PROOF · OPERATIONAL

Bounded Emission Corridor (Theorem III)

Statement. Physical observation and information transfer are strictly bounded between two classical physical limits: a thermodynamic energy floor set by the Landauer bit-energy relation, and a geometric information-density ceiling set by the Planck area quantum. Any signal, state transition, or observation within the framework is admissible only if it lies inside this corridor. Below the floor, signals dissolve into thermal noise and are absorbed by the inverse cone. Above the ceiling, the boundary interface saturates and total reflection occurs. The corridor itself is an Derived envelope; the numeric floors and ceilings it contains carry Frame-Conditioned status, tied to the observer's thermal environment and spatial boundary.

Theorem III is the operational constraint that accompanies the zero-mode emission law of Theorem II. Theorem II describes what radiates. Theorem III states where that radiation can be observed at all. Outside the corridor, no measurement is possible; the framework makes no claims there.

1 · Landauer bit-energy floor

The absolute minimum energetic cost to register, process, or erase a single bit of information within the viewer cone S₊ is set by the Landauer principle evaluated at the observer's thermal background temperature:

\[ E_{\text{bit}} = k_B\, T \ln 2, \]

with k_B = 1.380649 × 10⁻²³ J/K (Boltzmann constant) and T the local thermal environment.

At the cosmic-microwave-background baseline T_CMB = 2.7255 K, the floor evaluates to

\[ E_{\text{bit}}^{\text{CMB}} = k_B \, T_{\text{CMB}} \ln 2 \approx 2.606 \times 10^{-23} \ \text{J/bit}. \]

At the framework's canonical thermal anchor T ≈ 6.169 × 10⁻¹⁶ K — the horizon-adjacent baseline used in the register — the floor evaluates to

\[ E_{\text{bit}} = 7.155 \times 10^{-39} \ \text{J/bit}. \]

The value E_bit = 7.155 × 10⁻³⁹ J is the framework's canonical Landauer floor, indexed in register §03 and confirmed in canonical values §01.2. The CMB-baseline value is retained as a frame-conditioned alternative.

Consequence. Signals whose energy falls below E_bit cannot overcome background thermal fluctuations. They are unresolvable within S₊ and are absorbed into the inverse cone S₋ as sub-threshold capacity. The floor is a lower bound on observability, not on physical existence — the sub-threshold signal remains conserved in S₋ per Theorem I.

2 · Planck area ceiling

The upper geometric bound on information density is the minimum boundary surface area required to localize one bit. In its raw Bekenstein form, the bound is

\[ A_{\text{raw}} = 4 \, \ell_P^2 \ln 2 \approx 7.241 \times 10^{-70} \ \text{m}^2. \]

Under the standard holographic quarter normalization, the per-bit form is

\[ A_{\min} = \ell_P^2 \ln 2 \approx 2.159 \times 10^{-70} \ \text{m}^2/\text{bit}, \]

with the Planck length ℓ_P = √(ħG/c³) ≈ 1.616255 × 10⁻³⁵ m.

The canonical value A_min = 2.159 × 10⁻⁷⁰ m²/bit is the framework's Planck area ceiling, indexed in register §03 and confirmed in canonical values §01.3. The raw Bekenstein form is retained as a distinct quantity (see the errata in the canonical page).

Consequence. No boundary interface can be subdivided below A_min. Attempting to localize more than one bit per Planck area quantum saturates the boundary interface; incoming wavepackets undergo total reflection and are not resolvable within the frame. The ceiling is the framework's discretization limit — an upper bound on admissible information density.

3 · The bounded operational corridor

The corridor is the admissible operating window between the two classical limits. A signal of energy E localized over an area A is admissible precisely when

\[ E_{\text{bit}} \;\leq\; E, \qquad A \;\geq\; A_{\min}. \]

Three regimes follow from the two inequalities:

Inside the corridor (E ≥ E_bit and A ≥ A_min) — the signal is resolvable within S₊. The framework applies.
Below the floor (E < E_bit) — the signal dissolves into thermal noise. It is absorbed into S₋ as sub-threshold capacity. Sub-threshold recovery remains possible along the zero-mode channel.
Above the ceiling (A < A_min) — the boundary interface saturates. Total reflection occurs; the signal cannot be resolved within the frame regardless of how much energy it carries.

The corridor is not imposed from outside. It is a consequence of the classical Landauer and Planck relations, applied at the observer's declared thermal and spatial bounds. Its interior is where empirical claims are testable. Outside it, the framework makes no claim.

