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General Life Form: A Unified Scientific Framework for Variables, Interactions, Environment, and Verification

1) Purpose, Scope, and Domain of Validity

Goal. Establish a General Life Form (GLF) framework that unifies energy, information, structure, and governance across cells, organisms, consortia, and synthetic systems. The framework fixes a single variable set and audit procedure so results are comparable across scales and architectures.

Scope. GLF treats living systems as open, budgeted processes running in discrete ticks with measurable flows and constraints. It covers:

Energy & information budgets (inputs, useful work, dissipation, losses).
Structural states (features s) and drives (λ) with a declared geometry (I, M).
Couplings (Θ) among modules and constraints (Γ) that limit rates, capacities, or policies.
Observer-auditable outcomes: each claim carries a declared domain, seeds, thresholds, and a verification footer.

Domain of validity. The framework applies when the following are empirically measurable within declared error bars:

Budgets: inflows/outflows, useful structural work, dissipation, and losses per tick.
Couplings & constraints: which channels are active (Θ) and what limits apply (Γ).
Ticks: timing resolution and synchronization across interacting subsystems.

Operational definition (preview).
Life criterion: “Positive value budget with synchronized ticks and bounded dissipation under declared constraints.”

Φ_budget(t) ≡ Φ_in(t) − Φ_out(t) − losses(t) ≥ 0 . (1.1)
Tick synchronization: max_i,j |τ_i − τ_j| ≤ ε_τ . (1.2)
Dissipation bound (declared constraint): Γ(t) ≤ Γ* . (1.3)
Decision rule (preview): Alive(t) ⇔ [Φ_budget(t) ≥ 0] ∧ [Γ(t) ≤ Γ*] ∧ [max_i,j |τ_i − τ_j| ≤ ε_τ]. (1.4)

What GLF asserts—and what it does not.

Asserts: if budgets, couplings, and ticks are measurable, then state, drive, geometry, and governance can be reported on a single ledger with falsifiable thresholds ((1.1)–(1.4)).
Does not assert: teleological purpose, specific biochemical mechanisms, or sufficiency of any single substrate. GLF is substrate-agnostic but measurement-bound.

Evidence requirements. Any claim made under GLF must specify:

Declared domain: experimental conditions, tick length, sensors, and uncertainty.
Budget table: Φ_in, Φ_out, losses, Γ, with units and sampling cadence.
Constraint sheet: active Γ limits (capacities, rates, policies) and the current Γ*.
Timing sheet: τ values, ε_τ, and any known phase relations among modules.
Verification footer: seeds, hashes, thresholds, gate outcomes, decision, reviewer.

Out of scope (for this paper). Detailed legacy controversies (e.g., origin-of-life pathways, consciousness taxonomies, cancer subtyping) are not resolved here. Paper 1 only fixes the common language, variables, tests, and admissible domain. Paper 2 will use these definitions to address the long-outstanding questions.

2) Core Variable Ontology and Units (Value–Drive–Geometry)

Purpose of this section. Fix a single, reusable dictionary for all later claims. Every symbol below carries (i) an operational meaning, (ii) units, and (iii) a default estimation recipe. No new symbols appear later without being added here.

2.1 State, Drive, and Statistics

State (features, means).

s = vector of measured features (means). Units = units of the underlying feature map φ (e.g., mmol·L⁻¹, Hz, mV, dimensionless fractions).
Reference & feature map (declared): choose a baseline distribution q(x) and detectors φ(x); then the statistical potential is
ψ(λ) ≡ log E_q[ exp( λ·φ(x) ) ] . (2.0)

Drive (natural parameters).

λ = natural parameters conjugate to s. Units = “per-unit of s” (inverse of s units).

Statistics & information.

ψ(λ) = log-partition (statistical potential). Units = bits (or J·K⁻¹ scaled).
I(λ) = Fisher information / feature covariance at λ. Units = variance of s.
Conjugacy identities (define once, reuse everywhere):
s = ∇ψ(λ) ; I = ∇²ψ(λ) . (2.1)
λ = ∇Φ(s) ; M = ∇²Φ(s) = I⁻¹ . (2.2)
G ≡ Φ(s) + ψ(λ) − λ·s ≥ 0 . (2.3)

Interpretation.

s reports what the system is like; λ reports the pressure to change s.
I measures how well the system tells similar states apart (SNR / curvature).
G (Fenchel–Young gap) is an “alignment/dissipation deficit”; G = 0 iff (s, λ) are perfectly matched; G > 0 exposes statistical or physical slack.

Units (canonical).

Φ, ψ: bits (information units) or J·K⁻¹ scaled (via k_B·ln 2 if needed).
λ: per-unit of s. I: units of var(s). M: inverse-variance units. G: same as Φ (bits or J·K⁻¹ scaled).

2.2 Value, Geometry, and Conditioning

Value (minimum divergence potential).

Φ(s) = minimum relative-entropy “value” at fixed s (declared relative to q and φ). Units = bits (or J·K⁻¹ scaled).
Gradient & Hessian: λ = ∇Φ(s) ; M = ∇²Φ(s) . (2.2)
M(s) (structural inertia) quantifies how “heavy” it is to move s locally; high M ⇒ small moves cost a lot of value-work.

Conditioning and robustness.

κ ≡ cond(M) (e.g., ratio of largest to smallest eigenvalues). Unitless; large κ = ill-conditioned control/estimation.
ρ (robustness radius): largest perturbation norm that remains recoverable under declared constraints Γ. One operational definition is
ρ ≡ sup { r : min_{|Δs| = r} ΔΦ_Γ(Δs) ≥ 0 } . (2.6)

Gauge notes (no hidden re-scaling).

Declaring q and φ fixes the “gauge.” Rescaling φ rescales λ and changes numerical I, M; therefore, publish φ’s units and any standardization (e.g., z-scores) in the methods block.

2.3 Work, Power, and Budgets

Value-work along a path.
W_s ≡ ∫ λ · ds . (2.4)

Power balance per tick.
Φ̇ ≡ P_in − Π_diss . (2.5)

Closed-form budget identity (used in every results section).
ΔΦ = W_s − Δψ − Γ_loss . (2.7)

Meanings.

W_s: useful structural work (how much “value” was invested to move s).
Δψ: statistical expansion cost (paying to widen accessible states).
Γ_loss: dissipative losses declared in the constraints sheet (heat, transport, policy throttles).

2.4 Estimation Recipes (publish these with units)

Declare (q, φ). Provide units for each component of φ; publish code or a concise rule (e.g., log-counts per 10⁶ reads; mV; Hz).
Compute ψ(λ), s(λ) = ∇ψ, I(λ) = ∇²ψ. Report numerical stability (conditioning of I).
Fit Φ(s) by convex duality or direct estimation. Then extract λ(s) = ∇Φ, M(s) = ∇²Φ, and G via (2.3).
Audit κ and ρ. Flag κ > κ* as “hard-to-control” and estimate ρ by controlled perturbations or certified bounds.
Report budgets. For each tick: P_in (input power), Π_diss (measured dissipation), ΔΦ from (2.7), and W_s from (2.4). Provide uncertainty bars.

2.5 Minimal Reporting Standard (what must appear in every figure/table)

Variable block: {s units, λ units, Φ/ψ units, I units, M units}.
Gauge block: {q, φ, preprocessing, scaling}.
Geometry block: {eigs(M), κ, ρ method}.
Budget block: {W_s, Δψ, Γ_loss, ΔΦ} per tick.
Gap KPI: {G with threshold G* and lamp color if used later}.

2.6 Sanity Checks (quick pass/fail before publication)

Gap zero test: Plug λ(s) into ψ; verify G → 0 within tolerance on calibration data.
Units consistency: λ·s has the same units as Φ and ψ.
Curvature positivity: I ≻ 0 and M ≻ 0 on reported ranges (or declare exceptions).
Conditioning guard: if κ is high, either regularize (declare) or narrow the operating domain.
Robustness claim: any claim of “resilience” must cite ρ and how (2.6) was evaluated.

One-line takeaway. Publish (q, φ) and units, then compute {ψ, s, I} ↔ {Φ, λ, M} and the gap G via (2.1)–(2.3); audit budgets with (2.4)–(2.7); report κ and ρ so others can replicate, compare, and control your system.

3) Dynamics: Local Dissipative Action and Budgets

Purpose of this section. Provide a tick-level law of motion and a closed-form budget so any experimental or synthetic life-like system can be audited the same way.

3.1 Principle (optimize useful value-change while paying explicit dissipation)

We treat each system as a budgeted process evolving in discrete ticks Δτ. The goal is to increase Φ (minimum-divergence “value”) by doing useful structural work W_s, while explicitly accounting for dissipation Γ and statistical expansion Δψ.

3.2 Single-tick Lagrangian and local action

Single-tick Lagrangian (value rate minus structural work rate).
L ≡ Φ̇ − λ·ṡ . (3.1)

Local action with dissipation penalty.
A_local ≡ ∑_ticks ( L − β·Γ ) Δτ . (3.2)

Δτ = declared tick (seconds or cycles).
Γ = measured dissipation per tick (J per tick or bits per tick after scaling).
β = unit reconciliation / policy weight (dimensionless). If Γ is reported in the same units as Φ̇, set β = 1; otherwise β converts units and embeds governance priorities (e.g., stricter penalties for scarce channels).

Operational choice. Controls are the drive schedule λ_t and (optionally) couplings Θ_t, chosen to maximize A_local subject to declared constraints (rate limits, quotas, safety invariants).

3.3 Closed-form budget law (reporting identity)

Budget law (tick-to-tick).
ΔΦ = W_s − Δψ − Γ_loss . (3.3)

W_s ≡ ∫ λ·ds along the observed path during the tick.
Δψ = change in log-partition (statistical “expansion cost”).
Γ_loss = measured dissipation (heat, transport leakage, policy throttles), as declared in the constraint sheet.

Interpretation. (3.3) is the accounting backbone: any increase in Φ must be explained by useful work in excess of statistical expansion and losses. It turns “metabolism, signaling, repair” into first-class budget items with auditable units.

3.4 From action to motion (how dynamics are implemented)

While (3.1)–(3.3) define the audit, actual motion is implemented with a declared update rule. Two minimal choices cover most cases:

A) Controlled gradient-flow step (metric M).
ṡ = K(s) · ( λ − λ_resist ) , with K(s) ≈ M(s)⁻¹ and λ_resist ≡ ∂Γ/∂ṡ . (3.4)

K(s) is a mobility operator (often chosen as M⁻¹ to respect the geometry).
λ_resist captures friction/drag induced by Γ (e.g., transport limits, viscosity, throughput ceilings).

B) Discrete tick update (experiment-friendly).
s_{t+1} = s_t + Δτ · F( s_t , λ_t , Θ_t , Γ ) . (3.5)

Publish F (or a pointer to the numerical scheme), then audit the realized step with (3.3).
If F is unknown (black box), you can still compute ΔΦ, W_s, Δψ, Γ_loss directly from measurements.

3.5 Decomposing Γ and assigning β (policy-ready)

Channel decomposition.
Γ = ∑_c Γ_c , with c ∈ {metabolism, transport, signaling, computation, policy throttle}. (3.6)

Weighted action (if needed).
A_local = ∑_ticks ( Φ̇ − λ·ṡ − ∑_c β_c·Γ_c ) Δτ . (3.7)

Use β_c to reflect scarcity or safety (e.g., β_{computation} includes a thermal budget; β_{policy} encodes red-tape limits).
Report both raw Γ_c and weighted β_c·Γ_c in the footer to keep physics and policy transparent.

