Master register of proofs, values, and experiments
This is the single lookup table for the Focospectralics framework. Every parameter, invariant, theorem, and empirical result in the corpus appears here with its value, derivation path, and mandatory status tag. If a number appears anywhere in the manuals, proofs, or papers, it resolves to a row in this register.
Four sections precede the master table: foundational geometric invariants, the four conservation theorems, the thermal/holographic limits, and the projection/frame reconciliation constants. The master table at the end consolidates all 29 rows into one alphabetical/indexed list.
01 · Foundational geometric invariants
Seven invariants form the unconditioned base of the framework. All carry Derived status — they depend only on three-dimensional observer geometry and hold prior to any empirical calibration. Each remains strictly invariant under smooth diffeomorphisms (see diffeomorphism invariance).
| Invariant | Symbol | Value | Derivation path | Tag |
|---|---|---|---|---|
| Tetrahedral apex angle | \(\theta\) | 70.5288° | \(\theta = 2\arcsin(1/\sqrt{3})\), from \(\sin^2(\theta/2) = 1/3\) | Derived |
| Viewer cone fraction | \(S_+\) | 1/3 · 33.333% | \(\sin^2(\theta/2)\) solid-angle partition | Derived |
| Inverse cone fraction | \(S_-\) | 2/3 · 66.667% | \(\cos^2(\theta/2)\) complement | Derived |
| Boundary partition ratio | \(S_- / S_+\) | 2.000000 | \((2/3) / (1/3) = 2\) | Derived |
| Unit triangle area | \(A_{\text{area}}\) | 0.433013 | \(\sqrt{3}/4\) normalized inscribed equilateral | Derived |
| Alignment factor | \(D_{\text{align}}\) | 0.707107 | \(1/\sqrt{2}\) orthonormal frame projection | Derived |
| Machine-precision closure | \(R\) | 1.0000 | \((I_{\text{known}} + I_{\text{unknown}}) / I_{\text{total}}\) | Derived |
02 · The four conservation theorems
Four theorems carry the framework's conservation content. Each is proved in full in the proofs directory. Every one is Derived.
| # | Theorem | Statement | Verification | Tag |
|---|---|---|---|---|
| I | Universal slice invariance | \(\dfrac{dI(S_+)}{d(\delta t)} = -\dfrac{dI(S_-)}{d(\delta t)}\), so \(dI_{\text{total}}/dt = 0\). | GW150914 dual-readout | Derived |
| II | Asymptotic zero-mode transduction | \(Q_\pm(t) - Q_* = \pm A_Q (t-t_*)^{1/2} + B_Q (t-t_*)\), and \(P(t) = \kappa (dQ/dt)^2\). | MOTS numerical relativity | Derived |
| III | Bounded emission corridor | Energy bound: \(E_{\text{bit}} \le E_{\text{observed}}\) (both in J). Luminosity bound: \(P_{\text{observed}} \le c^5/G = L_{\max}\) (both in W). | Landauer floor + Planck ceiling | Derived |
| IV | Subcoherent persistence | \(\hat{\mathcal{C}}(S_+ \oplus S_-) = I_{\text{total}}\); \(S_{\text{thermo}} = (k_B \ln 2) H_{\text{BS}}\). | Entropy-bridge unification | Derived |
03 · Thermal and holographic limits
The operational corridor is bounded below by the Landauer bit-energy floor and above by the Planck area ceiling and the Planck/Dyson luminosity ceiling. These are the physical constraints that force unread capacity into the inverse cone \(S_-\).
