DFT Research Discipline

Formalization Targets

This registry decomposes Digital Fabrica Theory into explicit assumptions, theorem candidates, lemma dependencies, and proof obligations. It is a public roadmap for formalization, not a claim of completed proof.

Bibliography and Reference Graph Review Path Submit Formalization Review

Target	Domain	Status	Mechanization	Boundary
Infinite Stabilisation Formula — Ordinal Stabilization Target	logic	Mechanization target	Lean	Formalization target. Public description must not be treated as completed proof until mechanized and externally reviewed.
PI-DST — Fractal Routing Growth Target	fractal-geometry	Mechanization target	Lean	Formalization target. Scaling claims must remain conditional on stated assumptions and empirical implementation tests.
Spectral Expansion Routing Lemma	graph-theory	Verified reference	Lean	Established graph-theoretic references may be cited, but DFT-specific routing/security claims require separate modeling and validation.
Policy Knot Invariance Lemma	knot-theory	Verified reference	Coq	Knot invariance is an established mathematical concept; its DFT governance use is an authorial formalization target.
DFDF → FNS → IDFF → SIDS Composition Theorem Candidate	systems-architecture	Lemma map drafted	Model checker	Architecture model and theorem candidate. Requires implementation evidence and formal interface definitions.
Zeta-Indexed Economic Model Target	economics	Research extension	Simulation	Research extension. Not financial advice, token offering, or validated macroeconomic result.

Formalization Boundary

This route is a blueprint for formalization. It does not claim that ISF, PI-DST, fiber dynamics, or the 14D tensor ontology are proven, externally reviewed, endorsed, or operationally guaranteed. The route exists to expose the assumptions, definitions, lemmas, proof obligations, simulation obligations, and reviewer questions required before stronger claims can be made.

Proof-obligation blueprint

Candidate statements before theorem claims

Each candidate below must pass through definitions, assumptions, lemma dependencies, proof obligations, simulations, and external review before stronger public language is allowed.

P0theorem-candidate

ISF Stabilization Theorem Candidate

ISF is a candidate formalization for bounding recursive digital systems through regularization, ordinal transformation, and hierarchy constraints.

Formal statement draft

Given a recursive system S, a regularization operator R, an ordinal transformation T, and a hierarchy bound H_{ω1}, define P(S) = T(R(S)) ∩ H_{ω1}(S). Under declared admissibility and well-foundedness assumptions, P(S) is expected to admit either a terminating transition path or a fixed-point state.

Assumptions

• S has a declared state space
• S has a declared transition relation
• R is defined on the class of recursive flows considered
• T maps regularized flows into a well-founded ordinal index
• H_{ω1} supplies a hierarchy boundary or admissible rank constraint
• fixed-point and termination criteria are explicitly defined

Definitions needed

• recursive system
• regularization operator
• ordinal transformation
• hierarchy bound
• admissible transition
• fixed point
• termination

Proof obligations

• prove transition relation is well-founded under declared rank
• prove R preserves admissible structure
• prove T produces descending or non-increasing ordinal rank where required
• prove no forbidden divergence remains inside P(S)
• prove fixed-point case is distinguishable from nontermination

Simulation obligations

• toy recursive process with terminating path
• toy recursive process with fixed-point path
• toy divergent process rejected by hierarchy bound

Reviewer questions

• Are the operators R, T, and H sufficiently defined?
• Is the target theorem too broad and should it be restricted?
• What class of recursive systems should be admitted first?

Forbidden claims

• ISF proves all infinite systems terminate
• ISF is externally-validated
• ISF guarantees-universal-convergence

Boundary: This is a theorem candidate and formalization scaffold, not an accepted-proof.

P0theorem-candidate

PI-DST Digital Structure Theorem Candidate

PI-DST is a candidate formalization for stable recursive digital structures under explicit topology, growth, and message-depth assumptions.

Formal statement draft

Let {S_n} be a sequence of digital structures with admissible inclusions S_n ⊂ S_{n+1}. Under a declared graph/topology model, bounded fractal-growth condition, and invariant-preserving transition rules, prove or simulate that message-depth and structural coherence remain bounded by the stated growth model.

Assumptions

• digital structure S_n is explicitly defined
• inclusion relation S_n ⊂ S_{n+1} is admissible
• growth rule is specified
• topology/graph metric is specified
• Hausdorff/fractal dimension claim is measurable or replaced by a discrete analog
• message-depth metric is defined

Definitions needed

• digital structure
• admissible inclusion
• fractal growth condition
• message depth
• structural coherence
• invariant-preserving transition

Proof obligations

• prove admissible inclusions preserve defined invariants
• prove or bound message depth under the declared graph model
• prove dimension condition is meaningful for the selected digital structure
• show failure cases where the theorem does not apply

Simulation obligations

• graph growth simulation
• message-depth simulation
• invariant preservation simulation
• failure-mode simulation

Reviewer questions

• Should PI-DST use Hausdorff dimension directly or a discrete-network analog?
• Which graph families are first admissible?
• What constitutes structural collapse in the model?

Forbidden claims

• PI-DST proves-infinite-scalability
• PI-DST is accepted-proof
• all DFT systems scale without collapse

Boundary: This is a formalization target for digital structure, not proof of infinite operational scalability.

P0definition-candidate

Fiber Vector Formalization Candidate

The fiber vector defines a proposed modeling primitive for representing a fiber through tension, elasticity, length/scope, orientation, and density.

Formal statement draft

A fiber F is represented as F = [T, E, L, O, ρ] over a declared domain D, with each component assigned a measurement, normalization, or categorical interpretation. Interactions between fibers are defined only after inner product, distance, or relation operators are specified.

