Phase-Tuned Relationalism

Abstract

Phase-Tuned Relationalism (PTR) is a computational framework for asking when weighted, phase-bearing relational graphs exhibit organization that can be measured with geometric diagnostics. The program begins with a finite hypergraph, reduces it to tractable graph operators, and examines resistance distance, heat-kernel return, regional spectral concentration, cycle-phase statistics, and reconnection dynamics. These objects are mathematically well-defined on the discrete model; their interpretation as physical geometry is a separate and stronger claim.

The current evidence produces a mixed result. Global spectral and Hausdorff-like trends do not isolate a PTR-specific dimension and do not establish convergence to a finite-dimensional continuum. Paired regional diagnostics do identify localized changes in spectral concentration and cycle-phase organization after the tested dynamics. A finite one-dimensional locality control exhibits operationally geometric regimes, while the current reconstruction Hamiltonian does not itself generate a sufficient locality bias under seven tested readings. Adding an explicit term referenced to a stable external metric provides a positive control. Self-referential metric construction remains unresolved, and a later swap-move study is withheld from numerical publication because its generating code and raw data are not present in the current source set.

This paper reports those positive, negative, and corrected outcomes at their actual scope. It does not claim a derived continuum, physical spacetime emergence, quantum gravity, or a universal no-go theorem for self-reference.

Introduction

Geometry as a question, not a premise.

PTR starts from a conservative question: if the primitive object is relational rather than spatial, what measurable conditions would justify describing any resulting organization as geometric? The framework does not assign coordinates to its primary substrate. It represents degrees of freedom as nodes, multi-party relations as weighted hyperedges, and coherence information as phases. Geometry enters only through operators and diagnostics constructed from that relational data.

This separation matters because a graph can display short paths, clustered structure, or visually suggestive embeddings without approaching a continuum geometry. A serious test must compare the evolving system with bare and null controls, specify finite-size and fitting effects, and distinguish the observable from its physical interpretation. PTR therefore treats “emergent geometry” as a staged research program rather than a single conclusion.

Research question

Under what finite dynamics can relational structure develop persistent organization that satisfies explicit geometric diagnostics, and what additional evidence would be required before that organization could support a continuum or physical interpretation?

The work reported here addresses only the finite computational portion of that question. It includes formal construction, null-model comparison, localized graph response, one-dimensional locality controls, Hamiltonian ablation, and a correction record. Higher-dimensional recovery and continuum convergence are not established in the inspected evidence.

Claim discipline

Four tiers prevent evidence from carrying more weight than it has.

The framework uses four claim tiers. Each tier depends on the one above it and requires additional evidence. A graph-level measurement does not become a physical result because its vocabulary resembles geometry.

Tier	Claim class	Current standing	Boundary
1	Graph / operator	Formal construction and finite computational observables.	Directly defined on the model.
2	Geometric	Operational organization under stated diagnostics and controls.	Finite-model evidence; not automatically continuum geometry.
3	Physical	Conditional interpretation only.	Requires a validated bridge from graph observables to physical measurements.
4	Quantum gravity	Aspirational research direction.	No established result in the current program.

Formal framework

A phase-bearing relational substrate.

The PTR substrate is a weighted hypergraph G = (V, E, W, Φ). The node set V represents degrees of freedom; each hyperedge in E joins two or more nodes; W assigns positive correlation weights; and Φ assigns phase angles. The accompanying many-body state construction is a schematic computational ansatz, not a state derived as the ground state of a physical Hamiltonian.

From hyperedges to graph operators

For tractability, each hyperedge is expanded into weighted pairwise terms. The signed adjacency preserves the coherent phase contribution and divides the hyperedge weight across its pair count:

A_{i j}^{(sgn)} = \sum_{e \in E : v_{i}, v_{j} \in e} \frac{W (e) \cos (Φ (e))}{(\begin{matrix} | e | \\ 2 \end{matrix})}, i \neq j

The clique expansion makes standard spectral-graph tools available, but it collapses genuine multi-body structure into pairwise terms. That information loss is a modeling tradeoff, not a derived equivalence. The positive effective adjacency used for the Laplacian also includes a small connectivity floor in the original construction; the signed matrix remains a separate phase-sensitive observable.

The graph Laplacian and its Moore–Penrose pseudoinverse define the resistance-distance diagnostic:

L_{i j} = δ_{i j} \sum_{k} A_{i k} - A_{i j}

d (v_{i}, v_{j}) = L_{i i}^{+} + L_{j j}^{+} - 2 L_{i j}^{+}

Resistance distance is a valid finite-graph metric. Calling it “emergent distance” is Tier 2 language whose usefulness must be tested against scaling, locality, and null behavior. Its existence alone does not establish a continuum.