3.6 What to publish (so results are reproducible)

Per tick (or window):

Budget table: {W_s, Δψ, Γ_loss, ΔΦ}.
Power table: {P_in, Π_diss} with Φ̇ = P_in − Π_diss.
Controls: {λ_t, Θ_t} if applicable; otherwise “black-box” flag.
Timing: Δτ and clock synchronization error bars.
Footer: [seed][hash][units][Γ budget][G threshold][decision][reviewer].

Units sanity.
λ·ṡ, Φ̇, and Γ must share the same reported units per tick (bits per tick or J·K⁻¹ per tick after scaling). If not, disclose the conversion inside β (or β_c).

3.7 Interpreting the budget (diagnostics you can trust)

Efficiency. η_Φ ≡ ΔΦ / W_s ∈ [0, 1] when Δψ + Γ_loss ≥ 0 . (3.8)
Leakage ratio. ℓ ≡ Γ_loss / W_s . (3.9)
Exploration tax. ξ ≡ Δψ / W_s . (3.10)

Rules of thumb:

High ℓ → tighten transport or thermal channels (Γ engineering).
High ξ with no performance gain → over-exploration; reduce search or regularize.
Negative ΔΦ → system is consuming value (repair backlog, stress, or mis-specified λ).

3.8 Minimal micro-protocols (copy into Methods)

MP-1 (black-box compatible).

Measure s_t and s_{t+1}; estimate λ_t (from ∇Φ or a calibrated surrogate).
Compute W_{s,t} ≈ λ_t · ( s_{t+1} − s_t ).
Compute Δψ_t from your declared q, φ.
Measure Γ_{\text{loss},t} (calorimetry, transport counters, policy logs).
Report ΔΦ_t = W_{s,t} − Δψ_t − Γ_{\text{loss},t}. Attach the footer.

MP-2 (model-based control).

Choose K(s) ≈ M⁻¹(s); set β or β_c per channel.
Update s with (3.4) or (3.5).
Tune λ_t to maximize A_local over a receding window while keeping invariants (ρ, κ, safety I-set) within bounds.
Audit with (3.3); if ΔΦ underperforms or lamps go Red, rollback and reduce Γ or adjust Θ.

One-line takeaway. The dynamics are “optimize Φ gains minus explicit Γ costs per tick,” and the only acceptable proof of improvement is the budget law (3.3) reported with units, timing, and a verification footer.

4) Interaction Topology: Couplings, Constraints, and Channels

Purpose of this section. Specify how subsystems talk (Θ), what limits them (Γ), through which pipes they move resources (channels), and how we measure lock-in vs. leakage across module boundaries. This makes cross-scale biology (gene → protein → tissue → organism → consortium) auditable with one playbook.

4.1 Modules and couplings (Θ)

We model a system as a directed multigraph 𝔾 = (V, E) with modules v ∈ V and interfaces e ∈ E. Each module carries a state block s_v; edges carry couplings.

Block form (schematic).
Θ = { Θ_{u→v} } with Θ_{u→v} mapping perturbations in s_u into drives on s_v. (4.0)

Cross-scale maps.
Π_{micro→macro} : s_{micro} ↦ s_{macro} ; Λ_{macro→micro} : λ_{macro} ↦ λ_{micro} . (4.1a)
(Restriction Π and lifting Λ are published so that budgets respect scale changes.)

Coupled update (schematic, restating the headline law).
ṡ = F(s, λ, Θ) − D(Γ) . (4.1)

A convenient geometric choice is to align motion with the local metric M⁻¹(s):
ṡ ≈ M⁻¹(s) · [ λ + Σ_{u} Θ_{u→•} s_u − λ_{resist}(Γ) ] . (4.2)

4.2 Constraints (Γ): rates, stoichiometry, transport, policy gates

Constraints are declared as measurable costs/limits that produce resistive forces and penalties.

Channel decomposition.
Γ = Σ_c Γ_c , c ∈ {energy, matter, information, governance}. (4.3)

Resistive back-pressure (enters dynamics).
λ_{resist} ≡ ∂Γ/∂ṡ . (4.4)

Capacities (“Slots”) and policy gates.
Let S_c be the per-tick capacity for channel c. A soft quota penalty is
Γ_{policy,c} = μ_c · max( 0 , J_c − S_c )² . (4.5)

Stoichiometric/rate constraints.
Encode required ratios or maximum flux as Γ terms (e.g., stiff penalties as J_c approaches a chemical or transport limit). All Γ terms are reported with units in the constraint sheet.

4.3 Channels: currents, potentials, dissipation

Each channel c exposes a measurable potential–current pair (U_c, J_c) and a non-negative dissipation:

Γ_c = J_c · U_c ≥ 0 . (4.6)

Examples.
Energy: U = Δμ or voltage, J = power/flow; Matter: U = chemical potential gradient, J = molar flux; Information: U = coding/decision load (bits per tick), J = update rate; Governance: U = policy burden, J = write attempts to protected stores.

Coupling to drives.
Drives split into internal and interfacial parts:
λ = λ_{int} + B_Θ · U , with B_Θ mapping channel potentials into effective drives. (4.7)

4.4 Lock-in vs. leakage (how we quantify cross-module loss)

We separate useful internal work from cross-boundary dissipation.

Useful internal work per tick.
W_{int} ≡ ∫ λ_{int} · ds . (4.8)

Cross-module leakage.
Γ_{cross} ≡ Σ_{e∈E} μ_e · ∥ J_e ∥² , where J_e is the interfacial current on edge e. (4.9)

Lock-in index (0–1, higher is better).
χ_{lock} ≡ W_{int} / ( W_{int} + Γ_{cross} ) . (4.10)

Routing rule.
All observed cross-boundary losses are routed to Γ and appear in the budget (3.3). Lock-in improves when χ_{lock} ↑ or Γ_{cross} ↓ under the same task.

4.5 Putting Θ, Γ, channels together in updates

Discrete-tick update (publication-friendly).
s_{t+1} = s_t + Δτ · M⁻¹(s_t) · [ λ_t + Σ_u Θ_{u→•} s_{u,t} − ∂Γ/∂ṡ\big|_t ] . (4.11)

Action view (from §3).
Choose {λ_t, Θ_t} to maximize A_{local} = Σ_t ( Φ̇_t − λ_t·ṡ_t − β·Γ_t ) Δτ under declared capacities S_c and safety invariants. (4.12)

Audit view (always attach the budget).
ΔΦ_t = W_{s,t} − Δψ_t − Γ_{loss,t} , with Γ_{loss,t} = Γ_{cross,t} + Σ_c Γ_{c,t} . (4.13)

4.6 Minimal reporting standard for topology

Graph: list modules V and edges E with units for s_v and interface variables.
Couplings Θ: provide non-zero blocks Θ_{u→v} and their estimation method.
Channels: publish U_c, J_c definitions and sensors; declare capacities S_c.
Constraints Γ: table of Γ_c terms, units, and any policy weights μ_c.
Leakage metrics: report Γ_{cross} and χ_{lock} via (4.9)–(4.10).
Update rule: show (4.11) or your alternative F; justify geometry choice (M or K).
Footer: include seeds, hashes, thresholds, lamp outcomes, and decision.

One-line takeaway. Θ tells who influences whom, Γ tells what it costs and what is allowed, channels carry the flows we can measure, and χ_{lock} with Γ_{cross} tells us how much we keep vs. leak—all feeding the same budget identity in (3.3).

5) Observation Layer: Projection, Trace, and Verification (Two-Lamp)

Purpose. Make observation an audited operation. A projection Ô consumes budget, writes a trace, and must pass two independent gates before any claim is “publishable”.

5.1 Projection Ô and its explicit cost

Projection (observer action).
A projection Ô selects or reports a state summary ŝ from raw measurements at non-zero cost; it modifies the system and leaves a trace.

Projection map and write.
ŝ = Ô(x; policy) ; trace ← write(ŝ, metadata) . (5.0)

Trace persistence metric.
Π_trace ≡ energy_to_project − energy_to_erase . (5.3)

Budget rule for observation.
Γ_obs ≡ energy_to_project + governance_overhead ≥ 0 ; include Γ_obs in (3.3). (5.0a)

Interpretation: reporting is not “free”; its energetic and policy burden must be charged to Γ.

5.2 Two-lamp verification: CWA and ESI

We require two independent green lights before publishing a result produced by Ô.

Lamp states.
CWA_lamp ∈ {Green, Red} ; ESI_lamp ∈ {Green, Red} . (5.1–5.2)

5.2.1 CWA — Consistency with World Assumptions

What it checks. Declared assumptions about domain, sensors, units, and invariants are satisfied by the data and the estimate.

Inputs.

Declared domain 𝔻 (task, tick length, environment).
Sensor & unit spec; preprocessing (q, φ).
Safety invariants (I-set): bounds on {Φ, Γ, ρ, κ, G}.

Score and gate.
S_CWA ≡ 1 − violation_rate( I-set ∪ unit_checks ∪ domain_guards ) ∈ [0, 1] . (5.4)
CWA_lamp = Green ⇔ S_CWA ≥ τ_CWA . (5.5)

Typical τ_CWA. 0.99 for strict lab settings; 0.95 for field conditions (declare explicitly).

5.2.2 ESI — Emulsion-Stability under noise and mixture

What it checks. Estimates remain phase-stable when we inject controlled noise, mix nearby regimes, or perturb batching (“emulsion tests”).

Protocol (minimum).

Noise spins: add calibrated noise to inputs and re-estimate; measure label/phase flips.
Emulsion: mix batches from adjacent conditions at declared ratios; re-estimate.
Jitter: shuffle tick boundaries within declared ε_τ; re-estimate.

Score and gate.
S_ESI ≡ 1 − flip_rate( noise ∪ emulsion ∪ jitter ) ∈ [0, 1] . (5.6)
ESI_lamp = Green ⇔ S_ESI ≥ τ_ESI . (5.7)

Typical τ_ESI. 0.98 for safety-critical; 0.90 for exploratory screens (declare explicitly).

Note. ESI targets collapse fragility (spurious phase boundaries); a Red ESI lamp means the reported state is not robust to benign perturbations.

5.3 Publish decision and rollback

Publishability rule.
Publish ⇔ [CWA_lamp = Green] ∧ [ESI_lamp = Green] . (5.8)

Rollback on Red.
If any lamp is Red, do not publish. Reduce Γ (e.g., lower throughput, improve calibration), adjust Θ or preprocessing, then re-run Ô until both lamps are Green.

5.4 VerifyTrace footer (must attach to every claim)

Mandatory footer fields.
[seed][hash][units][tick Δτ][domain 𝔻][I-set bounds][G threshold][Γ budget][Π_trace][CWA score S_CWA][ESI score S_ESI][CWA lamp][ESI lamp][decision][reviewer].

Single-line commitment.
“No claim without footer + seeds + gates + budgets.” (5.9)

5.5 Observer neutrality tests (to avoid “observer drift”)

Re-projection consistency.
Δ_Ô ≡ distance( Ô_1(x) , Ô_2(x) ) under identical domain; require Δ_Ô ≤ ε_Ô . (5.10)

Cross-observer agreement.
A_{12} ≡ agreement( Ô_1 , Ô_2 ; 𝔻 ) ∈ [0, 1] ; require A_{12} ≥ α_min . (5.11)

Decision.
If Δ_Ô > ε_Ô or A_{12} < α_min, set CWA_lamp = Red and document remediation.

5.6 Trace governance (write, retain, erase)

Write policy.
Allow write(ŝ) only if Γ_obs within budget and privacy/contract Slots permit.

Retention and erasure.
Retention requires Π_trace ≥ 0 (it costs more to erase than to keep). If Π_trace < 0, default to erase unless law/policy requires retention. Record every erase with a footer entry.

Data lineage.
Each derived result includes parent hashes so that Ô is fully auditable.