| Limit | Value | Closed form & anchor | Tag |
|---|---|---|---|
| CMB thermal bit floor | 2.606 × 10⁻²³ J/bit | \(E_{\text{bit}}^{\text{CMB}} = k_B \cdot T \cdot \ln 2\) at \(T_{\text{CMB}} = 2.7255\) K | Frame-Conditioned |
| Horizon-adjacent bit floor | 5.9051 × 10⁻³⁹ J/bit | \(E_{\text{floor}} = k_B \cdot T \cdot \ln 2\) at \(T \approx 6.1706 \times 10^{-16}\) K | Frame-Conditioned |
| Planck area ceiling | 2.159 × 10⁻⁷⁰ m²/bit | \(A_{\min} = \ell_P^2 \ln 2\) (per-bit normalized form) | Derived |
| Planck/Dyson luminosity ceiling | 3.628 × 10⁵² W | \(L_{\max} = c^5/G\) | Derived |
| Critical squeezing threshold | 0.346573 · 3.0103 dB | \(r_* = \tfrac{1}{2} \ln 2\), \(\Delta X_1^2 = 0.125\), \(\Delta X_2^2 = 0.50\) | Derived |
04 · Projection & frame reconciliation
The Hubble tension is reconciled as a geometric projection factor between the 1964 CMB sound-horizon frame and the late-time local frame. No modified gravity, no systematic error, no new fields.
| Parameter | Value | Derivation path | Tag |
|---|---|---|---|
| Projection factor \(P_\theta\) | 1.08308 | \((13/12) / \sqrt{2/3}\), observer triangle at \(\theta = 70.5288°\) | Derived |
| Early-universe baseline \(H_{0,A}\) | 67.4 km/s/Mpc | 1964 CMB sound horizon (Planck 2018) | Frame-Conditioned |
| Late-universe projected \(H_{0,B}\) | 73.00 km/s/Mpc | \(H_{0,A} \times P_\theta = 67.4 \times 1.08308\) | Derived |
| Boundary flux gradient \(\Delta H\) | 5.60 km/s/Mpc | \(H_{0,B} - H_{0,A}\), drives back-reaction \(P(t) = \kappa (dQ/dt)^2\) | Fitted |
| Inner Hubble radius \(L_0\) | 13.90 Gly | \(c / H_{0,A}\) | Frame-Conditioned |
| Outer comoving horizon \(L_1\) | 47.90 Gly | Comoving particle boundary | Frame-Conditioned |
05 · Gravitational-wave telemetry
GW150914 provides the dual-readout reconciliation: discovery-frame strain telemetry vs. MOTS-frame numerical relativity. Coordinate-dependent parameters vary between frames; the invariants do not.
| Parameter | Value | Source | Tag |
|---|---|---|---|
| Remnant mass \(M_{\text{final}}\) | 62.0 M☉ | GW150914 strain telemetry | Frame-Conditioned |
| Horizon capacity \(I_{\max}\) | 1.514 × 10⁹³ bits | \(A_{\text{BH}} / (4 \ell_P^2 \ln 2)\) | Frame-Conditioned |
| Readable capacity \(I_{\text{readable}}\) | 1.315 × 10⁹² bits | Single-pass strain, 35–150 Hz | Frame-Conditioned |
| Strain readout efficiency \(\eta_{\text{strain}}\) | 0.0869 · 8.69% | \(I_{\text{readable}} / I_{\max}\) | Derived |
| Information deficit \(\Delta I\) | 1.383 × 10⁹³ bits | \(I_{\max} - I_{\text{readable}}\) | Derived |
06 · Cosmological energy budget
The ΛCDM budget is derived from a single geometric aperture and a single readout-efficiency parameter. Baryons, dark matter, and dark energy are geometric partitions of the same conic split — not new particle species.
