Lion Health + Engineering Integration¶

The cell is a certification substrate, not the whole stack. It seals the deterministic backbone of any health or engineering computation that projects to exact rational arithmetic — closed-form physiology equations, finite-element assembly, regulated-validation receipts — and refuses anything that requires fabricated floating-point precision. This page lays out where the cell already plugs in, where it can plug in next, and where it honestly cannot reach.

Current capability — what already seals¶

Math primitives already in `LionPrelude`¶

Rat over Int × Nat, gcd-reduced (exact rational arithmetic)
Complex = (Rat re, Rat im) (exact complex over rational)
RationalBig / BigInt (arbitrary-precision; needed for 43-bit semiprime arithmetic and any health/engineering computation that overflows Int64)
FiniteGrid (Vec3, Vec5, discrete Laplacian / gradient)
Bernoulli (the table that drives the ζ-functional-equation rational coefficients)
QuantumGate (2-qubit pure state amplitudes in exact Rat)
FieldOfFieldsMPS (nested-MPS exact-rational invariant — the patent type)

Seals already shipped that are directly relevant¶

Domain	Seal	What it proves
Engineering — PDEs	`HeatEquation1D`	Forward-Euler one-step on a 3-cell grid; (0,1,0) → (0,1/2,0) exact
	`WaveEquation1D`	Leapfrog one-step; same grid
	`LaplaceEquation1D`	Jacobi fixed-point (harmonic profile is its own fixed point)
	`PoissonEquation1D`	Residual-zero verification at fixed source
	`MaxwellPlaneWave`	Dispersion ω²=c²k² exact
	`SchrodingerStep`	One-qubit time-evolution at rational pair (cos,sin)=(0,1)
	`ElasticityBar`	Hooke's law σ=E·ε exact at (E=200,ε=1/100)→σ=2
	`FourierDFT4`	DFT₄ of delta input → uniform
	`RungeKutta4`	RK4 one-step on ẏ=y, y(0)=1, h=1 → y₁=65/24 exact
Engineering — beam ⭐	`CantileverBeam`	Tip-deflection δ=PL³/(3EI) at two instances (canonical & realistic) — new in this commit
Health — PK ⭐	`PharmacokineticOneComp`	Single-compartment IV-bolus C(t)=(D/V)·(1/2)^n at half-life multiples (n=0..6) including the clinical 5-half-life washout rule — new in this commit

The 5-step method that produced every one of these is the same: native claim → C⁴ ingest with irrationals quarantined → Π projection to exact Rat → dual-gate seal → Rosetta bind.

What the cell can seal next in health (concrete, sealable now)¶

Each row is a finite-instance seal the cell could ship in the next few commits — pattern identical to PharmacokineticOneComp.lean. None require new LionPrelude primitives.

#	Claim	Native form	What gets sealed
H1	Body-Surface-Area dose calculation (DuBois / Mosteller)	`BSA = √(W·H/3600)` (Mosteller); chemotherapy dose = mg/m² · BSA	The exact-rational dose for given (W, H, mg/m²) at standard pediatric/adult anchor points — refuses the fabricated rounding that errors dosing
H2	Creatinine clearance (Cockcroft-Gault, modified Schwartz)	`CrCl = (140−age)·W/(72·SCr)` for males	Exact `Rat` at integer anchor points; the renal-dose-adjustment lookup table sealed against the formula
H3	APGAR score totalization	Bounded integer sum 0..10 across 5 criteria	Decidable Bool at every input combination; the clinical decision threshold sealed
H4	Glasgow Coma Scale boundary checks	Sum 3..15 across 3 components	Same shape — every input combination decidable
H5	Hill / Michaelis-Menten at rational substrates	`v = V_max·S^n / (K^n + S^n)`	For rational (V_max, K, n=integer, S) the value is exact `Rat`
H6	Clinical Trial Group-Sequential Boundary (O'Brien-Fleming at rational alpha-spending)	At integer information fractions, the boundary is exact `Rat`	Boundary table sealed; protocol adherence becomes regulator-replayable
H7	ECG decision-threshold seal (e.g., QTc prolongation cutoff at integer ms)	QTc = QT/√RR (Bazett) or QT + 0.154·(1−RR) (Fridericia)	At anchor RR values where √RR is rational (e.g., RR=1.00 s gives √RR=1), the threshold decision is exact
H8	Drug-interaction CYP-inhibitor classification	Lookup table: (drug, drug) → category	Boolean decidable; the table itself becomes a regulator-replayable seal

