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Adjacency Theory markAdjacency Theory

Theory · Research architecture

Structure · Identity · Transformation · Constraints · Discovery

Five parts. Every experiment on this site belongs to one of them, declares which hypothesis it tests, and states what would falsify it.

Adjacency Theory hypothesisadjacency theory hypothesis · not established physics

Research architecture

  1. 01

    Structure

    What is the relational state, and what is it made of?

    A modelled system is a set of objects and a relational state R holding a coupling R_ij for every pair. Effective separation is derived from R, never stored alongside it. If a quantity cannot be computed from R, it is not part of the structure.

  2. 02

    Identity

    What must stay the same for an object to remain itself?

    Each tracked object carries an identity vector I. Modifying relations incident to that object applies strain to I. Identity deviation ΔI = ‖I′ − I‖₂ is the accounting for that strain, and every transformation is judged against a declared tolerance.

  3. 03

    Transformation

    Which changes of relational state are permitted, and at what cost?

    A transformation is an ordered list of edits R → R′. Its cost is C_A = Σ|R′_ij − R_ij| and its efficiency is Γ_A = |ΔD| / C_A. Transformations are recorded step by step so any claimed result can be replayed exactly.

  4. 04

    Constraints

    What would make a result invalid?

    Metric validity and numerical stability are audited and can genuinely fail. Causality, energy budget, geometric reconstruction and physical correspondence are reported NOT MODELED, because they are not modelled. A gate is never marked PASS by default.

  5. 05

    Discovery

    What does the model do that we did not tell it to do?

    Bounded automated search proposes transformations without being told which relation to change. Failures, infeasible populations and unexpected candidates are first-class results and stay on the page.

Core definitions

Relational address

R_X = (r_X1, r_X2, …, r_XN)
An object's position is not a coordinate. It is the list of its relations to everything else in the modelled system.

Two objects are close when their addresses make the derived separation small — not because they share a region of a background space. Changing an address means rewriting relations, not translating a point.

Relational neighbourhood

N(X) = { Y : r_XY ≥ θ }
The local relational structure around X: the objects whose couplings to X dominate its address.

Neighbourhood similarity measures how strongly X belongs to a given group. In AT-0021 the Traveler's coordinates never change; only its membership in N(Origin) and N(Destination) does.

Transition operator

T(R_i, R_j) → { allowed, cost }
An abstract statement of whether one relational state may become another, and what it costs.

T is not a dynamical law and does not evolve anything in time. Today it is defined only through declared cost and constraint functions. Its structure — corridors, forbidden regions, invariants — is what AT-0017 onward is meant to probe.

Hypothesis records

H1

Relational Distance Hypothesis

Adjacency Theory hypothesis
D(A, B) = F(R)
Claim
Effective separation between two objects is a function of the relational state of the system rather than an independent primitive.
How it is tested
Reproduce known separations from R alone; show that distance responds to R in a stable, path-consistent way; find where the mapping breaks.
What would falsify it
A modelled system where two configurations share an identical relational state but demand different separations.
H2

Identity Preservation Hypothesis

Adjacency Theory hypothesis
‖I′ − I‖₂ ≤ τ while N(X) changes
Claim
An object's identity state can be held within tolerance while its relational state — including its neighbourhood — is substantially rewritten.
How it is tested
AT-0016 identity failure boundary; AT-0021 relocation under tolerance; strain accounting per transformation step.
What would falsify it
Every transformation that meaningfully changes N(X) also drives ΔI beyond any declared tolerance.
H3

Discrete Transition Hypothesis

Adjacency Theory hypothesis
T(R_i, R_j) without a required intermediate chain
Claim
It is worth investigating whether a transition model necessarily requires every classical intermediate spatial state, or whether a relational description permits transitions that do not enumerate them.
How it is tested
AT-0023 continuity challenge; path-independence tests; comparison of transformation histories under different orderings.
What would falsify it
Every admissible transformation, under every ordering, is found to pass through the full intermediate chain.