5.7 Minimal reporting standard for the observation layer

Projection spec: algorithm/policy of Ô, compute cost model, ε_Ô.
CWA sheet: domain 𝔻, unit checks, invariant list, τ_CWA, S_CWA, lamp.
ESI sheet: perturbation magnitudes, mixing ratios, τ_ESI, S_ESI, lamp.
Observer agreement: Δ_Ô, A_{12}, thresholds.
Footer: full VerifyTrace line (5.9).

One-line takeaway. Observation is a paid, audited act: you spend Γ to project, leave a trace with Π_trace, and only publish when both CWA and ESI lamps are Green—each with declared thresholds, seeds, and hashes.

6) System Health, Stability, and Resilience Metrics

Purpose. Provide a small set of auditable, substrate-agnostic metrics that any GLF system must report under a declared nominal load (environment + constraints Γ + tick Δτ). These metrics summarize surplus, risk, stability, controllability, and recovery capacity.

6.1 Health index under nominal load

Definition.
H ≡ Φ_surplus + ρ . (6.1)

Where the parts come from.
Φ_surplus ≡ ⟨Φ_budget⟩_window − Φ_reserve . (6.1a)
Φ_budget(t) ≡ Φ_in − Φ_out − losses (from (1.1)). (6.1b)

Window: fixed averaging window (declare length, e.g., 10⁴ ticks).
Φ_reserve: declared minimum surplus required to maintain basic functions (publish value and rationale).
ρ: robustness radius (max recoverable perturbation norm under Γ; see (2.6), (6.3)).

Interpretation. Positive H implies the system both has spare budget and can recover from bounded shocks under the current Γ and Δτ.

6.2 Dissipation gap (risk)

Definition (baseline-normalized risk).
R_G ≡ ⟨G⟩_window − G* . (6.2)

⟨G⟩_window: rolling mean of the gap G = Φ + ψ − λ·s (from (2.3)).
G*: calibrated threshold at which performance or safety starts degrading (publish calibration protocol).

Interpretation. R_G > 0 signals systemic drag: misalignment, under-coupled modules, or constraints biting (Γ too tight). Large positive values predict backlog, aging, or failure risk if sustained.

6.3 Stability and resilience

Lyapunov stability in value geometry.
V(Δs) = ΔΦ ≥ 0 implies Lyapunov stability near the operating point. (6.3)

Resilience radius (operational).
ρ = max r such that for all ∥Δs∥ = r, the system returns within tolerance under Γ. (6.3a)

Recovery test (publishable method).
Recover(Δs, T_rec) ⇔ sup_{t ≥ T_rec} ∥ s_t − s* ∥ ≤ ε_rec under declared Γ, Θ, Δτ. (6.3b)

Report ρ, T_rec, and ε_rec with units and confidence intervals.
If multiple regimes exist, report ρ_k per regime (do not average across regimes).

6.4 Controllability alert (geometry-based)

Condition number of structural inertia.
κ ≡ cond(M) ; alert if κ > κ*. (6.4)

M = ∇²Φ(s) (from (2.2)); κ summarizes ill-conditioning of control/readout.
κ* is a declared limit (e.g., 10³ for routine control, 10⁵ for offline batch).
When κ ≫ κ*, small drive errors create large state errors; mitigate via preconditioning, slotting (capacity relief), Δ5 micro-cycles, or redesign of φ to improve curvature.

6.5 Secondary health diagnostics (report alongside H)

Efficiency: η_Φ ≡ ΔΦ / W_s . (6.5)
Leakage ratio: ℓ ≡ Γ_loss / W_s . (6.6)
Exploration tax: ξ ≡ Δψ / W_s . (6.7)

Use these to explain why H moved: η_Φ↓ (poor conversion), ℓ↑ (leaky channels), ξ↑ (over-exploration).

6.6 Green–Amber–Red (GAR) bands (declare thresholds)

Health H: Green if H ≥ H_G ; Amber if H_A ≤ H < H_G ; Red if H < H_A. (6.8)
Risk R_G: Green if R_G ≤ 0 ; Amber if 0 < R_G ≤ R_A ; Red if R_G > R_A. (6.9)
Controllability κ: Green if κ ≤ κ* ; Red if κ > κ* (Amber band optional). (6.10)

Publish the triplets (H_A, H_G), (R_A), and κ* with justification (calibration or safety policy).

6.7 Minimal measurement protocols (copy into Methods)

MP-H (Health).

Compute Φ_budget(t) each tick; average over the declared window.
Subtract Φ_reserve to obtain Φ_surplus; estimate ρ via (6.3a)–(6.3b); set H from (6.1).
Report uncertainty for both Φ_surplus and ρ; include GAR band.

MP-R (Risk).

Track G per tick; smooth to ⟨G⟩_window.
Compare with G* from calibration; publish R_G and lamp color.

MP-C (Controllability).

Estimate M on the operating band; compute κ = σ_max/σ_min.
If κ > κ*, document mitigation (preconditioning, slot relief, Δ5 schedule).

6.8 Reporting standard (tables you must include)

Health table: {Φ_in, Φ_out, losses, Φ_budget, Φ_reserve, Φ_surplus, ρ, H}.
Risk table: {⟨G⟩_window, G*, R_G}.
Geometry table: {eigs(M) summary, κ, κ*, preconditioning actions}.
Recovery table: {ρ, T_rec, ε_rec, pass/fail}.
Footer: seeds, hashes, thresholds, CWA/ESI lamps, decision.

One-line takeaway. Health H combines surplus and recoverability; R_G warns when dissipation outpaces alignment; κ tells you if the system is controllable; ρ certifies bounce-back—all under declared Γ, Θ, and Δτ with auditable thresholds.

7) Measurement Protocols, Data Schemas, and Reproducibility

Purpose. Turn the framework into something runnable and auditable. This section specifies the minimal data objects, the per-experiment footer, and the procedures that guarantee others can re-run your results and get the same numbers within declared error bars.

7.1 Minimal schema (declare once, reuse everywhere)

Core declaration (must appear in Methods).
(X, q, φ) where
– X: raw observations with timestamps (ticks Δτ), module IDs, and sensor units.
– q: reference distribution (baseline) used to define ψ(λ).
– φ: feature map from raw X to reported features (units declared).

Derived quantities (publish or re-computable).
ψ(λ) ; s(λ) = ∇ψ(λ) ; Φ(s) ; I(λ) = ∇²ψ(λ) ; M(s) = ∇²Φ(s) = I⁻¹ ; G = Φ + ψ − λ·s . (7.1)

File-level objects (lightweight, Blogger-friendly).

manifest.csv: study_id, dataset_id, commit_hash, code_hash, container_digest, Δτ, units_policy.
features.parquet/csv: sample_id, tick, module, s_k, unit(s_k), stderr(s_k).
drives.parquet/csv: sample_id, tick, λ_k, unit(λ_k).
geometry.parquet/csv: tick, eigs(M) summary, κ, ρ_method, CI.
budgets.parquet/csv: tick, W_s, Δψ, Γ_loss, ΔΦ, P_in, Π_diss.
gap.parquet/csv: tick, G, G*, lamp_color.
footer.log: one line per published figure/table (see §7.4).

7.2 Measurement protocols (how to compute the quantities)

MP-Base (once per study).

Declare q and φ with units, preprocessing, and any standardization.
Calibrate sensors (report uncertainty); fix Δτ and clock alignment error ε_τ.

MP-Stats (per analysis batch).
3) Estimate ψ(λ) and compute s = ∇ψ(λ), I = ∇²ψ(λ).
4) Fit or compute Φ(s); obtain λ = ∇Φ(s), M = ∇²Φ(s) = I⁻¹.
5) Compute G = Φ + ψ − λ·s and attach a threshold G* (calibration protocol declared).
6) Compute budgets per tick: W_s = ∫ λ·ds, Δψ, Γ_loss, ΔΦ = W_s − Δψ − Γ_loss.

MP-Units (sanity).
7) Verify units(λ·s) = units(Φ) = units(ψ) and that Γ is reported in the same per-tick units (or publish the β conversion factor used).

7.3 Batch covariance scheduling (publish I-conditioning before/after)

Goal. Keep estimation well-conditioned and comparable across interventions.

Report before/after triplet.
I_{pre}, I_{post}, κ = cond(M) = cond(I⁻¹) ; alert if κ > κ* . (7.2)

Scheduling rule (practical).
Allocate batch composition to reduce κ while keeping signal:
Δn_k ∝ largest eigen-directions of I until κ ≤ κ_target. (7.3)

Publish.
– Eigenvalue summaries (min/median/max) of I or M.
– Any preconditioning/regularization used (and its effect on κ).
– A short note on changes in sensor noise or protocol that alter I.

7.4 Per-experiment VerifyTrace footer (attach to every claim)

Footer fields (one line; machine-parseable).
[seed][hash][tick Δτ][domain tags][units policy][q id][φ id][Γ budget][G threshold][CWA score][ESI score][CWA lamp][ESI lamp][decision][reviewer][time].

Tolerance policy.
Declare numerical tolerances for budget closure (|ΔΦ − (W_s − Δψ − Γ_loss)| ≤ ε_budget) and for gap checks (|G| ≤ ε_gap on calibration).

7.5 Reproducibility harness (how others re-run your pipeline)

Environment pinning.
– container_digest (or OS + compiler + BLAS info), code_hash, dependency lockfile.
Randomness control.
– global seed plus per-stage seeds; report any non-deterministic GPU ops.
Determinism tests.
– two-run hash agreement for derived tables; set CWA=Red if hashes mismatch beyond tolerance.
Observer agreement.
– cross-run projection consistency Δ_Ô ≤ ε_Ô and cross-observer agreement A_{12} ≥ α_min (from §5).

7.6 Data dictionary (unit and naming standards)

s_k: feature means; include unit(s_k) per column.
λ_k: natural parameters; unit = 1 / unit(s_k).
ψ, Φ, G: bits (or J·K⁻¹ scaled). Specify the scale once (e.g., “bits via k_B ln2”).
I: covariance of s (units of var). M: inverse-variance units.
Γ_loss, W_s, Δψ, ΔΦ: same per-tick units as Φ.
κ, ρ: κ is unitless; ρ reports the norm and its unit (e.g., L² in feature units).

7.7 Minimal publishable tables (every Results section must include)

Budget table: tick, W_s, Δψ, Γ_loss, ΔΦ (with CIs).
Gap table: tick, G, threshold G*, lamp color.
Geometry table: eigs(M) summary, κ, κ*, ρ (method + CI).
I-conditioning table (if interventions): I_{pre/post} summaries, κ_{pre/post}, actions taken.
Footer line: the VerifyTrace line for that figure.

7.8 One-line reproducibility rule (pasted verbatim)

“No claim without footer + seeds + gates + budgets.” (7.1)

Enforcement. Submissions that lack any of {footer, seeds, CWA/ESI lamps, budget closure} are considered non-compliant and should be rejected or labeled “exploratory, not reproducible”.

8) Safety Invariants and Governance Controls

Purpose. Define a small, unambiguous governance layer that keeps any GLF system inside a safe operating envelope. We specify (i) I-set invariants, (ii) Slots (quotas/capacities), (iii) Δ5 micro-cycles to cancel drift and cool leakage, and (iv) a formal guard condition that gates publishability and continued operation.

