Extended Lemniscate

The extended lemniscate is the τ-categorical pushout of two SelfDesc agents' individual lemniscates along a shared signal channel S in their exterior lobes: L_ext(A,B) = L_A ∪_S L_B. It is the formal carrier of communication — the topological space across which one evaluator's signals reach another's wedge point — and the substrate on which language and social structure live.

Categorical invariant. Topological pushout of two figure-eight lemniscates along a sub-arc of their exterior lobes (VI.D70); fundamental group π₁(L_ext) is the amalgamated free product of π₁(L_A) ≅ ℤ * ℤ and π₁(L_B) ≅ ℤ * ℤ over the shared channel.

Primary registry anchor: VI.D70

Supporting items: VI.D27, VI.D15

1. I.K0 — Universe Postulate
2. VI.D15 — Life Sector — agents with τ-distinction and SelfDesc evaluator
3. VI.D27 — Temporal Lemniscate — individual figure-eight boundary
4. VI.D70 — Extended Lemniscate — pushout of two individual lemniscates along a shared signal channel S

Biomarker: Communication channels at all biological scales: bacterial autoinducer gradients (quorum sensing), pheromone plumes, acoustic signals (bird song, whale song, human speech), visual displays, the honeybee waggle dance, chemical cues in plant–mycorrhiza networks; cross-agent signal alignment measurable as mutual information.

Measurable range: Bandwidth: bacterial QS ~bits/min; mammalian vocal ~10–100 bits/s; human speech ~40 bits/s; sustained cross-species alignment (humans–dogs) on tens of emotional/spatial signals; quorum-sensing thresholds at ~10⁵–10⁹ cells/mL autoinducer concentration.

Observation method: Acoustic recording and spectral analysis, mass-spectrometry of pheromones, autoinducer concentration assays, mutual-information metrics on signal–response pairs, ethological observation of multi-agent coordination, fMRI of speech-comprehension overlap.

Calibration anchor: LG-Y02-kinetic-pseudoscalar-channel

Anchor chain:

1. VI.L18 chirality channel
2. stereospecificity of pheromone receptor binding (L-amino-acid GPCRs)
3. homochirality of signal molecules across the QS / pheromone / hormone classes

Manuscript reference: manuscript-sources/book-06/part08/ch52-language-lemniscate.tex

Status: Planned

Related glossary entries

• LG-T01-circadian-rhythm Circadian Rhythm
• LG-T03-sleep-cycle Sleep Cycle (NREM/REM)
• LG-Y02-kinetic-pseudoscalar-channel Kinetic Pseudoscalar Channel

Referenced by

• LG-T03-sleep-cycle Sleep Cycle (NREM/REM)