document: as is well known , the atom or atoms in the atomic clock are passive  they do not `` tick''so the clock needs an active oscillator in addition to the atom(s ) . in designing an atomic clock to realize the second as a measurement unit in the international system of units ( si ) , one encounters two problems : ( a ) the resonance exhibited by the atom or atoms of the clock varies with the details of the clock s construction and the circumstances of its operation ; in particular the resonance shifts depending on the intensity of the radiation of the atoms by the oscillator . 
   ( b ) the oscillator , controlled by , in effect , a knob , drifts in relation to the knob setting . 
   problem ( a ) is dealt with by introducing a wave function parametrized by radiation intensity and whatever other factors one deems relevant . 
   the si second is then `` defined '' by the resonance that `` would be found '' at absolute zero temperature ( implying zero radiation ) . for a clock using cesium 133 atoms , 
   this imagined resonance is declared by the general conference of weights and measures to be 9 192 631 770 hz , so that the si second is that number of cycles of the radiation at that imagined resonance @xcite . to express the relation between a measured resonance and the imagined resonance at 0 k 
   , a wave function is chosen . 
   problem ( b ) is dealt with by computer - mediated feedback that turns the knob of the oscillator in response to detections of scattering of the oscillator s radiation by the atom(s ) of the clock , steering the oscillator toward an aiming point . 
   a key point for this paper is that the wave function incorporated into the operation of an atomic clock can never be unconditionally known . 
   the language of quantum theory reflects within itself a distinction between ` explanation ' and ` evidence ' . for explanations it offers the linear algebra of wave functions and operators , while for evidence it offers probabilities on a set of outcomes . 
   outcomes are subject to quantum uncertainty , but uncertainty is only the tip of an iceberg : how can one `` know '' that a wave function describes an experimental situation ? 
   the distinction within quantum theory between linear operators and probabilities implies a gap between any explanation and the evidence explained . 
   @xcite :    [ prop : one ] to choose a wave function to explain experimental evidence requires reaching beyond logic based on that evidence , and evidence acquired after the choice is made can call for a revision of the chosen wave function .    because no wave function can be unconditionally known , 
   not even probabilities of future evidence can be unconditionally foreseen . 
   here we show implications of the unknowability of wave functions for the second as a unit of measurement in the international system ( si ) , implications that carry over to both digital communications and to the use of a spacetime with a metric tensor in explaining clock readings at the transmission and reception of logical symbols . 
   clocks that generate universal coordinated time ( utc ) are steered toward aiming points that depend not only on a chosen wave function but also on an hypothesized metric tensor field of a curved spacetime . 
   like the chosen wave function , the hypothesis of a metric tensor is constrained , but not determined , by measured data . 
   guesses enter the operations of clocks through the computational machinery that steers them . taking incoming data 
   , the machinery updates records that determine an aiming point , and so involves the writing and reading of records . 
   the writing must take place at a phase of a cycle distinct from a phase of reading , with a separation between the writing and the reading needed to avoid a logical short circuit . in sec . 
   [ sec : turing ] we picture an explanation used in the operation of a clock as a string of characters written on a tape divided into squares , one symbol per square . 
   the tape is part of a turing machine modified to be stepped by a clock and to communicate with other such machines and with keyboards and displays . 
   we call this modified turing machine an _ open machine_. the computations performed by an open machine are open to an inflow numbers and formulas incalculable prior to their entry . because a computer cycles through distinct phases of memory use , the most direct propagation of symbols from one computer to another requires a symbol from one computer to arrive during a suitable phase of the receiving computer s cycle . in sec . 
   [ sec : phasing ] we elevate this phase dependence to a principle that defines the _ logical synchronization _ necessary to a _ channel _ that connects clock readings at transmission of symbols to clock readings at their reception    recognizing the dependence of logic - bearing channels on an interaction between evidence and hypotheses about signal propagation engenders several types of questions , leading to a _ discipline of logical synchronization _ , outlined in sec . 
   [ sec : patterns ] . 
   the first type of question concerns patterns of channels that are possible aiming points , as determined in a blackboard calculation that assumes a theory of signal propagation . 
   [ sec : typei ] addresses examples of constraints on patterns of channels under various hypotheses of spacetime curvature , leading to putting `` phase stripes '' in spacetime that constrain channels to or from a given open machine . 
   an example of a freedom to guess an explanation within a constraint of evidence is characterized by a subgroup of a group of clock adjustments , and a bound on bit rate is shown to be imposed by variability in spacetime curvature . 
   [ sec : adj ] briefly addresses the two other types of questions , pertaining not to _ hypothesizing _ possible aiming points ` on the blackboard ' , but to _ using _ hypothesized aiming points , copied into feedback - mediating computers , for the steering of drifting clocks . after discussing steering toward aiming points copied from the blackboard , 
   we note occasions that invite revision of a hypothesized metric tensor and of patterns of channels chosen as aiming points . 
   computer - mediated feedback , especially as used in an atomic clock , requires logic open to an inflow of inputs beyond the reach of calculation . to model the logic of a computer that communicates with the other devices in a feedback loop , we modify a turing machine to communicate with external devices , including other such machines . 
