Core2 restarts the ancestry problem from minimal intrinsic properties and local relationships. Earlier Core results are not automatically imported. Geometry, particles, binding, atomic architecture and compression must re-emerge through permitted relationships—or fail.
Timothy L. Harvey · Independent Researcher
This candidate precursor model investigates whether persistent differences in internal states can produce enlargement of relational distances. Its exploratory phase is closed. The model supplies explicit dynamics and reproducible tests; a match to physical space or cosmic expansion has not yet been demonstrated.
Core2 explored expansion as a behavior of an underlying condition. The prototype gives that question a specified mathematical mechanism: local activation establishes differing states, the bridge constraint connects those differences to metric change, and internal references make enlargement measurable inside the model.
Sixteen graph identities carry internal states q. Fifteen relations carry positive metric lengths r, change rates v, and internal reference lengths s. The graph stays fixed. A measured path is the sum of r/s along its links. All quantities are dimensionless model quantities.
Δ = qⱼ − qᵢ K = 0.1; γ = 0.2
Bridge energy = K s Δ² / (2 r)
ṙ = v
v̇ = K s Δ² / (2 r²) − γ v
q̇ᵢ = qᵢ − qᵢ³ + Σⱼ K sᵢⱼ (qⱼ − qᵢ) / rᵢⱼ
ṡ = 1 − s − K Δ² / (2 r)
E = Σ v²/2 + Σ(q⁴/4 − q²/2)
+ Σ[K s Δ²/(2 r) + (s−1)²/2]
Ḋ = Σ q̇² + Σ ṡ² + γ Σ v²
E + D = constant while the model remains defined.
The inverse-length bridge, its outward sign, activation, drag, time, metric scale and reference dynamics are supplied ingredients. The energy identity and the conditional scaling law follow within those ingredients.
For persistent opposite states and drag-dominated inverse dynamics, r³ ≈ (6K/γ)t + constant, so boundary-link length grows approximately as t1/3. Four persistent boundary links in the full-system check reproduced the predicted cubic slope within 0.10%. This is a conditional model result, not a fitted cosmological expansion law.
A1–A15 examined local mediation, state retention, complete energy accounting, spacing, numerical resolution and local geometry response. These models use different state and metric rules. Their results explain assumptions and limits; they do not replace frozen v1.
| Test | Result | Specific limit |
|---|---|---|
| Uniform active intermediates · A5 | Mismatch forms a localized interface. | Adding identical sites did not preserve inverse cost dilution. |
| Positive-locking continuum · A9–A10 | A small finite-length stretching drive survives and decays exponentially. | Under an added constant mobility and slow settling, this supports a logarithmic-growth closure, separate from v1’s cube law. |
| Local response · A14 | Length 16 → 24.8078 at model time 1,000,000. | Finest-grid differences: 0.014% in final length, 0.17% in final rate. Grid-normalized mobility was supplied. |
| Local mobility · A15 | Length 16 → 25.6515 with ḣ = βhF, β = 1/16. | Finest-grid differences: 0.021% in final length, 0.27% in final rate. The constitutive mobility remains supplied. |
| Fast local branch · A13 | Extreme gaps develop; convergence fails. | Large growth totals cannot be promoted as a resolved continuum result. |
| Gate | Status | Finding |
|---|---|---|
| A1 | Conditional derivation | Serial quadratic mediation yields inverse effective cost; Core predicates alone do not select its sign or length dependence. |
| A2 | Conditional mechanism | Strictly convex penalties support cost dilution; the exact quadratic rule is not uniquely earned. |
| A3 | Bounded retention | Free endpoints can retain differing states under supplied threshold dynamics; formation depends on coupling and initial conditions. |
| A4 | Conditional coupling | Retention and serial dilution coexist when interior states are freely accommodating; finite settling matters. |
| A5 | Generalization failed | Equal reinforcement at all sites localizes mismatch into an interface; the inverse dilution route fails in that regime. |
| A6 | Bounded approximation | Weak interior locking permits inverse cost approximation; comparable accuracy across longer chains requires weaker locking. |
| A7 | Bounded energy reduction | Full local-plus-link energy decreases across sampled chains, but positive locking produces a plateau. |
| A8 | Promotion withheld | Declared spacing energy favors stretching in sampled chains, but coarse grids exaggerate the drive. |
