# Focospectralics — Quick Reference

A one-page summary of the framework's invariants, canonical values, four
conservation theorems, status tag taxonomy, and 4-step protocol.

Source of truth: https://[kb-url]/papers/quick-reference.html

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## 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]  |

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## 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]            |
| Landauer bit floor E_bit     | 7.155 × 10⁻³⁹ J                       | [Derived]           |
| 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]            |

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## 03 · The four conservation theorems

**Theorem I — Universal slice invariance.**
No net information flux across the frame interface on any Cauchy slice.
dI(S₊)/d(δt) = −dI(S₋)/d(δt).

**Theorem II — Asymptotic zero-mode transduction.**
What flows outward is transduced along the single principal zero-mode
channel with the square-root fold signature.

**Theorem III — Bounded emission corridor.**
The admissible operating window is bounded below by the Landauer floor
and above by the Planck area ceiling.

**Theorem IV — Subcoherent persistence.**
The residual content of the inverse cone is conserved indefinitely under
the Hermitian global conservation operator.

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## 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                     |

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## 05 · The four-step lock

Every application runs through the same four steps, in order.

1. **Pure geometric initialization** [Derived] — Establish θ = 70.5288°,
   S₊ = 1/3, S₋ = 2/3, partition ratio 2.000000.
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.
4. **Operational mode selection** [Derived / Fitted] — Mode A projects
   across frames. Mode B mines the conserved complement. Both return to
   measured values.

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## 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.

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Status: This quick reference is a portable summary of the Focospectralics
Knowledge Base. Every value carries its status tag. Every derivation
closes to R = 1.0000.