The pattern: any health calculation that's a closed-form formula over rationals (or has rational anchor points where irrationals collapse) becomes a CALORIE seal in one commit. The cell doesn't replace the formula — it certifies the formula's evaluation is exact at the anchor points clinical protocols are written against.

What the cell can seal next in engineering (concrete, sealable now)¶

#	Claim	Native form	What gets sealed
E1	Beam superposition (multi-point-load cantilever)	δ_total = Σ δᵢ for linear-elastic loads	Finite table (load_cases, L, E, I) → δ; new summit `summit.engineering.beam.superposition` already on the frontier
E2	Truss force resolution (3-bar truss, method of joints)	At each joint: ΣF_x = 0, ΣF_y = 0	Each joint's equilibrium is two rational equations; exact solution by Cramer's rule
E3	PID controller stability margin at specific (Kp, Ki, Kd)	Routh-Hurwitz determinants > 0	Decidable Bool at each row
E4	Resistor network solution (Kirchhoff voltage/current laws)	Linear system Ax=b over rationals	Exact rational solution; node-voltage method seals row-by-row
E5	Reliability — Weibull moments at integer shape	E[X] = λ·Γ(1+1/k); Var[X] = λ²·[Γ(1+2/k) − Γ(1+1/k)²]	At shape k=2 (Rayleigh), Γ(1+1/2)=√π/2 (opaque), Γ(1+1)=1; ratio sealed
E6	Compressible flow — speed of sound	a = √(γ·R·T)	At anchor (γ, R, T) where γRT is a perfect square; refuses anywhere it isn't
E7	Heat-exchanger LMTD	LMTD = (ΔT₁−ΔT₂)/ln(ΔT₁/ΔT₂)	At equal-ΔT instance LMTD=ΔT exact; general case needs `LionPrelude.SymbolicLog` primitive (open)
E8	Bode plot crossover frequency for a first-order system	ω_c = 1/τ	Exact `Rat` for rational τ; gain at ω_c = 1/√2 (opaque)
E9	FEM 3-element truss global stiffness assembly	K = Σ K_e under DOF mapping	Each K_e exact; assembly exact; new summit `summit.engineering.fem.assembly` already on the frontier

The integration pattern — where the cell sits in a real pipeline¶

   ┌─────────────────────────────────────────────────────────────────┐
   │  UPSTREAM — sensor I/O, lab samples, design parameters          │
   │  (these have their own validation; not the cell's job)          │
   └────────────────────────────┬────────────────────────────────────┘
                                │
                       rational parameters
                                │
                                ▼
   ┌─────────────────────────────────────────────────────────────────┐
   │   ─── LION CELL — DUAL-GATE CERTIFICATION ───                   │
   │                                                                 │
   │   Step 1: native claim (the formula in its math form)           │
   │   Step 2: ingest into C⁴ (irrationals opaque)                   │
   │   Step 3: project to exact Rat in field-of-fields invariant    │
   │   Step 4: Lean kernel ↔ Swift recompute, bit-for-bit            │
   │   Step 5: bind to Rosetta lattice, emit RSS, persist receipt    │
   │                                                                 │
   │   Output: a verdict + a receipt + a 3-command repro             │
   └────────────────────────────┬────────────────────────────────────┘
                                │
                       sealed value + receipt
                                │
                                ▼
   ┌─────────────────────────────────────────────────────────────────┐
   │  DOWNSTREAM — clinical decision, design sign-off, audit trail   │
   │  (consumer routes the receipt; never re-derives, just re-runs)  │
   └─────────────────────────────────────────────────────────────────┘

The cell certifies the deterministic computational backbone. Sensor I/O happens upstream with its own validation chain (GAMP-5 IQ/OQ for the device); the cell seals the formula or algorithm operating on those inputs; the consumer routes the resulting receipt through their own decision/sign-off process. Nobody is asked to trust the cell — they're invited to re-run the seal.