   the turing machine makes a record on a tape marked into squares , each square holding one character of an alphabet . operating in a sequence of ` moments ' interspersed by ` moves ' , at any moment the machine scans one square of the tape , from which it can read , or onto which it can write , a single character . a move as defined in the mathematics of turing machines consists ( only ) of the logical relation between the machine at one moment and the machine at the next moment @xcite , thus expressing the logic of a computation , detached from its speed ; however , in a feedback loop , computational speed matters . 
   let the moves of the modified turing machine be stepped by ticks of a clock . 
   a step occurs once per a period of revolution of the clock hand . 
   this period is adjustable , on the fly . 
   we require that the cycle of the modified turing machine correspond to a unit interval of the readings of its clock .    to express communication between open machines as models of computers , the modified turing machine can receive externally supplied signals and can transmit signals , with both the reception and the transmission geared to the cycle of the machine . in addition 
   , the modified turing machine registers a count of moments at which signals are received and moments at which signals are transmitted . at a finer scale , _ the machine records a phase quantity in the cycle of its clock , relative to the center of the moment at which a signal carrying a character arrives . 
   _ we call such a machine an _ open machine_. an open machine can receive detections and can command action , for instance the action of increasing or decreasing the frequency of the variable oscillator of an atomic clock .    calculations performed on an open machine communicating with detectors and actuators proceed by moves made according to a rule that can be modified from outside the machine in the course of operation . 
   these calculations respond to received influences , such as occurrences of outcomes underivable from the contents of the machine memory , when the open machine writes commands on a tape read by an external actuator . 
   the wider physical world shows up in an open machine as both ( 1 ) unforeseeable messages from external devices and ( 2 ) commands to external devices . 
   we picture a real - time computer in a feedback loop as writing records on the tape of an open machine . 
   the segmentation into moments interspersed by moves is found not just in turing machines but in any digital computer , which implies    the logical result of any computation is oblivious to variations in speed at which the clock steps the computer .     + corollary 2.1 . 
   _ no computer can sense directly any variation in its clock frequency . _ 
   although it can not directly sense variation in the tick rate of its clock , the logic of open machine stepped by an atomic clock can still control the adjustment of the clock s oscillator by responding to variations in the detection rate written moment by moment onto its turing tape . 
   a flow of unforeseeable detections feeds successive computations of results , each of which , promptly acted on , impacts probabilities of subsequent occurrences of outcomes , even though those subsequent outcomes remain unforeseeable . the computation that steers the oscillator depends not just on unforeseeable inputs , but also on a steering formula encoded in a program . 
   * remarks * :    1 .   to appreciate feedback , 
   take note that a formula is distinct from what it expresses . 
   for example a formula written along a stretch of a turing tape as a string of characters can contain a name @xmath0 for wave function as a function of time variable @xmath1 and space variables . 
   the formula , containing @xmath0 , once written , just `` sits motionless , '' in contrast to the motion that the formula expresses . 2 . 
   although unchanged over some cycles of a feedback loop , a feedback loop operates in a larger context , in which steering formulas are subject to evolution . 
   sooner or later , the string that defines the action of an algorithm , invoking a formula , is apt to be overwritten by a string of characters expressing a new formula . 
   occasions for rewriting steering formulas are routine in clock networks , including those employed in geodesy and astronomy . 
   logical communication requires clocking . 
   the reading of a clock of an open machine @xmath2an @xmath2-reading  has the form @xmath3 where @xmath4 indicates the count of cycles and @xmath5 is the phase within the cycle , with the convention that @xmath6 . 
   we define a channel from @xmath2 to @xmath7 , denoted @xmath8 , as a set of pairs , each pair of the form @xmath9 . 
   the first member @xmath3 is an @xmath2-reading at which machine @xmath2 can transmit a signal and @xmath10 is a @xmath7-reading at which the clock of machine @xmath7 can register the reception of the signal . define a _ repeating channel _ to be a channel @xmath8 such that @xmath11)(\exists m , n , j , k ) ( m+\ell j.\phi_{a,\ell},n+\ell k.\phi_{b,\ell } ) ] \in \abr,\ ] ] for theoretical purposes , it is convenient to define an _ endlessly repeating channel _ for which @xmath12 ranges over all integers . 
   again for theoretical purposes , on occasion we consider channels for which the phases are all zero , in which case we may omit writing the phases .    because they are defined by local clocks without reference to any metric tensor 
   , channels invoke no assumption about a metric or even a spacetime manifold . 
   for this reason evidence from the operation of channels is independent of any explanatory assumptions involving a manifold with metric and , in particular , is independent of any global time coordinate , or any `` reference system '' @xcite . 
   thus clock readings at the transmission and the reception of signals can prompt revisions of hypotheses about a metric tensor field . a record format for such evidence was illustrated in earlier work @xcite , along with the picturing of such records as _ occurrence graphs_.    from the beating of a heart to the bucket brigade , life moves in phased rhythms . for a symbol carried by a signal from an open machine @xmath2 to be written into the memory of an open machine @xmath7 , the signal must be available at @xmath7 within a phase of the cycle of @xmath7 during which writing can take place , and the cycle must offer room for a distinct other phase . 
   we elevate engineering commonplace to a principle pertaining to open machines as follows . 
   [ prop : three ] a logical symbol can propagate from one open machine to another only if the symbol arrives within the writing phase of the receiving machine ; in particular , respect for phasing requires that for some positive @xmath13 any arrival phase @xmath14 satisfy the inequality @xmath15    prop . 
