feat(corpus): enrich CS-07 + CS-08 + CS-09 + CS-10 + End-to-end construction view (Arc 3 — final)

corpus/construction-spine/index.md

@@ -100,6 +100,47 @@ The first three construction steps are supported by a foundational hinge route:
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+## End-to-end construction view
+
+The ten steps are not isolated essays. They glue into a single construction chain that starts at the kernel's primitive signature and closes at the boundary collapse where proof and commitment coincide. The chain has four manuscript-grounded arcs:
+
+### Arc I — Mathematics (CS-01 → CS-03; Books I–II)
+
+The kernel-and-mathematics arc earns the foundation:
+
+- **CS-01** posits the categorical kernel (K0–K6, five generators, one progression operator) and earns hyperfactorization, prime polarity, the split-complex boundary algebra, τ-holomorphy, and the τ-topos with four-valued internal logic.
+- **CS-02** recovers core mathematics under kernel discipline: an internal set theory from divisibility-as-membership (Cantor mirage refuted by K5); the earned number tower `ℕ_τ ⊆ ℤ_τ ⊆ ℚ_τ ⊆ ℝ_τ ⊆ ℂ_τ`; Tarski geometry as theorems II.T15–T18; earned transcendentals π, e, j, ι_τ; the Fork's five-mode comparison against orthodox mathematics.
+- **CS-03** internalizes self-enrichment: hom-objects as τ-objects; Yoneda as theorem (II.T36); the Central Theorem `O(τ³) ≅ A_spec(L)` (II.T40); Categoricity (II.T42) — moduli space `{pt}`, zero parameters, "τ³ is discovered, not constructed".
+
+By the end of Arc I, the framework has built a τ-internal mathematical layer that is **categoric**, **rigid**, and **without free parameters** — and the boundary↔interior holographic duality is established as a τ-internal theorem.
+
+### Arc II — Physics (CS-04 → CS-06; Books III–V)
+
+The physical arc earns the carrier and the bridge to measurement:
+
+- **CS-04** identifies the physical carrier as `E_1` — the structural layer where physics becomes definable. The 4+1 sector template is induced (not posited) by Langlands_0 functoriality; the Hinge Theorem (III.T20) establishes that all Books IV–VII results are sector instantiations of Book III; the Global Cartesian Gluing (III.T21) earns the 3+1 spacetime signature; the Eight Guarantees (Spatial / Harmonic / Regular / Discrete / Legible / Codable / Coherent / Predictive) close the physics layer.
+- **CS-05** populates the carrier with internal physical grammar: the Joint Core ontic sequence (vacuum → neutron → β-decay → hydrogen) with **`α = (121/225)·ι_τ⁴`** at 9.8 ppm; QM as address obstruction; time from τ¹ as proper-time = geodesic arc length; Gravity Earned with `G = (c³/ℏ)·ι_τ²`, the τ-Einstein equation as boundary-character identity, Lorentz Without Minkowski, and the gravitational closing identity `α_G = α¹⁸·√3·(1 − (3/π)·α)`. The Hermetic Principle declares fiber + base exhaust E_1.
+- **CS-06** builds the measurement bridges: the constants ledger (ten couplings as rational functions of `ι_τ`); the running-vs-readout distinction (no RG flow); the 10-link mass-ratio chain to sub-ppm; the neutron lifetime as crown of calibration; the Cosmic Stack API; and the **decisive falsification** — CMB-S4 r ≈ ι_τ⁴ ≈ 0.0136 (~2030).
+
+By the end of Arc II, the framework has earned a τ-internal physics layer with **zero free dimensionless constants** and a published falsification ledger committing the framework in advance to specific experimental outcomes.
+
+### Arc III — Life and Metaphysics (CS-07 → CS-10; Books VI–VII)
+
+The reflection-and-closure arc earns the upper enrichment layers and runs the final closure test:
+
+- **CS-07** recovers life as a structural class via two predicates: τ-Distinction ∧ SelfDesc; the Layer Separation Lemma (VI.T02) establishes life is genuinely E_2 (non-reducible to physics); the Parity Bridge Theorem (VI.T03) seeds self/non-self distinction from the weak sector; the 4+1 sector template instantiates as Archaea / Bacteria / Plants / Fungi / Animals; the Seven Hallmarks are derived as theorems; Predictions by Absence (virus / neutron / neutron star) establish falsifiability; the bridge to E_3 opens at consciousness as SelfDesc-of-SelfDesc.
+- **CS-08** recovers reflective structure: the four-register model (E / P / D / C); the 4+1 template at E_3 (S_E / S_D / S_P / S_C / S_L); the **Saturation Theorem** (`Enrich⁴ = Enrich³`) closes the enrichment ladder; minds as internal topoi (VII.D90) — self-models with story functors, consciousness as global section; the Categorical Imperative as fixed point on the commitment lattice.
+- **CS-09** self-hosts: ZFC, Lean-like kernels, and the τ-kernel itself as object theories inside τ; proof-as-act vs proof-as-static-relation; computation-as-process vs equation; the Logos sector S_L as the mediator where proof-validity = stance-stability (VII.T80); the boundary collapse lemma (VII.T81).
+- **CS-10** runs the closure test: Categorical Ontology (relations precede relata); the Inevitability Argument (VII.T55) — six ontic requirements converge uniquely to τ; the Metaphysical Problem Map (~17 problems classified resolved / reframed / open); ω-uniqueness (VII.T60); the boundary collapse lemma as the structural climax; the closing question — *what am I willing to live as true?*
+
+By the end of Arc III, the framework has run an end-to-end audit of the no-externalities discipline across all 9 prior steps; the residual commitment register is honestly disclosed as belonging to the reader.
+
+### The closing pattern
+
+The ten steps share a single voice: every later step inherits exactly what earlier steps earned, and hands forward exactly what the next step needs. The kernel's K0–K6 power the Central Theorem; the Central Theorem powers the Hinge Theorem; the Hinge Theorem powers the calibration cascade; the calibration cascade powers the falsification ledger; the falsification ledger powers life's predictions-by-absence; life's SelfDesc² powers reflective structure; reflective structure's Logos sector powers self-hosting; self-hosting's boundary collapse lemma powers ontic closure.
+
+**One algebraic seed (`ι_τ`); one empirical anchor (`m_n`); zero free parameters; one categorical kernel that hosts itself.**
+
 ## How to read this section
 