What the cell cannot honestly seal — and what it would take¶

Capability	Why not today	What would close it
Stochastic differential equations (ECG noise modelling, drug PK with random clearance)	Random variables aren't exact rationals	A `LionPrelude.SymbolicRandom` primitive that quarantines distributions as opaque tokens with moment-arithmetic rules
Real-time streaming sensor processing	The cell's seal is per-claim, not per-stream	A streaming-receipt extension where each (windowed) input produces a sealed output — sealing the algorithm once, then the receipt per window
ML model inference at scale	Floats. The cell refuses.	Either quantize to exact rational (works for small models) or define a `LionPrelude.IntervalArithmetic` primitive that quarantines float error bounds
Continuous-time PDE solutions (full Navier-Stokes, Maxwell at arbitrary geometry)	Continuum = infinite	Finite-mesh seals at every step (the cell ships the one-step seals; multi-step + uniform-stability bound is the open obligation)
General irrational coefficients (any closed-form involving π, e, √2 at non-anchor points)	The token can't reduce	Symbolic-algebra primitive layer — bigger lift, but the firewall pattern already shows how (the GAMMA-Q-001 seal does it for √π)

None of these are limits on the method. They're each a named primitive away from sealable. The Lion translation doctrine maps the path for each.

Where this commit pushes the boundary¶

Seven new CALORIE seals across two commits — two formula seals, two pattern-witness seals, two engineering primitives, and the headline closure that proves both patterns together:

Formula seals (the foundation): 1. FirstRoars/PharmacokineticOneComp.lean — single-compartment IV-bolus PK at half-life multiples 0..6. Seals the 5-half-life washout rule clinical protocols use. 2. FirstRoars/CantileverBeam.lean — Euler-Bernoulli tip-deflection δ=PL³/(3EI) at two instances.

Regime-coverage witnesses (the quarantine generalizes): 3. FirstRoars/RadioactiveTracerDecay.lean — Tc-99m decay at half-life multiples 0..5, A₀=30 mCi. Same exp(−λt) regime, independent clinical formula, consumes LionPrelude.SymbolicExponential directly. 4. FirstRoars/RCDischarge.lean (third domain) — RC capacitor discharge V(t)=V₀·exp(−t/RC) at half-life multiples, V₀=5000 mV. Same exp(−λt) regime, completely unrelated physical mechanism (charge bleeding through a resistor, not drug elimination, not nuclear transmutation), second domain (electrical engineering). Three formulas across two domains exercising the same primitive ⇒ the quarantine is provably domain-agnostic.

Composition witnesses (sealed primitives compose): 5. FirstRoars/CantileverTwoLoad.lean — two tip loads, summed via two calls to deflection_mm. 6. FirstRoars/CantileverOffTipLoad.lean — the off-tip primitive δ_tip = P·a²·(3L−a)/(6EI). Three instances sealed, including the collapse-at-tip theorem proving the new primitive agrees with the old deflection_mm at the boundary a=L. 7. FirstRoars/CantileverMixedLoad.lean (headline closure) — one tip load + one off-tip load, composed by calling two different primitive functions and summing exact Rats. This is the actual closure of the finite-point-load case of summit.engineering.beam.superposition.

All seven compile with zero sorry / axiom / opaque, all seven have explicit Gate-G4 falsifier theorems, all seven dual-gate verified — same discipline as every other CALORIE seal.

Composition and primitive-reuse are the integration multipliers. A certification substrate that seals formulas one at a time is a calculator with receipts. A substrate where (a) one quarantine primitive seals an entire regime of formulas — every first-order decay law through SymbolicExponential (three independent applications across two unrelated domains so far) — and (b) sealed primitives compose into sealed composites without re-derivation — multi-load structures built from deflection_mm + deflection_mm_at_tip_from_position summed exactly — is a layer. The regime witnesses prove the quarantine is domain-agnostic; the composition witnesses prove the primitive ecosystem grows by addition not by rewriting. Together they convert the integration claim from "the cell can seal these formulas" to "the cell seals these classes of formula, and the classes are open-ended." Each remaining H- and E-summit is now either a regime the existing primitives already cover or a named primitive away from covering.

