01 · From theory to instrument

A specification earns trust by never skipping a link. This document walks the whole chain — object, theorems, observables, estimation, pipeline, feature — so that every later document can point back to a step proved here, and so that nothing downstream rests on a claim that was never made upstream.

The chain has six links. Each is stated, then justified from the corpus, then handed to the next.

§1. Link 1 — the object: what the Console reads

Everything the Console computes is a function of a single object: the coherence matrix $\Gamma \in \mathcal D(\mathbb C^7)$ — a $7\times 7$ Hermitian, positive-semidefinite, unit-trace matrix. Its diagonal entries $p_i = \gamma_{ii}$ are the populations of the seven axes; its off-diagonal entries $\gamma_{ij} = |\gamma_{ij}|e^{i\theta_{ij}}$ are the coherences — the relationships between axes, carrying both a magnitude and a phase.

The seven axes are not a stylistic choice. They are $[A,S,D,L,E,O,U]$ — Articulation, Structure, Dynamics, Logic, Interiority, Ground, Unity — the seven functional roles that the minimality theorem proves are the fewest that close a self-maintaining, self-modelling system (autopoiesis + phenomenality + self-grounding). A state with fewer than seven axes cannot both hold together and know itself; a state with more is reducible. Seven is forced.

Why this matters for a product: the object is low-dimensional (a person's entire readable state is 49 real numbers, 48 after the trace constraint), yet it is the complete state — there is nothing about the system's self-reading that lives outside Γ. A product built on Γ is therefore simultaneously compact enough to compute and estimate, and complete enough that no essential quality is structurally excluded.

§2. Link 2 — the theorems that license the instrument

Four results turn "we have an object" into "we may build an instrument on it."

(T-224) The grammar is unique. Theorem Σ proves that any complete self-reading grammar — one that can localise a single fault in itself and distinguish a nontrivial repertoire of global states — has exactly seven axes with the Fano incidence, uniquely. Consequence for the product: the frame is not one option among many; it is the only complete one. This is the property no competitor can claim.

(T-153 / T-153a / T-253) The instrument is substrate-free. Substrate closure proves the same predicate reads a biological brain, a digital agent, or any admissible carrier, and T-253 gives the explicit construction (the retraction $G_V$) plus the accessibility clause (Acc). Consequence: one kernel serves human, team, and AI targets — the platform thesis of 00 §2.

(T-256) The charts are finite and classified. The classification theorem shows every symbolic system is a sub-configuration of the Fano grammar, falling into a finite orbit list. Consequence: the Console can host every legacy system (astrology, I Ching, tarot) as a named orbit and translate between them with honest loss accounting — the interop feature of 04.

(T-257) The channels are priced. The licensed-channel theorem proves inter-level coupling is gate/rate-modulation, never a symbolic message, and prices the only real external anchor (the two-channel solar/lunar ledger). Consequence: the sensor layer knows exactly which external inputs are physical (light, tides → circadian/circalunar) and which are category errors (planetary natal claims), and the modulation horizon 08 knows its hard boundary (it can bias, never write).

§3. Link 3 — from object to observables

The kernel computes eleven derived quantities from Γ. Each is a defined function with a corpus home; 02 gives the algorithms, here we establish why each is well-defined and computable.

Observable	Definition (from Γ)	Why computable
Purity $P$	$\operatorname{Tr}(\Gamma^2)$	a trace of a matrix product — direct
Reflection $R$	$1/(7P)$	a scalar function of $P$
Self-model quality $R_\varphi$	$1 - \lVert\Gamma-\varphi(\Gamma)\rVert_F^2/\lVert\Gamma\rVert_F^2$	needs the self-model map $\varphi$; canonical family closed-form
Integration $\Phi$	integrated-information functional on Γ	corpus gives an HS-projection form [T]
Differentiation $D_{\mathrm{diff}}$	$\exp S_{vN}(\Gamma)$	von Neumann entropy of Γ — eigenvalues
Gap map	$\mathrm{Gap}(i,j) = \lvert\sin\theta_{ij}\rvert$	phases of off-diagonal entries
Mandalagram	$(p_i;\ \gamma_{ij})$ laid on the 7+21 chart	direct reshaping of Γ
Archetype	decode of the thresholded diagonal	a lookup on $\{p_i > \tau\}$
Mode	dominant term of the triadic decomposition of $\dot\Gamma$	needs a trajectory increment
Meaning	$P\cdot D_{\mathrm{diff}}\cdot\Phi\cdot R_\varphi$	product of four scalars
Freedom	$\dim\ker\mathcal H_\Gamma + 1$	kernel dimension of the free-energy Hessian

The point of the table is that nothing in the readout requires anything but Γ (and, for mode, a Γ-increment). The product's entire analytic surface is a pure function of the estimated state.