 Each step page explains what the step builds, why it is required, the key constructions, related Registry items, TauLib modules, book locations, related Results, Verify surfaces, and what the step does not yet establish.

corpus/construction-spine/recover-life/index.md

@@ -93,29 +93,138 @@ construction_step_id: "CS-07"
 
 <div class="notice note"><strong>Status note.</strong> Build status reflects the current internal state of the Corpus. It does not imply external acceptance unless explicitly stated.</div>
 
-## What this step is required to do
+## 1. What this step must build
 
-The program must recover boundary, individuality, metabolism, encoding, heredity, reproduction, variation, evolution, morphogenesis, classification, and ecology as structural capacities.
+The program must build life as a **structural class** — not as Earth biology, not as DNA chemistry, not as a list of organisms. Life must be **categorically defined** so that what counts as life is *visible from the framework*, not labelled from outside.
 
-## What the corpus built
+By the end of this step:
 
-The Corpus frames life as a recoverable structural layer, with Earth life serving as calibration case rather than definition.
+- Two structural predicates — **τ-Distinction** and **SelfDesc** — must define life: their conjunction *is* life.
+- The **Layer Separation Lemma** (VI.T02) must establish that life is genuinely E<sub>2</sub>, **non-reducible to physics**.
+- The **Parity Bridge Theorem** (VI.T03) must identify the weak sector's parity violation as the unique polarity seed for self/non-self distinction (this is the Book III → Book VI bridge).
+- The **Life Loop Class** must formalize metabolism as a structural loop.
+- The **4+1 Sector Template** must classify all life forms (Archaea / Bacteria / Plants / Fungi / Animals).
+- The **Seven Hallmarks** of life must be derived as theorems (VI.T05–VI.T11), not posited.
+- **Predictions by Absence** — virus, neutron, neutron star — must establish falsifiability.
+- The **bridge to E<sub>3</sub>** (consciousness, language) must be opened — though resolution belongs to CS-08.
 
-## What this step builds
+What cannot yet be assumed: phenomenology / qualia (CS-08); proof-theoretic self-hosting (CS-09); ontic closure (CS-10).
 
-The seventh step treats life as a recoverable structural class. It asks what must hold for boundary, individuality, metabolism, heredity, reproduction, variation, evolution, morphogenesis, classification, and ecology to become intelligible inside the framework.
+## 2. The construction challenge
 
-## Why this step is required
+This step is hard for five interlocking reasons.
 
