Formula Reference & Builder

Algebra est cautio, non porta. — The algebra is a warranty, not a gate.

The complete formula structure of the GCD kernel — from the four primitive equations through the algebraic identities, frozen parameters, seam budget, regime gates, and 53 structural identities. Every formula derives from Axiom-0. The interactive builder at the bottom lets you evaluate any expression live.

Formula Structure

The kernel function K : [0,1]ⁿ × Δⁿ → (F, ω, S, C, κ, IC) takes a trace vector c ∈ [0,1]ⁿ and weights w ∈ Δⁿ and produces 6 outputs from 4 primitive equations + 2 derived values, with only 3 effective degrees of freedom (F, κ, C). S is asymptotically determined by F and C.

4 Primitive Equations

F (fidelity), κ (log-integrity), S (entropy), and C (curvature) are computed directly from the trace vector. They form the irreducible core of the kernel.

2 Derived Values

ω = 1 − F (drift) and IC = exp(κ) (integrity composite) are derived algebraically. They participate in the immutable identities.

3 Effective DOF

The statistical constraint S ≈ f(F, C) reduces 6 outputs to 3 independent degrees of freedom: F, κ, C. As n → ∞, corr(C, S) → −1.

Kernel Primitives

Tier-1 — immutable by construction. The mathematical function and its provable properties.

Fidelity (fidelitas)

F = Σ w_i c_i

Weighted arithmetic mean of the trace vector. Measures what survives collapse. Range: [0, 1]. Tag: K-F

Log-Integrity (log-integritas)

κ = Σ w_i ln(c_i,ϵ)

Logarithmic sensitivity of coherence. Guard band ϵ = 10⁻⁸ prevents pole. Range: ≤ 0. Tag: K-κ

Bernoulli Field Entropy (entropia)

S = −Σ w_i [ c_i ln c_i + (1−c_i) ln(1−c_i) ]

Entropy of the collapse field. Computed, but not free — asymptotically determined by F and C. Maximized at the equator c = ½. Range: ≥ 0. Tag: K-S

Curvature (curvatura)

C = stddev(c_i) / 0.5

Coupling to uncontrolled degrees of freedom. Normalized standard deviation. Range: [0, 1]. Tag: K-C

Drift (derivatio) derived from F

ω = 1 − F

How much is lost to collapse. Collapse proximity measure. Range: [0, 1]. Tag: K-ω

Integrity Composite (integritas composita) derived from κ

IC = exp(κ)

Multiplicative coherence — weighted geometric mean. Integrity cannot exceed fidelity: the limbus integritatis IC ≤ F. Range: (0, 1]. Tag: K-IC

Heterogeneity Gap

Δ = F − IC ≥ 0

Measures channel heterogeneity. Zero when all channels are equal (rank-1). Large when one channel is near dead (geometric slaughter). Taylor: Δ ≈ C²/(8F).

Algebraic Identities

Three identities always hold by construction. One statistical constraint tightens with n.

AI-1

Duality Identity (complementum perfectum)

F + ω = 1

Exact to machine precision — verified to 0.0e+00 residual across 10,000 traces. In Fisher coordinates: sin²θ + cos²θ = 1. Tertia via nulla — no third possibility.

AI-2

Integrity Bound

IC ≤ F

The solvability condition: for n = 2, the trace recovery c_1,2 = F ± √(F² − IC²) requires IC ≤ F for real solutions. Equality iff all c_i are equal. Derived independently from Axiom-0. Limbus integritatis.

AI-3

Log-Integrity Relation

IC = exp(κ)

Definition link between κ and IC. Bridges multiplicative (IC) and additive (κ) coherence.

SC-1

Entropy Determination (statistical constraint)

S ≈ f(F, C) | corr(C, S) → −1 as n → ∞

Reduces 6 kernel outputs to 3 effective degrees of freedom. CLT-based — tightens with channel count. S is computed but not free.

Frozen Parameters

Seam-derived, not prescribed. The unique values where seams close consistently across all 23 domains. Trans suturam congelatum — same rules both sides.

Symbol	Value	Role	Why This Value
ϵ	10⁻⁸	Guard band / ϵ-clamp	Pole at ω=1 does not affect measurements to machine precision
p	3	Exponent in Γ(ω)	Unique integer where ω_trap is a Cardano root of x³ + x − 1 = 0
α	1.0	Curvature cost: D_C = α·C	Unit coupling
λ	0.2	Auxiliary coefficient	Return-rate adaptation speed
tol_seam	0.005	\|s\| ≤ tol for PASS	Width where IC ≤ F holds at 100% across all 23 domains
c*	0.7822	Logistic self-dual fixed point	Maximizes S + κ per channel; solves ln((1−c)/c) + 1/c = 0
ω_trap	0.6823	Trapping threshold	Cardano root of x³ + x − 1 = 0
c_trap	0.3177	Channel trapping threshold	c_trap = 1 − ω_trap

Seam Budget

The seam is the verification boundary between outbound collapse and demonstrated return. The budget must reconcile — residual ≤ tol.

