Reciprocity without losing causality
Research preprint · version 1When parts of a system affect each other, can we simplify each part and still describe what the whole can do? We found a sharp rule for a finite class of systems — and a small example in which simplification invents an equilibrium that the original system cannot occupy.
7 November 2026 · Petri Mikael Autio · AI-assisted research and internal review. Priority and external mathematical review remain open.
The question drawn from Hegel
The project began with the Hegel Reader's discussion of formal causality. Its primary anchor is the end of Hegel's Doctrine of Essence. In the Encyclopaedia, §§154–156 and the Addition to §156, cause and effect develop into reciprocal action. Hegel then asks for more: saying that two things cause each other still leaves the relation unexplained. The sides must be understood through the whole of which they are parts.
That suggested a testable question. If a higher-level description claims to express the unity of interacting parts, does it preserve their actual causal possibilities? Or has it made the parts appear to agree by concealing the distinctions through which their interaction occurs?
The mathematics below is our finite interpretation of that question. Its state spaces, interventions and uniqueness assumptions are explicit modelling choices. A feedback graph does not by itself express Hegel's Concept, and a fixed point is not a mathematical definition of freedom.
Two parts, nine states, one false equilibrium
Each component has three states: 0, 1 and 2. A copies B. B responds to A by keeping 0 and swapping 1 with 2. Only (0, 0) satisfies both rules at once. Fixing either component externally also leaves exactly one solution.
Now report 0 as OFF and merge 1 and 2 into ON. Each simplified rule copies the other component. The simplified system therefore appears to allow both (OFF, OFF) and (ON, ON). The second has no fine equilibrium behind it.
An unchanging description can hide continuing change
- (1, 1)
- (1, 2)
- (2, 2)
- (2, 1)
The last state returns to the first. Every state in this cycle is described as (ON, ON).
Fine state (1, 1). Coarse description (ON, ON).
This displays one synchronous orbit. The theorem concerns simultaneous-equation solutions under interventions, not convergence of an update process.
Try an intervention or retain more detail
A point intervention fixes a chosen component and replaces its equation. The other equation stays in force. Coarse clamps use the labels of the selected fine values.
Fine equilibrium: (0, 0).
Coarse equilibria: (OFF, OFF); (ON, ON).
One coarse equilibrium has no fine lift. Local mechanism commutation holds, but the causal solution sets disagree.
Retaining only one component exposes a distinction whose input still merges 1 and 2. Its mechanism then ceases to be well-defined through that interface. In this example both components must be retained for a descending repair.
Two three-state components are the smallest possible failure in the stated class with no self-dependence. All minimum two-component failures are this example up to relabelling. That classification is proved in the paper.
The sharp rule
Assume that the fine system has exactly one solution after every point intervention, including no intervention. Assume also that each simplified rule agrees exactly with its fine rule through the component maps. Classify each component's map:
| Kind | Retained distinctions | Example |
|---|---|---|
| Fully retained | Every fine distinction | 0, 1, 2 → 0, 1, 2 |
| Wholly erased | No distinction | 0, 1, 2 → * |
| Partially simplified | Some retained, others merged | 0, 1, 2 → OFF, ON, ON |
For fixed component maps and a loopless allowed dependency graph, every compatible simple fine system preserves its unique solutions under all matched point interventions if and only if there is no directed cycle entirely among the partially simplified components.
The whole system may contain feedback cycles. A cycle of partial components means that the universal guarantee is unavailable; a particular installed system may still work. The graph condition transfers a fine solvability guarantee that must be established separately.
The proof starts from any proposed coarse solution, temporarily fixes the fully retained components, and uses fine simplicity to solve the rest. The partial components can then be evaluated in order because their graph is acyclic. Injectivity restores the temporarily replaced equations. Conversely, a partial cycle can hide one reversal around the loop, creating the obstruction.
A repair must retain the inputs its outputs need
Keeping more output detail can require keeping more input detail. The second result makes that requirement exact for upgrades that either retain a whole component or leave its map unchanged.
Draw a hidden-input edge from B to A when A's full output distinguishes two B values that the old interface merges. The upgraded coordinates must include every hidden-input predecessor of every upgraded output. Repeating this requirement gives the complete set of forced upgrades.
A certified repair must both satisfy that closure and break every cycle among the remaining partial components. With nonnegative costs, the paper proves an exact minimum-cost formulation. It is an optimum for these certificates and whole-coordinate upgrades; intermediate partitions and all semantically valid abstractions are broader problems.
What has been checked
| Family | Coverage | Result |
|---|---|---|
| Two components | 900 systems; every partition for sizes 1–3 | 5,206 simple descending pairs; 18 failures, all the minimum example up to relabelling |
| Three Boolean components | 4,096 systems; eight identity/erasure interfaces each | 2,928 simple descending pairs; all safe |
| Cycle construction | 64 graphs × 27 representative interface assignments | A simple counterexample verified in all 135 cases with a partial cycle |
| Retention | 23,276 full-coordinate upgrades | Direct factorization agrees exactly with hidden-predecessor closure |
Two separately written Python programs agree on all overlapping counts. They evaluate the actual equations before comparing the answers with the graph criteria. The Boolean search has no partial maps, so it tests the retained/erased cases. The cycle search checks constructed obstructions; it does not exhaust every three-component ternary system. The general result rests on the proofs.
What may be new, and what is already known
Cyclic causal abstraction and value merging already have a substantial literature, including Geiger and colleagues. Steggles's work on finite networks already recognizes that a coarse fixed point may represent hidden fine motion. Unique solvability and acyclic fixed-point arguments are also established.
The strongest candidate contribution is the exact universal classification by cycles of partial maps, together with the smallest-obstruction classification. We found no matching theorem in the targeted sources. That does not establish priority. The paper presents complete arguments and a reproducible record so that other researchers can assess the claim.
The promise for Hegelese
Hegelese asks what a change of representation preserves, what it relinquishes, and what evidence supports those claims. This research suggests a further check when components interact: does a revision preserve causal solutions after feedback connections are made?
A future checker could combine local mechanism equations, separate evidence of fine simplicity and the partial-cycle test. If an upgrade fails, it could return the two input states that expose the missing distinction. If a partial cycle remains, it should report that the universal guarantee is unavailable, rather than declaring the actual system incorrect.
This is where the Hegelian inspiration becomes operational. The proposed description is tested against its own preservation claim. Its failure identifies what the claim overlooked, and a repair must account for that content. This is a research proposal for Hegelese; the present package does not change the language or its checker.
The next question is how to certify an installed system when its partial graph has cycles, without enumerating all interventional equilibria. Another is how to optimize intermediate value partitions. Both remain open here.
I directed this research. AI assistants developed arguments, code, literature comparisons and separate internal critiques. The checks support the proofs; they do not replace them, establish novelty, or constitute external peer review.