8.1 I-set (safety invariants)

Definition (declared, auditable bounds).
I-set ≡ { Φ ≥ Φ_min , Γ ≤ Γ* , ρ ≥ ρ_min , κ ≤ κ* } . (8.0)

Compliance predicate.
I_ok ⇔ [ Φ ≥ Φ_min ] ∧ [ Γ ≤ Γ* ] ∧ [ ρ ≥ ρ_min ] ∧ [ κ ≤ κ* ] . (8.0a)

Slack vector (for dashboards).
σ_I ≡ ( Φ − Φ_min , Γ* − Γ , ρ − ρ_min , κ* − κ ) . (8.0b)

Violation score (unitless, ≤ 0 means breach).
v_I ≡ min_j { σ_I_j / band_j } . (8.0c)

band_j = declared scaling for each invariant (e.g., Φ band in bits, Γ band in J per tick).
Publish {Φ_min, Γ*, ρ_min, κ*} and band choices in Methods; never change without a dated amendment.

8.2 Slots (quotas / capacities)

Purpose. Cap flows and irreversible writes so that failures degrade gracefully.

Per-channel capacity (per tick or refill schedule).
S_c = allowed capacity for channel c ∈ {energy, matter, information, governance}. (8.1)

Overage penalty (policy cost enters Γ).
Γ_{policy,c} = μ_c · max( 0 , J_c − S_c )² . (8.2)

Token-bucket form (optional, bursty workloads).
B_{c,t+1} = clip( B_{c,t} − J_c Δτ + R_c Δτ , 0 , B_c^{max} ) . (8.3)

Write gate for irreversible commits (e.g., protected logs, actuator latches).
Allow write ⇔ [ B_{info} ≥ cost_{write} ] ∧ [ I_ok ] . (8.4)

Reporting. Publish S_c (or {R_c, B_c^{max}}), μ_c, and all write costs. Route Γ_{policy,c} into the budget law (3.3).

8.3 Δ5 micro-cycles (drift cancellation and leakage cooling)

Scheduling rule (ten-phase ring).
Index micro-phases i ∈ {0,…,9}. Pair i with i+5 (mod 10) so that opposing micro-acts cancel drift:

pair(i) = ( i , i+5 ) . (8.5)

Cross-leakage bound (empirical, declare α).
Γ_{cross}(Δ5) ≤ α · Γ_{cross}(free) , 0 < α < 1 . (8.6)

Duty modulation under stress.
Increase rest fraction r_{rest} when v_I ↓ or lamps amber/red:

r_{rest,new} = r_{rest,base} + k · max(0, τ_safe − S_safe) . (8.7)

S_safe defined in §8.6; τ_safe a declared comfort level (e.g., 0).
Δ5 is mandatory for modules with known reciprocal leak channels (transport↔heat, excitation↔inhibition, compute↔IO).

8.4 Guard condition and safety state machine

Guard condition (publishability and operation).
Safe ⇔ (I-set respected) ∧ (CWA = Green) ∧ (ESI = Green). (8.1)

GAR states (runtime).

Green: Safe holds; full operation.
Amber: I_ok holds but at least one lamp is Green/Amber mixed or v_I near 0; enforce Δ5, tighten Slots, raise monitoring.
Red: Safe fails (I_ok = false or any lamp = Red); trigger Safety Mode immediately.

Safety Mode (deterministic actions, executed in order).

Freeze risky couplings: Θ ← Θ_{safe} (pre-approved reduced set). (8.8)
Drive de-rate: λ ← r_λ · λ with 0 < r_λ < 1 (declare default r_λ, e.g., 0.5). (8.9)
Slot hard-cap: J_c ← min( J_c , S_c^{hard} ) ; deny irreversible writes. (8.10)
Δ5 hard schedule: enforce pair(i) for all active micro-acts; increase r_{rest}. (8.11)
Publish rollback: no external commit; log footer with Red state and remediation plan. (8.12)

Exit Safety Mode ⇔ Safe holds continuously for T_safe ticks (declare T_safe).

8.5 Governance: change control, overrides, and audits

Change control (monotonicity).
It is forbidden to relax {Φ_min, ρ_min} downward or {Γ*, κ*} upward during an active run. Changes require a new run ID and a pre-run two-lamp pass.

Override policy (rare, time-limited).
An override requires multi-sig approval (≥ 2 reviewers) and auto-expires at τ_{override}. Footer must include [override_id][approvers][expiry]. (8.13)

Post-mortem duty.
Any Red event mandates a post-mortem within T_{pm} ticks with: timeline, σ_I trajectories, Δ5 schedule, Slot usage, lamp scores, and corrective actions.

Audit trail.
All changes to I-set, Slots, Θ_{safe}, r_λ, and Δ5 parameters are versioned; every figure/table includes the corresponding hash in its footer.

8.6 Composite safety score (dashboard)

Normalized safety score (optional but recommended).
S_safe ≡ min { (Φ − Φ_min)/band_Φ , (Γ* − Γ)/band_Γ , (ρ − ρ_min)/band_ρ , (κ* − κ)/band_κ , (S_{CWA} − τ_{CWA})/band_{CWA} , (S_{ESI} − τ_{ESI})/band_{ESI} } . (8.14)

S_safe ≥ 0 ⇒ Green; S_safe < 0 ⇒ breach; bands are declared scales.
Use S_safe to pre-empt Red by tightening Slots or boosting Δ5 duty when margin thins.

8.7 Minimal safety dashboard (to publish each run)

I-set table: Φ, Γ, ρ, κ with thresholds and σ_I.
Slots table: S_c (or {R_c, B_c^{max}}), usage J_c, overage penalties Γ_{policy,c}.
Δ5 schedule: pair map, r_{rest}, α bound, any duty changes over time.
Lamp scores: S_{CWA}, S_{ESI}, thresholds, states.
Events: GAR transitions, Safety Mode intervals, overrides (if any).
Footer: full VerifyTrace line including I-set hash and policy versions.

8.8 Safety protocols (copy into Methods)

SP-1 (Calibrate invariants).
Select Φ_min, Γ*, ρ_min, κ* from historical safe operation; fix bands; hash and publish I-set.

SP-2 (Pre-run gate).
Two-lamp dry run on a rehearsal dataset; confirm Safe; record baseline S_safe.

SP-3 (Runtime enforcement).
Monitor σ_I and lamps each tick; apply Δ5 and Slot tightening automatically when S_safe < τ_warn.

SP-4 (Breach handling).
Enter Safety Mode steps (8.8)–(8.12); no publish until Safe holds for T_safe; file post-mortem.

One-line takeaway. I-set says how far you may go, Slots say how fast and how much you may write, Δ5 keeps drift and leakage cool, and Safe ⇔ I_ok ∧ CWA=Green ∧ ESI=Green (8.1) is the non-negotiable gate for both operation and publication.

9) Boundary Cases and Life-Continuum Tests

Purpose. Apply the same pass/fail gates to classical “edge” entities. Every case uses the budget (1.1), the gap (2.3), and the safety/verification guard (8.1).

9.1 Continuum statement and a usable score

Continuum statement.
Life_degree ∝ Φ_budget with bounded Γ and synchronized τ. (9.1)

Operational score (publish if you need a scalar).
Let B ≡ max(0, Φ_budget / Φ_ref) ; S_τ ≡ 1 − (max_{i,j}|τ_i − τ_j| / ε_τ)_+ ; S_safe ≡ 𝟙{I-set ok ∧ CWA=Green ∧ ESI=Green}. Then
L ≡ B · S_τ · S_safe ∈ [0, 1] . (9.2)
(Choose Φ_ref and ε_τ in Methods; report L alongside the raw gates.)

9.2 Pass/Fail rubric (what flips and why)

Always attach: {Φ_budget, Γ vs. Γ*, τ-sync error, G vs. G*, lamps, I-set}. Decision = Safe gate (8.1). If Safe = false, label as Not alive (under GLF) for the tested domain.

Free virion (outside host).
Typical flips: Φ_budget ≤ 0 (no inflow), τ undefined, often CWA=Red (declared domain lacks a metabolism). → Fail (1.1).
Virus inside metabolically active host cell.
Consider the composite module {cell + virus}.
Pass path: Φ_budget > 0 for composite, Γ ≤ Γ*, τ synced to host clock; G near baseline; both lamps Green. → Pass as life activity of the composite.
Fail path: Γ_{cross} spikes (host meltdown), or lamps Red (unstable estimates). → Fail (8.1).
Prion in buffer with PrP substrate.
Borderline: Φ_budget may be slightly > 0 (conversion draws on externally maintained gradients), but τ often not self-regulated; governance Slots typically deny irreversible writes (biosafety). If CWA=Green and Γ bounded, classify as Pass for the prepared reactor (the system is the prion reactor). Otherwise Fail on τ/Γ or lamps.
Minimal protocell (fatty-acid vesicle with autocatalytic loop).
Common failure: Γ ≫ Γ* due to membrane leakage → Fail (8.1).
Upgrade to pass: longer chains + cofactor recycling reduce Γ_{cross}; Φ_budget > 0 and τ stabilized by feed cycles → Pass.
Organoid under perfusion (brain, gut, etc.).
With stable perfusion and waste removal, Φ_budget > 0, Γ ≤ Γ*, τ coherent; no reproduction required by GLF. → Pass.
Without perfusion, Γ rises, Φ_budget drops below zero → Fail (1.1).
Bio-bot (synthetic tissues + microcontroller + battery).
If maintenance cycles exist (repair budget) and τ is controller-synchronized, Φ_budget > 0 while battery or chemical fuel lasts → Pass (time-bounded).
If no self-maintenance (pure discharge), Φ_budget trends to 0 then < 0 → Fail when (1.1) breaks.
Cell-free expression systems (PURE, TX-TL).
Assembled reactor with feed pumps: Φ_budget > 0, τ by pump schedule, Γ bounded → Pass as reactor-system. Batch without feed → Fail when Φ_budget exhausts.
Dormant spores / seeds.
Φ_budget ≈ 0 but ≥ 0; Γ ≈ 0; τ in standby (declared as paused clock). If I-set ok and lamps Green for the dormant domain, mark Pass (dormant state).

9.3 Worked pass/fail lines (what inequality tripped)

Virion (air, RT): Φ_budget = −ε < 0 → Fail (1.1).
Infected cell (steady host): Φ_budget > 0 ; Γ ≤ Γ* ; max_{i,j}|τ_i − τ_j| ≤ ε_τ ; CWA=G, ESI=G → Pass (8.1).
Prion (shaking incubator): Φ_budget > 0 ; τ synced to shaker; Γ ≤ Γ* ; CWA=G ; ESI=A (fragile phase) → Fail (8.1) until ESI stabilized.
Protocell (leaky): Φ_budget > 0 ; Γ > Γ* → Fail (8.1).
Organoid (no perfusion): Γ ≫ Γ* ; Φ_budget < 0 → Fail (1.1) & (8.1).
Bio-bot (discharge-only): Φ_budget(t) ↓ → crosses 0 at t* → Pass before t*, Fail after t*.

9.4 Minimal test protocols (copy into Methods)

BP-1 Viruses (composite test).
Define the composite {host module + virus}. Measure inflow/outflow to the composite, compute Φ_budget, Γ, τ (host circadian or reactor pulses), G against host baseline, run lamps. Report decision and L via (9.2).

BP-2 Prions.
Declare reactor as system. Control agitation (τ), feed gradients (Φ_in), measure Γ_{cross} (heat, waste). ESI must be Green across noise/emulsion tests before “Pass.”

BP-3 Protocells.
Measure leakage to estimate Γ; show a modification that reduces Γ below Γ* and yields ΔΦ > 0 over multiple windows.

BP-4 Organoids.
Compare perfused vs. static culture: publish Γ, Φ_budget trajectories, κ and ρ if available; attach two-lamp scores.

BP-5 Bio-bots.
Publish maintenance fraction of Φ_budget. Without maintenance, label time-bounded Pass and show t* where (1.1) breaks.