Γ(ω) — Drift Cost

Γ(ω) = ω^p / (1 − ω + ϵ)

Simple pole at ω = 1 with effective residue 1/2. Convex on [0, 1−ϵ) for p ≥ 2. At ω_trap = 0.6823, Γ = 1 exactly — single-step escape becomes impossible.

D_C — Curvature Cost

D_C = α · C

Linear curvature penalty. α = 1.0 (unit coupling).

Δκ — Conservation Budget

Δκ_budget = R · τ_R − (D_ω + D_C)

Credit (return) minus debits (drift + roughness). The ledger must reconcile.

Seam Residual

s = Δκ_budget − Δκ_ledger | |s| ≤ tol_seam for PASS

Budget identity minus observed log-integrity change. Sequential seam deltas telescope (L-20): Σ_i Δκ_i = κ(t_n) − κ(t₀). Forms a monoid — associative with identity, error 5.55×10⁻¹⁷.

Cost Elasticity

ϵ_Γ = ω(3 − 2ω) / (1 − ω)

Effective critical exponent for budget blowup. Approaches 4 near the pole.

Regime Gates

Four-gate criterion. Derived from gates, never asserted. Gate precedence: CRITICAL → COLLAPSE → WATCH → STABLE.

Stable 12.5% of Fisher space

ω < 0.038

F > 0.90

S < 0.15

C < 0.14

Conjunctive — ALL four gates must be met. Stability is rare.

Watch 24.4% of Fisher space

0.038 ≤ ω < 0.30

(or Stable gates not all satisfied)

Collapse 63.1% of Fisher space

ω ≥ 0.30

Dissolution, not failure — ruptura est fons constantiae.

Critical severity overlay

IC < 0.30

Not a regime — an overlay. Accompanies any regime when integrity is dangerously low.

Structural Identities

53 identities derived from Axiom-0 and verified to machine precision. Organized by derivation depth from the kernel.

Level A Foundation — 1 step from kernel

I-A1 Duality

F + ω = 1

sin²θ + cos²θ = 1 in Fisher coordinates

I-A2 Integrity Bound

IC ≤ F (equality iff all c_i equal)

Jensen: ln concave ⇒ κ ≤ ln F

I-A3 Fano-Fisher

h''(c) = −g_F(c) = −1/[c(1−c)]

Entropy curvature IS the Fisher metric

I-A4 Fisher Flatness

g_F(θ) = 1 (flat manifold)

All structure from embedding, not intrinsic curvature

I-A5 κ as Log-Sine

κ = 2Σ w_i ln|sin θ_i|

Log-integrity is logarithmic sine on the sphere

I-A6 Rank-2 Closed Form

IC = √(F² − C²/4)

Exact analytical solution for 2-channel systems

I-A7 Integral Conservation

∫₀¹ [h(c) + ln(c)] dc = −½

Average S + κ is exactly −1/2

Level B Structure — 2 steps from kernel

I-B1 Coupling Function

f(θ) = 2cos²θ · ln(tan θ)

S and κ are projections of ONE function (verified < 10⁻¹⁶)

I-B2 Solvability

c_1,2 = F ± √(F² − IC²)

Integrity bound IS the solvability condition

I-B3 Log-Integrity Correction

κ = ln F − C²/(8F²) + O(C⁴)

Taylor: heterogeneity correction always negative

I-B4 Jensen Entropy Bound

S ≤ h(F) (equality iff C = 0)

Entropy companion to integrity bound

I-B7 Fisher-Entropy Integral

∫₀¹ g_F(c)·S(c) dc = π²/3 = 2ζ(2)

Kernel geometry tied to the Basel constant

I-B9 Gap Taylor Expansion

Δ ≈ σ²/(2F) = C²/(8F)

Leading-order heterogeneity gap approximation

I-B10 Equator Quintuple

c = ½: S = ln 2, S+κ = 0, h' = 0, g_F = 4, θ = π/4

Five properties converge simultaneously at the equator

I-B11 Regime Partition

Collapse 63% / Watch 24% / Stable 12.5%

87.5% of the manifold is NOT stable

Level C Skeleton — 3 steps from kernel

I-C1 Logistic Self-Duality

c* = σ(1/c*) | c* ≈ 0.7822

Coupling peak is the logistic-reciprocal fixed point

I-C2 Reflection Formula

f(θ) + f(π/2−θ) = 2ln(tanθ)cos(2θ)

