The Space of Instruments Carries a Metric of Its Own: A Trajectory Has a Length, but Not a Proper Time – Two Geometries on Met(M).
Paper 119 identified a fourth logical type of “time” and left one cell of its table empty. dtau is a FUNCTIONAL OF A CURVE, but no counterpart was written for a Ricci flow trajectory. Paper 119’s reasoning ran: the very instrument of measurement is being deformed, so this cannot be a length measured by that instrument. THIS PAPER SHOWS THAT THE REASONING WAS CORRECT AND THE CONCLUSION PREMATURE. It is indeed not a length inside M. But what that sentence rules out is only lengths INSIDE M; it does not rule out lengths inside the SPACE OF METRICS Met(M). Whether the space in which the instruments are arrayed carries a metric of its own was never checked.
FIRST, THE CELL HAS A RESIDENT. The tangent space to Met(M) is the space of symmetric 2-tensor fields, and it carries one natural POSITIVE DEFINITE inner product. Clarke gave an explicit distance function for this L2 metric and showed that its completion is CAT(0) (non-positively curved) with a unique minimal path between any two points. TRAJECTORIES THEREFORE HAVE LENGTHS.
SECOND, THIS PAPER COMPUTES THAT LENGTH IN CLOSED FORM FOR A MODEL. For the family of round n-spheres g(t) = r^2(t) g_S, Ricci flow gives r^2(t) = r_0^2 – 2(n-1)t with extinction at t* = r_0^2/(2(n-1)). The speed is ||d g/dt||^2 = c^2 n r^n V_n with c = -2(n-1)/r^2 and V_n = Vol(S^n); changing variables to r and integrating gives L = 4 r_0^(n/2) sqrt(V_n/n), FINITE FOR EVERY n at least 2. Since the integrand blows up at the endpoint, the integral was verified independently by DOUBLE-EXPONENTIAL (tanh-sinh) QUADRATURE, which needs no knowledge of where the singularity sits. All fifteen cases n = 2,3,4,5,10 with r_0 = 0.5,1,2 agreed with the closed form (relative error 9.2e-9 at n=2, and at most 1e-12 for n at least 3). The model value: for n=2, r_0=1, with t*=1/2 and V_2=4pi, L = 4 sqrt(2 pi) = 10.0265130985.
THE MANNER OF THE RUNNING IS WHAT DESERVES ATTENTION. Approaching the extinction time THE SPEED DIVERGES – for n=2, r_0=1 it is 70.9 at t=0.49, 709.0 at t=0.4999, and 7.090e4 at t*-t = 1e-8. AND STILL THE DISTANCE RUN ENDS AT 10.0265. That is, THE SINGULARITY SITS AT FINITE DISTANCE. Since Clarke’s completion is complete, the flow converges to a point of that completion: the path does not break off midway; a path of finite length has a destination in the completion. What was computed, however, is ONE MODEL, the round sphere; nothing is claimed about general singularities such as neckpinches.
THIRD, HOWEVER, THERE IS NOT ONE RESIDENT BUT TWO, AND THEY DIFFER IN TYPE. The natural metric on Met(M) is not unique: an inner product on symmetric tensors admits the freedom of adding a term pairing the traces, giving the one-parameter family G_lambda(h,k) = tr(hk) – lambda tr(h) tr(k). Here lambda = 0 is Clarke’s L2 metric and lambda = 1 is the DEWITT SUPERMETRIC, which appears in the kinetic term of canonical gravity and is the metric general relativity itself specifies. These are two different geometries on the same space, and decisively THEIR SIGNATURES DIFFER.
THIS PAPER’S COMPUTATION. Splitting h into trace and traceless parts, h = (tau/n) delta + h_0, gives G_lambda(h,h) = tr(h_0^2) + tau^2 (1/n – lambda), verified to a maximum error of 5.3e-15 over 2000 random symmetric matrices. Since tr(h_0^2) is positive, the signature is decided by the coefficient of the trace direction alone, and THE SIGNATURE CHANGES EXACTLY AT lambda = 1/n: below 1/n it is positive definite (Riemannian), at 1/n it degenerates (the trace direction is null), and above 1/n it is Lorentzian with EXACTLY ONE TIMELIKE DIRECTION. For n=3, Sym(3) is 6-dimensional; computing the eigenvalues of the Gram matrix directly gives (0,0,6) at lambda=0, (0,1,5) at lambda=1/3, and eigenvalues -2,1,1,1,1,1 hence (1,0,5) at general relativity’s lambda=1 – that is, (-,+,+,+,+,+). Extracting the eigenvector of the negative eigenvalue, IT IS PURE TRACE (traceless part of norm exactly 0). THE TIMELIKE DIRECTION IS THE CONFORMAL, OR SCALE, DIRECTION, and the five spacelike directions are shear deformations. And general relativity sits at lambda = 1, on the FAR SIDE of the critical value 1/3.
FOURTH, THE CORE SORTING. Proper time differs from coordinate time NOT BECAUSE IT IS A “LENGTH” – distance in space is a length too. Proper time is special because it is THE ARC LENGTH OF A METRIC WITH A SIGNATURE: hence null directions exist (light has dtau = 0), causal structure arises, and timelike is distinguished from spacelike. THEREFORE PAPER 119’S EMPTY CELL SPLITS IN TWO RATHER THAN BEING FILLED ONCE. Clarke’s resident is NOT the analogue of dtau but the analogue of SPATIAL DISTANCE. The genuine analogue of dtau exists only on the DeWitt side.