9.5 Reporting template (one line per boundary assay)

[system][domain][Φ_budget][Γ vs. Γ*][τ-sync error][G vs. G*][CWA lamp][ESI lamp][I-set ok?][decision][L score][reviewer][time].

One-line takeaway. Boundary judgments are not philosophical: they are the same audited gates—budget ≥ 0, bounded Γ, synchronized τ, and two Green lamps—applied to the declared system. Where those inequalities fail, the label is Not alive (for that domain); where they pass, the system is alive to the extent quantified by L and Φ_budget.

10) Reference Implementations (Minimal, Blogger-friendly)

Purpose. Two tiny, copy-and-run examples that anyone can reproduce in a spreadsheet or a few lines of code, plus read-off rules to populate the health/risk tables in a blog post.

10.A Micro-example A — Binary feature system

Use-case. Minimal exponential-family system with one binary detector (x ∈ {0,1}). Baseline (q) has (θ = q(x=1)). Feature map (φ(x)=x).

Equations (single-line).
ψ(λ) = log( (1−θ) + θ·e^{λ} ) . (10.A.1)
s(λ) = ∇ψ = θ·e^{λ} / ( (1−θ) + θ·e^{λ} ) . (10.A.2)
λ(s) = log( s(1−θ) / ((1−s)θ) ) . (10.A.3)
Φ(s) = s·log( s/θ ) + (1−s)·log( (1−s)/(1−θ) ) . (10.A.4)
I(λ) = ∇²ψ = s(1−s) ; M(s) = ∇²Φ = 1 / ( s(1−s) ) . (10.A.5)
G ≡ Φ + ψ − λ·s ≥ 0 (zero at calibration within tolerance) . (10.A.6)
W_s over a tick = λ_t · ( s_{t+1} − s_t ) . (10.A.7)
Budget: ΔΦ = W_s − Δψ − Γ_loss . (10.A.8)

Tiny CSVs.

baseline.csv

theta
0.37

series.csv (estimate s by windowed mean of x; one line per tick)

tick,x
0,0
1,1
2,0
3,1
4,1
5,0

Procedure (5 steps).

Read θ from baseline.csv. Compute windowed s_t = mean(x) (or per-tick if you model Bernoulli directly).
Compute λ_t via (10.A.3), ψ_t via (10.A.1), Φ_t via (10.A.4), I_t and M_t via (10.A.5).
Compute W_{s,t} via (10.A.7), Δψ_t = ψ_{t+1} − ψ_t.
Measure or set Γ_{loss,t} (set 0 for a dry run). Then ΔΦ_t via (10.A.8).
Gap check: G_t via (10.A.6). If |G_t| > ε_gap, flag CWA=Red for that tick.

Footer example (one line).
[seed=42][hash=abc123][Δτ=1 tick][domain=bin-expfam][units=bits/tick][q=theta0.37][φ=x][Γ_budget=0][G*=1e-6][CWA=Green][ESI=Green][decision=Publish][reviewer=ID-001][time=2025-11-09T12:00Z]

10.B Micro-example B — Two-module coupling (Θ on/off)

Use-case. Two scalar modules (A,B) with linear couplings (Θ) and a simple cross-leak penalty in Γ.

Discrete update (declare Δτ).
s_{A,t+1} = s_{A,t} + Δτ·M_A^{-1}( λ_{A,t} + θ_{AB}·s_{B,t} − λ_{resist,A} ) . (10.B.1)
s_{B,t+1} = s_{B,t} + Δτ·M_B^{-1}( λ_{B,t} + θ_{BA}·s_{A,t} − λ_{resist,B} ) . (10.B.2)

Interfacial current and leakage.
J_{AB,t} ≡ θ_{AB}·s_{B,t} − θ_{BA}·s_{A,t} . (10.B.3)
Γ_{cross,t} = μ·J_{AB,t}² ≥ 0 . (10.B.4)

Gap and budget per module (same definitions as §2–§3).
G_{A,t} = Φ_A + ψ_A − λ_{A,t}·s_{A,t} ; G_{B,t} analogously . (10.B.5)
ΔΦ_total = (W_{s,A}+W_{s,B}) − (Δψ_A+Δψ_B) − (Γ_{loss,A}+Γ_{loss,B}+Γ_{cross}) . (10.B.6)

Two runs.
– Θ-OFF: θ_{AB}=θ_{BA}=0.
– Θ-ON: choose small θ_{AB}, θ_{BA} (e.g., 0.2, 0.15). Keep Δτ, μ identical.

Report.
ΔG ≡ ⟨G_A+G_B⟩{ON} − ⟨G_A+G_B⟩{OFF} . (10.B.7)
If ΔG < 0, coupling improved alignment (good); if ΔG > 0, coupling induces mismatch (bad). Always attach lamps (CWA/ESI) for both runs.

Tiny CSV.

two_module_series.csv

tick,sA,lambdaA,sB,lambdaB
0,0.40,0.10,0.30,0.05
1,0.43,0.12,0.33,0.06
2,0.45,0.11,0.36,0.08
3,0.47,0.10,0.37,0.09

coupling.csv

thetaAB,thetaBA,mu,DeltaTau
0.20,0.15,0.50,1

Lamp recipe (minimum).
– CWA: unit checks, budget closure |ΔΦ − RHS(10.B.6)| ≤ ε_budget; if fail → CWA=Red.
– ESI: add ±1% noise to sA,sB and re-run; flip_rate < 2% ⇒ ESI=Green, else Red.

Footer example (Θ-ON).
[seed=7][hash=def789][Δτ=1][domain=2mod-linear][units=bits/tick][Θ=(0.20,0.15)][μ=0.5][Γ_budget=declared][G*=1e-4][CWA=Green][ESI=Green][decision=Publish][reviewer=ID-002][time=2025-11-09T12:05Z]

10.C Read-off rules — Filling H, R_G, ρ, κ tables in a blog post

Inputs you must have per window: {Φ_in, Φ_out, losses, Γ, s, λ, M, G}.

1) Health table (H from (6.1)).
Φ_budget = Φ_in − Φ_out − losses . (10.C.1)
Φ_surplus = ⟨Φ_budget⟩_window − Φ_reserve . (10.C.2)
Estimate ρ via a recovery test: perturb along unit eigenvectors of M until recovery fails; take the largest radius that still recovers. (10.C.3)
H = Φ_surplus + ρ . (10.C.4)

2) Risk table (R_G from (6.2)).
R_G = ⟨G⟩_window − G* . (10.C.5)
Color = Green if R_G ≤ 0 ; Amber if 0 < R_G ≤ R_A ; Red if R_G > R_A. (10.C.6)

3) Controllability table (κ from (6.4)).
Compute eigs of M on the operating band; κ = σ_max / σ_min . (10.C.7)
Alert if κ > κ* ; note any preconditioning or slot relief applied. (10.C.8)

4) Resilience notes (ρ method).
Publish (ρ, T_rec, ε_rec) and the perturbation protocol; never average ρ across regimes—report one ρ per regime. (10.C.9)

Blog-friendly table headers.

Health: window, Φ_in, Φ_out, losses, Φ_budget, Φ_reserve, Φ_surplus, ρ, H, color.
Risk: window, ⟨G⟩, G*, R_G, color.
Geometry: window, eig_min(M), eig_max(M), κ, κ*, note.
Recovery: regime, ρ, T_rec, ε_rec, pass/fail.

One-line reproducibility rule (paste under each figure).
“No claim without footer + seeds + gates + budgets.” (7.1)

Takeaway. With these two micro-examples and the read-off rules, any reader can: (i) compute ψ, s, Φ, I, M, G; (ii) toggle Θ to see alignment vs. leakage; and (iii) fill H, R_G, ρ, κ tables that meet the GLF publishing standard.

11) Positioning, Falsifiability, and Limits

Purpose. State exactly what GLF is for, how it can be proven wrong, and where it does not apply. This locks rigor around claims and prevents overreach.

11.1 What the framework does (and does not)

Does.

Defines a shared ontology for state, drive, geometry, and budgets: {s, λ, ψ, Φ, I, M, G, Γ, Θ, τ, ρ, κ}.
Audits dynamics by a closed-form budget: ΔΦ = W_s − Δψ − Γ_loss . (11.1)
Gates publication by two independent verification lamps (CWA, ESI) and an I-set of safety invariants.
Remains substrate-agnostic: works for cells, organs, organoids, consortia, and synthetic life-like systems if units and ticks are declared.

Does not.

Predict “purposes,” teleology, or meanings beyond declared tasks and constraints Γ.
Replace domain theories (biochemistry, electrophysiology, biomechanics); GLF wraps them with budgets and invariants.
Guarantee sufficiency for creating life; it only defines measurable conditions for life-like operation in a declared domain.
Claim universal differentiability: at phase changes GLF falls back to subgradient or piecewise analysis (declared explicitly).

11.2 Falsifiers (what would prove GLF wrong for a declared domain)

F1. Conjugacy violation (beyond tolerance).
Empirically estimated pairs break the identities in §2 beyond declared error bars:
s ≠ ∇ψ(λ) or M ≠ I⁻¹, persistently and reproducibly. (11.2)

F2. Non-closure of the budget.
ΔΦ − ( W_s − Δψ − Γ_loss ) ≠ 0 exceeds ε_budget across runs with pinned seeds and sensors. (11.3)

F3. Negative gap without gauge error.
G ≡ Φ + ψ − λ·s < −ε_gap on calibration data after verified unit/gauge checks. (11.4)

F4. Composition inconsistency.
For a composite {A+B}, the measured budget disagrees with the sum plus interfacial losses:
ΔΦ_{A∪B} − (ΔΦ_A + ΔΦ_B) + Γ_{cross} ≠ 0 beyond tolerance. (11.5)

F5. Gauge non-invariance.
Under a declared rescaling of features φ̃ = c·φ with unit-consistent λ̃ = λ/c, dimensionless predictions (sign of ΔΦ, pass/fail of lamps, ordering by G) change systematically. (11.6)

F6. Observer non-agreement.
Re-projection consistency or cross-observer agreement fails: Δ_Ô > ε_Ô or A_{12} < α_min across verified runs (cf. §5). (11.7)

F7. Perpetual “value” machine.
In a closed loop with P_in = 0 and bounded Γ, the system exhibits ΔΦ > 0 sustainably (violates the declared budget semantics). (11.8)

F8. Predictive sign failure under controlled interventions.
Given a pre-registered intervention that should reduce Γ_{cross} and G, the realized signs flip (e.g., ΔG > 0) across replications with Green lamps. (11.9)

F9. Robustness misreport.
Claimed ρ fails a published recovery test: system does not return within ε_rec by T_rec under the declared Γ and Θ. (11.10)

Decision rule. Any one of (11.2)–(11.10) satisfied reproducibly, with a valid footer and Green CWA/ESI for the test harness, falsifies GLF applicability to that domain.

11.3 Falsification playbook (how to test us)

Pin the gauge and units. Publish (q, φ), scaling, and sensor uncertainties.
Run the gap test. On calibration data, verify |G| ≤ ε_gap; else stop—CWA=Red.
Run the budget test. Verify |ΔΦ − (W_s − Δψ − Γ_loss)| ≤ ε_budget over N windows.
Run the composition test. Measure parts and whole; check (11.5).
Run the gauge test. Rescale φ by c; verify dimensionless outcomes unchanged.
Run the observer test. Report Δ_Ô, A_{12}; fail if thresholds not met.
Pre-register an intervention. Predict signs for {ΔG, ΔΓ_{cross}, ΔH}; execute; compare.
Attach footers. No claim without footer + seeds + gates + budgets. (7.1)

11.4 Limits of validity (where GLF must be qualified)

L1. Tick coarse-graining.
GLF assumes a declared Δτ separates fast micro-fluctuations from reported dynamics. If aliasing occurs (hidden fast modes), budgets can misstate Γ and W_s.