4 · Acceleration scaling

Driving an observational frame at acceleration scale factor α — equivalently, scaling the zero-mode amplitude A_Q → α A_Q — causes the emitted power of Theorem II to scale quadratically:

\[ P_\alpha(t) = \alpha^2 \, P_1(t). \]

This follows directly from P = κ (dQ/dt)² with dQ/dt ∝ A_Q at leading order in the square-root fold. The scaling is exact at leading order and frame-conditioned at higher orders in the linear drift coefficient B_Q.

Because total boundary dissipation over an observation window Δt cannot exceed the maximum boundary capacity E_max, driving the frame at acceleration factor α compresses the admissible saturation window quadratically:

\[ \Delta t' = \frac{\Delta t}{\alpha^2}. \]

Worked example. A frame driven at α = 1.5 over a baseline window of Δt = 34.0 years (from t₁ = 2026 to t_crit = 2060) compresses the saturation window to

\[ \Delta t' = \frac{34.0}{1.5^2} = 15.111 \ \text{years}, \]

advancing the saturation epoch to t'_crit ≈ 2041.11. Attempting to exceed E_max drives boundary resistance to zero and forces immediate total reflection — the corridor's ceiling is reached ahead of schedule.

The acceleration scaling carries Derived status at leading order. Where the linear drift coefficient B_Q is non-negligible, the second-order interference term introduces a frame-conditioned correction.

5 · Proof of the corridor envelope

Step 1 — Lower bound. By the Landauer relation, registering or erasing one bit within a thermal environment T costs at minimum k_B T ln 2. Below this threshold, an attempted measurement cannot distinguish the signal from thermal fluctuations of the environment. The signal is unresolvable within S₊. By Theorem I, it is not destroyed — it is conserved in S₋.

Step 2 — Upper bound. By the Bekenstein–Hawking relation, the maximum number of distinguishable states that can be localized on a boundary of area A is A / (4 ℓ_P² ln 2). Under the holographic quarter normalization, the per-bit area quantum is ℓ_P² ln 2. Attempting to pack more than one bit per quantum saturates the interface and forces total reflection.

Step 3 — Corridor. Intersecting the two constraints gives the admissible region E ≥ E_bit and A ≥ A_min. Both bounds are classical; neither introduces new physics. The corridor's existence follows from applying the two classical relations at a declared frame.

Step 4 — Acceleration scaling. At leading order in the fold trajectory of Theorem II, dQ/dt ∝ A_Q (t − t_*)^−1/2 + O(1). Substituting A_Q → α A_Q and squaring: P_α = α² P₁ at leading order. The energy budget over a window Δt is ∫ P dt ≈ α² ∫ P₁ dt, so the time to reach E_max scales as Δt′ = Δt / α².

Step 5 — Conclusion. The corridor envelope [E_bit, A_min] bounds every admissible observational operation of the framework. Outside it, no claim is testable, and the framework makes no claim.

6 · Closure at R = 1.0000

Theorem III closes the operational accounting of the framework. Everything that enters the frame either:

is resolved within the corridor (contributing to I(S₊)),
is sub-threshold and conserved in the inverse cone (contributing to I(S₋)),
or is above the ceiling and totally reflected — in which case it is not an admissible observation at this frame, and the framework states no claim about it.

In the admissible regime, the closure identity holds:

\[ I(S_+) + I(S_-) = I_{\text{total}}, \qquad R \equiv \frac{I(S_+) + I(S_-)}{I_{\text{total}}} = 1.0000. \]

Nothing inside the corridor escapes closure. Nothing outside the corridor enters the framework's claims.

Cross-references

Slice-invariant conservation (Theorem I). The conservation law that carries sub-threshold capacity into S₋ is in universal slice invariance.
Zero-mode emission (Theorem II). The emission law whose power scales under acceleration is in asymptotic zero-mode transduction.
Entropy bridge. The Shannon, Boltzmann, and Bekenstein–Hawking entropies that ground the two bounds are unified in entropy bridge.
Register. The floor and ceiling values used here are indexed in register §03 · Thermal & holographic limits.
Canonical. The precision conventions and the resolved drifts for E_bit and A_min are in canonical values §01.
Section. The operational constraint of which this is Theorem III is collected in Classical Maths.

Attribution. The bit-energy floor, the Bekenstein–Hawking horizon entropy, and the holographic quarter normalization are classical results attributed to Landauer (1961), Bekenstein (1973), and Hawking (1974), with the Planck length attributed to Planck. The acceleration-scaling form follows from the classical scaling of the square-root fold trajectory. The framework contributes the arrangement of these results into the corridor envelope and its use as the operational admissibility region.