Computational methodology

Paired controls before interpretation.

The dimension program compares three modes using matched random seeds: a bare graph, the graph after PTR-style kernel embedding and reconnection dynamics, and a null perturbation. Global observables are evaluated across finite node counts. Regional diagnostics then compare the departure region, its wake, the target neighborhood, and a matched far region. This makes the relevant quantity a paired change—not an isolated dimension estimate.

Heat-kernel return probability is computed from the Laplacian spectrum:

P (t) = N^{- 1} \sum_{v} ⟨ v | e^{- t L} | v ⟩ = N^{- 1} \sum_{k} e^{- t λ_{k}}

d_{s} = - 2 \frac{d \log P (t)}{d \log t}

In a scale-free region, a linear fit of log P(t) against log t supplies a finite-window estimator for d_s. The protocol checks residual linearity, varies the fitting window, and reports medians, interquartile ranges, and replicate counts. A value that shifts substantially with the window is treated as a diagnostic, not a dimension measurement.

Experiment family	Primary comparison	Diagnostic	Main limitation
Global dimension	Bare / dynamics / null, paired by seed	Heat return and resistance-ball scaling	Narrow scaling regions; low replication at the largest N
Localized response	Affected regions / matched complement	Regional IPR, spectral and cycle-phase statistics	Finite region definitions; graph-level interpretation
Locality scan	Temperature and rewire fraction	Spectral estimate, shortcut density, fragmentation	Finite grid with analysis-defined regime labels
Hamiltonian ablation	Seven term readings	Effective locality bias and reattachment distance	N = 200 control study
Augmented control	Fixed and graph-derived metric references	Same operational criteria	Two replicates per coupling cell

Global dimension diagnostics

The global trend is a negative result for a clean continuum reading.

Initial bare-substrate runs measured heat-kernel and resistance-ball scaling at N = 50, 100, 500, and 1000. The return curves are smooth, but the selected log–log windows retain curvature. Apparent spectral dimension rises with graph size, while the Hausdorff-like estimate is more strongly affected by ball-volume saturation. At N = 1000, only three replicates were available.

Four heat-kernel return-probability plots for finite graphs with 50, 100, 500, and 1000 nodes, showing fitted scaling windows that retain visible curvature. — **Figure 1.** Heat-kernel return probability in the initial bare-substrate diagnostic. The curves remain mildly curved through the chosen fit windows, and the largest graph has only three replicates. These data motivate caution about single-number spectral-dimension claims; they do not show continuum convergence.

The three-mode comparison resolves the interpretation. The rising global trend and the separation between spectral and Hausdorff-like estimates occur in bare, dynamics, and null modes. The PTR-specific paired shift in global d_s decreases in magnitude as N grows. On the tested sizes, the global diagnostic therefore behaves like substrate background plus a localized finite perturbation—not a PTR-specific emergent dimension.

Localized relational response

Localized observables retain information that global averages erase.

The regional analysis asks where the dynamics change the spectrum. For a region R, the mode weight and inverse participation ratio are:

p_{k} (R) = {| R |}^{- 1} \sum_{v \in R} {| φ_{k} (v) |}^{2}

IPR (R) = \frac{\sum_{k} {p_{k}}^{2} (R)}{{(\sum_{k} p_{k} (R))}^{2}}

Paired dynamics-minus-bare comparisons show positive ΔIPR in the departure region across the tested finite sizes, while the whole-graph comparison stays near zero. The signal is therefore spatially localized in the graph partition used by the experiment.

Line plot of paired change in regional inverse participation ratio across graph sizes, with a consistently positive departure-region series and near-zero whole-graph series. — **Figure 2.** Paired regional ΔIPR after the tested dynamics. The departure region shows increased spectral concentration while the all-node aggregate remains near zero. Error bars and the anomalous far-region value at N = 100 are retained. This is a finite graph response, not a continuum invariant.

Cycle-phase organization

A complementary diagnostic sums phase around graph cycles and measures the concentration of the resulting angular distribution. The departure region shows a positive dynamics-minus-bare shift in angular resultant across the tested sizes, while other regions have larger uncertainty or approach zero. This establishes a change in a defined graph statistic. It does not establish curvature, topological holonomy in a continuum manifold, or a physical field observable.

Line plot of change in cycle-phase angular concentration for departure, wake, target, far, and whole-graph regions across finite graph sizes. — **Figure 3.** Change in the cycle-phase angular resultant, dynamics minus bare, paired by seed. The departure-region series is positive across the finite N grid; uncertainty grows in several other regions. The statistic characterizes graph-cycle phase organization only.

Continuum program

A one-dimensional control can test the machinery without proving a continuum.