| A9 | Small drive survives | Direct stationary solves and a continuum check retain a small finite-length drive; one coarse centered branch is unstable. |
| A10 | Conditional scaling | Positive-locking continuum drive decays approximately exponentially with length; constant mobility implies a logarithmic late-growth closure. |
| A11 | Bounded dynamics | Finite-response states and length evolve consistently; fast late trajectories fail the initial resolution comparison. |
| A12 | Coarse excess rejected | Shared and null perturbations agree; refinement removes large coarse extra growth. Precise fast-case late-rate convergence remains open. |
| A13 | Bounded local response | Independent link lengths enlarge in finite tests. Large fast gaps remain under-resolved. |
| A14 | Refinement passed | Slower local response shows 0.014% agreement in final length and 0.17% in final rate between the finest grids. |
| A15 | Refinement passed | A fixed local mobility rule retains enlargement; finest-grid agreement is 0.021% in final length and 0.27% in rate. |
The Core dependency map P/C/I/A/D does not yet determine these quantitative laws. Physical meanings and units for state, metric, reference, time and energy remain open. Causal propagation, dimensionality, isotropy, material coupling and observed expansion history must be tested explicitly. Closure preserves these debts.
Handoff: RS-EXP-V1-CLOSE-A15-2026-10-03
Later toy-model calculations found configurations that evolved into persistent positive-potential-energy states and supported accelerating expansion over the simulated interval. The origin of physical cosmic expansion remains unresolved.
The frozen v1 graph model remains the baseline. The subsequent branch explored a different mechanism: internal configuration evolves under specified dynamics, while supplied general-relativity equations connect its energy and pressure to changing geometry. These results extend the investigation rather than replace the frozen v1 equations.
Some weak initial configurations evolved into persistent states with retained potential energy. Selected runs then sustained expansion and acceleration over the tested interval.
CONDITIONAL MODEL RESULTOther configurations contracted. Outcomes depended on initial preparation and the relative timescales of internal evolution and geometric expansion.
DEPENDENCE IDENTIFIEDSubtracting the retained potential-energy level eliminated sustained acceleration in a tested successful branch. The internal force law remained unchanged, but the gravitational energy changed; initial curvature was adjusted to satisfy the model constraint.
ENERGY-BASELINE DEBTThe internal dynamics, geometry coupling, potential-energy baseline and initial preparation remain ingredients requiring deeper justification. These calculations do not identify dark energy or establish a physical expansion mechanism.
PHYSICAL ORIGIN OPENA conditional computational pathway from an evolving internal configuration to sustained accelerating expansion within a specified model, accompanied by failed comparison branches. Finite-duration success does not establish indefinite expansion.
Core is being used as an ancestry tool to examine the pathways behind expansion and acceleration. Both established cosmological proposals and the toy research are assessed by distinguishing supplied ingredients, derived consequences and unresolved origins.
Timothy L. Harvey · Independent Researcher · Prototype v1 preserved · Subsequent research branch recorded
The R31–R50 sequence developed a coherent candidate pathway through localized entities, particle-like organization, composites and higher-order organization. Those gates were reasoned through conceptually rather than generated by an explicit executable dynamical system. R51 exposed that missing layer. The investigation therefore moved beneath the earlier results to determine whether the required behavior can actually be derived from minimal dynamics.
The earlier pathway remains useful as a candidate ancestry map, but it is not treated as numerical simulation, empirical particle formation or experimental verification.
Core1 produced an extensive bottom-up ancestry pathway. Core2 asks a harder question: if the pathway is rebuilt under a stricter Real Space framing without importing mature structures, what actually reappears?
Begin with minimal intrinsic properties and local relationships rather than mature physical objects.