   [ prop : three ] serves as a fixed point to hold onto while hypotheses about signal propagation in relation to channels are subject to revision . 
   we call the phase constraint on a channel asserted by ( [ eq : main ] ) _ logical synchronization_. for simplicity and to allow comparing conditions for phasing with conditions for einstein synchronization , we take the engineering liberty of allowing transmission to occur at the same phase as reception , so that both occur during a phase interval satisfying ( [ eq : main ] ) . the alternative of demanding reception near values of @xmath16 can be carried out with little extra difficulty . + 
   * remarks : *    1 .   note that @xmath14 in the proposition is a phase of a cycle of a variable - rate clock that is _ not _ assumed to be in any fixed relation to a proper clock as conceived in general relativity . indeed , satisfying ( [ eq : main ] ) usually requires the operation of clocks at variable rates . 
   the engineering of communications between computers commonly detaches the timing of a computer s receiver from that of the computer by buffering : after a reception , the receiver writes into a buffer that is later read by the computer@xcite . in analyzing open machines we do without buffering , confining ourselves to character - by - character phase meshing as asserted in prop . 
   [ prop : three ] , which offers the most direct communication possible . 
   given the definition of a channel and the condition ( [ eq : main ] ) essential to the communication of logical symbols , three types of questions arise :    * type i : * what patterns of interrelated channels does one try for as aiming points ? 
   * type ii : * how can the steering of open machines be arranged to approach given aiming points within acceptable phase tolerances ? 
   * type iii : * how to respond to deviations from aiming points beyond tolerances ? 
   such questions point the way to exploring what might be called a _ discipline of logical synchronization_. so far we notice two promising areas of application within this discipline :    1 . 
   provide a theoretical basis for networks of logically synchronized repeating channels , highlighting 1 . 
   possibilities for channels with null receptive phases as a limiting case of desirable behavior , and 2 . 
   circumstances that force non - null phases . 2 . 
   explore constraints on receptive phases imposed by gravitation , as a path to exploring and measuring gravitational curvature , including slower changes in curvature than those searched for by the laser gravitational wave observatory @xcite . 
   answers to questions of the above types require hypotheses , if only provisional , about signal propagation . for this section 
   we assume that propagation is described by null geodesics in a lorentzian 4-manifold @xmath17 with one or another metric tensor field @xmath18 , as in general relativity . 
   following perlick @xcite we represent an open machine as a timelike worldline , meaning a smooth embedding @xmath19 from a real interval into @xmath17 , such that the tangent vector @xmath20 is everywhere timelike with respect to @xmath18 and future - pointing . 
   we limit our attention to worldlines of open machines that allow for signal propagation between them to be expressed by null geodesics . 
   to say this more carefully , we distinguish the _ image _ of a worldline as a submanifold of @xmath17 from the worldline as a mapping . 
   consider an open region @xmath21 of @xmath17 containing a smaller open region @xmath22 , with @xmath21 containing the images of two open machines @xmath2 and @xmath7 , with the property that every point @xmath23 of the image of @xmath2 restricted to @xmath22 is reached uniquely by one future - pointing null geodesic from the image of @xmath7 in @xmath21 and by one past - pointing null geodesic from the image of @xmath7 in @xmath21 . 
   we then say @xmath2 and @xmath7 are _ radar linkable _ in @xmath22 . 
   we limit our attention to open machines that are radar linkable in some spacetime region @xmath22 . 
   in addition we assume that the channels preserve order ( what is transmitted later arrives later ) . 
   indeed , we mostly deal with open machines in a gently curved spacetime region , adequately described by fermi normal coordinates around a timelike geodesic .    for simplicity and to allow comparing conditions for phasing with conditions for einstein synchronization , 
   we take the liberty of allowing transmission to occur at the same phase as reception , so that both occur during a phase interval satisfying ( [ eq : main ] ) . 
   the perhaps more realistic alternative of demanding reception near values of @xmath16 can be carried out with little difficulty .    to develop the physics of channels , 
   we need to introduce three concepts :    \(1 ) we define a _ group of clock adjustments _ as transformations of the readings of the clock of an open machine . 
   as it pertains to endlessly repeating channels , a group @xmath24 of clock adjustments consists of functions on the real numbers having continuous , positive first derivatives . 
   group multiplication is the composition of such functions , which , being invertible , have inverses . to define the action of @xmath24 on clock readings , we speak ` original clock readings ' as distinct from adjusted readings an adjustment @xmath25 acts by changing every original reading @xmath26 of a clock @xmath2 to an adjusted reading @xmath27 . 
   as we shall see , clock adjustments can affect echo counts . 
   \(2 ) to hypothesize a relation between the @xmath2-clock and an accompanying proper clock , one has to assume one or another metric tensor field @xmath18 , relative to which to define proper time increments along @xmath2 s worldline ; then one can posit an adjustment @xmath28 such that @xmath29 where @xmath30 is the reading imagined for the accompanying proper clock when @xmath2 reads @xmath26 . 
   \(3 ) we need to speak of positional relations between open machines . for this section 
   we assume that when an open machine @xmath7 receives a signal from any other machine @xmath2 then @xmath7 echoes back a signal to @xmath2 right away , so the echo count @xmath31 defined in sec . 