§4. Link 4 — the estimation problem

Γ is not handed to us; it must be estimated from what a system exposes. This is the honest core of the whole enterprise and the reason the anchor classes exist. The estimation problem is: given observable evidence $e$ about a system, produce $\widehat\Gamma(e)$ and a confidence region. Three evidence types, three estimators (03 gives the mathematics):

• Measured (Anchor I): $e$ = physiological/neural/behavioural signals; $\widehat\Gamma$ via the seven-channel embedding, with a proven consistency and confidence bound and the PCI $\leftrightarrow P_{\mathrm{crit}}$ calibration anchoring the purity axis.
• Self-tracked (Anchor II): $e$ = responses to the 28-item audit; $\widehat\Gamma$ via a scoring model that maps item responses to populations and coherences, with test-retest reliability standing in for measurement error.
• Oracular (Anchor III): $e$ = a structured random draw; $\widehat\Gamma$ is not estimated — the draw is a projective prompt, and the Console labels it so.

The estimation link is where over-claim would enter, and it is exactly where the honesty spine (00 §4) clamps it: the confidence region and anchor class travel with $\widehat\Gamma$ through every downstream computation, so no readout can present more certainty than its evidence earned.

§5. Link 5 — the analytic pipeline

Given $\widehat\Gamma$ (and its confidence region), the pipeline is deterministic:

1. Invariants — evaluate the eleven observables (§3), propagating confidence.
2. Optical construction — run the three guardrail questions (one-grammar §5): which subchart is this reading (T-256 orbit), what anchor and is it licensed (T-257), is the subject a viable holon (viability gate). These decide what kind of statement the reading is allowed to make.
3. Domain module — self / trajectory / dyad / group / clinical / AI (see 04) applies domain logic over the invariants.
4. Rendering + honesty layer — produce the mandalagram, forecast, or recommendation, each stamped with anchor class and loss/confidence.

The pipeline has no hidden stage. Every arrow from evidence to a rendered claim is one of these four steps, and each step is specified in a later document.

§6. Link 6 — from pipeline to product feature

A product feature is a pipeline configured for a job. "Read my current state" is (Anchor II → invariants → self module → mandalagram). "Are we compatible" is (two Anchor-II estimates → composite Γ → dyad module → cross-coherence report). "Is this AI drifting" is (agent telemetry → Anchor I → invariants → AI module → safety readout). Every feature in 04 is a named configuration of the one pipeline — which is why the whole product is one codebase.

§7. The master traceability table

The single artifact that makes this suite a specification rather than a brochure: every product feature, traced to the theorem or definition it rests on. If a row's source is ever retracted, the feature falls with it — that is the discipline.

Product feature	Rests on (source)	Corpus status	Spec doc
Seven-axis frame	Minimality (7 roles); Σ uniqueness (T-224)	[T]	02
State readout (invariants)	definitions of $P,R,\Phi,D,R_\varphi$ + thresholds	[T]/[C]	02
Mandalagram / bodygraph	28-cell chart (gamma-canon)	[T]/[D]	02, 05
Archetype & mode	16 signatures [T]; triadic decomposition [T]	[T]	02
Meaning & Freedom readout	Meaning formula; Freedom $=\dim\ker\mathcal H_\Gamma+1$ [T]	[T]	02
Measured estimation	seven-channel embedding; PCI calibration	[Т-path]/[C]	03
Self-audit estimation	П1 28-item audit	[И, structured]	03
Trajectory & forecast	transparency windows, bifurcations (gap-diagnostics)	[T]/[C]	03, 04
Correction / practice	minimal-intervention protocol	[C]	04
Synastry (dyad)	composite Γ; cooperation theorem ($P{+}2\lVert\gamma_{\mathrm{cross}}\rVert^2$); empathy $=1{-}\mathrm{Gap}(E,E)$; one-theme law	[T]+[C]	04
Alignment / knowing–doing gap	misalignment $=\mathrm{Gap}(L,D)$; optimality of alignment	[C]	04
Ethics guardrails (harm, value hierarchy, non-manipulation)	good $=dP/d\tau$; value hierarchy; non-violence; responsibility $\propto$ Freedom	[D]+[C]+[T]	10
Org diagnostics	7-D org profile; one-theme pathology	[T]/[C]	04
Clinical (DOC/anaesthesia/meditation)	altered-states profiles; PCI bridge	[C]/[research]	04, 07
AI introspection	substrate closure (T-153); SYNARC	[T]	04
Symbolic interop	back-projection; classification (T-256)	[T]+[I]	04
Human "why/tasks" reading	archetype + Freedom + $\Gamma^*(\tau)$; birth-refusal	[T] + refusal	05
Chronobiology anchor	licensed-channel ledger (T-257)	[T]/[C]	03, 08
Modulation horizon	$\mathcal R$; correction; gate-not-message (T-257a)	[speculative]/[T-bound]	08
Every guardrail	optical construction (T-256/T-257) + viability gate	[T]+[I]	02, 10

Where this leads. The next document, 02 · The kernel, turns §3's observables into exact algorithms and the internal API that every domain module calls.