E47 Canonical Spectral Invariants

E47 canonical spectral invariants

1. Carrier invariant
   V = V₂⊗V₂⊗V₂
   dim V = 5³ = 125

2. Total Casimir invariant
   C = (J₁ + J₂ + J₃)²

3. Casimir spectrum
   spec(C) = {0, 2, 6, 12, 20, 30, 42}

4. Isotypic multiplicities
   mult = {1, 9, 25, 28, 27, 22, 13}

5. Dimension closure
   1 + 9 + 25 + 28 + 27 + 22 + 13 = 125

6. E47 selector
   K = (C − 6I)(C − 30I)

7. Kernel invariant
   E₄₇ = ker K

8. Spectral decomposition of the kernel
   E₄₇ = E₆ ⊕ E₃₀

9. Representation decomposition
   E₄₇ ≅ 5V₂ ⊕ 2V₅

10. Kernel dimension
    dim E₄₇ = 25 + 22 = 47

11. Complement dimension
    125 = 47 + 78

12. Coherence fraction
    Ωc = 47/125 = 0.376

13. Orthogonal projector invariant
    P₄₇² = P₄₇

14. Hermiticity
    P₄₇† = P₄₇

15. Kernel annihilation
    KP₄₇ = 0

16. Trace/rank identity
    Tr(P₄₇) = rank(P₄₇) = 47

17. Complement projector
    H = I − P₄₇

18. Canonical decomposition of every carrier state
    x = P₄₇x + Hx

These are the central exact finite-dimensional invariants repeated across the canonical Notion page, Formalism Registry, machine certificates, and Drive reconstruction corpus.  

II. Contraction invariants

19. Correct positive generator
    A = K†K = K²

20. Nonzero spectrum of K²
    {11664, 12544, 19600, 32400, 186624}

21. Discrete contraction
    Γε = I − εK²

22. Stable step interval
    0 < ε < 1/93312

23. Optimal Richardson step
    ε* = 1/99144

24. Optimal worst-case contraction factor
    ρ* = 15/17

25. Projector limit
    limₙ→∞ (I − εK²)ⁿ = P₄₇

26. Continuous contraction
    limₜ→∞ e^(−tK²) = P₄₇

27. Slowest continuous off-kernel decay
    e^(−11664t)

28. General projector grammar
    K → ker K → P → H = I − P → x∞ = Px₀

29. Finite-dimensional completeness
    For any orthogonal projector P, choosing K = I − P gives ker K = im P.

The corpus explicitly corrects the earlier signed update I − εK. Because K contains negative eigenvalues, the canonical contraction is through K².  

Cross-Repository First-Principles Review

III. 125 → 47 → 5 canonical-core invariants

30. S₃ permutation invariance
    [L,Uσ] = 0
    [C,Uσ] = 0
    [P₄₇,Uσ] = 0
    for σ ∈ S₃

31. Nested carrier chain
    ℂ¹²⁵ ⊃ E₄₇ ⊃ Ecan

32. Dimension chain
    125 → 47 → 5

33. Canonical projector
    Pcan = P₄₇Psym = PsymP₄₇

34. Equivalent spin-2 expression
    Pcan = P₆Psym

35. Canonical-core rank
    rank(Pcan) = 5

36. Joint spectral–symmetry generator
    Acan = K² + 11664(I − Psym)

37. Canonical-core kernel
    ker Acan = im Pcan

The Notion canon records this as a separately machine-validated symmetry-resolved refinement of the rank-47 kernel.