   [ sec : phasing ] involves no delay at @xmath7 . in this case , 
   evidence in the form of an echo count becomes explained , under the assumption of a metric tensor field @xmath18 , as being just twice the radar distance @xcite from @xmath2 to the event of reception by @xmath7 . 
   questions of type i concern constraints on channels imposed by the physics of signal propagation . here 
   we specialize to constraints on channels imposed by spacetime metrics , constraints obtained from mathematical models that , while worked out so to speak on the blackboard , can be copied onto turing tapes as aiming points toward which to steer the behavior of the clocks of open machines . 
   questions of types ii and iii are deferred to the sec . 
   [ sec : adj ] . 
   we begin by considering just two machines . assuming an hypothetical spacetime @xmath32 , 
   suppose that machine @xmath2 is given as a worldline parametrized by its clock readings : what are the possibilities and constraints for an additional machine @xmath7 with two - way repeating channels @xmath8 and @xmath33 with a constant echo count ? 
   we assume the idealized case of channels with null phases , which implies integer echo counts . 
   for each @xmath2-tick there is a future light cone and a past light cone .     with tick events indicated ; 
   ( b ) light cones associated to ticks of @xmath2 ; ( c ) ticks of @xmath34 and @xmath35 at light cone intersections corresponding to @xmath36 . 
   , width=470 ]    the future light cone from an @xmath2-reading @xmath37 has an intersection with the past light cone for the returned echo received at @xmath38 . 
   [ fig:3 ] illustrates the toy case of a single space dimension in a flat spacetime by showing the two possibilities for a machine @xmath7 linked to @xmath2 by two - way channels at a given constant echo count . in each solution , the clocking of @xmath7 is such that a tick of @xmath7 occurs at each of a sequence of intersections of outgoing and incoming light cones from and to ticks of @xmath2 . 
   note that the image of @xmath7 , and not just its clock rate , depend on the clock rate of @xmath2 . 
   determination of the tick events for @xmath7 leaves undetermined the @xmath7 trajectory between ticks , so there is a freedom of choice . 
   one can exercise this freedom by requiring the image of @xmath7 to be consistent with additional channels of larger echo counts . 
   a clock adjustment of @xmath2 of the form @xmath39 for @xmath40 a positive integer increases the density of the two - way channel by @xmath40 and inserts @xmath41 events between successive @xmath7-ticks , thus multiplying the echo count by @xmath40 . as @xmath40 increases without limit , @xmath7 becomes fully specified . 
   turning to two space dimensions , the image of @xmath7 must lie in a tube around the image of @xmath2 , as viewed in a three - dimensional space ( vertical is time ) . 
   so any timelike trajectory within the tube will do for the image of @xmath7 . for a full spacetime of 3 + 1 dimensions , the solutions for the image of @xmath7 fall in the corresponding `` hypertube . '' 
   the argument does not depend on flatness and so works for a generic , gently curved spacetime in which the channels have the property of order preservation .     and @xmath7 freely chosen ; ( b ) light signals lacing @xmath2 and @xmath7 define tick events ; ( c ) interpolated lacings of light signals added to make @xmath36.,width=470 ]    a different situation for two machines arises in case only the image of @xmath2 s worldline is specified while its clocking left to be determined . in this case the image of @xmath7 can be freely chosen , after which the clocking of both @xmath2 and @xmath7 is constrained , as illustrated in fig .  [ fig:4 ] for the toy case of flat spacetime with 1 space dimension . to illustrate the constraint on clocking 
   , we define a `` lacing '' of light signals to be a pattern of light signals echoing back and forth between two open machines as illustrated in fig . 
   [ fig:4 ]  ( b ) . for any event chosen in the image of @xmath2 , 
   there is a lacing that touches it . 
   in addition to choosing this event , one can choose any positive integer @xmath40 to be @xmath31 , and choose @xmath41 events in the image of @xmath2 located after the chosen event and before the next @xmath2-event touched by the lacing of light signals . 
   the addition of lacings that touch each of the @xmath41 intermediating events corresponds to a repeating channel @xmath8 with echo count @xmath42 , along with a repeating channel @xmath33 with the same echo count @xmath43 . 
   this construction does not depend on the dimension of the spacetime nor on its flatness , and so works also for a curved spacetime having the property of order preservation . 
   evidence of channels as patterns of clock readings leaves open a choice of worldlines for its explanation . in the preceding example of laced channels between open machines @xmath2 and @xmath7 , part of this openness can be reflected within analysis by the invariance of the channels under a subgroup of the group of clock adjustments that `` slides the lacings , '' as follows . 
   suppose that transmissions of an open machine @xmath2 occur at given values of @xmath2-readings . 
   we ask about clock adjustments that can change the events of a worldline that correspond to a given @xmath2-reading . 
   if a clock adjustment @xmath28 takes original @xmath2-readings @xmath26 to a revised @xmath2-readings @xmath27 , transmission events triggered by the original clock readings become triggered when the re - adjusted clock exhibits the _ 
   same readings_. as registered by original readings , the adjusted transmission occurs at @xmath44 . based on this relation 
   we inquire into the action of subgroups of @xmath45 on the readings of the clocks of two open machines @xmath2 and @xmath7 . in particular 
   , there is a subgroup @xmath46 that expresses possible revisions of explanations that leave invariant the repeating channels with constant echo count @xmath40 . 
