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Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/COMMON-COVER-COMPOSITUM.txt
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000001 | CANONICAL COMMON COVER OF THE OLD ROOT COORDINATE AND THE v117 SIGN LINES
000002 | ===========================================================================
000003 | 
000004 | 26 October 2026.  Exact finite coefficient and holomorphic transport result.
000005 | No Atlas, paper, ledger, or canonical contract was changed.
000006 | 
000007 | Verdict
000008 | =======
000009 | 
000010 | PROVED.  Let
000011 | 
000012 |   f(z)=(z^2-1)/(z^2+1),
000013 | 
000014 |   C: w^3=f(z),       eta=z(1-w^3),       eta^2=1-w^6,
000015 | 
000016 |   D: H^12=f(z),      w=H^4,              g=H^3,
000017 | 
000018 | and let Y_q -> C be the accepted unramified V4 cover which trivializes the
000019 | three actual negative coefficient lines
000020 | 
000021 |   h_a^2=(w+omega^a)/(w+omega^(a+1)),      h_0 h_1 h_2=1.
000022 | 
000023 | The normalization
000024 | 
000025 |   X = normalization(D times_C Y_q)                         (1)
000026 | 
000027 | is the minimal connected cover which simultaneously carries the original g
000028 | coordinate and trivializes all three v117 coefficient lines.  Its exact
000029 | invariants are
000030 | 
000031 |   [X:C]=16,       [X:P1_z]=48,
000032 | 
000033 |   Gal(X/C)=C4 times V4,
000034 | 
000035 |   Gal(X/P1_z)=C12 times_C3 A4 congruent A4 times C4,
000036 | 
000037 |   g(X)=41.                                                (2)
000038 | 
000039 | No extra cover or line character was selected.  Equation (1) is the
000040 | canonical normalization of the two already fixed function fields.
000041 | 
000042 | The actual geometric quarter lift also combines canonically on X, once its
000043 | accepted marked lift on Y_q is retained.  It still has order four.  It is
000044 | not the order twelve compiler clock from JOINT-CLOCK.txt: here the quarter
000045 | map inverts the C12 deck generator, whereas the compiler construction uses a
000046 | different root cover on which base quarter and deck commute.
000047 | 
000048 | 1. Existing results and source search
000049 | =====================================
000050 | 
000051 | The inputs are already established in:
000052 | 
000053 |   research-20261026-real-boundary/SIGN-LINE-COVER-DICTIONARY.txt
000054 |   SHA256 555f75a2d3c67347fa5f8e3c4f7f46e79d94cb4172176c84010224585dc842e7;
000055 | 
000056 |   research-20261026-real-boundary/CLOCK-Q-PACKET-COMPATIBILITY.txt
000057 |   SHA256 fe37f20bbb2d959dfe678ca1bf435a69067cfcb859a3d36769294bbe9bf61c32;
000058 | 
000059 |   paper-source:10bdafe2930b29584f91db06, lines 203-327,
000060 |   SHA256 574058876ebe23d693a9fce061f68da6fdec63419af78243ac9d457d7858f74b.
000061 | 
000062 | The September source constructs C, the signed permutation point transport,
000063 | and the geometric quarter map.  SIGN-LINE-COVER-DICTIONARY identifies Y_q
000064 | as the actual A4 kernel and proves that D does not by itself contain its
000065 | unramified sign lines.  CLOCK-Q-PACKET-COMPATIBILITY constructs the marked
000066 | order-four action on those lines.
000067 | 
000068 | A targeted Atlas and literal source search found the individual covers,
000069 | their earlier fiber products, and the separate quarter actions.  It did not
000070 | find the normalized compositum (1), the degree 48 group calculation, or the
000071 | genus 41 statement in an older source.  The result here is a direct
000072 | reconciliation of those existing fields.  It is not a new freely fitted
000073 | cover.
000074 | 
000075 | For comparison, the separate compiler cover contract is:
000076 | 
000077 |   experiments/theory-contracts/e-series-compiler-closure-20261023/COVERS.txt
000078 |   SHA256 cdd04f5f38819b60ddd8488eb1c3c316c6849829387205e78ec5e4bf767f80bf;
000079 | 
000080 |   experiments/theory-contracts/e-series-joint-source-20261023/JOINT-CLOCK.txt
000081 |   SHA256 02a15050dc0d8f7ad69ad1d867318d7a9abfa4bee274e14f1f06a12cf16c9628;
000082 | 
000083 |   experiments/theory-contracts/e-series-joint-source-20261023/
000084 |     COMPILER-CLOSURE.txt
000085 |   SHA256 de73510dc40b968940c68662ab07252995f96a98363e6cb1e9436937ecf62b3c.
