Recorded data and provenance for the associated calculation.
| experiment | smooth-cubic-mass-and-coupling-candidate |
|---|---|
| scope | One six-depth algebra, retained-tower, frozen-field, and conditional parent-embedding decision batch. The relative cubic coefficient remains an input and the old spatial binding is unchanged. |
View complete data
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"analytic_result": "H=Tr[C(C-(Tr C)/r I)^2] is polynomial, zero with zero occupied source on rank-r coherent current. For a prepared two-ray current it is exactly r(rho+sigma)rho sigma(1-s^k), and a fixed-trace partner split costs 2rho^3 epsilon^2. H is nonnegative for C>=0 but sign-indefinite for general Hermitian C. The linear trace mass term and cubic H carry distinct coefficients unless an origin fixes their relative scale.",
"smooth_nonnegative_boundary": "H4=Tr[C^2(C-(Tr C)/r I)^2] is nonnegative for every Hermitian C and is exactly 2r(rho sigma)^2(1-s^k)^2 on a prepared pair. More generally, if a smooth globally nonnegative correction vanishes on every coherent rank-r current, its gradient and Hessian along tangent coherent-ray rotations vanish there. A prepared pair has the same first-order tangent, so its angular excess is o(theta^2). A positive quadratic pair stiffness in this construction requires a correction that is sign-indefinite off the positive-current cone or a different coherent self law.",
"scope": "Matrix-current algebra and complete noncommuting differential; no full spatial solution, collision, or cross-depth embedding."
},
"retained_tower": {
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"scope": "Complete grade-two and grade-one retained mass-current maps at one spatial point, all six maintained depths. The dimensionless cubic coefficient is set to one only for algebraic checking; its physical scale is not derived."
},
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{
"depth": 6,
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"strength": 1.0,
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"max_imaginary_frequency": 8.172715976195732e-15
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{
"depth": 6,
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{
"depth": 6,
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"momentum": 10.0,
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{
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{
"depth": 6,
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},
{
"depth": 6,
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}
],
"maxima": [
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"strength": 0.05,
"max_imaginary_frequency": 0.0
},
{
"depth": 2,
"strength": 0.25,
"max_imaginary_frequency": 0.0
},
{
"depth": 2,
"strength": 1.0,
"max_imaginary_frequency": 0.9807603839918115
},
{
"depth": 4,
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"max_imaginary_frequency": 3.555092147316391e-15
},
{
"depth": 4,
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"max_imaginary_frequency": 4.630193797816595e-15
},
{
"depth": 4,
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{
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},
{
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{
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],
"scope": "Complete real-linear constant-coefficient Dirac/spinor/partner control at depth two, four, six. This does not test a nonconstant spatial core, high oscillator grades, or cross-depth coupling."
},
"parent_embedding": {
"experiment": "smooth-cubic-parent-embedding",
"rows": [
{
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"parent_depth": 3,
"lower_rank": 2,
"parent_rank": 2,
"isolated_parent_excess_at_rho_one": 0.0,
"parent_excess_per_trace_at_rho_one": 0.0,
"occupied_source_shift_at_rho_one": 0.0,
"rotation_error": 7.703719777548945e-33
},
{
"lower_depth": 2,
"parent_depth": 4,
"lower_rank": 2,
"parent_rank": 4,
"isolated_parent_excess_at_rho_one": 0.5,
"parent_excess_per_trace_at_rho_one": 0.25,
"occupied_source_shift_at_rho_one": 0.75,
