Established in scope
The actual fourth-order frame has a phase-sensitive sixty-mode dual
In the question: Common clock sectors, quartic geometry and operational reconstruction
Finite duality of the same supplied frame; no physical sixty-site world or coupling. Only the magnitude replacement is excluded,not all classical/nonlinear self-organization. Central J acts trivially in quartic degree.
Read the exact result, assumptions and evidence
Result
Columns z_a^tensor4 define F and K=F*F with spectrum3^5,(3/2)^30,0^25. F/sqrt3 literally identifies its top five with the native code. Each row contains1 once,1/16 thirty-two times,−1/4 twelve times and0 fifteen times. Actual triangle0,1,2 has invariant product−1/64. Magnitude replacement |K| instead has a unique Perron mode at6;|K|² is strictly positive.
Required assumptions
- Existing even E8 lattice, Gaussian J and sign lifts
Given existing even unimodular E8 with actual J(x0,x1,...)=(-x1,x0,...) and J²=-I, and the original ordered sign lattice cocycle. Sign automorphisms have kernel Hom(L,F2). Retain original integral ordered basis and generator phases; no enlargement to U1 characters except explicitly defined scalar pushouts.
- Actual clock/complex-reflection subgroup and native quartic space
R=diag(-1,1,1,1) on Gaussian C4 is the complex reflection of root line(1+i)e0; G0=<J,R>=C4×C2. P5 is the existing five-dimensional symmetric quartic subspace in Sym4(V4), W6=exterior²V4. R is not a real codim1 E8 Weyl reflection; P5 is not the full fourth tensor power.
- Symmetric quartic and actual erasure intertwiner
Use Sym^4 of the native four-representation, the actual five codeword sums, and the existing one-register erasure intertwiner. The symmetric tensor rule and its physical selection are not silently inferred from the full tensor product.
Original evidence
- COLLECTIVE-DYNAMICS.txt lines 1–130
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/collective-native-marginal-origin-20261026/COLLECTIVE-DYNAMICS.txt
ea4eaebb6d70a81df5d6adf0f570a4bbfcd021429712117b0dde96cd073d761b
- check_collective_dynamics.py lines 1–120
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/collective-native-marginal-origin-20261026/check_collective_dynamics.py
b2a4d111830a2fc85489d0da033eef1bd1feb464edbada38231700db4e128dc6
- dynamics_results.json lines 1–99
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/collective-native-marginal-origin-20261026/dynamics_results.json
4d6ab1df704cb1ace891a30a87e4cad61317265cf7cfaa5ff4e704dbb78a4fd0
finding:native-fourth-root-frame-has-signed-collective-sixty-mode-dual