Equator θ=π/4 is double zero; bridges c* and c_trap

I-C3 Composition Law

IC₁₂ = √(IC₁·IC₂) | F₁₂ = (F₁+F₂)/2

IC geometric, F arithmetic

I-C5 Cubic Trapping

Γ(ω_trap) = 1 | x³ + x − 1 = 0

Below c_trap, budget cannot close without R > Γ

I-C7 Sandwich Theorem

IC ≤ F ∧ S ≤ h(F) (both exact iff C = 0)

Kernel sandwiched: integrity below, entropy above

I-C8 Dimension Collapse

8 channels → 4 dimensions (PCA)

Kernel halves the dimensionality of the closure algebra

Level D Convergence — 4 steps from kernel

I-D1 Coupling Maximum

max(S+κ) = (1−c*)/c* ≈ 0.278

Odds ratio equals exponential at c*

I-D7 p=3 Cardano

x³ + x − 1 = 0 | Δ = −31 < 0

p = 3 is the UNIQUE integer with Cardano structure

I-D8 Composition Invariance

Δ(n copies) = Δ(1 copy) exactly

Heterogeneity gap invariant under replication

I-D9 Gap Composition

Δ₁₂ = (Δ₁+Δ₂)/2 + (√IC₁−√IC₂)²/2

Hellinger-like correction; gap grows for unequal IC

I-D6 Omega Hierarchy

ω_stable < ω* < ω_collapse < ω_trap

0.038 < 0.218 < 0.300 < 0.682 strict ordering

I-D11 Geodesic Partition

{c_trap, ½, c*, 1−ϵ} partition [0, π]

Structural constants partition the Fisher half-circle

Level E Discovery — 5 steps from kernel

I-E1 Cost Cross-Product

Γ(ω)·Γ(1−ω) = [ω(1−ω)]^p−1

Cost function reflection law; p=3: product = square

I-E2 Trapping Echo

(S+κ)(1−c*) = −1 exactly

Coupling at reflected fixed point = exactly −1 (integer resonance)

I-E3 Departure Half

1 − IC/F = Var(c)/(2F²) with β = ½

Fractional IC deficit has coefficient exactly ½

I-E6 Log-Variance Gap

Δ ≈ IC · Var(ln c) / 2

Alternative gap formula — 1.45× more accurate than Taylor

I-E8 Algebraic Signature

F,ω,S,κ: arithmetic | IC: geometric | C,Δ: none

Complete composition classification of all 7 kernel outputs

I-E9 Sensitivity Divergence

∂κ/∂c_i = w_i/c_i → ∞ as c_i → 0

Geometric slaughter mechanism — dead channels dominate

Composition Laws

How kernel outputs compose when subsystems are combined. IC geometric, F arithmetic, C does not compose.

Output	Composition Rule	Type	Tag
F	F₁₂ = (F₁ + F₂) / 2	Arithmetic	I-C3
ω	ω₁₂ = (ω₁ + ω₂) / 2	Arithmetic	I-C3
S	S₁₂ = (S₁ + S₂) / 2	Arithmetic	I-E4
κ	κ₁₂ = (κ₁ + κ₂) / 2	Arithmetic	I-C3
IC	IC₁₂ = √(IC₁ · IC₂)	Geometric	I-C3
C	Does not compose	None	I-E7
Δ	Δ₁₂ = (Δ₁+Δ₂)/2 + (√IC₁−√IC₂)²/2	Hellinger-like	I-D9

Fixed Points & Fisher Metric

Geodesic Partition (I-D11)

ϵ = 10⁻⁸ θ/π ≈ 0

Guard boundary — no channel fully dies

c_trap = 0.3177 θ/π = 0.191

Γ(ω_trap) = 1 — trapping threshold (Cardano root)

c = ½ θ/π = 0.250

Equator — max entropy, S + κ = 0 (quintuple fixed point)

c* = 0.7822 θ/π = 0.354

Self-dual — maximizes S + κ per channel

c ≈ 1 θ/π ≈ 0.5

Perfect fidelity — upper clamp

Fisher Metric Results

g_F(c)

g_F(c) = 1 / [c(1 − c)]

Fisher metric in c-parameterization

g_F(θ)

g_F(θ) = 1 (flat!)

