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?

The system emits three distinct quantities:

Quantity	Meaning
`P(c	E,A,I)`
`P0(c	E)`
Conf(c)	confidence after uncertainty decomposition

A claim can be probable while still record-contingent. A claim can be strictly proven inside a narrow reference set while its broader interpretation remains low-confidence. A claim can be plausible yet unsuitable for promotion because access-state uncertainty remains material.

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:

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

Evidence atoms are immutable within a run. Later corrections or productions create new atoms and preserve the earlier state as part of the audit trail.

Record-Access States

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

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

The access_k value may be available, inaccessible, sealed, withheld, destroyed, not_generated, unknown, produced_late, partial, contradicted, or unavailable_at_time_t.

This lets the system distinguish:

• absence of evidence;
• evidence of absence;
• inaccessible evidence;
• sealed evidence;
• withheld evidence;
• destroyed evidence;
• not-generated evidence;
• partial or nonresponsive production;
• evidence unavailable at the relevant decision time.

Absence can affect ranking only when record-generation duty, expected observability, ownership/control, collection path, production state, and access state justify that effect.

Claims And Statuses

A claim is:

$$ c_j = (p_j, frame_j, refs_j, contingencies_j, \sigma_j) $$

where p_j is a proposition, frame_j is an argument frame, refs_j is an evidence-reference set, contingencies_j is a record-contingency set, and \sigma_j is one of proven, probable, plausible, record_contingent, conflicted, unsupported, rejected, or insufficient_evidence.

Relations among claims are typed through a relation set R, including supports, refutes, implies, specializes, generalizes, qualifies, depends_on, undercuts, 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 Residuals

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 have high chirality if its logical or access structure falls apart under grounding. In GCTS, chirality is a diagnostic residual:

• 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;
• rendering chirality, when generated language drops proof or access contingencies.

Chirality does not prove falsity. It identifies mismatch that must be resolved by evidence, rules, access modeling, or explicit uncertainty.

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:

$$ \mathcal{E}(W_k;E,A,I) = \alpha C(W_k) + \beta X(W_k) + \gamma G_w(W_k) + \delta K(W_k)

• \eta S_r(W_k) - \lambda S_e(W_k) $$

where C is contradiction, X is access mismatch, G_w is weak grounding, K is unsupported complexity, S_r is source risk, and S_e is evidence support.

World posterior mass is:

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

Lower energy worlds are better supported. Contradictions, unsupported complexity, access mismatch, weak grounding, and source risk 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. LLM confidence has no role in the runtime truth value.

Strict proof support is emitted separately:

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

Confidence is a function of grounding quality, world entropy, access-state uncertainty, source risk, and residual conflict:

$$ Conf(c) = f(q_g(c), H(W), u_A(c), r_s(c), \delta(c)) $$

The system must emit P(c | E,A,I), P0(c | E), and Conf(c) separately.