000086 | 
000087 | 2. Why the two extensions have trivial intersection over C
000088 | ===========================================================
000089 | 
000090 | Over C the old root cover is simply
000091 | 
000092 |   D/C: H^4=w.                                             (3)
000093 | 
000094 | The function w has two simple zeros, at (w,eta)=(0,+1) and (0,-1), and two
000095 | simple poles, the two points at infinity of the even hyperelliptic model.
000096 | Thus (3) is a cyclic C4 cover totally ramified at exactly four points.
000097 | Its only proper nontrivial intermediate cover is the quadratic cover
000098 | obtained from H^2.  That cover is ramified at the same four valuations.
000099 | 
000100 | By contrast, Y_q/C is the connected unramified V4 cover defined by the
000101 | three two-torsion lines.  Let E be the intersection of the function fields
000102 | of D and Y_q over C.  Since E is a subfield of Y_q, E/C is unramified.  Since
000103 | it is also a subfield of the cyclic C4 extension D/C, it can only be C, the
000104 | ramified quadratic intermediate, or D.  The latter two are ramified.
000105 | Therefore
000106 | 
000107 |   C(D) intersect C(Y_q) = C(C).                         (4)
000108 | 
000109 | This proves linear disjointness, connectedness of the normalized fiber
000110 | product, and
000111 | 
000112 |   [X:C]=4 times 4=16,       Gal(X/C)=C4 times V4.       (5)
000113 | 
000114 | The argument uses ramification type rather than coprime degrees.  Both
000115 | relative covers have degree four, so a degree count alone would not prove
000116 | (4).
000117 | 
000118 | 3. Absolute group and the shared cubic clock
000119 | =============================================
000120 | 
000121 | Both D and Y_q are Galois over the z sphere:
000122 | 
000123 |   Gal(D/P1_z)=C12,
000124 |   Gal(Y_q/P1_z)=A4.
000125 | 
000126 | Their common intermediate C has
000127 | 
000128 |   Gal(C/P1_z)=C3.
000129 | 
000130 | The quotient C12 -> C3 is the action on w=H^4.  The quotient A4 -> C3 has
000131 | kernel V4 and is the same marked rotation w->omega w.  Therefore the deck
000132 | group of X over z is the fiber product
000133 | 
000134 |   G = C12 times_C3 A4,       |G|=12*12/3=48.            (6)
000135 | 
000136 | This is not C12 times A4 of order 144.  The common cubic motion is identified
000137 | once, rather than copied twice.
000138 | 
000139 | The Chinese remainder isomorphism C12 congruent C3 times C4 gives an exact
000140 | marked isomorphism
000141 | 
000142 |   G -> A4 times C4,
000143 |   (n,a) |-> (a,n mod 4),                                  (7)
000144 | 
000145 | where n mod 3 is the A4/V4 class of a.  Given (a,k), the inverse chooses the
000146 | unique n mod 12 satisfying
000147 | 
000148 |   n=q(a) mod 3,       n=k mod 4.
000149 | 
000150 | Thus
000151 | 
000152 |   Gal(X/P1_z) congruent A4 times C4.                     (8)
000153 | 
000154 | The C4 factor in (8) is the relative root deck of D/C.  It is not a second
000155 | family C3 clock and does not duplicate the A4/V4 quotient.
000156 | 
000157 | 4. Minimality in the requested object class
000158 | ============================================
000159 | 
000160 | The phrase “carries the original g” is already strong enough to require D.
000161 | Indeed, on C one has
000162 | 
000163 |   g=H^3,       w=H^4,       H=w/g.                      (9)
000164 | 
000165 | Hence C(C,g)=C(D).  A connected cover carrying w and the given g coordinate
000166 | therefore factors through D.
000167 | 
000168 | Likewise, a connected cover on which all three flat coefficient characters
000169 | become trivial has fundamental group contained in the intersection of their
000170 | kernels.  That kernel is precisely the connected V4 cover Y_q.  Equivalently,
000171 | the cover contains two independent Kummer roots h_0,h_1, with h_2 fixed by
000172 | h_0 h_1 h_2=1, and therefore factors through Y_q.
000173 | 
000174 | Any connected cover satisfying both requirements contains both function
000175 | fields.  It consequently contains their compositum and maps to X.  By (4)
000176 | its degree is at least 16 over C and 48 over z.  The normalization (1)
000177 | attains both bounds.  Minimality holds among all connected covers, not only
000178 | among Galois or cyclic candidates.
000179 | 
000180 | 5. Genus