-Life cannot be used as an unexplained external input if the framework claims full-stack scope. The Corpus must show how life becomes a valid structural category.
+**2.1 Cannot define life by rabbits, DNA-only, or Earth catalogue.** Life must be a structural class, not an Earth-biology inventory. "Living things have DNA" is a calibration observation about a specific carrier, not a structural definition.
 
-## Key constructions
+**2.2 Must distinguish life from non-life structurally.** The boundary cannot be empirical labelling. If the framework says "life is what we call living," it has accomplished nothing.
 
-- Life as recoverable category.
-- Boundary and individuality.
-- Metabolism and throughput.
-- Encoding and heredity.
-- Evolution, morphogenesis, classification, and ecology.
+**2.3 Must account for boundary, metabolism, encoding, heredity, evolution, development, ecology — but as derived structures, not as primitives.** Each canonical hallmark of life must be a *consequence* of the structural definition, not a separate posit.
+
+**2.4 Earth life is calibration, not definition.** Specific Earth organisms are sector instantiations; the structural class admits substrate-independent realizations (Book VI Part VII Cosmic Life).
+
+**2.5 Must connect to physics without collapse.** Life is at E<sub>2</sub> (above physics' E<sub>1</sub>); the Layer Separation Lemma must establish *non-reducibility* without losing the bridge from physics' carrier.
+
+## 3. What Panta Rhei builds
+
+The Corpus frames life as a recoverable structural layer, with Earth life serving as calibration case rather than definition. Step 7 treats life as a recoverable structural class — asking what must hold for boundary, metabolism, heredity, reproduction, evolution, morphogenesis, classification, and ecology to become intelligible inside the framework.
+
+### Two structural predicates: τ-Distinction + SelfDesc (Book VI Part I)
+
+Life is defined by the conjunction of two predicates:
+
+> **Life = τ-Distinction ∧ SelfDesc.**
+
+- **τ-Distinction** — the predicate that establishes self/non-self differentiation through the boundary algebra. Built on the **Parity Bridge Theorem** (CS-04 carrier inheritance): the **weak sector's unique parity violation** is the *sole polarity seed* for self/non-self distinction.
+- **SelfDesc** — the predicate that the system models its own boundary structure. Inherited from CS-03's self-description (II.D54).
+
+The conjunction is what makes life *visible from the framework*: any structure that admits τ-Distinction and carries SelfDesc *is* life, regardless of substrate (carbon, silicon, plasma).
+
+### The Layer Separation Lemma (Book VI Part I; VI.T02)
+
+Life is genuinely E<sub>2</sub> — **non-reducible to physics**. The Layer Separation Lemma establishes that no E<sub>1</sub> (physics) construction satisfies the τ-Distinction ∧ SelfDesc conjunction by itself; the conjunction requires E<sub>2</sub>-level enrichment.
+
+This is what blocks naive reductionism. Life is not "complex physics that crosses a threshold" — it is a *structurally distinct enrichment layer*.
+
+### The Parity Bridge Theorem (Book III ↔ Book VI; VI.T03)
+
+The Book III → Book VI bridge: the **weak sector's unique parity violation** (CS-04 §4+1 Template) is the *sole polarity seed* for self/non-self distinction at E<sub>2</sub>. Without parity violation, τ-Distinction cannot be earned. With it, the polarity required for "self vs non-self" emerges as a structural readout of the weak sector.
+
+This is why the Parity Bridge Theorem already appeared in CS-04 as load-bearing for the carrier's later instantiations — its full payoff lands here.
+
+### The 4+1 Sector Template at E₂ (Book VI Parts II–VI)
+
+The 4+1 template (CS-04) instantiates at E<sub>2</sub> as the canonical taxonomy of life kingdoms — *not* as a coincidence with the physics-side instantiation, but as a structural readout at a different enrichment level:
+
+| Sector | Kingdom | Axis | Book VI Part |
+|---|---|---|---|
+| α (Persistence) | Archaea | Temporal axis | II |
+| π (Agency) | Bacteria | Spatial axis | III |
+| γ (Source) | Plants | Production fiber | IV |
+| η (Closure) | Fungi | Recycling fiber | V |
+| ω (Consumer) | Animals | Mixed sector | VI |
+
+### The Life Loop Class (Book VI Part I)
+
+Metabolism is formalized as the **Life Loop Class** — a τ-categorical loop structure where the system continuously regenerates its own boundary through interaction with non-self. This makes metabolism a *structural property*, not a biochemical one.
+
