The trace-free (unimodular) form of Einstein's equations is exactly blind to vacuum energy: shifting T_μν → T_μν + ρ_vac·g_μν leaves all 16 components of the trace-free field equation IDENTICALLY zero on a fully general metric — while the standard form picks up a residual of −κρ_vac·g_μν. In FRW the trace-free equation reduces to Ḣ = −4πG(ρ+p), which depends on matter only through (ρ+p), so a vacuum with p = −ρ contributes nothing at all.

lassical split into 'pure trace' and 'trace-free' parts is
        therefore scheme-dependent once loops are included, which is
        precisely the split the immunity theorem relies on.
    (3) In canonical unimodular formulations Lambda becomes conjugate
        to a global time variable, so quantum mechanically it can
        FLUCTUATE rather than being a sharp constant (Henneaux &
        Teitelboim 1989; Unruh & Wald 1989; Ng & van Dam).  A
        fluctuating Lambda is a different theory from a fixed one.
    (4) The boundary condition itself.  If the initial state of the
        universe is eventually COMPUTED rather than chosen, the
        integration constant stops being free and the fine-tuning
        question returns in that computation.

  This script verifies none of (1)-(4) and takes no side on any of
  them.  It establishes exactly one thing: the classical algebra.

------------------------------------------------------------------------------
C5. THREE FURTHER STRUCTURAL CAVEATS
------------------------------------------------------------------------------
  (i)  CONSERVATION IS AN EXTRA POSTULATE HERE.  In standard GR,
       nabla^m T_mn = 0 is FORCED: it follows from the Bianchi
       identity applied to G_mn = kappa T_mn.  In the trace-free
       theory it does not follow (B4) and must be assumed separately.
       The trace-free theory is therefore strictly WEAKER at this
       point.  It is common to see this presented as a bonus
       ('conservation is no longer required!'); it is a cost.
  (ii) THE FIXED VOLUME ELEMENT.  Deriving the trace-free equations
       from an action normally requires constraining sqrt(-g), which
       reduces the symmetry group from full diffeomorphisms to
       volume-preserving ones.  Nothing above needed an action -- all
       of PART A and PART B is equation-level algebra -- but any
       claim about quantization or about coupling to matter does need
       one, and then this caveat is live.
  (iii) NOTHING HERE IS NEW.  Einstein wrote the trace-free equations
       in 1919; Anderson & Finkelstein identified Lambda as an
       integration constant in 1971; Weinberg's 1989 review already
       states in section V that this does not solve the cosmological
       constant problem, for exactly the reason recomputed in C1-C2.
       This file is a verification of a known argument and a
       quantification of its known limitation.  REPRODUCED, NOT
       DISCOVERED.

==============================================================================
VERDICT -- tracefree_lambda.py
==============================================================================

  QUESTION: does the trace-free (unimodular) form dissolve the
  vacuum-energy problem?

  ANSWER: it dissolves HALF of it, exactly, and replaces the other
  half with a problem we cannot currently formulate, let alone solve.

  WHAT IS ESTABLISHED (exact symbolic algebra, no tolerances):
   1. IMMUNITY, HEADLINE.  Under T_mn -> T_mn + rho_vac g_mn the
      trace-free field equations change by IDENTICALLY ZERO -- all 16
      components, arbitrary symbolic metric, arbitrary symbolic
      rho_vac, both sign conventions.  Confirmed component-by-
      component on flat FRW, on Schwarzschild-de Sitter, and on an
      inhomogeneous anisotropic metric with ten free source
      functions.  Verified: yes
   2. THE SAME SHIFT IS *NOT* INVISIBLE to the standard Einstein
      form, where it leaves -kappa rho_vac g_mn.  That one line is
      the entire difference between the two formulations.
   3. THE BLIND DIRECTION IS EXACTLY ONE-DIMENSIONAL.  The trace
      projector has rank 9 on the 10-dimensional space of symmetric
      tensors, and its kernel is spanned by g_mn alone.  Vacuum
      energy is the ONLY stress-tensor modification the equations
      cannot see; everything else still gravitates in full.
   4. LAMBDA IS A CONSTANT OF INTEGRATION.  Bianchi (verified
      symbolically on four metrics, up to inhomogeneous and off-
      diagonal) plus assumed conservation gives d_nu C = 4 kappa
      nabla^m T_mn with C = R + kappa T; conservation makes C
      constant, C = 4 Lambda, and G_mn + Lambda g_mn = kappa T_mn
      follows.  dC/dt = 0 verified on-shell EXACTLY on FRW and on
      anisotropic Bianchi I.  Schwarzschild-de Sitter solves the
      trace-free vacuum equation for every M and every Lambda.
   5. AT RUNTIME: rho_vac = 0 and rho_vac = 1e60 give BIT-IDENTICAL
      trajectories (np.array_equal: True), while in standard GR a
      rho_vac of only 1e-3 already shifts a(t_end) by 3.79e-04.

  WHAT IS *NOT* ESTABLISHED, AND THIS IS THE POINT:
   A. NO PREDICTION OF LAMBDA.  Five target values spanning
      -0.5 to 1e60 were each realised exactly by choosing the initial
      expansion rate; the field equations are identical in all five.
      No measurement of Lambda can falsify this formalism and no
      measurement of Lambda supports it.
   B. THE DIGITS DID NOT GO AWAY.  Landing on the observed density
      still requires selecting a number ~10^-121 of the Planck scale
      (computed here: M_Pl^4/rho_Lambda = 10^122.9 raw, 10^120.7 at one
      loop).  A Lagrangian fine-tuning became a boundary condition.
      Whether that is 'better' cannot be decided without a measure on
      initial data, which the formalism does not supply.  This file
      asserts neither that the tuning survives nor that it is gone.
   C. THE QUANTUM QUESTION IS UNTOUCHED.  Everything here is
      classical.  Whether radiative corrections to Lambda remain
      immune is NOT settled by this algebra; the standard objections
      (quantum inequivalence to GR, the trace anomaly, Lambda as a
      fluctuating conjugate variable) are listed in C4 and none is
      adjudicated here.
   D. CONSERVATION BECAME AN ASSUMPTION.  In GR it is a theorem.
      That is a loss of content, not a gain.

  ONE SENTENCE: the trace-free form makes vacuum energy provably,
  exactly irrelevant to the field equations -- and buys that with
  exactly one arbitrary constant, so the cosmological constant
  problem is not solved but relabelled 'initial condition'.

  RELATION TO THE REST OF THE SEAM: wave 3's vacuum_energy_ladder.py
  showed the famous exponent is assumption-laden but that measured
  masses alone still overshoot by ~10^55.  This file shows the
  leading structural escape from that overshoot is algebraically
  sound and predictively empty.  Neither file solves anything.

  STATUS: REPRODUCED, NOT DISCOVERED.  Einstein (1919); Anderson &
  Finkelstein (1971); Unruh (1989); Henneaux & Teitelboim (1989);
  Weinberg RMP 61, 1 (1989) section V; Ellis, van Elst, Murugan &
  Uzan (2011).  The contribution of this file is that the argument
  is CHECKED symbolically rather than quoted, on metrics general
  enough that the checks could have failed, and that its limitation
  is quantified in the same run.
==============================================================================

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