Because the bare ensemble does not supply a clean finite-dimensional target, the next experiment imposes a latent one-dimensional locality reference and asks whether rewiring preserves or destroys it. The diagnostic classifies finite runs using spectral dimension, shortcut density, resistance growth, and fragmentation. These labels are operational analysis categories.

The broad scan used N = 350, 100 steps, four replicates, a finite set of locality temperatures, and two rewire fractions. A degree-floor variant prevents the graph from disconnecting. Across the sampled cells, the final spectral estimate remains in a roughly one-dimensional range, while shortcut behavior and the imposed floor determine whether the run is classified as geometric, small-world, or fragmented.

Two-panel finite phase scan showing operational regime labels and final spectral-dimension values across sampled locality temperatures and rewire fractions. — **Figure 4.** Broad finite scan with degree floor 2. Parameters: N = 350, 100 steps, four replicates. “Geometric” is an operational classification based on analysis thresholds, not evidence of a thermodynamic phase transition. Sparse sampling means the figure does not locate an exact critical temperature.

The experiment supports a narrower conclusion: strict locality is a workable positive control in finite 1D, and unconstrained reconnection tends to preserve shortcuts or destroy locality. It does not support current public claims of 2D or 3D recovery. It also does not prove that the same regime survives increasing N, vanishing lattice scale, or a change of graph ensemble.

Dynamical locality and self-reference

The present Hamiltonian does not supply the missing locality bias.

The reconstruction dynamics combine an inertia term with phase-alignment coupling between a kernel boundary and target boundary. In the tested implementation:

H_{recon} = λ_{L} \sum_{e \in \partial K} W (e) {(1 - n_{e})}^{2} - λ_{A} \sum_{e \in \partial K, e^{'} \in \partial T} M (e, e^{'}) W (e) W (e^{'}) \cos (Φ (e) - Φ (e^{'}))

Seven readings of the available terms were sampled at N = 200 with 800 candidate reattachments. None produced the sub-threshold locality preference required by the finite 1D control. The reattachment-distance ratio remained close to the uniform baseline, and the effective locality temperature remained above the operational geometric threshold used by that study.

Horizontal bar charts comparing effective locality temperature and reattachment-distance ratios for seven tested Hamiltonian configurations. — **Figure 5.** Locality test for seven readings of the current reconstruction Hamiltonian, using N = 200 and 800 samples. All tested configurations remain on the non-geometric side of the study’s operational threshold and near the no-bias reattachment baseline. The result is specific to these implemented readings.

Explicit locality as a positive control

The follow-up study adds a locality cost to the existing PTR term:

H_{aug} = H_{PTR} + H_{loc}

H_{loc} = λ_{G} \sum_{(i, j)} W_{i j} d_{eff} (i, j)

With a stable external 1D metric, increasing λ_G strengthens locality. In the saved two-replicate-per-cell sweep, the strongest coupling tested (λ_G = 1000) is classified as geometric, with d_s approximately 1.0, shortcut density 0.46, and bounded mean resistance near 15. This is a useful positive control: an explicit stable reference can drive the finite system toward the study’s operational target.

λ_G	Induced T_eff	d_s	Shortcut density	R_mean	Classification
2	0.58	1.14	0.96	5.1	Small-world
10	0.11	1.08	0.91	4.9	Small-world
50	0.026	1.15	0.76	5.1	Small-world
200	0.0067	1.17	0.58	6.8	Small-world
1000	0.0023	≈1.0	0.46	≈15	Geometric

The fixed reference is intentionally a control, not an emergence result. The graph does not generate that metric. Resistance- and diffusion-based references were also explored, but the current record does not support a general self-reference conclusion at the same provenance level. Recovered implementations or fresh reruns are required before a stronger statement is published.

Swap-move experiment

A consequential hypothesis awaiting a reproducible source chain.

A later study replaces unconstrained rewiring with degree-preserving swaps and asks whether a stable reference can sustain partial local order under continuous churn, and whether graph-derived metrics progressively legitimize existing shortcut defects. The surviving findings memo and composite chart agree internally on the proposed mechanism.

The study remains part of the research timeline because it sharpens an important question: can a metric that is updated from the same evolving graph distinguish historical defects from legitimate local structure? Publication requires either recovery of the original run package or a controlled rerun with saved seeds, raw trajectories, pair-sampling details, and uncertainty estimates.

Corrections and withdrawn interpretations

The final interpretation replaces the earlier one.

PTR’s correction record is part of the result, not a footnote. Several early readings were narrowed after stronger controls or implementation audits became available.

