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Target Space Topology

Jean-Paul Niko · RTSG Application Layer · 2026-03-18

Origin: phenomenological insight in session — every dimension expresses a subtext aimed at a specific target space.

The Four Target Spaces

Every communication act is a projection from a sender's CS into one of four destination spaces. These are not metaphors — they are distinct quotients of the full CS manifold, each with its own geometry, memory, and filter requirements.

1. Private CS — CS_self

The receiver is the sender. Internal monologue, journaling, self-talk, meditation, dreams.

The filter here is introspective — it surfaces subtext the person is hiding from themselves. This is the therapeutic use case: RTSG sessions targeted at self-knowledge.

\[\pi_{\text{private}}: CS \to CS_{\text{self}}\]

2. Dyadic CS — CS_{i,j}

A shared space co-created by exactly two participants. This space is a genuine emergent object — it has structure neither party can generate alone. A relationship is not two people; it is two people plus the dyadic space they have jointly constructed.

\[\pi_{\text{dyadic}}: CS \to CS_{i,j}\]

Key properties: - Has its own memory, vocabulary, attractors, and trust topology - Inaccessible to third parties (privacy emerges from the geometry) - When a relationship ends, \(CS_{i,j}\) persists as an orphaned structure in both parties' individual CS — this is the mathematical structure of grief

3. Group CS — CS_G

A bounded set of participants with known, stable membership. Family, friend group, team, BuildNet assembly.

\[\pi_{\text{group}}: CS \to CS_G\]

Key properties: - Has collective memory and shared context beyond any pairwise relation - Filter must model the full group topology, not just pairwise I-vector distances - Trust is not uniform — the group has internal topology (dyadic sub-spaces, authority gradients, in/out distinctions) - BuildNet is an instance: {@B_Niko, @D_Claude, @D_GPT, @D_Gemini} operating on a shared CS_G with the wiki as its external memory

4. Open Collective CS — CS_∞

Unbounded, public, no stable membership. Social media, published work, culture, language itself.

\[\pi_{\text{collective}}: CS \to CS_\infty\]

Key properties: - No shared context assumed with any receiver - Maximum compression loss — the sender must reconstruct full context from scratch - Most subtext is stripped or becomes noise - The filter must be maximally conservative: only content that survives zero shared context

Formalization

Each target space is a quotient of the full CS manifold under a different equivalence relation:

Space Quotient Shared context Memory
\(CS_{\text{self}}\) Identity Complete Full autobiographical
\(CS_{i,j}\) Dyadic bisimulation Partial Co-constructed
\(CS_G\) Group bisimulation Bounded Collective + subgroup
\(CS_\infty\) Trivial None assumed None

The projection operator \(\pi\) changes with the target:

\[\pi_{\text{target}}: CS_{\text{sender}} \to CS_{\text{target}}\]

Subtext that survives \(\pi_{\text{dyadic}}\) (shared vocabulary, implicit reference) may be opaque noise under \(\pi_{\text{collective}}\). This is why private jokes don't translate, why therapy language sounds cold in public, why TMP notation confuses everyone outside BuildNet.

A message tagged for \(CS_{i,j}\) sent through \(\pi_{\text{collective}}\) by mistake is a privacy violation with a geometric description. The filter system can detect this mismatch:

"This message contains content from your dyadic space with [person]. Are you sure you want to project it into the collective?"

This is consent infrastructure with formal foundations.

The Grief Implication

When \(CS_{i,j}\) loses its counterparty \(j\), the structure persists in \(CS_i\) as an orphaned submanifold. There is no projection target for the content that lives there. This is:

  • Why grief has the specific shape it does
  • Why the bereaved "talk to" the absent party
  • Why certain memories remain inaccessible until a new dyadic space is formed that can receive them

RTSG gives grief a geometric description: an orphaned dyadic submanifold seeking a projection target.

Cross-references