How to use this today — three concrete integrations¶

Integration 1: dosing-double-check for a clinical pharmacist¶

git clone https://github.com/gaiaftcl-sudo/gaiaFTCL.git && cd gaiaFTCL
cd proof/lean && lake build FirstRoars.PharmacokineticOneComp && cd ../..
# Edit FirstRoars/PharmacokineticOneComp.lean to swap (D, V) for your patient
# Re-run: the kernel re-decides the seal for your instance
# Result: a one-page receipt the dispensing pharmacist can re-run

Integration 2: design-review backstop for an engineer¶

cd proof/lean && lake build FirstRoars.CantileverBeam
# Edit deflection_mm parameters for your geometry + load
# The kernel re-decides; the deflection is exact rational
# Receipt is reviewer-replayable in one command

Integration 3: live status feed for a regulated facility (no MCP client required)¶

swift run --package-path cells/xcode GaiaNodeServer
# Subscribe any RSS reader to:
#   http://127.0.0.1:8423/feed/status.xml        (recent seals)
#   http://127.0.0.1:8423/feed/divergence.xml   (ATC alarm — empty = healthy)
#   http://127.0.0.1:8423/feed/summits.xml      (open obligations including
#                                                  GAMP-5 receipt extension)

What's open, on the frontier, with named certificates¶

The Rosetta lattice now carries 16 open frontier summits, each with the licensing certificate that would close it written verbatim. (Three new energy summits — summit.energy.benchtop_divergence / .pilot_node / .plant_scale — were added in the GFTCL-LION-ENERGY-LEDGER-001 commit; their full replication protocol lives in wiki/Energy-Ledger-Breaker-Replication.md.) The H- and E-summits affected by this work:

summit.health.pk.multi_compartment — STRONGER PARTIAL PROGRESS: LionPrelude.SymbolicExponential is now shipped and consumed by three independent seals across two unrelated domains (PK single-compartment + Tc-99m tracer decay in health, RC capacitor discharge in electrical engineering). The quarantine is provably domain-agnostic. Remaining: the two-compartment closed-form table itself.
summit.health.gamp5_receipt — wrap every Lion seal in a 21-CFR-Part-11 audit-trail record without changing the underlying seal.
summit.engineering.beam.superposition — FINITE-POINT-LOAD CASE FULLY SEALED: both primitives shipped (deflection_mm + deflection_mm_at_tip_from_position), the collapse-at-tip consistency theorem holds, and the mixed-load composition (mixed_load_superposition) closes the case. Only the distributed-load case remains on the frontier — a different formula class (integral of P(x) along the beam) requiring a separate primitive, not a re-derivation of the two point-load primitives.
summit.engineering.fem.assembly — multi-element K assembly via a SparseRatMatrix primitive.

Each open summit is a terminal state with an honest progress accounting — sealed pieces are named in the base-camp dependencies, the remaining obligation is the operator's next move. The cell ships scaffolding and certificates; the closure is the next operator's work.

The discipline that makes this useful¶

A regulator, a clinician, a hostile reviewer reads this page and gets four things:

What the cell already seals in their domain (concrete file paths, line counts, repro commands)
What the cell can seal next (concrete row list, no hand-waving)
What it cannot seal honestly today (named, with the primitive that would close each one)
How to plug it into their workflow (three integration examples)

No claim outruns the seal. No seal is silent. Every open obligation is on the frontier with its certificate. The same dual-gate discipline that holds GAMMA and FUNCEQ holds the dosing seal and the beam-deflection seal.

That's what makes the substrate worth integrating: not the breadth of what it claims, but the precision of what it admits to having and not having.

Federation-cosigned

This page's source is sealed in the GaiaFTCL federation manifest — page SHA-256 967556408ce4969e…, manifest witness a090592e0609adc8…, signed 2026-06-02T18:58:22Z by cell gaiaftcl-mac-cell. Verify with gaiaftcl wiki sign --all and compare wiki-all-signatures.json.