Protective Switching in Ordinal-Sum Systems

What the paper studies

A logical readout can remain positive while coming arbitrarily close to its threshold. This paper examines that distance in one explicit two-node dynamical realization. Two modes select different product-ordinal-sum operations while sharing the second-coordinate update. The observation margin is the smallest coordinate distance to the threshold along the complete exact trajectory, including the starting state.

The questions are global as well as asymptotic: which states have zero margin, when a first protective switch is enough, and which switching schedule is worst under a bound on consecutive beta updates.

Main results

• The constant-beta zero-margin basin has a unique continuous, strictly decreasing, involutive critical boundary.
• For beta/2 <= alpha < beta, the switched and constant-beta zero basins agree. A first alpha use at any beta-critical point gives a positive margin under every subsequent switching schedule. This is a sufficient range, not a maximality claim.
• In the stated initial region, the periodic word consisting of K-1 beta updates followed by one alpha update minimizes both coordinates at every finite horizon among words with no K consecutive betas.
• At fixed parameters and a fixed exact beta-critical start, the late one-use margin is asymptotic to (2 beta / (1-beta)) times n to the power -3. The periodic margin has the coefficient (2 beta / (1-beta)) times zeta(3), multiplying K to the power -3. Outside the sufficient global range, identifying that periodic margin with the constrained infimum requires the stated initial-region condition.
• An exact certificate at alpha = 1/100 and beta = 3/5 gives a critical input with zero margin even under alternating switches. It also disproves global basin equality outside the sufficient range. This single obstruction does not classify the full remaining parameter region.

The paper distinguishes the positive margin of each individual fair word from the infimum over all fair words. Those are not interchangeable guarantees.

Scope and relation to B-Theory

Evolutionary Logic explores how continuous change relates to logical descriptions. This paper supplies one explicitly chosen dynamical realization relevant to evolutionary correspondence-matrix/logical-matrix representations. The state domain, ordinal sums, duality and switching contracts are additional assumptions. The results do not automatically extend to every CM/LM system, arbitrary monotone maps, or a new matrix-product algebra.

The margin concerns observation error applied to exact trajectories. It is not a guarantee about disturbances injected into the recurrence, stochastic noise, approximate critical initialization, cognition or consciousness. Every asymptotic limit fixes the parameters and critical initial state. Singular parameter limits and the maximal good region remain outside the claims.

Established nilpotent-parabolic theory supplies the local antecedent. The paper examines the model-specific global schedule comparison, terminal-margin transfer and certified obstruction. Historical priority remains unresolved.

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