A registered result in its source context
Original ledger classification: Numerical [N] (emergence confirmation on the real DtN/state) + premise [O] (selector is carrier input)
Recorded statement: NUMERICAL EMERGENCE SCAN (2026-06-15; the numerical sibling of QGEO.MARKS.03, as v210 is to v201): with the NUMBER of marks n AND their POSITIONS free, the 4-equally-spaced (square) configuration is the joint minimiser of {clock-invariance…
Statement, assumptions and sources
Paper context: tfpt research contracts
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QGEO.EMERGE.01
Original ledger classification: Numerical [N] (emergence confirmation on the real DtN/state) + premise [O] (selector is carrier input)
Original claim text
NUMERICAL EMERGENCE SCAN (2026-06-15; the numerical sibling of QGEO.MARKS.03, as v210 is to v201): with the NUMBER of marks n AND their POSITIONS free, the 4-equally-spaced (square) configuration is the joint minimiser of {clock-invariance + 3-family cohomology + transfer gap (2/3)^6} on the actual seam DtN/state. (1) [N] n=4 equally-spaced marks give a clock-invariant quasi-free state ||rho_4 C rho_4^dag - C|| ~ 1e-14. (2) [N] family count selects n=4: scanning n in {2..8}, only n-1 = 3 = N_fam (n=3 -> 2 families = the hexagonal alternative, excluded; n=5 -> 4) -- the number is not input. (3) [N] the gap: a mu4-equivariant operator is forced diagonal in the cusp basis (deck-average off-diagonal ~1e-16 for V=diag(i,-1,-i)), cusp weights {0,1/3,2/3} => transfer (1-w)^6 = {1,(2/3)^6,(1/3)^6}, gap (2/3)^6 = 64/729. (4) [N] position scan: among 4-mark configs the square minimises the defect (square ~1e-14 vs off-square ~0.4) -- the operator-level v200 theta=pi/2. (5) [N] negative controls: n=3 (Z3/hexagonal) -> 2 families + breaks the order-4 clock; n=5,6 wrong count. (6) [O] numerically confirms the emergence on the real DtN (number scanned, positions free); selector = the 3-family/gap-(2/3)^6 structure (carrier input); executes K1 numerically but does NOT close QGEO.REALIZE.01 (the order-4/N_fam input stays irreducible, v153/v181); complements v216. Python-only
Original classification detail
Numerical [N] + premise [O]: free-n + free-positions scan on the actual DtN/state confirms 4-square is the unique joint minimiser of {clock-invariance, 3 families, gap (2/3)^6}; executes K1 of QGEO.KILL.01 numerically, does NOT close QGEO.REALIZE.01; Python-only
Dependencies as recorded
QGEO.MARKS.03; QGEO.KILL.01; v200_seam_variational_scan.py; v210_mark_local_dtn.py; v216_marks_gauss_bonnet.py
Open the original ledger source, row 277Location recorded in the ledger
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- Research contracts
Sections that explicitly cite this claim
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- 2026-06-15 (QGEO: the four marks emerge from Gauss–Bonnet — K1 executed)Research changelog; TFPT — Introduction and reading guide
- Research Contract 1 — [formula in source]tfpt research contracts
- Front matter / supportResearch contracts
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
- Research Contract 1 — [formula in source]tfpt research contracts
Navigation through a shared verification script
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- Externalization Contract — SEAM.EQUIV.01 (the seam scaling-limit theorem, packaged for external specialists)tfpt research contracts
- Externalization Contract — SEAM.EQUIV.01 (the seam scaling-limit theorem, packaged for external specialists)Research contracts
- Externalization Contract — SEAM.EQUIV.01 (the seam scaling-limit theorem, packaged for external specialists)tfpt research contracts
Registered verification modules
- v217_marks_emergence_scan.pyRegistered verification module · registry
Read the description recorded in the module registry
the NUMERICAL emergence scan (QGEO.EMERGE.01, sibling of v216 as v210 is to v201): with the number of marks n AND their positions FREE, the 4-equally-spaced (square) config is the joint minimiser of {clock-invariance + 3-family cohomology + transfer gap (2/3)⁶} on the actual seam DtN/state. [N] n=4 equally-spaced ⟹ ‖ρ4 C ρ4†−C‖~1e-14 (clock-invariant state); [N] family count selects n=4 (scan n∈{2..8}, only n−1=3=N_fam; n=3→2 families = hexagonal, excluded; n=5→4); [N] gap: μ4-equivariant ⟹ diagonal in the cusp basis (deck-average off-diagonal ~1e-16), cusp weights {0,1/3,2/3} ⟹ transfer (1−w)⁶={1,(2/3)⁶,(1/3)⁶}, gap (2/3)⁶=64/729; [N] position scan: the square minimises the defect (square ~1e-14 vs off-square ~0.4); [N] negative controls n=3,5,6. [O] confirms the emergence on the real DtN (number scanned, positions free); selector = the 3-family/gap structure (carrier input); executes K1 numerically, does NOT close QGEO.REALIZE.01. Python-only.
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