| Component | Value | Closed form | Tag |
|---|---|---|---|
| Effective aperture \(\Omega_{\text{eff}}\) | 31.73% | 33.33% − 1.60% thermal back-reaction | Fitted |
| Cosmo readout efficiency \(\eta_{\text{budget}}\) | 0.15442 · 15.44% | Baryon / dark matter split | Fitted |
| Baryonic share \(\Omega_b\) | 4.90% | \(\eta_{\text{budget}} \cdot \Omega_{\text{eff}}\) | Derived |
| Dark matter share \(\Omega_c\) | 26.83% | \((1 - \eta_{\text{budget}}) \cdot \Omega_{\text{eff}}\) | Derived |
| Dark energy share \(\Omega_\Lambda\) | 68.27% | \(1 - \Omega_{\text{eff}}\) | Derived |
| Closure sum | 100.00% | 4.90% + 26.83% + 68.27% = 100.00% (R = 1.0000) | Derived |
07 · Sub-threshold recovery
Mode B operates on the inverse cone \(S_-\). It converts theoretical information deficits into targeted search coordinates for existing archival data.
| Result | Value | Derivation | Tag |
|---|---|---|---|
| Information deficit \(\Delta I\) | \(I_{\max} - I_{\text{readable}}\) | Boundary capacity minus resolved capacity | Derived |
| Zero-mode filter \(S_{\text{negative}}(t)\) | \(A_Q (t-t_*)^{1/2} + B_Q (t-t_*)\) | Sub-threshold residual fit along fold bifurcation | Derived |
| Net SNR accumulation | \(\sqrt{\sum_{k=1}^{N} \text{SNR}_k^2}\) | Coherent cross-frame integration of sub-noise residuals | Derived |
| NEO 2019 OK warning gain | +9.0 days · 5σ | Sub-threshold optical residuals, t = −9.6 d vs. −0.6 d | Derived |
08 · Master parameter table
The consolidated index. All 29 rows from the framework. Numbers are formatted for reading; canonical precision is on the canonical values page.
| # | Parameter / metric | Value | Derivation path / source | Tag |
|---|---|---|---|---|
| 1 | Tetrahedral apex angle \(\theta\) | 70.5288° | \(\Omega_+ = \sin^2(\theta/2) = 1/3\) | Derived |
| 2 | Viewer cone fraction \(S_+\) | 0.333333 · 1/3 | Solid-angle partition \(S_+\) | Derived |
| 3 | Inverse cone fraction \(S_-\) | 0.666667 · 2/3 | Solid-angle partition \(S_-\) | Derived |
| 4 | Orthonormal alignment \(D_{\text{align}}\) | 0.707107 · 1/√2 | Orthonormal frame projection | Derived |
| 5 | Unit triangle area \(A_{\text{area}}\) | 0.433013 · √3/4 | Unit equilateral geometry | Derived |
| 6 | Boundary partition ratio \(S_- / S_+\) | 2.000000 | \((2/3) \div (1/3)\) | Derived |
| 7 | Machine closure \(R\) | 1.0000 | \((I_{\text{known}} + I_{\text{unknown}}) / I_{\text{total}}\) | Derived |
| 8 | Horizon-adjacent bit floor \(E_{\text{floor}}\) | 5.9051 × 10⁻³⁹ J/bit | \(k_B \cdot T \cdot \ln 2\) at \(T \approx 6.1706 \times 10^{-16}\) K | Frame-Conditioned |
| 9 | Planck luminosity ceiling \(L_{\max}\) | 3.628 × 10⁵² W | \(c^5 / G\) | Derived |
| 10 | Hubble projection factor \(P_\theta\) | 1.08308 | Observer triangle at \(\theta = 70.5288°\) | Derived |
| 11 | Projected Hubble \(H_{0,B}\) | 73.00 km/s/Mpc | \(67.4 \times 1.08308\) | Derived |
| 12 | Remnant mass \(M_{\text{final}}\) | 62.0 M☉ | GW150914 merger telemetry | Frame-Conditioned |
| 13 | Horizon capacity \(I_{\max}\) | 1.514 × 10⁹³ bits | \(A_{\text{BH}} / (4 \ell_P^2 \ln 2)\) for 62 M☉ | Frame-Conditioned |
