The Cosmological Corollary

Structural connection. The cosmological channel of section 5 reads the CMB as leakage at one boundary in cosmic time — the recombination surface. AP43 D5 (debt D5) reads the cosmological residue as an ongoing structural process: every actualization-state instant produces one unpaired 1×ε that does not return to symmetry. This section names the corollary that AP06 carries into Notebook IV: the two readings are not two phenomena. They are one process read at two boundaries.

6.1 — The accumulated 1×ε

The 420 Code’s governing axiom is 1:1 + 1×ε @ AS. The @ AS reads at the actualizing now — the surface from which all records are written (AP01). Each AS-instant is the actualization of the axiom at one point in cosmic time. AP43 D5 establishes that each such actualization produces one unpaired ε that persists: the break does not heal.

Across cosmological time, these unpaired 1×ε accumulate. The cosmological residue is the integrated quantity. AP43 D5 carries the formal account; the corpus reading is that the cosmological residue is what the corpus’s axiom must continue to produce while AS-instants continue to actualise.

6.2 — Connection to AP06’s leakage structure

AP06 section 5 reads the CMB as leakage at one boundary — the recombination surface, ∼380,000 years post-expansion. The CMB is the historical record of leakage at that boundary: thermal radiation at 2.72548 K, anisotropies ∼10⁻⁵, blackbody to one part in 10⁴.

AP43 D5 reads the cosmological residue as the accumulated 1×ε across all AS-instants. The two readings meet at the structural fact: the CMB is the leakage that was observable at one specific historical boundary; the cosmological residue is the leakage that has continued to accumulate since.

AP06’s contribution is the identification that these are one process. The Leakage Theorem (3.1) establishes that finite c forbids perfect absorption at every boundary in the analysed channels. Identification 6.1 (section 7) reads the ε of the axiom as the leakage ratio η. Read forward to cosmological scales: every AS-instant is a boundary at which the axiom actualises; finite c forbids perfect absorption at that boundary; the unpaired ε produced is the leakage that AP43 D5 names. The CMB is the snapshot at one specific historical boundary. The cosmological residue is the integrated process.

6.3 — What AP06 carries forward

AP06 carries the corollary into Notebook IV as a structural commitment, not a new derivation. The formal account of the cosmological 1×ε integration lives in AP43 D5. AP06’s contribution is the structural reading that grounds the Notebook IV treatment of the cosmological channel: the channel is open, ongoing, and structurally necessary for the same reason the gravitational and electromagnetic channels are open — finite c forbids closure.

The corollary connects to the rest of Notebook IV as follows. AP24 (section 2 Six Faces) formalises the multi-projection structure of ε; the cosmological projection is one face. AP28 derives G via channel-counting across six faces × three spatial dimensions plus three actualization couplings; the actualization couplings are the AS-instant integration this corollary names. AP14 (Correction) handles the Planck-scale corrections in the gravitational sector; the cosmological residue connects to AP14’s one-loop finiteness at the high-energy end of the same structural process.

6.4 — Scope of the corollary

The corollary is structural, not quantitative. AP06 does not predict the magnitude of the cosmological residue, does not derive the cosmological constant Λ, and does not claim that the CMB-anisotropy figure ∼10⁻⁵ quantifies the integrated residue. Those quantitative questions live in AP43 D5 and in Notebook VI (Cosmology).

What section 6 carries is the structural commitment: the cosmological channel of section 5 and the cosmological residue of AP43 D5 are the same process read at two boundaries. The reading is load-bearing for Notebook IV’s coherence — without the corollary, the cosmological channel of section 5 sits as an isolated structural match. With the corollary, it grounds the cosmological line that Notebook VI develops.

Claim C6a (Structural Corollary). The cosmological channel of section 5 (CMB as boundary radiation) and AP43 D5’s cosmological 1×ε integration are one structural process read at two boundaries: the CMB is the historical leakage at the recombination surface; the cosmological residue is the integrated leakage across all AS-instants. The Leakage Theorem (3.1) forbids closure of either reading.