Assumptions

• fiber domain is declared
• component units or categories are declared
• normalization scheme is defined
• interaction operator is defined
• meaning of resonance is restricted to the declared operator

Definitions needed

• fiber
• tension
• elasticity
• length/scope
• orientation
• density
• fiber interaction
• resonance

Proof obligations

• prove components are comparable only under declared normalization
• show when dot-product style resonance is valid
• show when metrics are semantic-only rather than operational

Simulation obligations

• example governance fiber
• example evidence fiber
• example contract fiber
• fiber interaction worked table

Reviewer questions

• Are T, E, L, O, and ρ measurable, categorical, or hybrid?
• Should the vector model be domain-specific rather than universal?
• What examples best demonstrate the primitive without overclaiming?

Forbidden claims

• fiber dynamics are validated-physics
• fiber vector proves smart-contract behavior
• fiber resonance is externally-validated

Boundary: The fiber vector is a modeling primitive requiring measurement and example specification.

P1review-needed

14D Tensor Ontology Conformance Candidate

The 14D tensor ontology is a semantic architecture map for classifying spatial, topological, governance, economic, and cross-dimensional relations.

Formal statement draft

Given a digital fabric object X, define a mapping Φ(X) into dimension bands B = {spatial, topological, governance, economic, cross-dimensional}. Conformance requires each mapped component to include role, tensor-like class, evidence state, and boundary.

Assumptions

• dimension bands are semantic classes
• mapping Φ is classification-oriented unless formal semantics are supplied
• each component has route/evidence traceability
• no physical-theory claim is inferred from the semantic map

Definitions needed

• dimension band
• tensor-like class
• semantic mapping
• conformance
• projection
• traceability

Proof obligations

• show mapping is internally consistent
• show each dimension band maps to reviewable architecture content
• show ontology does not imply physics finality

Simulation obligations

• map YellowChain to 14D ontology
• map CitizenSolar to 14D ontology
• map Stitchia to 14D ontology
• map GFE to 14D ontology

Reviewer questions

• Does the 14D ontology improve reviewability?
• Which mappings are too speculative?
• What evidence is needed before a mapping becomes implementation-supported?

Forbidden claims

• 14D ontology validates Geometric Unity
• 14D ontology is accepted-physics
• tensor ontology proves implementation correctness

Boundary: The 14D ontology is a semantic architecture model, not accepted-physics.

Shows that mapped ontology records preserve route, evidence, and boundary traceability.

Supports: semantic-band-classification-lemma

Required for: tensor-ontology-conformance

Review Boundary

Digital Fabrica Theory is presented as an authorial framework and formalization program. Public review tracks identify what must be formalized, simulated, implemented, or independently reviewed before stronger claims can be made. The presence of a review path does not mean peer review has been completed.

Review Discipline

Formalization Review Path

These review tracks identify what would strengthen, weaken, or revise DFT claims. They are designed to make the framework testable, challengeable, and externally reviewable.

Formalization neededmathematical-logic

ISF / Well-Founded Stabilization Review

Can the ISF stabilization claim be reduced to a precise well-founded transition theorem with explicit state space, rank function, and fixed-point semantics?

Would Strengthen

Lean/Coq formalization of the ordinal rank argument
Explicit definition of transition relation and admissible states
Proof that non-terminal transitions cannot generate infinite descending chains
External review by mathematical logician or proof engineer

Would Weaken

Undefined state space
Circular rank function
Unbounded recursion without descent measure
Claims of completeness incompatible with Gödelian limits

Boundary: This review track evaluates a formalization target, not a completed theorem.

Simulation neededgraph-theory

Spectral Expansion / Ramanujan Routing Review

Which DFT routing, robustness, or partition-resilience claims follow from expander graph properties, and which require separate empirical simulation?

Would Strengthen

Explicit graph family and routing model
Adversarial partition model
Simulation under node/link failure
Separation of graph robustness from cryptographic security

Would Weaken

Equating spectral expansion with cryptographic security
No adversarial model
No implementation topology
Overclaiming quantum-security from graph properties alone

Boundary: External graph theory supports topology design; DFT-specific network claims require modeled evidence.

Review neededeconomics-game-theory

Zeta-Indexed Economics Review

Can the zeta-indexed economic model be expressed as a simulation-bounded mechanism without implying financial validity or investment return?

Would Strengthen

Explicit economic variables
Convergence proof where used
Agent-based simulation
Legal review for token/financial implications

Would Weaken

Valuation claims without methodology
Investment language
Unbounded token-stability claims
No simulation

Boundary: This is a research extension and must not be represented as financial advice, offering, or validated macroeconomic result.

Claim-Level Source Trace

Formalization Claim Trace

Major claims on this page are mapped to source routes, bibliography records, formalization targets, review records, and public boundaries.

formalization target#formalization-roadmap

Formalization Roadmap

DFT theorem candidates are decomposed into assumptions, dependencies, and proof obligations before stronger claims are allowed.

Sources: dft-whitepaper-2026

Bibliography: godel-incompleteness-1931, turing-computable-numbers-1936

Formalization: isf-ordinal-stabilization, pidst-fractal-routing, dfdf-fns-idff-sids-composition

Review: isf-well-foundedness-review

Boundary: Formalization roadmap. Not a completed proof registry.

Review Rule

A DFT claim moves from public explanation to formal result only when symbols, assumptions, lemmas, dependencies, proof obligations, mechanized proof artifacts, and independent review status are all explicit.