   an element @xmath47 is a pair of clock adjustments that leaves the channels invariant , and such a pair can be chosen within a certain freedom . for the adjustment @xmath28 
   one is free to : ( a ) assign an arbitrary value to @xmath48 ; and ( b ) , if @xmath49 , then for @xmath50 , choose the value of @xmath51 at will , subject to the constraints that @xmath52 and @xmath53 is less than the original clock reading for the re - adjusted first echo from @xmath54 . with these choices , @xmath55 is then constrained so that each lacing maps to another lacing . 
   the condition ( a ) slides a lacing along the pair of machines ; the condition ( b ) nudges additional lacings that show up in the interval between a transmission and the receipt of its echo . in this way a freedom to guess within a constraint is expressed by @xmath56 .      moving to more than two machines , 
   we invoke the    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ * definition : * an _ arrangement of open machines _ consists of open machines with the specification of some or all of the channels from one to another , augmented by proper periods of the clock of at least one of the machines . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    ( without specifying some proper periods , the scale of separations of one machine from another is open , allowing the arrangement to shrink without limit , thus obscuring the effect of spacetime curvature . ) 
   although gentle spacetime curvature has no effect on the possible channels linking two open machines , curvature does affect the possible channels and their echo counts in some arrangements of five or more machines , so that the possible arrangements are a measure spacetime curvature . 
   the way that spacetime curvature affects the possible arrangements of channels is analogous to the way surface curvature in euclidean geometry affects the ratios of the lengths of the edges of embedded graphs . 
   the effect on ratios shows up in mappings from graphs embedded in a plane to their images on a sphere . 
   for example , a triangle can be mapped from a plane to a generic sphere , in such a way that each edge of the triangle is mapped to an arc of the same length along a great circle on the sphere . 
   the same holds for two triangles that share an edge , as illustrated in fig . 
   [ fig : sphere ] , panel ( a ) ; however , the gauss curvature of the sphere implies that the complete graph on 4 vertices generically embedded in the plane , shown in panel ( b ) , can not be mapped so as to preserve all edge lengths . the property that blocks the preservation of edge ratios is the presence of an edge in the plane figure that can not be slightly changed without changing the length of at least one other edge 
   ; we speak of such an edge as `` frozen . ''    in a static spacetime , which is all we have so far investigated , a generic arrangement of 4 open machines , is analogous to the triangle on the plane in that a map to any gently curved spacetime can preserve all the echo counts .    [ 
   prop : nine]assume four open machines in a static spacetime , with one machine stepped with a proper - time period @xmath57 , and let @xmath40 be any positive integer . 
   then , independent of any gentle riemann curvature of the spacetime , the four open machines can be arranged , like vertices of a regular tetrahedron , to have six two - way channels with null phases , with all echo counts being @xmath58 . 
   _ proof : _ assuming a static spacetime , choose a coordinate system with all the metric tensor components independent of the time coordinate , in such a way that it makes sense to speak of a time coordinate distinct from space coordinates ( for example , in a suitable region of a schwarzschild geometry ) . 
   let@xmath59 denote the machine with specified proper period @xmath60 , and let @xmath61 , @xmath62 , and @xmath63 denote the other three machines . for @xmath64 , @xmath65 , we prove the possibility , independent of curvature , of the channels @xmath66 require that each of four machines be located at some fixed spatial coordinate . because the spacetime is static , 
   the coordinate time difference between a transmission at @xmath59 and a reception at any other vertex @xmath67 ( a ) is independent of the value of the time coordinate at transmission and ( b ) is the same as the coordinate time difference between a transmission at @xmath67 and a reception at @xmath59 . 
   for this reason any one - way repeating channel of the form ( [ eq : vs ] ) can be turned around to make a channel in the opposite direction , so that establishing a channel in one direction suffices . for transmissions from any vertex to any other vertex , the coordinate - time difference between events of transmission equals the coordinate time difference between receptions . a signal from a transmission event on @xmath59 propagates on an expanding light cone , while an echo propagates on a light cone contracting toward an event of reception on @xmath59 . under the constraint that the echo count is @xmath58 , ( so the proper duration from the transmission event to the reception event for the echo is @xmath68 ) 
   , the echo event must be on a 2-dimensional submanifold  a sphere , defined by constant radar distance @xmath69 of its points from @xmath59 with transmission at a particular ( but arbitrary ) tick of @xmath59 . 
   in coordinates adapted to a static spacetime , this sphere may appear as a `` potatoid '' in the space coordinates , with different points on the potatoid possibly varying in their time coordinate . 
   the potatoid shape corresponding to an echo count of @xmath58 remains constant under evolution of the time coordinate . 
   channels from @xmath59 to the other three vertices involve putting the three vertices on this potatoid . put @xmath61 anywhere on the `` potatoid '' . put @xmath62 anywhere on the ring that is intersection of potatoid of echo count @xmath58 radiated from @xmath61 and that radiated from @xmath59 . put @xmath63 on an intersection of the potatoids radiating from the other three vertices . 
