The Fold of Pride Status-Gated Attribution, the Negative-Attribution Regime, and the Identifiability of Channel-Gated Agents

Abstract Traditional ethics places pride first among the sins because it is the most hidden and the hardest to repent of. This paper writes the computable side of these two reasons as a minimal model: the agent’s belief bias x is updated by an asymmetric rule, and the share a of failures that are booked falls exponentially with the height of the agent’s public commitment, a(x) = a₀·e^{−χx} + a_ext, where χ is the number of nats by which the cost of admission rises per noise unit of claimed overstatement, and a_ext is an external channel not gated by status. First, a threshold theorem. Under Gaussian feedback, let h(x) be the restoring elasticity, the negative log-slope of the ratio of expected upward and downward deviations. The paper proves that h is even, strictly increasing on the positive half-line, and satisfies h(x) ≥ √(2π)/σ with equality only at x = 0 (Theorem 1). Hence under exponential gating a saddle-node with hysteresis appears if and only if χ > √(2π)/σ; near the threshold the fold location scales as (χ − χ_c)^{1/2} and the width of the bistable window as (χ − χ_c)^{3/2}, with closed-form coefficients in which π − 3 appears. Second, two versions of gating side by side. Symmetric gating gives two mirror wells—a pride branch and a self-abasement branch—with a bistable window symmetric about a₀ = 1 and an upper fold strictly above 1: once the fold is crossed, returning to “fully honest booking at zero commitment” is not enough to leave the pride branch. One-sided gating is more pessimistic: above the threshold, the pride branch is the unique attractor whenever a₀ ≤ 1, the lower edge of the bistable window is a non-smooth corner at the onset of gating rather than a saddle-node, and the low well is shallow; in simulation, at a₀ = 1.15 every replica falls onto the pride branch. There are only two ways out: temporary over-booking (a₀ > 1), or a sufficiently strong ungated external channel. At χ = 3, a_ext ≥ 0.065 suffices to leave at the honest baseline, and for a_ext ≥ 0.164 the fold disappears. Third, heavy-tailed feedback. The existence of the threshold is a property of light tails. Under Laplace and Student-t feedback, h attains its maximum rather than its minimum at zero, and the geometry flips: instead of two bounded wells there is a calibrated well plus an unbounded creep of pride. Under t feedback any positive gating suffices to produce this creep, and no baseline a₀ removes it; the creep rate is set by the tail of the feedback, x ~ t^{1/ν} under t(ν) (simulated slopes 0.303 and 0.189 against theoretical 1/3 and 1/5). Adding a_ext makes the pride branch bounded again and restores bistability. Under heavy tails the external channel changes from “helpful” to “a necessary condition for boundedness”. Fourth, the negative-attribution regime. Between a = 0 and a < 0 there is a boundary of sign, not of degree: at a = 0 belief creeps upward ever more slowly, while for a < 0 it runs away at least as e^{α|a|t}. This writes the statement of [3]’s fifth component, that a sign flip is worse than a decline to zero, as a growth law. Fifth, a negative identifiability result and its repair. The second scope clause of The Law of Action [4] separates estimation problems from value problems by asking whether stating the probabilities clearly would change the choice. For an agent with a → 0, this test classifies it as a value case with probability one: the Kullback–Leibler divergence between the observation distributions under the two hypotheses is identically zero. A two-sided probe on the same channel separates them by the asymmetry index Γ = (1 − a)/(1 + a), and Γ is precisely the order parameter of the model in the first part. Six falsifiable predictions are registered, the sharpest being the threshold prediction: in populations where the elasticity of the probability of admitting error with respect to the size of public commitment exceeds √(2π)/σ, the cross-sectional distribution of Γ should be bimodal. Number of executions on external data: zero.