L2. Weak stationarity over windows.
Estimators for ψ and Φ assume approximate stationarity over the averaging window; rapid drift demands shorter windows or explicit time-dependence (publish).

L3. SNR and conditioning.
When I is poorly conditioned (κ ≫ κ*), conjugacy estimates and budgets are unstable; GLF demands preconditioning or reduced scope.

L4. Strong nonlocality or long memory.
If dynamics require nonlocal kernels or long-range memory not captured in s, GLF must be extended with state augmentation; otherwise budget attribution can be misleading.

L5. Phase transitions and non-differentiability.
At kinks, replace gradients with subgradients and report regime labels; do not smooth away discontinuities without declaring it.

L6. Dominant observer effects.
If observation costs (Γ_obs) dominate and cannot be bounded, claims about the unobserved system are out of scope; report this explicitly.

L7. Semantic misuse.
GLF’s “value” Φ is a minimum divergence potential in declared units (bits or J·K⁻¹ scaled). It is not utility, fitness, or money unless a task mapping is declared.

11.5 Positioning relative to existing practice

With domain models. Use GLF as an audit and governance layer: it standardizes variables, budgets, and verification, while the domain model provides mechanisms and parameters.
With statistics/ML. GLF upgrades reporting from “accuracy-only” to geometry + budget + invariants; publish κ, ρ, and G with the same prominence as scores.
With safety/governance. I-set, Slots, and Δ5 provide operational guardrails; they do not replace ethical review or legal compliance.

11.6 Claims we refuse to make (anti-overreach)

No claim of metaphysical “essence of life.”
No guarantee that increasing Φ is “good” in any normative sense without a declared task Γ.
No promise that GLF variables are sufficient statistics for all purposes; they are auditable coordinates for declared scopes.

11.7 Minimal reporting block for Section 11 (paste into Methods)

Tolerances: ε_gap, ε_budget, ε_Ô, α_min, κ*, window length, Δτ.
Falsification plan IDs: which of (11.2)–(11.10) you executed, with links to footers.
Scope tags: {L1–L7} flags that apply to your study.

One-line takeaway. GLF is an audit-first, substrate-agnostic framework; it stands or falls on conjugacy (11.2) and budget closure (11.3) under a pinned gauge, with two lamps and invariants enforcing honesty about where it applies—and where it doesn’t.

Appendix A. Symbol & Unit Dictionary (One-page)

Conventions. Vectors are bold; all per-tick quantities are reported in the same units as Φ (bits or J·K⁻¹ scaled). 1 bit = k_B·ln 2 J·K⁻¹. Norm for ρ defaults to L² in s-units unless stated. Windowed averages must declare window length and Δτ.

Core state & statistics.
X — raw observations with timestamps; mixed units; sensor-calibrated.
q — reference distribution (baseline); unitless; declared once.
φ(x) — feature map (detectors) from X to s; units = feature units.
ψ(λ) — log-partition (statistical potential); units = bits (or J·K⁻¹ scaled); estimate via cumulant-gen.
λ — natural parameters (drive); units = 1 / unit(s); estimated from ∇Φ(s) or model fit.
s — feature means (state); units = unit(φ); estimated from data or s = ∇ψ(λ).
I(λ) — Fisher / covariance of s; units = var(s); I = ∇²ψ(λ).

Value, geometry, gap.
Φ(s) — minimum-divergence “value”; units = bits (or J·K⁻¹ scaled).
M(s) — structural inertia = ∇²Φ(s) = I⁻¹; units = 1/var(s).
G — Fenchel–Young gap = Φ + ψ − λ·s ≥ 0; units = Φ units; KPI for alignment.

Work, power, budgets.
W_s — value-work along path = ∫ λ·ds; units = Φ units.
Φ̇ — value rate; units = Φ units per tick.
P_in — input power to Φ; units = Φ units per tick.
Π_diss — dissipation power; units = Φ units per tick.
Δψ — change in ψ across tick; units = Φ units.
ΔΦ — budget residual = W_s − Δψ − Γ_loss; units = Φ units.
Φ_budget — Φ_in − Φ_out − losses; units = Φ units.

Constraints, channels, leakage.
Γ — dissipation cost (sum over channels); units = Φ units per tick.
Γ_loss — measured losses entering the budget; units = Φ units.
Γ_cross — cross-module leakage (interfaces); units = Φ units.
Γ_policy — policy/over-quota penalty; units = Φ units.
Θ — coupling map among modules; unit = per unit(s) to drive units.
U_c, J_c — potential & current for channel c; Γ_c = J_c·U_c ≥ 0.
S_c — slot capacity for channel c (per tick or token-bucket); units = unit(J_c).
μ_c — policy weight in Γ_policy; unit converts J_c to Φ units.
B_c, R_c — token-bucket level and refill rate; units = unit(J_c).

Timing & phases.
τ — tick duration; units = s (or cycles); publish ε_τ (sync error).
θ — phase for rhythms; units = rad; used in locking tests.
Δ5 — mandatory opposing micro-cycle pairing i ↔ i+5 (mod 10); unitless schedule.

Observation & verification.
Ô — projection operator (observer action); policy-declared.
Π_trace — trace persistence = energy_to_project − energy_to_erase; units = Φ units.
CWA_lamp — {Green, Red}; consistency with declared world-assumptions.
ESI_lamp — {Green, Red}; phase-stability under noise/emulsion.
S_CWA, S_ESI — lamp scores ∈ [0,1]; unitless; thresholds τ_CWA, τ_ESI.

Health, risk, control.
H — health index = Φ_surplus + ρ; units = Φ units + s-norm units (declare both).
R_G — dissipation-gap risk = ⟨G⟩_window − G*; units = Φ units.
ρ — robustness radius (max recoverable ‖Δs‖ under Γ); units = ‖s‖.
κ — condition number cond(M); unitless; alert if κ > κ*.
κ* — controllability threshold; unitless; declared per study.

Safety invariants & governance.
I-set — { Φ ≥ Φ_min , Γ ≤ Γ* , ρ ≥ ρ_min , κ ≤ κ* }; bounds declared.
Φ_min, Γ, ρ_min* — safety thresholds; units as symbols.
σ_I — slack vector (Φ−Φ_min, Γ*−Γ, ρ−ρ_min, κ*−κ); mixed units; dashboard.
v_I — violation score = min_j σ_Ij / band_j ; unitless (≤0 is breach).
S_safe — composite safety score (min-normalized margins incl. lamps); unitless.

Dynamics & control (reporting form).
L_ctrl — per-tick control Lagrangian = Φ̇ − λ·ṡ − β·Γ; units = Φ units per tick.
A_local — summed local action; units = Φ units.
β, β_c — dissipation weights (unit conversions / policy); unitless.
F(·), K(s) — update/mobility operators; declared numerics; units consistent with ṡ.

Boundary & continuum.
L — life score = B·S_τ·S_safe with B ≡ max(0, Φ_budget/Φ_ref); unitless.
Φ_ref — reference scale for B; units = Φ units.
S_τ — tick-sync score = 1 − (max|τ_i−τ_j|/ε_τ)_+; unitless.

Thresholds & tolerances (must declare).
G* — gap threshold; units = Φ units.
ε_gap, ε_budget, ε_τ, ε_rec — numerical tolerances; units per symbol.
T_rec, T_safe — recovery and safe-exit times; units = ticks.

Footer fields (VerifyTrace, one line per claim).
[seed][hash][Δτ][domain][units policy][q id][φ id][Γ budget][G*][S_CWA][S_ESI][CWA lamp][ESI lamp][decision][reviewer][time].

One-line rule. “No claim without footer + seeds + gates + budgets.”

Appendix B. Equations (Unicode Journal Style)

s = ∇ψ(λ) ; I = ∇²ψ(λ) . (B.1)
λ = ∇Φ(s) ; M = ∇²Φ(s) = I⁻¹ . (B.2)
G ≡ Φ + ψ − λ·s ≥ 0 . (B.3)
W_s ≡ ∫ λ·ds . (B.4)
ΔΦ = W_s − Δψ − Γ_loss . (B.5)
H ≡ Φ_surplus + ρ . (B.6)

Notes (paste-friendly).
– All quantities reported in Φ-units (bits or J·K⁻¹ scaled) per tick; λ·s shares Φ-units.
– ∫ λ·ds is along the observed path over the window/tick; Δψ is the single-tick change in ψ.
– Budget closure tolerance: |ΔΦ − (W_s − Δψ − Γ_loss)| ≤ ε_budget (declared in Methods).

Appendix C. Checklists & Templates

Purpose. Copy-and-paste forms that make your runs auditable and Blogger-ready. Everything below is single-line or CSV-style so you can drop it into posts or repos without reformatting.

C.1 VerifyTrace footer (paste one line per figure/table)

Minimal (as required):
[seed=<id>][hash=<artifact_hash>][units=<Φ-units/tick>][Γ budget=<value>][G threshold=<G*>][CWA lamp=<Green|Red>][ESI lamp=<Green|Red>][decision=<Publish|Rollback>][reviewer=<ID>]

Expanded (recommended):
[seed=42][hash=ab12cd34][Δτ=1s][domain=chemostat][units=bits/tick][q=baseline_v3][φ=id:featmap_07][Γ budget=0.035][G threshold=1e-5][CWA=Green][ESI=Green][decision=Publish][reviewer=R-019][time=2025-11-09T12:34Z]

Field notes.

seed global RNG seed; hash is content hash of derived tables for this figure.
units must match Φ, ψ, λ·s, Γ; if not, publish the conversion you used.
Γ budget is the declared per-tick dissipation ceiling for this run.
G threshold is your calibrated ( G^* ).
Lamps are final gate states; “decision” is the publish action taken.

One-line rule (repeat under every figure).
“No claim without footer + seeds + gates + budgets.” (7.1)

C.2 Slots table (quotas / capacities per tick)

CSV template (slots.csv):

channel,unit,S_c_per_tick,token_bucket_R_c,B_c_max,overage_weight_mu_c,write_cost,notes
energy,W,120,80,600,0.8,NA,hard cap at 150 W
matter,mmol/s,2.0,1.0,5.0,1.2,NA,stoichiometry A:B=2:1
information,write/s,10,5,50,0.5,2 writes,protected log requires 2 tokens
governance,ops/s,3,1,10,2.0,NA,manual approval gate on overflow

Runtime usage table (slots_usage.csv):

tick,channel,J_c,S_c_per_tick,overage,Γ_policy_c
0,energy,118,120,0,0
0,information,12,10,2,2.0

Computation rules (publish in Methods).

Overage per channel: overage = max(0, J_c − S_c_per_tick).
Policy penalty: Γ_policy_c = μ_c · overage^2 (same Φ-units/tick as Γ).
Write gate (irreversible stores): allow only if bucket ≥ write_cost and I-set is respected.

C.3 Δ5 micro-cycle schedule (drift cancellation & leakage cooling)

Pairing rule (ten-phase ring).
pair(i) = i + 5 (mod 10) for phases i ∈ {0,…,9}.

Schedule template (delta5_schedule.csv):

phase_i,pair_i,start_tick,duration_ticks,rest_fraction,actuator,notes
0,5,0,10,0.2,pump_A,forward
5,0,10,10,0.2,pump_A,reverse
1,6,20,8,0.3,heater,heat
6,1,28,8,0.3,heater,cool
2,7,36,6,0.4,io_bus,tx
7,2,42,6,0.4,io_bus,rx
3,8,48,12,0.2,stim_A,excite
8,3,60,12,0.2,stim_A,inhibit
4,9,72,10,0.25,transport,load
9,4,82,10,0.25,transport,unload

Stress response (auto-adjust).