   + q.e.d . 
   according to prop [ prop : nine ] the channels , and in particular the echo counts possible for a complete graph of four open machines in flat spacetime are also possible for a spacetime of gentle static curvature , provided that three of the machines are allowed to set their periods not to a fixed proper duration but in such a way that all four machines have periods that are identical in coordinate time . 
   the same holds if fewer channels among the four machines are specified . 
   but for five machines , the number of channels connecting them matters . 
   five open machines fixed to space coordinates in a static spacetime are analogous to the 4 vertices of a plane figure , in that an arrangement corresponding to an incomplete graph on five vertices can have echo counts independent of curvature , while a generic arrangement corresponding to a complete graph must have curvature - dependent relations among its echo counts .    [ prop:9.5 ] assuming a static spacetime , consider an arrangement of five open machines obtained by starting with a tetrahedral arrangement of four open machines with all echo counts of @xmath58 as in prop . 
   [ prop : nine ] , and then adding a fifth machine : independent of curvature , a fifth open machine can be located with two - way channels having echo counts of @xmath58 linking it to any three of the four machines of tetrahedral arrangement , resulting in nine two - way channels altogether .    _ 
   proof : _ the fifth machine can be located as was the machine @xmath63 , but on the side opposite to the cluster @xmath59 , @xmath61 , @xmath62 . + q.e.d . 
   in contrast to the arrangement of 9 two - way channels , illustrated in fig . 
   [ fig:5pt ] ( a ) consider an arrangement of 5 open machines corresponding to a complete graph on five vertices , with has ten two - way channels , as illustrated in fig . 
   [ fig:5pt ] ( b ) . for five open machines in a generic spacetime , not all of the ten two - way channels can have the same echo counts . 
   instead , channels in a flat spacetime as specified below can exist with about the simplest possible ratios of echo counts . 
   label five open machines , @xmath70 , @xmath71 , @xmath72 , @xmath34 , and @xmath35 . 
   take @xmath34 to be stepped by a clock ticking at a fixed proper period @xmath57 , letting the other machines tick at variable rates to be determined . 
   let @xmath73 be any machine other than @xmath34 . 
   for a flat spacetime it is consistent for the proper periods of all 5 machines to be @xmath57 , for the echo counts @xmath74 to be @xmath75 and for the echo counts @xmath76 to be @xmath77 , leading to twenty channels , conveniently viewed as in fig . 
   [ fig:5pt ] ( b ) as consisting of ten two - way channels . 
   [ prop : ten ] consider 5 open machines each fixed to space coordinates in a static curved spacetime in which the machines are all pairwise radar linkable , with 10 two - way channels connecting each machine to all the others ; then :    1 .   allowing for the periods of the machines other than @xmath34 to vary , 
   it is consistent with the curvature for all but one of the ten two - way channels to have null phases and echo counts as in a flat spacetime , but at least one two - way channel must have a different echo count that depends on the spacetime curvature . 
   2 .   suppose @xmath4 of the 10 two - way links are allowed to have non - zero phases . 
   if the spacetime does not admit all phases to be null , in generic cases the least possible maximum amplitude of a phase decreases as @xmath4 increases from 1 up to 10 . 
   the periods of the clocks of the open machines can be taken to be the coordinate - time interval corresponding to the proper period @xmath57 at @xmath34 .    _ 
   proof : _ reasoning as in the proof of prop . 
   [ prop : nine ] with its reference to a static spacetime shows that the same echo counts are possible as for flat spacetime _ with the exception _ that at least one of the two - way channels must be free to have a different echo count . for @xmath78 
   , similar reasoning shows that allowing @xmath79 machines vary in echo count allows reduction in the maximum variation from the echo counts in a flat spacetime , compared to the case in which only @xmath4 machines are allowed to vary in echo count . 
   + q.e.d .    adding the tenth two - way channel to an arrangement of five open machines effectively `` freezes '' all the echo counts . to define `` freezing '' 
   as applied to echo counts , first take note an asymmetry in the dependence of echo counts on clock rates . 
   consider any two machines @xmath2 and @xmath7 ; unlike echo count @xmath80 , which @xmath7 can change by changing it clock rate , the echo count @xmath31 is insensitive to @xmath7 s clock rate . 
   an echo count @xmath31 will be said to be _ to _ @xmath7 and _ from _ @xmath2 . 
   _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ * definition : * an arrangement of open machines is _ frozen _ if it has an echo count to a machine @xmath7 that can not be changed slightly without changing the length of another echo count to @xmath7 . 
   _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    the property of being frozen is important because of the following . 
   whether or not a frozen arrangement of open machines is consistent with an hypothesized spacetime depends on the weyl curvature of the spacetime . 
   for example , think of the 5 open machines as carried by 5 space vehicles coasting along a radial geodesic in a schwarzschild geometry . in this example 
   the variation of echo counts with curvature is small enough to be expressed by non - null phases of reception . in fermi 
   normal coordinates centered midway between the radially moving open machine @xmath34 and @xmath35 one has the metric with a curvature parameter @xmath81 , where @xmath82 is the schwarzschild radial coordinate to the origin of the fermi normal coordinates , @xmath83 is the radial distance coordinate from from the center point between @xmath34 and @xmath35 , and @xmath84 and @xmath85 are transverse to the radial direction along which @xmath86 coasts@xcite . 
   the speed of light is @xmath87 . 
   we make the adiabatic approximation which ignores the time dependence of @xmath82 , so that in calculations to first order in curvature we take advantage of the ( adiabatically ) static spacetime by locating open machines at fixed values of @xmath88 . 
   the metric is symmetric under rotation about the ( radially directed ) @xmath83-axis . 
   let @xmath34 and @xmath35 be located symmetrically at positive and negative values , respectively , of the @xmath83-axis , and let @xmath89 , @xmath70 , and @xmath71 be located on a circle in the plane @xmath90 . with the five machines so located , the coordinate - time difference between transmissions is then the same as the coordinate - time difference between receptions , and the coordinate - time delay in one direction equals that in the opposite direction ( as stated in the proof of prop [ prop : nine ] ) . 
   we construct seven two - way channels as above with null phases and show that the remaining 3 two - way channels can have the equal phases , but that this phase must be non - null with a curvature dependent amplitude @xmath91 . 