Earlier interpretation	Later test	Current interpretation
Global dimension trends suggested a PTR-specific emergent geometry.	Paired bare / dynamics / null comparison.	The trend is primarily substrate-level; the PTR response is localized at tested N.
Refinement preserved the signatures as an invariant.	Basis and replicate review.	Distribution-level pure subdivision showed no detected degradation, but invariance is not established.
A connectivity floor sustained self-referential organization.	Move-count audit.	100% of attempted rewires were blocked in that configuration. The sustain claim is withdrawn as a frozen-dynamics artifact.

The corrected conclusions supersede the old readings. They do not coexist as competing public claims.

Discussion

What the program has learned.

Three conclusions survive the present evidence. First, global dimension-like measurements on sparse, phase-weighted graphs are not reliable without bare and null controls, fit-window diagnostics, and replication. The most visually compelling trend can belong to the substrate rather than the dynamics.

Second, locality is the more discriminating level of analysis in the current finite models. Regional IPR and cycle-phase concentration detect structured changes that a whole-graph average suppresses. This makes localized response a legitimate Tier 1 result and a candidate Tier 2 diagnostic, while leaving physical meaning open.

Third, locality does not emerge automatically from the implemented reconstruction terms. A stable external reference can supply it, demonstrating that the operational target is reachable. The unresolved scientific problem is to construct a reference that is both endogenous and informative: smooth enough to avoid runaway response, yet non-degenerate enough to guide dynamics from a disordered state.

Negative results are central here. The absence of clean global convergence, the Hamiltonian ablation, the withdrawn frozen-floor result, and the provenance failure of the swap package constrain the program more usefully than a generalized claim of success would.

Limitations and open questions

The finite program does not yet close the conceptual loop.

Continuum convergence. No sequence of increasing graph sizes and shrinking effective scale has been shown to converge to a continuum operator or metric space.
Higher dimensions. The current verified controls are finite and one-dimensional. Supporting artifacts for newer 2D or 3D recovery claims were not located.
Ensemble dependence. Results may depend on sparse-graph construction, degree distribution, kernel size, phase initialization, and connectivity-floor choices.
Statistical power. Several largest-N points and augmented-sweep cells have low replicate counts.
Refinement. Basis changes prevent a strong per-seed invariance conclusion from the current refinement package.
Self-reference. A memory-bearing or history-sensitive metric could behave differently, but no reproducible current experiment establishes that path.
Physical bridge. No validated map currently connects the finite graph observables to laboratory observables, spacetime dynamics, or quantum gravity.
Reproducible swap package. The degree-preserving experiment requires source recovery or rerun before its proposed mechanism can enter the evidence record.

Conclusion

A disciplined finite-graph result.

PTR currently establishes a formal relational construction and a set of reproducible finite computational findings. The global diagnostics reject a simple continuum reading of the tested bare substrate. The regional experiments identify localized spectral and phase-organization responses. The one-dimensional controls show how locality changes operational behavior, and the Hamiltonian study identifies a missing locality mechanism. An explicit fixed-reference term supplies a positive control, while endogenous self-reference remains an open problem.

The strongest public position is therefore neither success nor failure of a theory of spacetime. PTR is an active computational research program whose value lies in separating what is defined, what is measured, what has been corrected, and what remains conjectural.

References

Scientific context and computational record.

S. Ryu and T. Takayanagi, “Holographic Derivation of Entanglement Entropy from the anti–de Sitter Space/Conformal Field Theory Correspondence,” Physical Review Letters 96, 181602 (2006). doi
J. Maldacena and L. Susskind, “Cool horizons for entangled black holes,” Fortschritte der Physik 61, 781–811 (2013). arXiv
G. Vidal, “Entanglement Renormalization,” Physical Review Letters 99, 220405 (2007). doi
P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter, and Z. Yang, “Holographic duality from random tensor networks,” Journal of High Energy Physics 2016, 9 (2016). doi
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2000). publisher
R. D. Sorkin, “Causal Sets: Discrete Gravity,” in Lectures on Quantum Gravity (2005). arXiv
S. Wolfram, “A Class of Models with the Potential to Represent Fundamental Physics,” Complex Systems 29, 107–536 (2020). doi
A. Almheiri, X. Dong, and D. Harlow, “Bulk locality and quantum error correction in AdS/CFT,” Journal of High Energy Physics 2015, 163 (2015). doi
L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, SIAM (2017). publisher
Korelis Labs, PTR dimension and three-mode diagnostics, unpublished computational record (May 2026).
Korelis Labs, PTR regional spectral and cycle-phase diagnostics, unpublished computational record (May 2026).
Korelis Labs, PTR finite-locality, Hamiltonian-ablation, and augmented-control studies, unpublished computational record (June 2026).

Research provenance

The web paper draws on formal manuscripts, implementation bundles, machine-readable result files, generated figures, and the final correction record. Empirical statements are limited to the scope supported by those materials.