Earlier results may provide comparison targets but do not automatically become Core2 ingredients.
Downstream work may not redesign an earlier candidate simply to make the next organizational level succeed.
Failed branches remain part of the ancestry record and constrain what may legitimately follow.
This diagram records the earlier conceptual development from R31 toward particle and composite organization. It is retained as research history and must be interpreted in light of the later R51 formalization audit.
These gates formed the original conceptual route toward the R38 particle-like candidate. Their historical status is preserved while their executable foundation remains under audit.
A candidate bounded concentration regime was identified conceptually. R51 later showed that its executable dynamical foundation had not been established.
CONCEPTUAL CANDIDATEThe candidate organization was reasoned to follow its own organization rather than remain attached to a privileged substrate location.
CONCEPTUAL / BOUNDEDContinued directional propagation with organizational continuity was developed conceptually.
CONCEPTUAL / BOUNDEDOrganizational identity was tested conceptually against changes in motion, direction, stopping, restarting and reversal.
CONCEPTUAL / BOUNDEDThe candidate was audited for localization, characteristic scale, organizational identity, rest and propagation behavior.
CONCEPTUAL CENSUSA reproducible graded response to changes in propagation state was interpreted cautiously as inertia-like behavior.
CONCEPTUAL / BOUNDEDConceptual entities were tested for mutual response, encounter survival and systematic exchange behavior.
CONCEPTUAL / BOUNDEDWithin the conceptual ancestry criteria used at that stage, the R31–R37 candidate crossed the primitive particle-like gate. Its conceptual properties were frozen so later composite work could not retrofit the candidate to obtain a desired higher-level result.
Multiple identical frozen candidates were conceptually capable of persistent configurations within a limited interaction regime.
The conceptual system developed preferred separation, characteristic composite scale, internal timescale, collective center, perturbation recovery and a dissociation threshold.
Identical candidates could occupy different organizational roles. Roles followed relationships rather than permanent particle identity.
These remain conceptual ancestry results pending successful executable reconstruction from the lower-level dynamics.
Identity was associated with continuous relational organization and organizational history rather than fixed position or constituent labels.
CONCEPTUAL / BOUNDEDComposite architecture became a candidate source of architecture-specific response and selective association without introducing a new fundamental force.
CONCEPTUAL / BOUNDEDAssociated composites were conceptually capable of higher-order organizational identity and characteristic architecture.
CONCEPTUAL / BOUNDEDA recurring organizational construction pattern was identified, while unlimited recursion and formation accessibility remained unearned.
CONCEPTUAL / OPEN LIMITSR51 stopped the progression of conceptual higher-level gates and asked whether the earlier behavior could be generated by an explicit dynamical rule:
The earlier conceptual sequence had not established executable interaction functions sufficient to reproduce localization, binding and composite behavior. R31–R50 therefore remain candidate conceptual ancestry rather than numerical demonstration.
The simplest conservative local equalization rule was diffusive. Localized disturbances dispersed.
LOCALIZATION FAILState–relationship feedback became possible, but neutral positive adaptation still strengthened smoothing.
PARTIAL / LOCALIZATION FAILBounded node and relationship states limited redistribution but did not create localization.
FAILThe smoothing family could alter rates but could not create a stable nonuniform localized state.
PASS-STRONG / NECESSITYThe frozen smoothing dynamics could not spontaneously generate a second dynamical regime.
FAILAn independent relational state could carry history and support propagating relational disturbances, but localization remained unearned.
PARTIALTopology was explored as a persistence route, but protection required additional conditions and no finite localization scale was derived.
CONDITIONAL / SCALE FAILLocality, conservation and symmetry do not themselves require differences to decay. Diffusion was one dynamical family, not the entire foundation.
FOUNDATIONAL CORRECTIONA proto-state can alter relationships that subsequently affect its evolution. This established feedback architecture, not cohesive feedback.
CANDIDATE / BOUNDEDFormation history can matter only if it leaves information in the present system state.
