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000027 | Geometric Self Consistency on the Orientable Double Cover
000028 | Independent Note
000029 | September 18, 2025
000030 | 1 Geometric self consistency on the orientable double cover
000031 | 1.1 Starting point and notation
000032 | We use the invariants
000033 | 1 1 3 41
000034 | c = , ϕ = , δ = = 48c4, b = .
000035 | 3 8π tree 6π top 256π4 3 1 10
000036 | The cubic fixed point equation (CFE) is written in the normal form
000037 | b 1
000038 | α3 Aα2 Ac2κ(ϕ ) = 0, A = 2c3, κ(ϕ ) = 1 ln , (1)
000039 | − − 3 0 3 0 2π ϕ
000040 | 0
000041 | 󰀓 󰀔
000042 | with the baseline geometric fixpoint ϕold = ϕ +δ .
000043 | 0 tree top
000044 | 1.2 Backreaction on the double cover: derivation
000045 | The same electromagnetic energy that produces the logarithm in κ(ϕ ) couples back to the
000046 | 0
000047 | geometry that defines ϕ . On the orientable double cover M there are two identical boundary
000048 | 0
000049 | contributions. Consider the stationary total action S [α,ϕ] = S [ϕ]+U [α,ϕ], where we
000050 | tot geom EM
000051 | retain the ϕ dependent parts, 󰁩
000052 | 2
000053 | S [ϕ] = 6πϕ = 12πϕ, U [α,ϕ] = const ακ(ϕ),
000054 | geom EM
000055 | ·
000056 | s=1
000057 | 󰁛
000058 | with κ as in (1). Stationarity requires ∂ S = 0 and ∂ S = 0. The first condition reproduces
000059 | α tot ϕ tot
000060 | (1). Variation with respect to ϕ gives
000061 | ∂S ∂κ ∂κ b 1
000062 | tot 1
000063 | 0 = = 12π + const α , = . (2)
000064 | ∂ϕ · ∂ϕ ∂ϕ 2π ϕ
000065 | Linearisingaroundthebaselinevalueϕold = ϕ +δ andwritingϕ = ϕold+∆ϕ,theresponse
000066 | 0 tree top 0 0
000067 | ∆ϕ is proportional to α. By symmetry and additivity on the two sheets the topological addend
000068 | is reduced by αδ on each sheet. Hence
000069 | top
000070 | ϕnew(α) = ϕ +δ (1 2α) (3)
000071 | 0 tree top −
000072 | with no new free number. The factor two is fixed by the double cover.
000073 | Remark. Equation (3) is the minimal allowed correction built from the same ingredients that
000074 | define the CFE. It is the linear first order response in α. Higher geometric corrections with
000075 | unambiguous normalisation only shift the numerical effect slightly.
000076 | 1
000077 | 
000078 | ===== Page 2 of 2 =====
000079 | 1.3 Implicit solution of the CFE with ϕ (α)
000080 | 0
000081 | Insert (3) into (1) and solve implicitly for α. Numerically we obtain
000082 | α o−ld 1 = 137.036501464885822..., α n−e 1 w = 137.035990390121546...,
000083 | which lowers the deviation to CODATA 2022 from +3.665372 parts per million to 0.064121
000084 | −
000085 | parts per million. The associated geometric shift is ∆ϕ = 2αδ 1.7558 10 6, about
000086 | 0 top −
000087 | − ≈ − ×
000088 | thirty three parts per million relative on ϕ .
000089 | 0
000090 | The sensitivity from the implicit derivative of the cubic,
000091 | dα
000092 | =
000093 | Ac2
000094 | 3 2
000095 | b
000096 | π
000097 | 1
000098 | ϕ
000099 | 1
000100 | 0 ,
000101 | dϕ − 3α2 2Aα
000102 | 0
000103 | −
000104 | evaluated at the baseline, reproduces the observed shift to better than a few parts in ten to the
000105 | eight.
000106 | 1.4 Physical background
000107 | 1. Same cause, same effect. Thefieldenergythatdrivesthelogarithminκ(ϕ )couplesto
000108 | 0
000109 | the same geometry that sets ϕ . Self consistency therefore requires a response ∆ϕ α.
000110 | 0 0
000111 | ∝
000112 | 2. Double cover enforces a factor two. On M the two boundary sheets contribute
000113 | equally. The electromagnetic response adds up, which yields the factor two in (3).
000114 | 󰁩
000115 | 3. Parsimony. Neither c nor ϕ is altered, and δ remains a fixed topological invariant.
000116 | 3 tree top
000117 | Only its effective strength is slightly reduced by field energy on the two sheets.
000118 | 4. Side effects are tiny. Cosmic birefringence β = ϕ /(4π) changes by about eight micro
000119 | th 0
000120 | degrees, well below present uncertainties.
000121 | 1.5 Old versus new
000122 | Aspect Old: ϕ = ϕ +δ New: ϕ = ϕ +δ (1 2α)
000123 | 0 tree top 0 tree top
000124 | −
000125 | Extra number no no
000126 | Geometric structure constant double cover acts
000127 | Form of κ κ(ϕ ) κ(ϕ (α))
000128 | 0 0
000129 | Solution type direct implicit fixed point
000130 | α 1 137.0365014649 137.0359903901
000131 | −
000132 | Deviation to CODATA 2022 +3.665372 ppm 0.064121 ppm
000133 | −
000134 | Physical plausibility missing backreaction closed self consistency
000135 | Impact on other tests unchanged only micro degree in birefringence
000136 | 1.6 Conclusion
000137 | The form ϕ (α) = ϕ +δ (1 2α) closes the only open loop in the normal form. It is math-
000138 | 0 tree top
000139 | −
000140 | ematically minimal, physically required by the same energy that sources κ, and fully justified
000141 | by the orientable double cover. The resulting improvement in the prediction is already below