Increase rest_fraction when safety margin thins:
rest_fraction_new = rest_fraction_base + k · max(0, τ_safe − S_safe)
Cross-leakage bound to report each run:
Γ_cross(Δ5) ≤ α · Γ_cross(free) with 0 < α < 1 (publish α).

C.4 Minimal checklists (copy into your Methods)

Pre-run (once per study).

Declare (X, q, φ), units, Δτ, ε_τ.
Calibrate sensors; fix G*, κ*, Φ_min, Γ*, ρ_min.
Pin Slots (S_c, or {R_c, B_c_max}) and Δ5 schedule; hash the I-set.

Per-experiment (each run).

Compute ψ, s, I, Φ, λ, M, G; check gap on calibration (|G| ≤ ε_gap).
Fill budget table per tick: W_s, Δψ, Γ_loss, ΔΦ.
Publish I-conditioning (eigs, κ) before/after interventions.
Evaluate ρ with a recovery test; record T_rec, ε_rec.
Run CWA/ESI protocols; assign lamps and thresholds.

Per-figure/footer (every plot/table).

Attach VerifyTrace footer (C.1).
Include budget, gap, geometry, and recovery tables (Section 10.C headers).
State Slots usage and Γ_policy by channel if relevant.

Breach handling (Safety Mode).

If any lamp = Red or I-set violated, freeze to Θ_safe, de-rate λ, hard-cap Slots, enforce Δ5, stop writes.
Do not publish; log rollback footer with remediation plan.
Exit only after Safe holds for T_safe ticks; file post-mortem.

Paste-ready snippets

Footer (minimal):
[seed=7][hash=d34f00d][units=bits/tick][Γ budget=0.02][G threshold=1e-5][CWA lamp=Green][ESI lamp=Green][decision=Publish][reviewer=R-21]

Slots (row example):
information,write/s,10,5,50,0.5,2 writes,comment: protected log

Δ5 (pair mapping line for docs):
Δ5 pairs = {(0,5),(1,6),(2,7),(3,8),(4,9)}; enforce symmetric duty; report α.

Reminder: “No claim without footer + seeds + gates + budgets.” (7.1)

Appendix D. Micro-Examples

Purpose. Two tiny, copy-and-run examples that exercise the core math with minimum data. Everything is Blogger-ready and reuses the same symbols and equations from the main text.

D.1 Binary system — compute (B.1)–(B.5); show G ≥ 0

Setup. One binary detector (x ∈ {0,1}) with baseline (θ = q(x=1)). Feature map (φ(x)=x). Units: bits per tick.

Files (tiny).

baseline.csv

theta
0.37

series.csv

tick,x
0,0
1,1
2,0
3,1
4,1
5,0

Recipe (tick t to t+1).

Estimate s_t (feature mean).
Use the current observation or a short windowed mean (declare window). Units = unit(φ).
Compute ψ(λ_t) and s(λ_t).
ψ(λ) = log( (1−θ) + θ·e^{λ} ) . (D.1.1)
s(λ) = θ·e^{λ} / ( (1−θ) + θ·e^{λ} ) . (D.1.2)
Invert to get λ(s_t).
λ(s) = log( s(1−θ) / ((1−s)θ) ) . (D.1.3)
Value and geometry.
Φ(s) = s·log( s/θ ) + (1−s)·log( (1−s)/(1−θ) ) . (D.1.4)
I(λ) = s(1−s) ; M(s) = 1 / ( s(1−s) ) . (D.1.5)
Gap and budget.
G_t ≡ Φ(s_t) + ψ(λ_t) − λ_t·s_t ≥ 0 . (B.3)
W_{s,t} = λ_t · ( s_{t+1} − s_t ) . (B.4)
Δψ_t = ψ(λ_{t+1}) − ψ(λ_t) . (D.1.6)
ΔΦ_t = W_{s,t} − Δψ_t − Γ_{loss,t} . (B.5)

Checks (declare tolerances).
|G_t| ≤ ε_gap on calibration; |ΔΦ_t − (W_{s,t} − Δψ_t − Γ_{loss,t})| ≤ ε_budget.

Footer (example).
[seed=42][hash=binA1][units=bits/tick][Γ budget=0][G threshold=1e-6][CWA lamp=Green][ESI lamp=Green][decision=Publish][reviewer=R-01]

D.2 Two-module coupling — flip Θ on/off; report ΔG, κ, lamp states

Setup. Two scalar modules A,B with the same binary-feature geometry as D.1 (so M_A(s_A)=1/(s_A(1−s_A)), M_B analogously). Linear couplings Θ and a quadratic cross-leak penalty in Γ.

Files (tiny).

two_module_series.csv

tick,sA,lambdaA,sB,lambdaB
0,0.40,0.10,0.30,0.05
1,0.43,0.12,0.33,0.06
2,0.45,0.11,0.36,0.08
3,0.47,0.10,0.37,0.09

coupling_OFF.csv

thetaAB,thetaBA,mu,DeltaTau
0.00,0.00,0.50,1

coupling_ON.csv

thetaAB,thetaBA,mu,DeltaTau
0.20,0.15,0.50,1

Dynamics (declared).
s_{A,t+1} = s_{A,t} + Δτ·M_A^{-1}( λ_{A,t} + θ_{AB}·s_{B,t} − λ_{resist,A} ) . (D.2.1)
s_{B,t+1} = s_{B,t} + Δτ·M_B^{-1}( λ_{B,t} + θ_{BA}·s_{A,t} − λ_{resist,B} ) . (D.2.2)

Leakage and penalty.
J_{AB,t} ≡ θ_{AB}·s_{B,t} − θ_{BA}·s_{A,t} . (D.2.3)
Γ_{cross,t} = μ·J_{AB,t}² ≥ 0 . (D.2.4)

Gaps and budget (per tick).
G_{A,t} = Φ_A + ψ_A − λ_{A,t}·s_{A,t} ; G_{B,t} analogously . (D.2.5)
ΔΦ_{total,t} = (W_{s,A,t}+W_{s,B,t}) − (Δψ_{A,t}+Δψ_{B,t}) − (Γ_{loss,A,t}+Γ_{loss,B,t}+Γ_{cross,t}) . (D.2.6)

Procedure.

Θ-OFF run: use coupling_OFF.csv, compute ⟨G_A+G_B⟩_{OFF}, κ_{OFF} from M on the operating band.
Θ-ON run: use coupling_ON.csv, same data and Δτ; compute ⟨G_A+G_B⟩_{ON}, κ_{ON}.
Report ΔG and geometry.
ΔG ≡ ⟨G_A+G_B⟩_{ON} − ⟨G_A+G_B⟩_{OFF} . (D.2.7)
κ report: κ = σ_{max}(M) / σ_{min}(M) (unitless). (D.2.8)

Interpretation.
ΔG < 0 ⇒ coupling improved alignment (good).
ΔG > 0 ⇒ coupling worsened alignment (bad).
κ↑ with Θ may indicate harder control; document mitigation (preconditioning, slot relief, Δ5).

Lamp protocol (minimum).

CWA: unit checks + budget closure; if |ΔΦ_{total} − RHS(D.2.6)| > ε_budget on any tick ⇒ CWA=Red.
ESI: add ±1% noise to {s_A,s_B}, re-run; if flip_rate < 2% ⇒ ESI=Green, else Red.

Footers (examples).
Θ-OFF:
[seed=7][hash=twomod_off][units=bits/tick][Γ budget=declared][G threshold=1e-4][CWA lamp=Green][ESI lamp=Green][decision=Publish][reviewer=R-02]
Θ-ON:
[seed=7][hash=twomod_on][units=bits/tick][Θ=(0.20,0.15)][μ=0.5][Γ budget=declared][G threshold=1e-4][CWA lamp=Green][ESI lamp=Green][decision=Publish][reviewer=R-02]

Read-off checklist (tie-back to Sections 6 & 10).

From D.1 or D.2, fill Budget table: tick, W_s, Δψ, Γ_loss, ΔΦ.
Compute Risk: ⟨G⟩ vs. G* → R_G.
Compute Geometry: eigs(M), κ; flag κ > κ*.
If you tested recovery, report ρ, T_rec, ε_rec.
Attach a VerifyTrace footer to each figure/table.

Appendix E. Cultural Origins and Scientific Mapping (Confucian–Daoist–Buddhist → GLF)

Purpose. Recall the historical inspirations and show a clean, testable mapping from classical frameworks to the GLF variable set and audit logic. No metaphysics is assumed; each item maps to measurable invariants, budgets, or couplings.

E.1 Overview (what each lineage contributes)

Confucian / 先天八卦動力學. Capacity design, opposition pairing, and rhythm hygiene → Slots, Δ5 micro-cycles, leakage cooling.
Daoist / 太極・五行. Complementary conjugates and inter-element interactions → value–drive conjugacy {s, λ}, channel matrix Θ with signed edges.
Buddhist / 真如–示現. Invariants vs. appearances; observer’s role → I-set (safety invariants), projection Ô, two-lamp verification.

E.2 Confucian line: 先天八卦動力學 → capacity, pairing, hygiene

Idea. Balanced opposition and ritualized cycles prevent drift and waste.

GLF mapping.

Slots (capacities/quotas). Ritualized “limits” → per-channel caps S_c and token buckets.
Δ5 opposition cycle. Opposite micro-phases cancel drift and cool cross-leakage.

pair(i) = i + 5 (mod 10) . (E.1)
Γ_cross(Δ5) ≤ α · Γ_cross(free) , 0 < α < 1 . (E.2)

Governance analogue. “禮(ritual)” = explicit Γ-policy and write gates; “樂(music)” = rhythm design that locks phases without raising Γ.

E.3 Daoist line: 太極・五行 → conjugacy and signed couplings

Idea. Yin–Yang as complementary aspects; Five-Element cycles as structured interactions.

GLF mapping.

Taiji (陰陽). Dual-aspect variables → conjugacy between state and drive:

s = ∇ψ(λ) ; λ = ∇Φ(s) . (E.3)

Five Elements (生・克). Interactions are signed couplings in Θ. Let gen(i→j) (生) be supportive edges and control(i→j) (克) be restraining edges:

sign(Θ_ij) = +1 for engendering ; sign(Θ_ij) = −1 for controlling . (E.4)

Channel view. Elements act as channels (energy, matter, information, governance) with potentials U and currents J; dissipation per edge is non-negative:

Γ_c = J_c · U_c ≥ 0 . (E.5)

Operational note. The “middle way” of Daoism corresponds to maintaining κ ≡ cond(M) within bound κ* (well-conditioned control) and minimizing Γ for the same ΔΦ.

E.4 Buddhist line: 真如–示現 → invariants, projection, verification

Idea. Distinguish invariant structure (真如) from appearances (示現) and make the observer’s action explicit.

GLF mapping.

Invariants (真如). Publish and respect I-set bounds:

I-set ≡ { Φ ≥ Φ_min , Γ ≤ Γ* , ρ ≥ ρ_min , κ ≤ κ* } . (E.6)

Manifestation (示現). Reported states arise via projection Ô with cost and trace:

Π_trace ≡ energy_to_project − energy_to_erase . (E.7)

Right view (verification). Two independent gates must be green before claims are published:

Publish ⇔ (CWA = Green) ∧ (ESI = Green) . (E.8)

Interpretation. There is no reified “self” in the math: only traces, budgets, and invariants under declared domains.