+### The Seven Hallmarks as theorems (Book VI Part I; VI.T05–VI.T11)
+
+The traditional seven hallmarks of life — boundary, metabolism, sensitivity, heredity, growth, response, reproduction — are derived as **theorems** from τ-Distinction + SelfDesc + Life Loop Class. Each hallmark is a *consequence*, not a definition.
+
+### Predictions by Absence (Book VI Part I; VI.T15)
+
+Falsifiability is established through **negative predictions**: structures that look life-like but lack the τ-Distinction ∧ SelfDesc conjunction must be classified as **non-life**:
+
+- **Virus** — has SelfDesc (genetic encoding) but no τ-Distinction (no metabolic boundary); not life.
+- **Neutron** — has boundary (defect-bundle from CS-05) but no SelfDesc; not life.
+- **Neutron star** — has stable boundary structure but no SelfDesc; not life.
+
+If any of these were classified as life by the framework, the definition would be wrong. **The framework is committed in advance.**
+
+### The Bridge to E₃ — Consciousness and Language (Book VI Part VIII)
+
+The mixed sector (Animals; ω-coupling) opens the door to E<sub>3</sub> via two structural facts:
+
+- **Consciousness** is defined as the consumer sector's **SelfDesc applied to its own modeling** — not qualia (that is Book VII / CS-08), but the formal requirement that the self-model includes a model of itself.
+- **Language** extends the lemniscate between agents — *life as nature's expression of computation*.
+- **The ω-germ question** — whether the consumer's self-modeling capacity reaches all the way to ontic ground — *cannot be resolved diagrammatically*. This is the **principled science–faith boundary**: a category-theoretic statement about what proof can and cannot reach.
+
+Six export contracts close Book VI; Book VII opens at consciousness (CS-08).
+
+## 4. Why this matches the required answer-shape
+
+Step 7 builds life as a recoverable structural class. Its admissibility is evaluated against the obligation to make life *structurally visible* — neither labelled from outside nor reduced to physics.
+
+**Gluing.** CS-07 inherits CS-04's 4+1 template (now instantiated at E<sub>2</sub>); CS-04's Parity Bridge Theorem (now load-bearing for self/non-self); CS-03's self-description (now SelfDesc); CS-05's defect-bundle formalism (now the metabolic boundary's substrate).
+
+**No-externalities discipline.**
+
+- **No DNA primitive.** Encoding is a derived structure; specific DNA is calibration data.
+- **No Earth catalogue.** Kingdoms are sector instantiations; specific organisms are bridge calibration.
+- **No vital force.** Life is τ-Distinction ∧ SelfDesc — no élan vital, no irreducible vitalism.
+- **No reductionist collapse.** The Layer Separation Lemma (VI.T02) blocks any "life is just complex physics" claim.
+
+**Earned language.** Life is *defined* by τ-Distinction ∧ SelfDesc, not posited. Hallmarks are *theorems* (VI.T05–VI.T11). Predictions by Absence (VI.T15) make the definition *falsifiable*.
+
+**Internal standpoint.** Life is τ-categorical at E<sub>2</sub>; the bridge to Earth biology is *calibration*, surfaced explicitly.
+
+**Step gluing — what later steps does it enable.**
+
+- **CS-08 Recover Reflective Structure** opens at consciousness: SelfDesc-of-SelfDesc (VI.D80) is the bridge to E<sub>3</sub>.
+- **CS-09 Self-Host Formal Systems** uses the principled science–faith boundary (VI.P50) as a model for self-hosting limits — what the framework can prove vs what remains a commitment.
+- **CS-10 Test Ontic Closure** uses the Layer Separation Lemma to test whether the no-externalities discipline holds across enrichment layers.
+
+**Bridge status.** Earth biology is calibration data; the kingdom taxonomy is structural; the mapping between specific organisms and the structural predicates is explicit bridge work.
+
+**Unresolved boundaries.** CS-07 does not by itself settle:
+
+- Empirical adequacy at the species level (extremophiles, syncytia, coral-like collectives — edge cases pending).
+- Cosmic-life specifics (Book VI Part VII remains conjectural in places).
+- The ω-germ question, which is *principled-non-resolution* by design — the science–faith boundary is part of the construction's discipline, not a gap.
+
+These are explicit handoffs / honest limits.
+
+**This is an internal construction claim, not external acceptance.** Step 7 builds life as a structural class under τ-discipline; reviewer scrutiny is invited via Book VI's Layer Separation proof, the kingdom-as-sector mapping, and the Predictions by Absence falsification surface.
 
 ## Registry spine