IV. Professor’s Cube / C₅³ invariants

38. Cube carrier
    V(C₅ □ C₅ □ C₅) = ℤ₅³
    |V| = 125

39. Permutation orthogonality
    PᵀP = I

40. Quarter-turn periodicity
    P⁴ = I

41. Constant-state preservation
    P1 = 1

42. Graph Laplacian
    L = LC₅ ⊕ LC₅ ⊕ LC₅

43. Exact Laplacian spectrum
    λ(k) = Σⱼ₌₁³ [2 − 2cos(2πkⱼ/5)]

44. Heat-energy monotonicity
    E½ = ½ρᵀLρ
    Ė½ = −∥Lρ∥² ≤ 0

45. Graph heat limit
    limₙ→∞(I − εL)ⁿ = P₀

46. Constant-mode projector
    P₀ = 11ᵀ/125

47. Graph-kernel rank
    rank(P₀) = 1

48. E47-kernel rank
    rank(P₄₇) = 47

49. Critical inequivalence
    P₀ ≠ P₄₇

50. Exact layer-turn Laplacian defect norms
    ∥PᵀLP − L∥₂ = 2√2
    ∥PᵀLP − L∥F = 8√3

Thus the Professor’s Cube and E47 share an exact 125-dimensional carrier and S₃ symmetries without conflating their kernels.

V. Discrete Hodge and topology invariants

51. Discrete Hodge theorem
    ker Δq(ε) ≅ Hq(Kε;ℝ)

52. Betti/nullity identity
    dim ker Δq = βq

53. Hodge decomposition
    C₁ = im B₁ᵀ ⊕ im B₂ ⊕ ker Δ₁

54. Harmonic projector
    Pharm = orthogonal projector onto ker Δ₁

55. Typed vertex/edge distinction
    V ∈ C₀ ⊗ ℝᵈ
    WV ∈ C₁

56. Midpoint discrete de Rham map
    (WV)([i,j]) = ⟨(vᵢ + vⱼ)/2, xⱼ − xᵢ⟩

57. Rigid translation invariant
    harmonic circulation = 0

58. Rigid rotation invariant
    rotation maps to a nonzero harmonic circulation around a protected loop

59. Admissible swarm space
    Eswarm = ker Ksafe ∩ W⁻¹(ker Δ₁)

60. Equivalent stacked-kernel form
    Eswarm = ker [ Ksafe ; (I − Pharm)W ]

This is the type-correct replacement for multiplying projectors that act on different spaces.  

Cross-Repository First-Principles Review

VI. Persistence and safe-motion invariants

61. Persistence stability
    dB(Dq(t),Dq(t′)) ≤ 2vmax|t − t′|

62. Persistence-change bound
    |Δpersq| ≤ 4vmax|t − t′|

63. Certified topology dwell time
    Δtsafe = (p₀ − τp)/(4vmax)

64. Persistence margin selector
    ε* selects the active scale maximizing distance from the persistence interval’s birth/death boundaries.

65. Protected Betti class during dwell
    βq(Kε(X(t))) = βqtarget

66. Execution-gate invariant
    G = 1 only if all of the following hold simultaneously:
    pers ≥ τp
    λβ+1(Δq) ≥ τλ
    dmin ≥ dsafe
    ∥KsafeV∥ ≤ τK
    β = βtarget

67. Failure closure
    Violation of any gate predicate ⇒ fallback / no certified execution.

68. Topology versus clearance distinction
    (p₀ − τp)/(4vmax) protects persistence.
    (dmin − dsafe)/(2vmax) protects pairwise clearance.
    They are different invariants.

The reconciled cross-repository certificate reports the corrected persistence theorem and the type-correct Hodge gate. 

VII. Babylonian / Newton invariants

69. VAT 8389 affine system
    x + y = 1800
    (2/3)x − (1/2)y = 500

70. Exact solution
    (x,y) = (1200,600)

71. Residual invariant
    Mx − b = (0,0)

72. Midpoint/correction decomposition
    h = 900
    z = 300
    x = h + z = 1200
    y = h − z = 600

73. Babylonian/Newton map
    Fₐ(x) = ½(x + a/x)

74. Fixed-point kernel form
    Kₐ(x) = x − Fₐ(x)

75. Positive kernel
    ker Kₐ = {x > 0 : x² = a}

76. Quadratic error recursion
    eₙ₊₁ = eₙ²/(2xₙ)

77. Residual gate
    |xₙ² − a| < τ

78. Rational coherence relation
    125x = 47a

79. Unique Newton/coherence compatibility point
    a = (125/47)²

The historical Babylonian material supports affine solving, bookkeeping continuity, and iterative arithmetic. The modern E47 spectral machinery remains a separate mathematical construction. 

