=== STATUS === CORRIGENDUM (T284). ZERO new physics claims. Corrects a Lie-algebra branching misstated in three filed papers (five occurrences). T167 and T168: unaffected in every result. T169: the number 137 survives (its inputs are intrinsic algebra facts), but its photon-environment JUSTIFICATION now requires an explicit exclusion rule it does not supply — a gap NAMED, not resolved. The phrase "exact and canonical in E8 Lie theory" is RETRACTED. Provenance: raised provisionally in T283 §7; confirmed and extended here. Tags: CORRECTION / UNAFFECTED / NAMED-GAP / NOT CLAIMED. === DESCRIPTION (abstract) === Photonic Universe Hypothesis (PUH) — Corrigendum. THE ERROR. The decomposition of E8 under E7 × U (1) is misstated in three filed papers, in five places: T167 (eqs. 1 and 6, the latter in its Proposition 2), T168 (eqs. 2 and 5, the latter in its Proposition 1), and T169 (eq. 3, in its Definition 1). In every instance it reads 248 = (133, 0) ⊕ (1, 0) ⊕ (56, +1) ⊕ (56, −1). The dimensions sum to 133 + 1 + 56 + 56 = 246, not 248. Two states are omitted. NOT A TRANSCRIPTION ARTIFACT. Three features establish the error is in the papers: it recurs five times, identically, across three separate documents; T167 accompanies it with an explicit bullet enumeration of exactly the three (incomplete) sector types; and T167 asserts of it, "This decomposition is exact and canonical in E8 Lie theory. " The enumeration and the assertion are themselves part of the error. That phrase is RETRACTED. THE CORRECTION. Via the standard route (Slansky 1981): E8 ⊃ E7 × SU (2) gives 248 = (133, 1) + (1, 3) + (56, 2) = 133 + 3 + 112. Restricting SU (2) to its U (1) Cartan splits the SU (2) adjoint (1, 3) into 1+₂ + 1₀ + 1−₂, and the doublet (56, 2) into 56+₁ + 56−₁. Hence: 248 = 133₀ ⊕ 1₀ ⊕ 1+₂ ⊕ 1−₂ ⊕ 56+₁ ⊕ 56−₁ = 133 + 1 + 1 + 1 + 56 + 56 = 248. The filed papers keep the Cartan generator 1₀ (the photon) and DROP the raising and lowering operators 1±₂ that accompany it — the three together forming the adjoint of the SU (2) in E8 ⊃ SU (2) × E7. The omitted pair are E7 singlets carrying twice the U (1) charge of the (56, ±1) sector. T167 — UNAFFECTED. I ts Theorem 1 (photon = the (1, 0) U (1) Cartan generator) survives: that generator is present and unchanged. Its rank decomposition 8 = 4grav + 1EM + 3colour is a statement about the CARTAN SUBALGEBRA — a different object from the 248 adjoint decomposition — and is untouched. (This matters: that rank decomposition is what forces the 4-component Dirac spinor in T203, and hence the Clifford structure from which T283 derives the spin algebra. None of that chain is affected. ) The graviton (shape fluctuation of the degree-2 Casimir) does not use the branching. T168 — UNAFFECTED. Its mediating sector, the (56, ±1), is present and unchanged. (Its uniqueness claim survives, but now on the strength of the E7 index: the restored 1±₂ also carry U (1) charge, but are E7 singlets, so the (56, ±1) remain the unique bifundamentals. ) Its normalisation N = 1/540 = T (56) /h∨ (E7) × 1/h∨ (E8) = (1/18) × (1/30) is built from the Dynkin index and the dual Coxeter numbers — no state count enters — as T168's own addendum (10. 5281/zenodo. 19425304) independently confirms. T169 — THE NUMBER SURVIVES; THE JUSTIFICATION HAS A NAMED GAP. The formula 1/α₀ = dim (E7) + rank (E8) /2 = 133 + 4 = 137 uses only dim (E7) = 133 and rank (E8) = 8 — intrinsic facts about the algebras, not counts of states in the branching. The value 137 stands, as does 137. 036 = 137 (bare) + 0. 036 (QED vacuum polarisation). BUT T169's Definition 1 introduces the photon's "E8 neighbourhood" BY MEANS OF the incomplete decomposition, and partitions it into: Sector A (the (133, 0) E7 gauge adjoint), the (56, ±1) matter fields (the coupling channel), and Sector B (the four graviton-polarisation Cartan generators, imported from the rank decomposition). The two omitted states fall into NONE of these: they are E7 singlets (not in the adjoint), not in the 56, and not Cartan generators. T169's taxonomy has no slot for them and supplies NO RULE excluding them from the photon's environment — it could not, not knowing they were there. Had they been counted, the sum would be 133 + 4 + 2 = 139, and the derivation would not produce 137. Since α = 1/137. 036 is an experimental fact, the exclusion must be CORRECT; what T169 lacks is a STATED REASON for it. A plausible candidate exists (the neighbourhood is counted over the E7 GAUGE sector and the GRAVITON sector, and the charged singlets of the broken SU (2) belong to neither) — but T169 does not say this, and this corrigendum does not adopt it: supplying a justification would be a new claim, and this note makes none. THE GAP IS NAMED, NOT RESOLVED, and recorded as an open item against T169: the photon-environment count requires an explicit exclusion rule for the 1±₂ states. The number is not in doubt; the derivation's completeness is. CORRECTED TEXT for any future revision: wherever the four-term equation appears, it should read 248 = 133₀ ⊕ 1₀ ⊕ 1+₂ ⊕ 1−₂ ⊕ 56+₁ ⊕ 56−₁. T167's enumeration should gain a fourth entry for the 1±₂. The "exact and canonical" sentence should be deleted or applied to the corrected form. T169's Definition 1 additionally requires the exclusion rule above. NOT CLAIMED: any new physics; that α₀ = 1/137 is wrong (it is not — the number survives) ; that T169 is falsified (it is not — its stated justification is incomplete, a lesser and repairable thing) ; that the exclusion rule is hereby supplied (it is NOT — naming the gap is the whole of the claim) ; or that any other PUH result depends on the omitted states — in particular the rank decomposition 8 = 4grav + 1EM + 3colour and everything downstream of it (T203's spinor structure, T283's spin algebra) is untouched.
Brian Martell (Sun,) studied this question.