Conviction Systems: Openness, Hysteresis and Lock-in Dynamic Percolation under Attribution Gating, Screening Length, and the Topology of Error-Correction Channels

Whether a consensus forms, and whether the framework itself is right, are not this paper's questions. Its question is: once a group has formed a consensus around a framework, can that consensus dissolve when the drive is withdrawn? Model. The paper writes three quantities already in the series as the microscopic rates of a single percolation process. A “bond” between two individuals means that their readings are coupled through the same framework; an uncoupled tie is called an independent tie, and a reading can pass along it without being rewritten. Bonds form from two sources: the external drive G (threat, promotion, testimony), and the confirmation s₀ produced by the cluster itself, which is the self-referential shaping of [S6]. Bonds break through failures that can be attributed to the framework, at a rate gated by the attribution rate a: the larger the cluster, the larger the veto set and the higher the price of admitting error h_org = max h_i, so the fewer failures can be attributed; the strength of this gating is written χ. Confirmation and gating are broadcast within the cluster: the rate of a bond depends on the overall size of the cluster it belongs to, not only on its nearest neighbours. This assumption is natural for organisations with broadcast channels such as group chats and a lecture stage; the geometric conclusions of Section 5 depend on it, and the monotonicity of the lock-in threshold does not. Classification. By the position of the lower fold, conviction systems fall into three classes: open (monostable), reversibly hysteretic (lower fold G_down > 0) and closed (G_down ≤ 0). Gating χ and self-referential confirmation s₀ form a two-by-two arrangement, and the quasi-static mean-field analysis and the full dynamic simulation agree. Two of its entries are direct consequences of the model's structure: when both are zero, consensus formation is ordinary percolation, continuous and reversible; without self-referential confirmation the formation rate is zero once the drive is withdrawn, so gating alone can produce only reversible hysteresis, and consensus necessarily dissolves when the drive is lowered far enough. What carries information is the lock-in condition: self-referential confirmation exceeds the lock-in threshold s₀*(χ). This threshold is finite even without gating, so self-referential confirmation alone suffices for lock-in; gating does not by itself produce lock-in, but it lowers the threshold. That the threshold falls with χ is a theorem: in the quasi-static approximation it has a closed form and decreases strictly; the dynamic rule is attractive, and a monotone coupling built with common random numbers proves that it is non-increasing at any finite size and any criterion time. The numbers serve only as examples: in the dynamic simulation the threshold on the two-dimensional square lattice falls from about 2.0 to about 0.5 as χ rises from 0 to 4, and on the random network from about 1.7 to about 0.4. In time units, the lock-in condition is s₀/(r_f·a₀) > s₀*(χ): lock-in is a joint property of self-referential confirmation, the attribution rate and the price of admitting error, and cannot be read from s₀ alone. Whether gating takes the population share of a cluster or its absolute size decides whether “large organisations lock in more easily”: with the share, the threshold does not depend on system size; with absolute size (for example ln n), both the lock-in threshold and the lower fold tend to zero as the system grows. Geometry. With rates broadcast within clusters, the order-parameter exponent of two-dimensional percolation is 5/36 and its slope diverges at the critical point, so the same gating produces much wider hysteresis in two dimensions: in the dynamic simulation, at gating strength 1 the two-dimensional hysteresis has width 0.14 and the random network only 0.03; if the rates take only local, nearest-neighbour quantities, the order parameter no longer appears in the quasi-static equation and this geometric difference disappears. Under independent, identically distributed occupation, in geometries whose bond-percolation threshold is not below 1/2, a spanning consensus cluster cuts the independent ties into islands; on the two-dimensional square lattice, whose threshold is exactly 1/2, the implication runs both ways, so that spanning consensus and islanded dissent are one geometric event, and the statement of [S5] that “atomisation and homogenisation are the same thing under the maximum operator” becomes a geometric fact; in geometries whose threshold exceeds 1/2 there is also an interval in which neither spans, with atomisation but no homogenisation. The exclusion comes from the threshold, not from planarity: on the triangular lattice, also planar, the threshold is about 0.347 and the two can coexist over an interval; so too on a random network. Point checks on dynamic steady states support these conclusions within the measured range. A longer screening length does not simply make lock-in easier. Identification. In a crisis, threat and failure signals rise together, so “reality worsens and consensus does not fall” cannot identify lock-in; lock-in can be read only after the drive falls. The dynamic simulation also shows that in finite systems the sweep-rate halving test of The Law of Action [S16] can read a system that is bistable at the measured size and time as metastable. The test must therefore be combined with size comparison and fixed-drive tests. The paper registers six predictions. Executions on external data: zero.