Manifold is flat in Fisher coordinates — all structure from embedding

Fano-Fisher

h''(c) = −g_F(c)

Entropy curvature IS the Fisher metric (I-A3)

∫ g_F·S

∫₀¹ g_F(c) · S(c) dc = π²/3

= 2ζ(2) — the Basel constant (I-B7)

Rank Classification

Rank-1: all c_i=c₀ → IC=F, C=0 (1 DOF)

Rank-2: 2-channel → IC=√(F²−C²/4) (2 DOF)

Rank-3: general → F, κ, C independent (3 DOF)

Interactive Formula Builder

Type any expression using the kernel symbols. Supports exp, log, sqrt, abs, sin, cos, and arithmetic. Adjust sliders to update all expressions in real time. Sweep any variable over a range.

Formula Builder

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Variable Bindings

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Preset Expressions

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Example Expressions

Try these in the builder above. Organized by category.

Core Identities

F + omega Duality identity (AI-1) — always exactly 1

exp(kappa) - IC Log-integrity relation (AI-3) — always 0

F - IC Heterogeneity gap Δ (AI-2) — always ≥ 0

1 - F - omega Residual check — should be exactly 0

Coherence & Gap Analysis

IC / F Coherence ratio — 1.0 = homogeneous, low = geometric slaughter

1 - IC / F Fractional IC deficit (I-E3) — = Var(c)/(2F²)

C**2 / (8 * F) Gap Taylor approximation (I-B9) — compare with F − IC

sqrt(F**2 - C**2/4) Rank-2 IC closed form (I-A6) — exact for 2-channel systems

Entropy & Coupling

S + kappa Coupling function — zero at equator (c = ½)

-F * log(F) - (1-F) * log(1-F) Homogeneous entropy h(F) — upper bound for S (I-B4)

log(F) - C**2 / (8 * F**2) κ Taylor approximation (I-B3) — compare with kappa

S - (-F*log(F) - (1-F)*log(1-F)) Entropy deficit — always ≤ 0 (Jensen bound I-B4)

Drift Cost & Seam Budget

omega**3 / (1 - omega + 1e-8) Γ(ω) — drift cost with Cardano exponent p = 3

omega * (3 - 2*omega) / (1 - omega) Cost elasticity ϵ_Γ (I-C6) — approaches 4 near pole

omega**3/(1-omega+1e-8) + 1.0 * C Total cost Γ(ω) + D_C — drift + curvature

omega**3/(1-omega+1e-8) * (1-omega)**3/((omega)+1e-8) Cost cross-product (I-E1) — = [ω(1−ω)]² for p=3

Fisher Geometry

1 / (F * (1 - F)) Fisher metric g_F(F) = 1/[F(1−F)] (I-A3)

-1 / (F * (1 - F)) - 1 / F**2 Coupling curvature f''(c) = −g_F − 1/c² (I-B5)

2 * kappa κ as log-sine: 2Σ w_i ln|sin θ| (I-A5)

log(4 / (F * (1-F))) Fisher volume proxy: ln(4·g_F) (I-B13)

Regime Detection

omega < 0.038 Stable drift gate (1 if met)

omega ≥ 0.30 Collapse gate (1 if in collapse)

IC < 0.30 Critical overlay (1 if critical)

omega**3/(1-omega+1e-8) ≥ 1.0 Trapped (1 if Γ ≥ α, no single-step escape)

Key Lemma Expressions

IC * (1/F) L-6 sensitivity: ∂IC/∂c_k ∝ IC·w_k/c_k

(1 - IC/F) * 2 * F**2 Computes Var(c) from departure formula (I-E3)

(F - IC) / F Normalized heterogeneity — fraction of F lost to gap

exp(log(F) - C**2/(8*F**2)) IC from κ Taylor (I-B3): exp(ln F − C²/(8F²))

Variable Reference

Variable	Symbol	Range	Description	Formula
F	F	[0, 1]	Fidelity — what survives collapse	Σ w_ic_i
omega	ω	[0, 1]	Drift — what is lost	1 − F
S	S	≥ 0	Bernoulli field entropy	−Σ w_i[c_iln c_i + (1−c_i)ln(1−c_i)]
C	C	[0, 1]	Curvature — coupling to DOF	stddev(c_i) / 0.5
kappa	κ	≤ 0	Log-integrity	Σ w_i ln(c_i,ϵ)
IC	IC	(0, 1]	Integrity composite	exp(κ)

All variables are computed from the trace vector c ∈ [0,1]ⁿ with weights w ∈ Δⁿ. The kernel has 3 effective degrees of freedom: F, κ, C.

Numeri sunt intellectus. — The numbers are the understanding.