Part of CNS 8.0 / Grounded Dialectical Orthesis

04 — Mathematical Specification

3 min read •Updated Jun 3, 2026

04 — Mathematical Specification

1. Spaces and maps

Let $L$ be a language manifold or representation space.

Let $\mathcal{T}$ be a tensor-logic space containing atoms, predicates, rules, proof traces, and constraint states.

Let:

$$ G: L \to \mathcal{T} $$

be grounding, and:

$$ S: \mathcal{T} \to L $$

be synthesis/rendering.

The closure map in logic space is:

$$ C = G \circ S: \mathcal{T} \to \mathcal{T} $$

The CNS loop searches for stable structured states under $C$, subject to evidence and proof constraints.

2. Fiber-bundle interpretation

For each language state $l \in L$, let $\mathcal{T}_l$ be the fiber of admissible logical interpretations over $l$. The total space is:

$$ B = \{(l,t): l\in L,\ t\in \mathcal{T}_l\} $$

with projection $\pi:B\to L$.

A CNS narrative path is a path through $B$, not only through $L$. Chirality appears when language movement and logic movement fail to commute.

3. Curvature / holonomy diagnostic

Let $\Gamma$ be a closed dialectical loop:

$$ T_0 \xrightarrow{S} L_0 \xrightarrow{\text{antagonist/reframe}} L_1 \xrightarrow{G} T_1 \xrightarrow{\text{proof closure}} T_2 \xrightarrow{S} L_2 \xrightarrow{G} T_3 $$

The holonomy residual is:

$$ \mathrm{Hol}(\Gamma) = \|T_3 - T_0\|_\Omega $$

A large holonomy residual marks unstable narrative transport.

4. Zero-temperature closure

Let $F$ be grounded facts and $R_0$ be zero-temperature rules. A rule $r$ has the form:

$$ Y[\mathbf{i}] = \mathrm{step}\left(\sum_{\mathbf{j}} \prod_k X_k[\mathbf{i}_k,\mathbf{j}_k]\right) $$

The closure is the least fixed point:

$$ Cl_0(F;R_0)= \mu X.\; F \cup \bigcup_{r\in R_0} r(X) $$

Assumptions for soundness:

  • monotone rules;
  • no unsafe negation;
  • all variables range over finite domains;
  • all premises originate from grounded evidence or previously derived proof atoms.

5. Soundness sketch

If $R_0$ is monotone and every rule application records a proof trace, then every atom in $Cl_0(F;R_0)$ is reachable by finite rule applications from grounded facts. Unsupported atoms cannot be promoted because promotion requires a proof trace rooted in $F$.

This gives zero-temperature hallucination rate:

$$ \mathrm{ZTHR}= \frac{ |\{c \in C_{\mathrm{strict}}: \neg \exists \pi(c)\}| }{ |C_{\mathrm{strict}}|+\epsilon } $$

Target: $\mathrm{ZTHR}=0$.

6. Residual contradiction tensor

Let $X,Y,Z,C$ be subject, predicate, object, and context index sets. Define residual tensor:

$$ R[x,y,z,c] = m_{\mathrm{support}}[x,y,z,c] - m_{\mathrm{refute}}[x,y,z,c] $$

or, for unresolved mass:

$$ R_{\mathrm{unres}}[x,y,z,c] = \min(m_{\mathrm{support}}, m_{\mathrm{refute}}) \cdot (1 - m_{\mathrm{resolved}}) $$

This tensor identifies where proof closure cannot settle support/refute conflict.

7. Predicate invention by factorization

A low-rank approximation:

$$ R_{\mathrm{unres}} \approx \mathcal{C} \times_1 M_X \times_2 M_Y \times_3 M_Z \times_4 M_C $$

proposes latent factors. A latent context predicate $\lambda_k$ is accepted only if it improves residual energy while passing evidence gates:

$$ \mathrm{PIU}(\lambda_k) = \frac{ E_R(\text{before}) - E_R(\text{after}) }{ \mathrm{Complexity}(\lambda_k)+1 } $$

Acceptance requires:

$$ \mathrm{PIU} > \theta_{\mathrm{PIU}} \quad \land \quad \mathrm{GroundingScore}(\lambda_k) \geq \theta_G $$

8. Multiverse views as auxiliary posterior

Possible worlds $W_i$ are candidate structured states containing facts, predicates, access assumptions, and proof status. They are ranked after synthesis constraints are applied:

$$ P(W_i\mid E,A) \propto P(E\mid W_i,A)P(W_i)\exp(-\alpha E_R(W_i)-\beta \chi_{LL}(W_i)) $$

World posterior mass reports uncertainty. It does not replace the synthesis operator.

9. Calibration

For confidence bins $B_m$:

$$ \mathrm{ECE}= \sum_m \frac{|B_m|}{n} |\mathrm{acc}(B_m)-\mathrm{conf}(B_m)| $$

CNS reports ECE for promoted strict claims, likely claims, and latent-predicate proposals separately.

10. Orthesis acceptance

A synthesized SNO is accepted as an orthesis candidate when:

$$ \mathrm{CitationValidity}=1 $$
$$ \mathrm{MeanEntailment}\geq \theta_E $$
$$ \mathrm{ZTHR}=0 $$
$$ \chi_{LL}\leq \theta_{\chi} $$
$$ E_R \leq \theta_R $$
$$ \Delta \beta_1 \geq \theta_\beta \quad \text{or residual contradiction is explicitly preserved} $$
Step 5 of 39 in CNS 8.0 / Grounded Dialectical Orthesis
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