"rotation_error": 4.440892098500626e-16
},
{
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"occupied_source_shift_at_rho_one": 0.75,
"rotation_error": 0.0
},
{
"lower_depth": 2,
"parent_depth": 6,
"lower_rank": 2,
"parent_rank": 8,
"isolated_parent_excess_at_rho_one": 1.125,
"parent_excess_per_trace_at_rho_one": 0.5625,
"occupied_source_shift_at_rho_one": 1.6875,
"rotation_error": 4.440892098500626e-16
},
{
"lower_depth": 2,
"parent_depth": 10,
"lower_rank": 2,
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"isolated_parent_excess_at_rho_one": 1.7578125,
"parent_excess_per_trace_at_rho_one": 0.87890625,
"occupied_source_shift_at_rho_one": 2.63671875,
"rotation_error": 8.881784197001252e-16
},
{
"lower_depth": 3,
"parent_depth": 4,
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"isolated_parent_excess_at_rho_one": 0.5,
"parent_excess_per_trace_at_rho_one": 0.25,
"occupied_source_shift_at_rho_one": 0.75,
"rotation_error": 2.220446049250313e-16
},
{
"lower_depth": 3,
"parent_depth": 5,
"lower_rank": 2,
"parent_rank": 4,
"isolated_parent_excess_at_rho_one": 0.5,
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"occupied_source_shift_at_rho_one": 0.75,
"rotation_error": 0.0
},
{
"lower_depth": 3,
"parent_depth": 6,
"lower_rank": 2,
"parent_rank": 8,
"isolated_parent_excess_at_rho_one": 1.125,
"parent_excess_per_trace_at_rho_one": 0.5625,
"occupied_source_shift_at_rho_one": 1.6875,
"rotation_error": 4.440892098500626e-16
},
{
"lower_depth": 3,
"parent_depth": 10,
"lower_rank": 2,
"parent_rank": 32,
"isolated_parent_excess_at_rho_one": 1.7578125,
"parent_excess_per_trace_at_rho_one": 0.87890625,
"occupied_source_shift_at_rho_one": 2.63671875,
"rotation_error": 8.881784197001252e-16
},
{
"lower_depth": 4,
"parent_depth": 5,
"lower_rank": 4,
"parent_rank": 4,
"isolated_parent_excess_at_rho_one": 0.0,
"parent_excess_per_trace_at_rho_one": 0.0,
"occupied_source_shift_at_rho_one": 0.0,
"rotation_error": 8.334702575578817e-31
},
{
"lower_depth": 4,
"parent_depth": 6,
"lower_rank": 4,
"parent_rank": 8,
"isolated_parent_excess_at_rho_one": 1.0,
"parent_excess_per_trace_at_rho_one": 0.25,
"occupied_source_shift_at_rho_one": 0.75,
"rotation_error": 3.3306690738754696e-16
},
{
"lower_depth": 4,
"parent_depth": 10,
"lower_rank": 4,
"parent_rank": 32,
"isolated_parent_excess_at_rho_one": 3.0625,
"parent_excess_per_trace_at_rho_one": 0.765625,
"occupied_source_shift_at_rho_one": 2.296875,
"rotation_error": 8.881784197001252e-16
},
{
"lower_depth": 5,
"parent_depth": 6,
"lower_rank": 4,
"parent_rank": 8,
"isolated_parent_excess_at_rho_one": 1.0,
"parent_excess_per_trace_at_rho_one": 0.25,
"occupied_source_shift_at_rho_one": 0.75,
"rotation_error": 4.440892098500626e-16
},
{
"lower_depth": 5,
"parent_depth": 10,
"lower_rank": 4,
"parent_rank": 32,
"isolated_parent_excess_at_rho_one": 3.0625,
"parent_excess_per_trace_at_rho_one": 0.765625,
"occupied_source_shift_at_rho_one": 2.296875,
"rotation_error": 0.0
},
{
"lower_depth": 6,
"parent_depth": 10,
"lower_rank": 8,
"parent_rank": 32,
"isolated_parent_excess_at_rho_one": 4.5,
"parent_excess_per_trace_at_rho_one": 0.5625,
"occupied_source_shift_at_rho_one": 1.6875,
"rotation_error": 0.0
}
],
"analytic_result": "For a rank-s positive lower-sector current embedded by zero padding in a parent rank-r current, H_parent=s*rho^3(1-s/r)^2 and its occupied gradient is 3*rho^2(1-s/r)^2. Both are nonzero whenever s<r, so parent excess terms change the isolated lower-sector mass source unless the embedding or action has further structure. This is a conditional zero-padding embedding, not a derived physical nested-sector map.",
"scope": "All nested pairs from depths 2,3,4,5,6,10 using their representation ranks; no spatial embedding or cross-depth scattering."
}
},
"component_sha256": {
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"scope": "One six-depth algebra, retained-tower, frozen-field, and conditional parent-embedding decision batch. The relative cubic coefficient remains an input and the old spatial binding is unchanged.",
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