| 14 | Readable capacity \(I_{\text{readable}}\) | 1.315 × 10⁹² bits | GW150914 single-pass strain | Frame-Conditioned |
| 15 | Strain passband efficiency \(\eta_{\text{strain}}\) | 0.0869 · 8.69% | \(I_{\text{readable}} / I_{\max}\) across 35–150 Hz | Derived |
| 16 | GW150914 invariant lock | \(\xi = 0.3333,\; R = 1.0000\) | Dual-readout reconciliation | Derived |
| 17 | Inner Hubble radius \(L_0\) | 13.90 Gly · 1.315 × 10²⁶ m | \(c / H_{0,A}\) | Frame-Conditioned |
| 18 | Outer comoving horizon \(L_1\) | 47.90 Gly · 4.532 × 10²⁶ m | Comoving particle boundary | Frame-Conditioned |
| 19 | Outside horizon volume fraction | 97.56% | \(1 - (13.90/47.90)^3\) | Derived |
| 20 | Thermal back-reaction correction | −1.60% | Integral of \(P(t) = \kappa (dQ/dt)^2\) | Fitted |
| 21 | Effective thermal aperture \(\Omega_{\text{eff}}\) | 31.73% | 33.33% − 1.60% | Fitted |
| 22 | Cosmo readout efficiency \(\eta_{\text{budget}}\) | 0.15442 · 15.44% | 4.90% / 31.73% | Fitted |
| 23 | Baryonic energy share \(\Omega_b\) | 4.90% | \(\eta_{\text{budget}} \cdot \Omega_{\text{eff}}\) | Derived |
| 24 | Dark matter energy share \(\Omega_c\) | 26.83% | \(\Omega_{\text{eff}} - 4.90\%\) | Derived |
| 25 | Dark energy share \(\Omega_\Lambda\) | 68.27% | \(1 - \Omega_{\text{eff}}\) | Derived |
| 26 | Cosmological closure sum | 100.00% | 4.90% + 26.83% + 68.27% | Derived |
| 27 | Asteroid 2019 OK zero-mode fit | \(S_{\text{negative}}(t)\) | \(A_Q (t-t_*)^{1/2} + B_Q (t-t_*)\) | Derived |
| 28 | Asteroid 2019 OK warning gain | +9.0 days | Multi-sensor residual cross-correlation | Derived |
| 29 | Detection statistical confidence | 5σ | Noise-floor residual rejection | Derived |
09 · Status tag taxonomy
Every parameter in this register carries exactly one tag. The taxonomy is load-bearing: it prevents geometric identities from being confused with empirical calibrations, and prevents both from being mistaken for frame-conditioned artifacts.
| Tag | Definition | Examples |
|---|---|---|
| Derived | Pure unconditioned geometry, trigonometry, or differential identity. Dimensionless, scale-invariant, valid prior to empirical calibration. | \(S_+ = 1/3\), \(S_- = 2/3\), \(\theta = 70.5288°\), \(P_\theta = 1.08308\), \(A_{\text{area}} = \sqrt{3}/4\), \(R = 1.0000\) |
| Frame-Conditioned | Parameter bound to a specific instrument aperture, observation epoch, or spatial boundary. | \(H_{0,A} = 67.4\) km/s/Mpc, \(L_0 = 13.90\) Gly, \(L_1 = 47.90\) Gly, \(T_{\text{CMB}} = 2.7255\) K, \(I_{\max} = 1.514 \times 10^{93}\) bits |
| Fitted | Empirical calibration mapping verified telemetry onto the geometric engine. | \(\Omega_{\text{eff}} = 31.73%\), \(\Omega_b = 4.90%\), \(\Omega_c = 26.83%\), \(\Omega_\Lambda = 68.27%\), \(\eta_{\text{budget}} = 0.15442\) |
Cross-references. Full derivations for each theorem live in proofs/. Full narrative treatment lives in manuals/. The consolidated submission documents live in papers/. Canonical numeric precision and errata are on the canonical values page.