GCTS Theory

GCTS separates three questions that are often collapsed:

1. What is strictly proven?
2. What is likely true across admissible worlds?
3. What uncertainty remains because of evidence quality, missing records, access conditions, source incentives, or contradiction structure?

Evidence Atoms

An evidence atom is:

$$ e_i = (u_i, s_i, t_i, q_i, a_i, m_i) $$

where u_i is a stable source identifier, s_i is a span, observation, record, or structured datum, t_i is temporal scope, q_i is source/evidence quality, a_i is access path, and m_i is metadata.

The available evidence set is:

$$ E = \{e_1,\dots,e_n\} $$

Record-Access States

GCTS models missing evidence as structured information. A record-access state is:

$$ r_k = (id_k, type_k, owner_k, duty_k, expected_k, access_k, production_k, q_k) $$

The access_k value is one of available, inaccessible, sealed, withheld, destroyed, not_generated, or unknown.

This lets the system distinguish:

• absence of evidence;
• evidence of absence;
• inaccessible evidence;
• withheld evidence;
• not-generated evidence.

Absence can penalize a claim only when record-generation duty, expected observability, collection path, and access path justify that penalty.

Claims And Statuses

A claim is:

$$ c_j = (p_j, a_j, \rho_j, \kappa_j, \sigma_j) $$

where p_j is a proposition, a_j is an argument frame, rho_j is a reference set, kappa_j is a record-contingency set, and sigma_j is one of proven, probable, plausible, record_contingent, conflicted, unsupported, or rejected.

Relations among claims are typed through a relation set R, including supports, refutes, implies, specializes, generalizes, qualifies, depends_on, and independent.

Language, Logic, And Access

Let L be the language/concept manifold, T the logic/proof space, and A the access/missingness space. A grounding map extracts proof and access structure:

$$ G: L \rightarrow \mathcal{T} \times \mathcal{A} $$

A rendering map turns structured worlds back into language:

$$ S: \mathcal{T} \times \mathcal{A} \rightarrow L $$

The orthesis is the stable structured state:

$$ (\mathcal{T}^{\ast},\mathcal{A}^{\ast}) = G(S(\mathcal{T}^{\ast},\mathcal{A}^{\ast})) $$

The orthesis is the structured state that survives language rendering without losing proof support, likely-truth support, access-state coherence, or uncertainty.

Chirality

Round-trip chirality measures whether a structured state survives rendering and re-grounding:

$$ \delta(X) = d_{\mathcal{T},\mathcal{A}}(X, G(S(X))) $$

A fluent narrative can still have high chirality if its logical or access structure falls apart under grounding.

GCTS also measures:

• graph chirality, based on edge-incidence differences between claim graphs;
• residual tensor chirality, based on unresolved support/refutation mass;
• access chirality, when structured modeling breaks narrative access assumptions.

Possible Worlds

A world view is:

$$ W_k = (F_k, R_k, Z_k, \Pi_k, A_k, M_k, H_k) $$

where F_k contains accepted facts and likely-truth claims, R_k is a rule subset, Z_k are latent context predicates, Pi_k are proof traces, A_k are assumptions, M_k is a record-access model, and H_k is an institutional-incentive hypothesis set.

Worlds are scored by energy:

$$ Q(W_k \mid E,A,I) = \frac{\exp(-\mathcal{E}(W_k;E,A,I))} {\sum_\ell \exp(-\mathcal{E}(W_\ell;E,A,I))} $$

Lower energy worlds are better supported. Contradictions, unsupported complexity, access mismatch, and weak grounding raise energy; evidence support lowers it.

Likely-Truth Ranking

For a claim c:

$$ P(c \mid E,A,I)= \sum_k Q(W_k\mid E,A,I)\,\mathbf{1}[c \in Cl(W_k)] $$

The score reports posterior mass across structured worlds rather than direct LLM confidence.

Strict proof support is emitted separately:

$$ P_0(c \mid E)= \sum_k Q(W_k\mid E,A,I)\,\mathbf{1}[c \in Cl_0(W_k)] $$

The system must emit P(c | E,A,I), P_0(c | E), and Conf(c) separately. A claim can be likely but low-confidence, strictly proven but narrow, or plausible but record-contingent.