The Arithmetic of §10

Inputs: α = 1/137.035999177 (CODATA 2022); G = 6.67430(15) × 10⁻¹¹, relative 2.2 × 10⁻⁵; the age 13.787 ± 0.020 Gyr (Planck 2018); the corpus's realised G = 6.6719 × 10⁻¹¹ (−0.036%); the corpus's cycle 13.830 Gyr (+0.31%); the proton row 1836.152673445 against 1836.152673426(32), +0.59σ. Every figure below was recomputed from the formulas at a second desk, not carried across.

Road Three. α from the age: α = [(21/18)·τ_C/T]^(1/18), so δα/α = δT/(18T). Planck's 0.145% gives 80 ppm. The corpus's cycle sits 0.311% above the measured age, so α inferred from the measured age sits 0.311%/18 = 173 ppm above the measured α. The proton row: dD/dα = 21 in the leading term, so one sigma (32 × 10⁻⁹) needs δα/α = 2.1 × 10⁻⁷, 0.2 ppm. 80/0.2 = 400.

Road Four. α from G: the gravitational object goes as α²¹, so δα/α = δG/(21G). CODATA's 22 ppm gives 1 ppm. The realised G sits 357 ppm below measured, so α inferred from measured G sits 357/21 = 17 ppm above the measured α. On CODATA's bar the G row is −15.9σ; the same 17 ppm in α's units.

Road Five. Eliminating α: G goes as T^(−21/18). Fed the measured age, q_T = α(1 + 173 ppm), and the returned G is 6.696 × 10⁻¹¹, +0.33% on measured; on the age's Planck bar (0.145% × 21/18 = 0.17%) that is 1.9σ; on KS-STRETCH.3's lane (±0.66 Gyr, 4.8%) it is 0.06σ. Fed measured G, q_G = α(1 + 17 ppm) and the returned age is 13.826 Gyr, +0.28% on measured, the age row unchanged.

Road Six. C(q) = 21q/(1 − 16q/1836); D(q) = 1836 + C(q) − q/4π; D(α) = 1836.152673445, +0.59σ. The series form 1836 + 21α(1 − 1/84π) + α²·21·16/1836 = 1836.152673444.

Road Two. The event graph K₇,₃ has adjacency spectral radius √21 = 4.583; the held graph K₆,₃ has √18 = 4.243; reciprocals 0.218 and 0.236; the line-graph radii are m+n−2, so 8 and 7, reciprocals 0.125 and 0.143. None is near 0.0073.

Road Nine, the composition. Write the six faces as G, c, s, α, mₑ, t. AP10 §5.1's axis pairing is the involution (G c)(s α)(mₑ t). AP45 §6's balance pairing is (mₑ G)(α c)(s t). Their composition is (G α t)(c mₑ s): two three-cycles, not one six-cycle. Each closes, which is what the argument needs.

The withdrawn bound, and why it fails. For a polynomial with coefficients a₀ … aₙ, every root satisfies |x| ≤ 1 + max|aᵢ|/|aₙ| for i < n, and |x| ≥ |a₀|/(|a₀| + max|aᵢ|) for i > 0. Both bounds are of order one only when every coefficient is. The picture’s coefficient list contains 1836, and with a₀ = 1 the lower edge is then 1/1837 = 5.44 × 10⁻⁴, below α = 7.297 × 10⁻³; with 84π it is 3.78 × 10⁻³, still below. The annulus admits α and the bound excludes nothing. The bound excludes α whenever the largest coefficient is below 1/α − 1 = 136.04 — at 84 the edge is 1.18 × 10⁻² and at 21 it is 4.5 × 10⁻², both above α. Of the picture’s own numbers only 1836 and 84π defeat it, and the picture’s list contains 1836.

The field, counted. Take as coefficients the picture’s counts 1, 2, 3, 4, 6, 16, 18, 21, 84, 1836 and π, together with all their pairwise products and ratios — a pool of 135 values. Over quadratics alone, with either sign on the linear and constant terms, that pool yields 5,883 positive roots within one per cent of α, 35 within a hundred parts per million, and 2 within one. Normalising each quadratic by its leading coefficient and comparing exactly — every pool element carried as a rational times a power of pi, never as a float — collapses those to 3,763, 17 and 1 distinct equations. The pair inside one part per million is one quadratic and eight times it. A float comparison gives 3,765 in the widest band; the exact figure is the one printed, and the dedup rule is stated here so the count can be reproduced rather than trusted. Cubics and higher were not enumerated and would only enlarge the field. The closest single root sits 0.64 parts per million from α, outside the proton row’s bar of 0.21 ppm; no root in the enumeration falls inside that bar. This is the measured look-elsewhere denominator of §11, and the reason a landing carries no information.

What the five closures do say. Ratios of two single counts: 16/1836 = 0.008715 is α × 1.194; 1/84π = 0.003789 is α × 0.519; 6/21 is α × 39. No such ratio is at α. The five roads as built used coefficients of exactly this kind, which is why each returned a size of order one rather than α’s — a fact about those five constructions and not about the class.

The per-cycle ledger, checked. One window makes α⁻¹⁸ cuts per cycle at α-worth each, α⁻¹⁷ of possibility spent, against one grain shed per cycle. Setting α⁻¹⁷ equal to a structural number N gives α = N^(−1/17), which is 0.84 at N = 21 and 0.64 at N = 1836. For the root to be at α, N would have to be 2.1 × 10³⁶. No count in the picture is within thirty orders of magnitude of that, which is the whole of Road Twelve’s closure and does not rest on the withdrawn bound.

§6. ΔE_G/ℏ with ΔE_G = Gmₑ²/ξ and ξ = ℏ/mₑc gives (Gmₑ²/ℏc)(mₑc²/ℏ) — the dimensionless gravitational object at one per Compton tick. Numerically the object is 1.752 × 10⁻⁴⁵ on measured G, and the tick's reciprocal is 7.76 × 10²⁰ per second. From G, mₑ, ℏ and c exactly one dimensionless quantity can be formed, which is why the agreement carries no information.