   [ prop : eleven ] under the stated conditions , if @xmath92 is small enough so that + @xmath93 , then @xmath94      for a fixed separation @xmath95 between @xmath34 and @xmath35 , an adiabatic change in curvature imposes a constraint on bit rate possible for the channels , stemming from a lower bound on clock periods . 
   suppose the cluster of 5 open machines is arranged so that the proper radar distance @xmath95 from @xmath34 to @xmath35 is 6,000 km , suppose the cluster descends from a great distance down to a radius of @xmath96 km from an earth - sized mass @xmath97 kg . for simplicity , 
   assume that the positions and clock rates are continually adjusted to maintain null phases for all but the three channels @xmath98 . 
   because @xmath99 , prop . 
   [ prop : eleven ] implies @xmath100 , which with ( [ eq : main ] ) implies that @xmath101 .    substituting the parameter values 
   , one finds that for the phases for the channels @xmath98 to satisfy ( [ eq : main ] ) , it is necessary that @xmath102 s. if an alphabet conveys @xmath103 bits / character , the maximum bit rate for all the channels in the 5-machine cluster is @xmath104 bits / s . 
   turning from type i to questions of type ii , we look at how the preceding `` blackboard modeling '' of clocks , expressed in the mathematical language of general relativity , get put to work when models are encoded into the open machines that manage their own logical synchronization . for questions of type ii 
   ( and type iii ) both models that explain or predict evidence and the evidence itself , pertaining to physical clocks , come into play . models encoded into computers contribute to the steering of physical clocks in rate and relative position toward an aiming point , generating echo counts as evidence that , one acquired , can stimulate the guessing of new models that come closer to the aiming point . 
   to express the effect of quantum uncertainty on logical synchronization , specifically on deviations from aiming points , one has to bring quantum uncertainty into cooperation with the representation of clocks by general - relativistic worldlines . this bringing together hinges on distinguishing evidence from its explanations . 
   timelike worldlines and null geodesics in explanations , being mathematical , can have no _ mathematical _ connection to physical atomic clocks and physical signals . to make such a connection them one has to invoke the logical freedom to make a guess . within this freedom 
   , one can resort to quantum theory to explain deviations of an atomic clock from an imagined proper clock , represented as a worldline , without logical conflict . 
   because of quantum uncertainty and for other reasons , if an aiming point in terms of channels and a given frequency scale is to be reached , steering is required , in which evidence of deviations from the aiming point combine with hypotheses concerning how to steer @xcite . to keep things simple , 
   consider a case of an aiming point with null phases , involving two open machines @xmath2 and @xmath7 , as in the example of sec . 
   [ sec : typei ] , modeled by a given worldline @xmath2 with given clock readings @xmath26 , where @xmath7 aims to maintain two - way , null - phase channel of given @xmath105 . for this 
   @xmath7 registers arriving phases of reception and adjusts its clock rate and its position more or less continually to keep those phases small . 
   deviations in position that drive position corrections show up not directly at @xmath7 but as phases registered by @xmath2 , so the steering of machine @xmath7 requires information about receptive phases measured by @xmath2 . 
   the knowledge of the deviation in position of @xmath7 at @xmath106 can not arrive at @xmath7 until its effect has shown up at @xmath2 and been echoed back as a report to @xmath7 , entailing a delay of at least @xmath80 , hence requiring that machine @xmath7 predict the error that guides for @xmath80 prior to receiving a report of the error . 
   that is , steering deviations by one open machine are measured in part by their effect on receptive phases of other open machines , so that steering of one machine requires information about receptive phases measured by other machines , and the deviations from an aiming point must increase with increasing propagation delays that demand predicting further ahead .    as is clear from the cluster of five machines discussed in sec . 
   [ sec : typei ] , the aiming - point phases can not in general all be taken to be zero . for any particular aiming - point phase @xmath107 
   there will be a deviation of a measured phase quantity @xmath91 given by @xmath108 whatever the value of @xmath107 , adjustments to contain phases within tolerable bounds depends on phase changes happening only gradually , so that trends can be detected and responded to on the basis of adequate prediction ( aka guesswork ) . + 
   * remarks : *    1 .   unlike cycle counts of open machines , which we assume are free of uncertainty , measured phases and deviations of phases from aiming points are quantities subject to uncertainty . for logic to work in a network , transmission of logical symbols 
   must preserve sharp distinctions among them ; yet the maintenance of sharp distinctions among transmitted symbols requires responses to fuzzy measurements . 2 . 
   the acquisition of logical synchrony in digital communications involves an unforeseeable waiting time , like the time for a coin on edge to fall one way or the other @xcite . 
