03 — Core Theory

03 — Core Theory

1. Structured Narrative Objects

An SNO is the unit of CNS reasoning.

$$ \mathcal{S} = (H, G, E, A, P, R, U, M) $$

where:

• $H$ is the central hypothesis or account embedding.
• $G=(V,\mathcal{E}_G,\kappa,\rho)$ is a typed reasoning graph.
• $E$ is the evidence set attached to claims and relations.
• $A$ is the record-access state set.
• $P$ is a proof-trace bundle.
• $R$ is a residual contradiction tensor.
• $U$ is calibrated uncertainty metadata.
• $M$ is source, time, lineage, and domain metadata.

SNOs are structured narrative objects. They preserve the account being synthesized, not only the truth value of isolated claims.

2. Chiral opposition

A pair of SNOs $\mathcal{S}_a,\mathcal{S}_b$ is chiral when the accounts are oriented against one another while sharing a basis.

CNS 8.0 uses three compatible chirality estimators.

2.1 Graph chirality

Let $B_a$ and $B_b$ be signed incidence matrices over the aligned reasoning graph. Let $W_E$ weight edges by evidence quality.

$$ \chi_G(a,b) = \| W_E^{1/2}(B_a - B_b) \|_F $$

This measures structural asymmetry in reasoning flow.

2.2 Evidence-polarity chirality

Let $s_a(e,c)$ be the signed stance of evidence item $e$ toward claim $c$ in SNO $a$, with support $+1$, refute $-1$, neutral $0$.

$$ \chi_E(a,b) = \frac{ \sum_{e,c} w(e) |s_a(e,c)-s_b(e,c)| }{ \sum_{e,c} w(e) + \epsilon } $$

This captures same-evidence / opposite-interpretation tension.

2.3 Language–logic chirality

Let $G: L\rightarrow \mathcal{T}$ be grounding from language to logic and $S:\mathcal{T}\rightarrow L$ be rendering/synthesis from logic to language. For logic state $T$:

$$ \chi_{LL}(T) = \|G(S(T)) - T\|_{\Omega} $$

where $\Omega$ weights proof-critical predicates and evidence-linked atoms more heavily than cosmetic phrasing.

High $\chi_{LL}$ means the language rendering does not preserve the logic state when re-grounded.

3. Evidential Entanglement

Evidential Entanglement measures whether two SNOs argue over the same evidentiary substrate.

$$ \mathrm{Ent}(a,b) = \frac{ \sum_{e \in E_a \cap E_b} w(e) }{ \sum_{e \in E_a \cup E_b} w(e) + \epsilon } $$

High entanglement without chiral opposition is agreement or redundancy. High chirality without entanglement is often unrelated disagreement. High values of both identify productive synthesis targets.

4. Productive Conflict Score

$$ \mathrm{PCS}(a,b) = \sigma(\alpha \chi_G +\beta \chi_E +\gamma \chi_{LL} +\delta \mathrm{Ent} +\lambda \chi_E\mathrm{Ent} -\eta \mathrm{AccessGap}) $$

The interaction term $\chi_E\mathrm{Ent}$ is central: CNS cares about conflict over shared evidence.

5. Orthesis

Orthesis is the stable synthesis candidate in logic space.

Given an SNO pair and a synthesis operator $\Phi$, CNS produces a candidate logic state $T_c$. It is an orthesis candidate if:

$$ \|G(S(T_c)) - T_c\|_\Omega \leq \epsilon_{\mathrm{roundtrip}} $$$$ \mathrm{ZTHR}(T_c) = 0 $$$$ \Delta \beta_1 = \beta_1(G_a \cup G_b) - \beta_1(G_c) \geq \theta_{\beta} $$$$ \mathrm{ResidualEnergy}(T_c) \leq \theta_R $$

Orthesis is a stability condition. It does not assert metaphysical truth. It says the synthesized narrative object survives the CNS consistency, grounding, and re-rendering loop.

6. Synthesis as creation

The Synthesizer does not choose the most likely input account. It constructs a new SNO:

$$ \mathcal{S}_c = \Phi(\mathcal{S}_a,\mathcal{S}_b, P_0, R, \Lambda) $$

where $P_0$ is zero-temperature proof closure, $R$ is residual contradiction, and $\Lambda$ is the set of accepted latent predicates. The output can preserve unresolved contradiction when evidence does not support a stronger resolution.

7. Failure conditions

CNS returns no synthesis or a partial synthesis when:

• citations fail;
• evidence does not entail promoted claims;
• no productive conflict exists;
• contradictions require a latent predicate that cannot be grounded;
• possible worlds remain too diffuse;
• round-trip chirality remains above threshold;
• proof-critical claims lack proof traces.