PAPERS · QUICK REFERENCE

Quick Reference

A one-page summary of the Focospectralics framework: the geometric invariants, canonical values, four conservation theorems, status tag taxonomy, and 4-step protocol. Every value carries its status tag. Every derivation closes to R = 1.0000.

This page is a companion to the papers index. A plain-text version is downloadable as quick-reference.md.

01 · Geometric invariants

Quantity	Symbol	Value	Status tag
Tetrahedral apex angle	θ	70.5288°	Derived
Viewer cone fraction	S₊	1/3 · 33.333%	Derived
Inverse cone fraction	S₋	2/3 · 66.667%	Derived
Boundary partition ratio	S₋ / S₊	2.000000	Derived
Unit triangle area	A_area	√3/4 · 0.433013	Derived
Alignment factor	D_align	1/√2 · 0.707107	Derived
Closure ratio	R	1.0000	Derived

02 · Canonical values

Quantity	Value	Status tag
Projection factor P_θ	1.08308	Derived
Hubble baseline H₀,A	67.4 km/s/Mpc	Frame-Conditioned
Hubble projected H₀,B	73.00 km/s/Mpc	Derived
Flux gradient ΔH	5.60 km/s/Mpc	Fitted
Effective aperture Ω_eff	31.73%	Fitted
Readout efficiency η_budget	0.15442	Fitted
CMB thermal bit floor	2.606 × 10⁻²³ J/bit	Frame-Conditioned
Horizon-adjacent bit floor	5.9051 × 10⁻³⁹ J/bit	Frame-Conditioned
Planck area ceiling A_min	2.159 × 10⁻⁷⁰ m²/bit	Derived
Dark sector budget	Ω_b 4.90% · Ω_c 26.83% · Ω_Λ 68.27%	Fitted

03 · The four conservation theorems

Theorem	Statement	Reference
I	No net information flux across the frame interface on any Cauchy slice. dI(S₊)/d(δt) = −dI(S₋)/d(δt).	Universal slice invariance
II	What flows outward is transduced along the single principal zero-mode channel with the square-root fold signature.	Asymptotic zero-mode transduction
III	The admissible operating window is bounded below by the Landauer floor and above by the Planck area ceiling.	Bounded emission corridor
IV	The residual content of the inverse cone is conserved indefinitely under the Hermitian global conservation operator.	Subcoherent persistence

04 · Status tag taxonomy

Tag	Meaning	Examples
Derived	Pure unconditioned geometry, trigonometry, or differential identity. Valid prior to empirical calibration.	θ = 70.5288°, S₊ = 1/3, P_θ = 1.08308
Frame-Conditioned	Parameter bound to a specific instrument aperture, observation epoch, or spatial boundary.	H₀,A = 67.4 km/s/Mpc, L₀ = 13.90 Gly, T_CMB = 2.7255 K
Fitted	Empirical calibration mapping verified telemetry onto the geometric engine.	Ω_eff = 31.73%, η_budget = 0.15442, dark sector budget

05 · The four-step lock

Step	Action	What happens
1	Pure geometric initialization Derived	Establish θ = 70.5288°, S₊ = 1/3, S₋ = 2/3, partition ratio 2.000000. No empirical content.
2	Empirical boundary pinning Frame-Conditioned	Anchor the geometry to measured baselines. Declare aperture, epoch, archive.
3	Thermodynamic aperture coupling Fitted	Apply the calibration. Landauer floor, Planck ceiling, Bekenstein–Hawking capacity enter here.
4	Operational mode selection Derived Fitted	Mode A projects across frames. Mode B mines the conserved complement. Both return to measured values.

Attribution. Every classical result retains its originator's name: Landauer, Planck, Bekenstein, Hawking, Shannon, Boltzmann, von Neumann, Fredholm, Lyapunov, Schmidt, Euclid, Hubble. The framework contributes the arrangement of these results into the conic partition and its conservation theorems.