   aiming points are not forever , and here we say a few words about questions of type iii , in which an aiming point based on a hypothesized metric tensor appears unreachable , and perhaps needs to be revised . 
   we have so far looked at one or another manifold with metric @xmath32 as some given hypothesis , whether explored on the blackboard or coded into an open machine to serve in maintaining its logical synchronization . in this context 
   we think of @xmath32 as `` given . '' but deviations of phases outside of tolerances present another context , calling for revising a metric tensor field . in this context 
   one recognizes that a metric tensor field is hypothesized provisionally , to be revised as prompted by deviations outside allowed tolerances in implementing an aiming point . 
   drawing on measured phases as evidence in order to adjust a hypothesis of a metric tensor is one way to view the operation of the laser interferometer gravitational - wave observatory ( ligo ) @xcite . while ligo sensitivity drops off severely below 45 hz , the arrangement of five open machines of prop . 
   [ prop : ten ] has no low - frequency cutoff , and so has the potential to detect arbitrarily slow changes in curvature . 
   99 b. n. taylor and a. thompson , eds , _ the international system of units ( si ) _ , nist special publication 330 , 2008 edition , national institutes of science and technology 
   . j. m. myers and f. h. madjid , `` a proof that measured data and equations of quantum mechanics can be linked only by guesswork , '' in s. j. lomonaco jr . and h.e . 
   brandt ( eds . ) _ quantum computation and information _ , contemporary mathematics series , vol . 
   305 , american mathematical society , providence , 2002 , pp . 
   . f.  h. madjid and j.  m. myers , `` matched detectors as definers of force , '' ann . 
   physics * 319 * , 251273 ( 2005 ) . 
   j.  m. myers and f.  h. madjid , `` ambiguity in quantum - theoretical descriptions of experiments , '' in k. mahdavi and d. koslover , eds . 
   , _ advances in quantum computation _ , contemporary mathematics series , vol .  482 ( american mathematical society , providence , i , 2009 ) , pp .  107123 . j. m. myers and f. h. madjid , `` what probabilities tell about quantum systems , with application to entropy and entanglement , '' in a. bokulich and g. jaeger , eds . , _ quantum information and entanglement _ , cambridge university press , cambridge uk , pp . 
   127150 ( 2010 ) . 
   a.  m. turing , `` on computable numbers with an application to the entscheidungsproblem , '' proc . 
   london math . 
   soc . , series 2 , * 42 * , 230265 ( 193637 ) . 
   m. soffel et al . , `` the iau resolutions for astrometry , celestial mechanics , and metrology in the relativistic framework : explanatory supplement , '' the astronomical journal , * 126 * , 26872706 ( 2003 ) . j. m. myers and f. h. madjid , `` rhythms essential to logical communication , '' in quantum information and computation ix , e. donkor , a. r. pirich , and h. e. brandt , eds , proceedings of the spie , * 8057 * , pp . 
   80570n112 ( 2011 ) . 
   j. m. myers and f. h. madjid , `` rhythms of memory and bits on edge : symbol recognition as a physical phenomenon , '' arxiv:1106.1639 , 2011 . 
   h. meyr and g. ascheid , _ synchronization in digital communications _ , wiley , new york , 1990 . 
   the ligo scientific collaboration ( http://www.ligo.org ) `` ligo : the laser interferometer gravitational - wave observatory , '' rep . 
   prog . phys . * 
   72 * , 076901 ( 2009 ) v. perlick , `` on the radar method in general - relativistic spacetimes , '' in h. dittus , c. lmmerzahl , and s. turyshev , eds . , _ lasers , clocks and drag - free control : expolation of relativistic gravity in space _ , ( springer , berlin , 2008 ) ; also arxiv:0708.0170v1 . f. k. manasse and c. w. misner , j. math phys . 
   , `` fermi normal coordinates and some basic concepts in differential geometry , '' * 4 * , 735745 ( 1963 ) . t. e. parker , s. r. jefferts , and t. p. heavner , `` medium - term frequency stability of hydrogen masers as measured by a cesium fountain , '' 2010 ieee international frequency control symposium ( fcs ) , pp . 318323 ( 
   ( available at http://tf.boulder.nist.gov/general/pdf/2467.pdf ) j. levine and t. parker , `` the algorithm used to realize utc(nist ) , '' 2002 ieee international frequency control symposium and pda exhibition , pp . 537542 ( 2002 )
summary: a clock steps a computer through a cycle of phases . for the propagation of logical symbols from one computer to another , each computer must mesh its phases with arrivals of symbols from other computers . 
   even the best atomic clocks drift unforeseeably in frequency and phase ; feedback steers them toward aiming points that depend on a chosen wave function and on hypotheses about signal propagation . 
   a wave function , always under - determined by evidence , requires a guess . 
   guessed wave functions are coded into computers that steer atomic clocks in frequency and position  clocks that step computers through their phases of computations , as well as clocks , some on space vehicles , that supply evidence of the propagation of signals . 
   recognizing the dependence of the phasing of symbol arrivals on guesses about signal propagation elevates ` logical synchronization . ' from its practice in computer engineering to a dicipline essential to physics . within this discipline 
   we begin to explore questions invisible under any concept of time that fails to acknowledge the unforeseeable . in particular , variation of spacetime curvature 
   is shown to limit the bit rate of logical communication .