PARTIAL / NOT EXECUTABLEIndependent relational state can retain formation information, but the persistence law was initially underdetermined.
UNDERDETERMINEDAutomatic reset requires an extra preferred state and restoration rule. In the absence of further cause, persistence of the current relational state is the weaker assumption.
PASS-BOUNDEDPersistent relational imprint does not determine whether feedback is stabilizing, destabilizing or neutral.
UNDETERMINEDThe investigation then moved away from searching for an assumed attractive “glue.” The question became whether formation could alter the future transition possibilities of its own relational environment.
Persistent relational imprint can create locally altered transition possibilities, supplying a candidate boundary mechanism without earning confinement.
PARTIAL PASSA dynamically active relational state must be capable of changing subsequent local transition possibilities.
PASS-BOUNDEDLocalized persistent relational modification creates an inside/outside interface without inserting a prebuilt wall.
PASS-BOUNDEDDynamically different relational regions can respond differently to the same disturbance.
PASS-BOUNDEDNothing earned so far selects suppressed outward propagation. Choosing that sign would manufacture confinement.
FAILFormation can produce a causally asymmetric relational environment through persistent present-state differences.
PASS-BOUNDEDFormation-generated relational occupation can reduce locally accessible transitions in some candidate states.
PASS-BOUNDEDDifferent relational constraints can yield different organizational lifetimes, permitting stability filtering.
PASS-BOUNDEDDifferent local histories can leave different persistent relational organizations under the same primitive rules.
PASS-BOUNDEDFormation-generated constraints can accidentally overlap with transitions that would otherwise destroy the organization. Such self-preservation is possible but not guaranteed.
PASS-BOUNDEDThis sequence asks the harder question: can differential persistence become genuinely bounded organization rather than merely slow decay?
Long lifetime is not localization. Slowly dispersing or collapsing organization cannot be promoted to a persistent bounded entity.
FAILExcessive spreading can destroy relational continuity while excessive concentration can encounter finite local capacity. Opposing failure limits therefore define a finite admissible organizational range without specifying a preferred radius.
PASS-BOUNDEDStability filtering can retain trajectories that already remain inside the finite window, but it cannot manufacture a bounded trajectory if none exists.
PARTIAL PASSChanging relational state can change dynamics without forcing the direction of structural evolution to reverse.
FAILThe present relational configuration need not exhaust the dynamical state. A separate change-state can distinguish otherwise identical configurations changing in different directions or at different rates.
PASS-BOUNDEDLocal interaction can modify the relational change-state without prescribing its sign. Reversal therefore becomes dynamically possible, but the appropriate stabilizing correlation has not been derived.
PASS-BOUNDEDR51-R19 established a candidate finite organizational survival window by excluding excessive spreading and excessive concentration. R51-R20 and R51-R21 then showed that neither stability filtering nor state-dependent dynamics alone forces an organization to remain inside that window. A finite permitted range is therefore not yet a persistent localized entity.
R51-R22 and R51-R23 added another ingredient: relational change can itself carry state and that change-state can itself be modified. This makes reversal dynamically possible without yet explaining why the correct reversal should occur near either survival limit.
The ancestry does not independently select reduced outward propagation at the relational interface.
FAIL PRESERVEDSlow dispersal or slow collapse cannot be relabeled as persistent bounded organization.
FAIL PRESERVEDChanging relational state changes dynamics, but no earned rule yet links approach toward a destructive limit with reversal away from it.
FAIL PRESERVED“Complexity must earn its ancestry.”
The historical R31–R50 pathway remains preserved, including the R38 conceptual particle freeze. R51 changed the evidentiary standard by requiring an explicit foundation beneath that pathway. The present work therefore does not assume that the old particle result must survive.
The new foundational route has progressed from formation-generated relational imprint through dynamically relevant interfaces, constraint, candidate diversity, differential persistence and a finite survival window. However, persistent bounded localization remains unearned.
The parked foundational checkpoint is R51-R24 — Boundary-Proximity / Change-State Coupling Gate . No later localization or particle result is assumed in advance.