FIFTH, THE RESULTING CONCLUSION. One does not get to choose which metric to measure with. The statement that Ricci flow is a gradient flow is made with respect to the POSITIVE DEFINITE lambda = 0 metric; Perelman’s gradient-flow structure for the F and W functionals is a gradient with respect to that inner product. DeWitt’s lambda = 1, by contrast, is the metric specified by canonical quantum gravity (the Wheeler-DeWitt equation). HENCE: A RICCI FLOW TRAJECTORY HAS A LENGTH (lambda = 0, finite, 4 sqrt(2 pi)) BUT NO PROPER TIME (lambda = 0 has no signature). The geometry with a signature belongs to a different problem on the same space. This is not a correction to Paper 119 but A REPLACEMENT OF THE REASON: Paper 119 could be read as saying “it is not dtau because there is no length”; the correct statement is “there IS a length; it is not dtau because there is NO SIGNATURE”. Same conclusion, different root.
SIXTH, AN INDEPENDENT CORROBORATION. Ricci flow is NOT naively a gradient flow on Met(M): it differs from the gradient of the F functional by an INFINITESIMAL DIFFEOMORPHISM ACTION, and only on the quotient Met(M)/Diff(M) is the (normalised) Ricci flow exactly the gradient flow of the mu functional. The construction itself says it does not go through until the redundant freedom has been divided out. WHAT CARRIES GEOMETRIC MEANING IS THE TRAJECTORY, NOT THE TICK MARKS ALONG IT – what Paper 119 argued from types, Perelman’s construction says by another route. The same caveat reaches this paper’s length: L = 4 sqrt(2 pi) is a length IN Met(M), not in the quotient. For the round spheres the two agree, since the family does not move in the diffeomorphism directions, but for a general trajectory the distinction matters.
SEVENTH, A ROLL-CALL OF THE WORD. “L2 metric” itself has distinct roots. By Michor-Mumford, on SHAPE SPACES (immersions modulo diffeomorphism) and on DIFFEOMORPHISM GROUPS with the right-invariant L2 metric, THE GEODESIC DISTANCE VANISHES IDENTICALLY: any two points can be joined by arbitrarily short paths. The same phrase yields a genuine CAT(0) distance on Met(M) and annihilates distance on shape spaces. Hence “trajectories have lengths” does not follow automatically from the word L2; it can be said only once the arena is fixed to Met(M), and shifting the arena by one step makes it false.
EIGHTH, A RHYME INSIDE THIS SERIES, AND ITS BRAKE. Paper 111 asked how far light determines the geometry of spacetime and answered: up to conformal structure (nine of ten components in four dimensions) – what is left undetermined is THE OVERALL SCALE. This paper computed that in superspace THE TIMELIKE DIRECTION IS EXACTLY THE CONFORMAL (SCALE) DIRECTION. The one component light cannot determine is the one that plays the role of time in superspace. THESE MUST NOT BE JOINED. The arenas differ (four-dimensional spacetime versus a six-dimensional space on a three-dimensional slice; 10 = 9+1 and 6 = 5+1 are different counts), the roles differ (“light does not determine it” versus “the supermetric is negative there”, with no common mechanism exhibited), and this series has its own discipline (Paper 100: “a resonance is a signature, not a proof”; Paper 101’s record of Eddington). What is recorded is one line and no more: THE SAME COMPONENT – THE OVERALL SCALE OF THE METRIC – CARRIES A DISTINGUISHED ROLE IN TWO SEPARATE CONTEXTS, AND THE RELATION BETWEEN THEM HAS NOT BEEN ESTABLISHED. This is not even future work; for now it is an observation.
IMPORTANT (scope): a structural interpretation of established standard facts, and the filling of a cell in this series’ own table (Layer 2). Every ingredient is known – Clarke’s L2 distance and CAT(0) completion, the Michor-Mumford vanishing of geodesic distance, the DeWitt supermetric and its signature (1967), Perelman’s gradient-flow structure with its diffeomorphism caveat, Hamilton’s sphere solution. NOT ONE NEW THEOREM IS PROVED. What is not claimed: DeWitt’s timelike direction is not identified with physical time or the time of the universe (the problem of time in quantum gravity is unsolved) / Clarke’s length is not called a proper time (no signature) / the two metrics are not merged (lambda = 1/n separates them) / no claim is made that trajectory length is finite at a general singularity / lengths in Met(M) are not confused with lengths in Met(M)/Diff(M) / W is not identified with thermodynamic entropy (Papers 88, 103, 119) / Paper 111’s scale component is not joined to the supermetric’s timelike direction (a rhyme, not a proof; Papers 100, 101) / the Poincare and geometrization conjectures are not touched. Layer-3 lenses are quarantined under a non-scientific label.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). The Space of Instruments Carries a Metric of Its Own. A Trajectory Has a Length, but Not a Proper Time: Two Geometries on $\mathrm{Met}(M)$ [Paper 121]. Zenodo.
DOI: 10.5281/zenodo.22038594
ORCID: 0009-0006-8587-3450