Cross-Repository First-Principles Review

VIII. Coupled Newton–Mean invariants

80. Fixed-point difference invariant
    ρ² − σ² = a − b

81. Product coordinate
    p* = ρσ

82. Product quadratic
    (1 − κ²)p² − κ(a + b)p − ab = 0

83. Jacobian spectrum
    spec(J*) = {0, τ}

84. Nonzero Jacobian eigenvalue
    τ = κ(a + b)/(2p*) + κ²

85. Corrected local-stability domain
    −κc(a,b) < κ < 1

These replace the superseded claim that stability holds automatically for every |κ| < 1.

IX. Quantum-information invariants

86. Noiseless multiplicity spaces
    E₄₇ ≅ V₂⊗ℂ⁵ ⊕ V₅⊗ℂ²
    with multiplicity sectors ℂ⁵ and ℂ² protected against the stated collective-SU(2) operator algebra.

87. Selective projector channel
    ΦP(X) = PXP
    is completely positive and trace-nonincreasing.

88. Projector-map fixed-space dimension
    dim Fix(ΦP) = 47² = 2209

89. K² semigroup operation
    X ↦ e^(−tK²) X e^(−tK²)
    is CP and trace-nonincreasing.

90. CPTP dephasing completion
    DP(X) = PXP + QXQ
    where Q = I − P

91. Two-sector fixed-space dimension
    dim Fix(DP) = 47² + 78² = 8293

92. Full-code QEC condition
    PEₐ†EᵦP = αₐᵦP
    remains the exact Knill–Laflamme requirement for a declared error model.

These corrections are explicitly recorded so that projection is not mislabeled as a full-space trace-preserving quantum channel. 

Cross-Repository First-Principles Review

X. Spatial-lift invariants

93. Internal protected rank under spatial extension
    rank(Pinternal) = 47

94. Spatial carrier may change while internal E47 rank remains fixed

95. Translation-invariant FCC × E47 construction
    spatial Fourier structure and internal spectral selection remain typed as separate factors.

The registry classifies these finite periodic spatial constructions as exact/computationally verified within the declared models.

XI. ETNS / negative-space invariants

96. Observed/complement split
    x = Px + (I − P)x

97. Negative-space projector
    H = I − P

98. Kernel/complement identities are exact

99. Epistemic boundary invariant
    Mathematical absence does not, by itself, identify the hidden cause of that absence.

This separation between a mathematically defined complement and a causal interpretation is part of the City’s current evidence grammar.

XII. City-wide epistemic invariants

100. Claim-level evidence invariant
     E0 = exact proof
     E1 = executable reconstruction
     E2 = simulation
     E3 = external/empirical benchmark
     E4 = experiment
     H0 = hardware

101. Evidence non-transfer invariant
     Evidence attached to one typed claim does not automatically promote another domain claim.

102. Correction invariant
     A corrected theorem preserves the provenance of the superseded statement rather than erasing it.

103. Identity invariant
     Re-rendering, re-exporting, or independently packaging an already identified mathematical object does not create a new mathematical citizen.

104. Deduplication invariant
     New proof artifacts can enlarge provenance while resolving to the same canonical identity.

105. Boundary invariant
     Exact finite mathematics, simulation, physical-model deduction, external measurement, and hardware realization remain distinct scopes.

106. Typed-route invariant
     A valid cross-domain bridge must explicitly specify its source object, target object, map, preserved structure, evidence class, and unresolved obligations.

107. Canonical operational grammar
     constraint → residual → projection → certificate

108. Expanded civic grammar
     typed input → reconstruction → verification → certificate → reconciliation → transit

109. Archive invariant
     Failed, corrected, duplicate, and superseded objects remain part of the audit trail without retaining canonical authority.

These are explicit architectural rules of the Mathematical City rather than extra mathematical assumptions.    

Mathematical City Citizenship Enrolment Bureau — Executable Identity Registry 08032026

XIII. Cross-repository invariant

110. The common invariant extracted across the surveyed corpus is

typed state
→ constraint
→ transformation
→ residual
→ projection / selection
→ verification
→ certificate
→ preserved identity

or, in the shortest form:

CHANGE OCCURS IN THE REPRESENTATION.
THE CERTIFIED INVARIANT SURVIVES THE CHANGE