Experiment registry · AT-0011 → AT-0023

idexperimentgatestatuspurpose
AT-0011Adjacency Threshold DetectionAT-2implementedLocate the relational state at which effective separation first satisfies D ≤ ε, and record the cost of reaching it.ε is a chosen number in a graph model. Crossing it means nothing physical.
AT-0012Adjacency StabilityAT-2prototypeDetermine whether a state satisfying D ≤ ε survives bounded perturbation of the relational state, or collapses immediately.Stability of a simulated configuration is not stability of anything real.
AT-0013Reversal and HysteresisAT-2specifiedReverse an accepted transformation step by step and test whether the return path costs the same as the outbound path.Path dependence in a greedy search is a property of the search, until shown otherwise.
AT-0014Adjacency DebtAT-2specifiedMeasure whether reducing D between one pair systematically increases separation elsewhere in the same relational state.Any conserved-looking quantity here is a bookkeeping artefact until derived, not assumed.
AT-0015Multiple Simultaneous AdjacenciesAT-2specifiedAttempt to satisfy D ≤ ε for several pairs at once and find where the demands become mutually unsatisfiable.Infeasibility in a toy optimiser is a statement about the optimiser.
AT-0016Identity Failure BoundaryAT-2implementedPush the transformation until ΔI exceeds tolerance, and characterise where identity preservation fails.Identity here is a stipulated vector. Its failure boundary is a property of that stipulation.
AT-0017Conservation Surface SearchAT-3specifiedSearch the space of transformations for any quantity that stays invariant across accepted moves.An invariant of a model is not a conservation law. Deriving one would require the model to mean something first.
AT-0018Path IndependenceAT-2specifiedTest whether different transformation orders reaching the same final R also incur the same adjacency cost.Order effects are combinatorics, not thermodynamics.
AT-0019Self-Organised AdjacencyAT-3specifiedLet local update rules act on the relational state without a global objective, and see whether low-separation structure emerges unbidden.Emergence in a cellular rule set is common and cheap. It carries no explanatory weight by itself.
AT-0020Discovery SweepAT-2implementedRun the bounded optimiser across seeds and collect the candidate transformations, the failures, and the unexpected results.A ranked candidate list is a ranked list of edits to numbers.
AT-0021Relational RelocationAT-2implementedChange a tracked object's relational neighbourhood from Origin to Destination while preserving its identity vector. The object never moves.MATHEMATICAL / COMPUTATIONAL MODEL. PHYSICAL EVIDENCE: NONE. This is not relocation of anything.
AT-0022Minimum Relocation CostAT-2implementedGrid-search the cheapest relational transformation that reaches the destination-similarity target inside the identity tolerance.A minimum over a chosen grid, under a chosen cost function.
AT-0023Continuity ChallengeAT-2implementedInspect the accepted transformation history and report whether the modelled transition passed through intermediate relational states.Absence of intermediate states in a graph model is not evidence that a physical transition can be discontinuous.
AT-0024Spectral Connectivity ResponseAT-3implementedMeasure how algebraic connectivity λ₂ of the weighted Laplacian responds to bounded single-relation changes.λ₂ is a whole-graph observable, not a separation between two objects.
AT-0025Topology Robustness SweepAT-3implementedRun one perturbation protocol across random, small-world, scale-free and lattice families to see what survives a change of topology.Robustness across toy families is not robustness across physical systems.
AT-0026Observable CompetitionAT-3implementedRank geodesic, resistance, efficiency, λ₂ and spectral radius by measured response and noise robustness for one identical transformation.A high rank makes an observable responsive, not real.
AT-0027Scale SweepAT-3implementedRepeat the probe protocol across bounded graph sizes and fit a scaling trend only when the log–log diagnostic supports one.Five bounded sizes cannot distinguish a power law from a logarithm.
AT-0028Noise and Adversarial StabilityAT-3implementedAdd bounded noise and randomised control changes, then check whether candidates persist, disappear or reverse sign.Survival is measured inside one generator and one observable.
AT-0029Invariant SearchAT-3implementedLook for quantities that stay fixed under accepted transformations, separating invariants imposed by construction from invariants actually discovered.No invariant here is a conserved physical quantity; energy and causality stay NOT MODELED.
AT-0030Missing-Rule SearchAT-3implementedScore candidate response laws O = F(R, I, T, Q) on response, cost, identity deviation, invariance and stability minus complexity, then attack the winner with seven countermodels.A promoted candidate is a candidate response law inside a toy model. It is not new physics.