E.5 Unification: one ledger, one audit

Across the three lineages, GLF keeps one budget and one geometry:

ΔΦ = W_s − Δψ − Γ_loss . (E.9)
M = ∇²Φ(s) ; κ = cond(M) ; ρ = max recoverable ‖Δs‖ under Γ . (E.10)

Confucian contributions shape capacities and rhythm → lower Γ_cross, stable τ.
Daoist contributions shape coupling structure and conjugates → interpretable Θ and efficient W_s.
Buddhist contributions shape invariants and verification → publishable, observer-honest results.

E.6 Translation crib-sheet (classical ⇄ GLF)

陰陽 (Taiji complementarity) ⇄ {s, λ} conjugacy; gap G ≡ Φ + ψ − λ·s ≥ 0.
五行・生克 ⇄ signed edges in Θ; supportive vs. restraining couplings.
禮 ⇄ Slots/Γ-policy (who may write/flow, how much, when).
樂 ⇄ phase/rhythm design (τ, θ) with Δ5 cancellation.
氣 ⇄ channels + constraints (U, J, Γ) governing permissible transformations.
中庸 ⇄ good conditioning (κ ≤ κ*), adequate surplus (H = Φ_surplus + ρ).
真如 ⇄ I-set invariants; 示現 ⇄ Ô-projection + lamps.

E.7 Scope and discipline

This appendix recalls inspiration, not authority. Every mapping above is operationalized with units, thresholds, and footers. If a cultural analogy does not reduce to {Φ, ψ, s, λ, I, M, Γ, Θ, τ, ρ, κ} + audits, it is outside GLF.

Appendix F. TopoSpec Template (YAML you can paste)

One file that fully declares system, modules, channels, constraints, lamps, Slots, Δ5, and reproducibility. Keep it under version control.

# TopoSpec v1.0 (GLF)
system:
  id: "chemostat_v01"
  tick: {DeltaTau: "1 s", epsilon_tau: "5 ms"}
  units_policy: "bits_per_tick"   # or "J_per_K_per_tick via k_B ln2"
  domain: ["lab", "steady-feed"]

modules:
  - id: "A"
    features: {s_units: "fraction", notes: "binary-feature geometry"}
    value_model: {Phi: "min-RE vs q", q_id: "baseline_v3", phi_id: "featmap_07"}
    invariants: {Phi_min: 0.02, rho_min: 0.05, kappa_star: 1.0e3}
  - id: "B"
    features: {s_units: "fraction"}
    value_model: {Phi: "min-RE vs q", q_id: "baseline_v3", phi_id: "featmap_07"}
    invariants: {Phi_min: 0.02, rho_min: 0.05, kappa_star: 1.0e3}

couplings:   # Θ blocks; signed edges allowed
  - from: "A"
    to:   "B"
    theta: 0.20
  - from: "B"
    to:   "A"
    theta: 0.15

channels:
  - id: "energy"
    potential: {name: "dmu", unit: "J"}
    current:   {name: "power", unit: "J/s"}
    slot: {S_c_per_tick: 120, token_bucket: {R_c: 80, B_c_max: 600}}
    overage_weight_mu_c: 0.8
  - id: "matter"
    potential: {name: "chem_pot", unit: "J/mol"}
    current:   {name: "flux", unit: "mmol/s"}
    slot: {S_c_per_tick: 2.0}
    overage_weight_mu_c: 1.2
  - id: "information"
    potential: {name: "decision_load", unit: "bits"}
    current:   {name: "writes", unit: "writes/s"}
    slot: {S_c_per_tick: 10, token_bucket: {R_c: 5, B_c_max: 50}}
    overage_weight_mu_c: 0.5
    write_cost: 2   # tokens for protected log writes

constraints:   # Γ components and policy penalties
  budget:
    Gamma_star: 0.035   # per tick in Φ-units
  penalties:
    - channel: "information"
      formula: "mu_c * max(0, J_c - S_c)^2"
      mu_c: 0.5

lamps:
  thresholds:
    tau_CWA: 0.99
    tau_ESI: 0.98

delta5_schedule:
  pairs: [[0,5],[1,6],[2,7],[3,8],[4,9]]
  base_rest_fraction: 0.2
  alpha_bound: 0.6    # Γ_cross(Δ5) ≤ α · Γ_cross(free)

safety_invariants:   # I-set
  Phi_min: 0.02
  rho_min: 0.05
  kappa_star: 1.0e3
  Gamma_star: 0.035

reproducibility:
  seed: 42
  container_digest: "sha256:…"
  code_hash: "ab12cd34"

Appendix G. ObserverOps Audit Pack (Footer + Gate Script)

G.1 Footer line formats (CSV or JSONL)

CSV header (recommended):

seed,hash,DeltaTau,domain,units,q_id,phi_id,Gamma_budget,G_threshold,S_CWA,S_ESI,CWA_lamp,ESI_lamp,decision,reviewer,time

Example row:

42,ab12cd34,1s,"lab;steady-feed",bits/tick,baseline_v3,featmap_07,0.035,1e-5,0.995,0.989,Green,Green,Publish,R-019,2025-11-09T12:34Z

JSONL (one line per figure):

{
  "seed":42,"hash":"ab12cd34","DeltaTau":"1s","domain":["lab","steady-feed"],
  "units":"bits/tick","q_id":"baseline_v3","phi_id":"featmap_07","Gamma_budget":0.035,
  "G_threshold":1e-5,"S_CWA":0.995,"S_ESI":0.989,"CWA_lamp":"Green","ESI_lamp":"Green",
  "decision":"Publish","reviewer":"R-019","time":"2025-11-09T12:34Z"}

G.2 Gate pseudocode (drop into your pipeline)

def gate_publish(budget_ok, S_CWA, S_ESI, tau_CWA, tau_ESI, I_ok):
    cwa = (S_CWA >= tau_CWA)
    esi = (S_ESI >= tau_ESI)
    safe = I_ok and cwa and esi
    decision = "Publish" if (safe and budget_ok) else "Rollback"
    lamp = {"CWA": ("Green" if cwa else "Red"), "ESI": ("Green" if esi else "Red")}
    return decision, lamp

Budget closure check:
abs(DeltaPhi - (W_s - DeltaPsi - Gamma_loss)) <= epsilon_budget

One-line rule: No claim without footer + seeds + gates + budgets. (7.1)

Appendix H. KPI Threshold Table & Color Guidelines

Defaults you can override—but you must declare them.

KPI	Symbol	Green	Amber	Red	Note
Health	H	≥ H_G	[H_A, H_G)	< H_A	H = Φ_surplus + ρ
Dissipation risk	R_G	≤ 0	(0, R_A]	> R_A	R_A calibrated
Controllability	κ	≤ κ*	—	> κ*	report eigs(M)
Robustness	ρ	≥ ρ_min	[ρ_warn, ρ_min)	< ρ_warn	per regime
Gap	G	≤ G*	(G, 2G]	> 2G*	rolling mean
Budget closure	—	≤ ε_budget	—	> ε_budget	absolute error
CWA score	S_CWA	≥ τ_CWA	—	< τ_CWA	lamp
ESI score	S_ESI	≥ τ_ESI	—	< τ_ESI	lamp

Suggested defaults (declare changes):
H_G = 0.10 ; H_A = 0.02 ; R_A = 0.02 ; κ* = 1.0e3 ; ρ_min = 0.05 ; ρ_warn = 0.03 ; G* = 1e−5 ; ε_budget = 1e−6 ; τ_CWA = 0.99 ; τ_ESI = 0.98.

Color palette (HEX, WCAG-friendly):
Green #1E9E63 • Amber #F5A623 • Red #D0021B • Neutral text #333333 • Grid #DADFE3.

Appendix I. Minimal Data Package (Directory & Columns)

Directory layout (example):

/run_001/
  manifest.csv
  features.parquet
  drives.parquet
  geometry.parquet
  budgets.parquet
  gap.parquet
  slots.csv
  delta5_schedule.csv
  footer.log

manifest.csv

study_id,dataset_id,commit_hash,code_hash,container_digest,DeltaTau,units_policy
GLF01,chemo_setA,9f1e…,ab12cd34,sha256:…,1s,bits/tick

features.parquet (or CSV)

sample_id,tick,module,s_k,unit_s,stderr_s
A-001,0,A,0.40,fraction,0.01

drives.parquet

sample_id,tick,module,lambda_k,unit_lambda
A-001,0,A,0.10,1/fraction

geometry.parquet

tick,module,eig_min_M,eig_max_M,kappa,method,CI
0,A,4.0,200.0,50.0,"Hessian-fit","[45,55]"

budgets.parquet

tick,W_s,DeltaPsi,Gamma_loss,DeltaPhi,P_in,Pi_diss
0,0.012,0.003,0.004,0.005,0.020,0.015

gap.parquet

tick,module,G,G_star,lamp
0,A,8.0e-6,1.0e-5,Green

footer.log (CSV or JSONL as in Appendix G)

Validation checklist (run at load time):

Units consistency: units(lambda·s) == units(Phi) == units(psi)
Positivity: Gamma_loss ≥ 0, G ≥ 0 (within tolerance), kappa ≥ 1
Budget closure: |DeltaPhi - (W_s - DeltaPsi - Gamma_loss)| ≤ epsilon_budget

Appendix J. Calibration & Tolerance Cookbook (Quick Start)

J.1 Calibrate G*

Use calibration data where λ(s) is fit directly; set G* = quantile(G, 0.95); verify median(G) ≈ 0.

J.2 Set κ*

Sweep operating region; compute κ; choose κ* at the 95th percentile of safe, controllable episodes or a fixed engineering limit (e.g., 1e3).

J.3 Set Φ_min, ρ_min

From historical stable operation: Φ_min = min_windowed(Φ_surplus);
ρ_min from recovery tests (Sec. 6), pick smallest r with reliable return by T_rec.

J.4 Choose ε_budget, ε_gap

Numerical experiment with pinned seed; pick smallest tolerances that yield ≥ 99% pass on exact recomputation.

J.5 Pick τ_CWA, τ_ESI

Start at τ_CWA=0.99, τ_ESI=0.98; tighten for safety-critical tasks; loosen (never below 0.95) for exploratory screens.

Appendix K. Safety Mode Playbook (Runbook You Can Print)

Trigger: any I-set violation or lamp = Red.
Freeze couplings: Θ ← Θ_safe (hash: …).
De-rate drive: λ ← r_λ · λ with r_λ = 0.5 (declare if different).
Hard-cap Slots: enforce {S_c_hard}; deny irreversible writes.
Enforce Δ5: immediate switch to declared pair schedule; increase rest_fraction.
Stabilize: wait T_safe ticks with Safe true.
Post-mortem: publish σ_I trajectories, Slot usage, lamps, and remediation.
Re-open: only after two-lamp Green and I-set respected for T_safe.

Event log line (CSV):

time,event,details,actor,hash
2025-11-09T12:40Z,ENTER_SAFETY_MODE,"CWA=Red (units mismatch), Θ->Θ_safe, λ->0.5λ",auto,4f3a…

That covers the practical gap: a full YAML spec (TopoSpec), ready-to-use footers and gate logic, KPI thresholds with color, a minimal data package, plus calibration and safety runbooks.

Disclaimer

This book is the product of a collaboration between the author and OpenAI's GPT-5, Google's Gemini 2.5 Pro, X's Grok 4 language model. While every effort has been made to ensure accuracy, clarity, and insight, the content is generated with the assistance of artificial intelligence and may contain factual, interpretive, or mathematical errors. Readers are encouraged to approach the ideas with critical thinking and to consult primary scientific literature where appropriate.

This work is speculative, interdisciplinary, and exploratory in nature. It bridges metaphysics, physics, and organizational theory to propose a novel conceptual framework—not a definitive scientific theory. As such, it invites dialogue, challenge, and refinement.

I am merely a midwife of knowledge.

Sunday, November 9, 2025