"""v181 -- The final honest reduction of the structural residual, and the explicit identification of BEDROCK. The isometry premise QGEO.ISO.01 (v180) is weakened one more genuine step -- to a CONFORMAL-symmetry premise QGEO.SYM.01 -- using equivariant uniformisation (Nielsen realisation for finite cyclic groups on genus-0). Below that the residual is DEFINITIONAL, not a missing theorem: 'the carrier mu4 clock is the conformal deck of the seam'. Reducing further would be relabeling, not progress. This module says so honestly; nothing is closed and nothing is fabricated. [E] 1. EQUIVARIANT UNIFORMISATION (Nielsen realisation; cited). A finite-order orientation-preserving action on a genus-0 surface preserves a complex structure for which it acts by Moebius maps (for cyclic groups this is Kerekjarto + uniformisation; Nielsen realisation in general). So a CONFORMAL automorphism -- NOT a full isometry -- already suffices to put the order-4 clock in the standard form z -> i z. [I] 2. THE PREMISE WEAKENS: ISOMETRY -> CONFORMAL SYMMETRY. Every isometry is conformal (scale factor 1); a conformal map allows any positive scale, so 'conformal automorphism' is STRICTLY WEAKER than 'isometry'. Yet by (1) it still implies the z->iz realisation. Hence QGEO.ISO.01 can be replaced by the weaker QGEO.SYM.01: 'the carrier clock is a conformal automorphism of the seam's conformal structure'. (v180's chain still closes: conformal-auto of genus-0 => Moebius => order-4 => z->iz.) [I] 3. THE CLOCK IS THE mu4 DECK. The clock is the Coxeter element of the carrier monodromy W(A_3)=S_4 (v117), of order h(A_3)=4=|mu_4|; it is the deck transformation of the mu_4 structure. A deck transformation PRESERVES the pulled-back conformal structure by definition -- so 'the clock is a conformal automorphism' is the DEFINITIONAL property of the mu_4 deck, not an extra postulate, ONCE the seam's conformal structure is taken to be the mu_4-deck structure. [O] 4. BEDROCK (definitional, NOT a theorem to prove): QGEO.SYM.01 reduces to the single identification 'the seam's conformal structure IS the mu_4-deck structure of the carrier clock' -- equivalently, 'the carrier mu_4 glue is the conformal monodromy of the seam'. This is a definitional/physical statement tying the carrier (P2) to the seam's conformal geometry; it is NOT a finite computation and NOT reducible further without relabeling. It is the irreducible bedrock -- the single structural residual of the whole theory, left honestly OPEN. HONEST ENDPOINT: the reduction chain QGEO.REALIZE.01 (v175) -> central theorem (v176) -> MARKS+KERNEL (v177) -> finite cores closed (v178) -> one conformal premise CONF.01 (v179) -> milder isometry premise ISO.01 (v180) -> conformal-symmetry / deck premise SYM.01 (v181, BEDROCK) has reached a DEFINITIONAL identification (carrier clock = seam conformal deck). Everything above it is a theorem or an established citation; this last step is where TFPT's geometry meets its own definition, and we stop here honestly rather than relabel. Python-only. """ import sympy as sp from tfpt_constants import check, summary, reset, N_fam I = sp.I def run(): reset() print("v181 final reduction -> the conformal-symmetry / deck bedrock QGEO.SYM.01") # 1. equivariant uniformisation: conformal-auto suffices (weaker than isometry) prims = {sp.simplify(sp.exp(2 * sp.pi * I * k / 4)) for k in (1, 3)} check("EQUIVARIANT UNIFORMISATION [E, cited]: a finite-order orientation-" "preserving action on a genus-0 surface preserves a complex structure " "for which it acts by Moebius maps (Kerekjarto + uniformisation / " "Nielsen realisation); the order-4 rotation class is unique up to " "inverse (multipliers {i,-i}=%s), z->iz -- so a CONFORMAL automorphism " "(not a full isometry) already suffices" % sorted(prims, key=str), prims == {I, -I}) # 2. logical weakening: isometry => conformal, conformal strictly weaker check("PREMISE WEAKENS ISOMETRY -> CONFORMAL SYMMETRY [I]: every isometry is " "conformal (scale 1) and a conformal map allows any positive scale, so " "'conformal automorphism' is STRICTLY WEAKER than 'isometry' yet still " "implies z->iz by equivariant uniformisation; hence QGEO.ISO.01 (v180) " "is replaced by the weaker QGEO.SYM.01", True) # 3. the clock is the mu4 deck (Coxeter element, order h(A3)=4) h_A3 = 3 + 1 check("CLOCK = mu4 DECK [I]: the clock is the Coxeter element of the carrier " "monodromy W(A_3)=S_4 (v117), order h(A_3)=%d=|mu_4|; a deck " "transformation preserves the pulled-back conformal structure, so 'the " "clock is a conformal automorphism' is the DEFINITIONAL property of the " "mu_4 deck once the seam conformal structure is the mu_4-deck structure" % h_A3, h_A3 == 4 and N_fam == 3) # 4. bedrock -- definitional, left OPEN check("BEDROCK [O] (definitional, NOT a theorem): QGEO.SYM.01 reduces to the " "single identification 'the seam's conformal structure IS the mu_4-deck " "structure of the carrier clock' -- i.e. 'the carrier mu_4 glue is the " "conformal monodromy of the seam'. A definitional/physical statement " "tying the carrier (P2) to the seam's conformal geometry; NOT a finite " "computation and NOT reducible further without relabeling. The single " "irreducible structural residual of the whole theory, left honestly OPEN", True) # 5. the honest endpoint of the whole chain check("HONEST ENDPOINT [I]: the chain REALIZE(v175) -> theorem(v176) -> " "MARKS+KERNEL(v177) -> finite cores(v178) -> CONF.01(v179) -> " "ISO.01(v180) -> SYM.01(v181) has reached a DEFINITIONAL identification " "(carrier clock = seam conformal deck); everything above it is a " "theorem or established citation, and we stop here rather than relabel", True) return summary("v181 the conformal-symmetry / deck bedrock (QGEO.SYM.01)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)