Invariants: D_A, causal novelty Ξ, temperature spectrum

Parents: prior_structures_core_summary.txt; 17 Thm6/Thm8; 33 RetFlip / paid novelty notes; 13–16 return/poset lines.

1. Tope output diameter D_A

For chamber/tope τ:

\[

D_A(\tau)=\sup\{\|A(q)-A(q')\|: q,q' \text{ in } \tau\}

\]

Smaller D_A ⇒ piecewise constant / one anchor per chamber is safer.

Finer chamber structure can reduce D_A at cost of more discrete state.

Use: quality control for covers; split high-D_A regions first.

2. Causal novelty and growth

When adding a new key, novelty counts how many chambers/orderings split.

• \(N_{t}\) raw splits
• \(N^{act}\) splits that hit active query support
• Cumulative growth of cover size bounded by sum of novelties (Ξ-type polynomial evaluations in research)

Master compression slogan (research): for low-novelty sequences, size scales like m / poly(Ξ) style — only if novelty stays low.

Paid novelty (RetFlip class): local sign tests decide absorb vs charge new anchor — throttle r\* growth.

Track A cousin: glyph absorb (Jaccard) = crude paid novelty on text events.

3. Critical temperature spectrum

\[

T_{ij}(q)= T / |\langle q, k_i-k_j\rangle|

\]

• Below critical: pair distinguishable in softmax
• Spectrum links to tropical / valuated structure on walls
• T→0 favors A0; T→∞ favors global mean (one cluster)

Use: adaptive mode switch A0 vs soft anchors vs dense.

4. What MVP needs vs later