive memory and the investment model—sorting them into “only a change of language”, “already supported by external evidence” and “a different prediction”, and addresses head-on two external findings that constrain this paper: conflict-avoiding couples can be stable over the long term, and marital satisfaction on average rises after the children leave home. Sixteen falsifiable predictions are registered with a tiered execution route; executions on external data: zero.
Author: Li, Qinfu
Language: English
Subfield: Social Psychology
Field: Psychology
Domain: Social Sciences
Open Access status: green
OpenAlex ID: W7225757555
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