I need to adjust rocker arm clearance on a 15.5 briggs engine

RTM? google it find and and read it, if lost it,
they are not adjustable.
OHV, 1cyc, rocker screw set and vast engine made,and not the same spec.
briggs on line charge$ for that manual.
so ask them for it,
104M02-0001-F1

It's not a side valve engine. OHV means overhead valves. Side valve engines are the old style flatheads such as used in '30-'50 era Ford and Mercury cars.
I suggest you look in the owner's manual for whatever equipment this engine powers and it should give you the correct valve clearances in thousandths of an inch, and whether you need to adjust the valves when hot or cold. Cold is the most common.

You'll need at least a feeler gauge, a wrench the same size as the tightening nut on the valve pushrod, and probably a medium size flathead screwdriver to do the actual adjustment. Removing the valve cover may require a wrench or screwdriver depending on the specific engine. All of these tools are available at most hardware stores, Home Depot, Lowe's, Auto Zone, etc. Valve clearances are usually given in a range, such as: .005" -.007" which translates to 5 to 7 thousandths of an inch between the pushrod and rocker arm. Feeler gauges are so marked. Aim for the middle of the range, like .006"

If you've never done this before, I'd suggest you get a competent auto mechanic to do it and watch carefully as it's done. It's not rocket science but does require a certain touch. Once you've seen it done you should be able to to it yourself. This type of maintenance is generally required perhaps once a year if the equipment is not used every day.

The following is assuming it is a OHV version and that is a 280000 series.

NOTE: Check valve clearance while the engine
is cold.

1. Turn crankshaft counterclockwise until
piston is at top dead center on the
compression stroke. This prevents the
compression release from holding the
valves open.
2. Insert a narrow screwdriver or rod into the
spark plug hole as a gauge, then slowly
turn crankshaft counterclockwise until the
piston has moved down the bore by 1/4"
(6mm).
3. Using a feeler gauge, adjust rocker nut to obtain the clearance.
Intake .003" - .005" Exhaust .005" - .007"
4. Hold rocker nut and tighten the rocker ball
setscrew to the torque valve. 45 lb-in. (5 Nm)
5. Check clearance again and re-adjust, if necessary.
6. Repeat for other valve.

use a business card for the gap get the valves all the way up loosen the nut on the rocker arm snugg rocker arm bolt til you have slight resistance pulling the card out tighten nut while holding bolt

its the clearance between the rocker arm and the valve on briggs engines i set them to .006

Usually there is a reason why the rocker arm came off the push rod so you will need to check a few things before putting it back together. First make sure the push rod is not bent, if it is replace it, do not try to straighten it. Make sure the rocker arm stud did not loosen, if it did back out, re-tighten. If you're ready to put it back together, you will need to set the valve clearance between the rocker arm and the push rod. You will need to know the engine model number to get this clearance since it varies. Once you get the clearance values, make sure the piston is at TDC on a compression stroke and adjust the valves per the setting. In some rare instances, the valve can "stick" in the valve guide and this will cause excessive gap. However, usually when this occurs the engine will not start anyway. Hope this gets you in the right direction.

INtake valve adjustment needed. at least on one cylinder. maybe both sides. while your in there adjust all. must have some clearance. .004 on intake .006 on exhust.

I think you have the rocker arm clearance set wrong on the one that you worked on.
See if this video will help -

Lawn Mower Repair Valve Adjustment Twin

remove the valve covers and pull the spark plugs. so you can find the compression stroke. when you fill air coming out of the head. make the piston come all the way to top. watch rocker arms so that they the same hieght. use a filler gauge ( .005 ) put betwen the rocker arm and the valve. tighten rocker arms to the gauge. make sure you cheek again after tighting the rocker nuts

If that's a Briggs & Stratton engine, try this:

Valve adjustment for 28 and 31 Series B&S OHV engines done with engine cold.

The best thing to do is buy an Overhaul Manuel for this series of engine,
between $15 - $20..

Remove the valve cover. Remove the spark plug and turn the engine to TDC
(Top Dead Center). Now some people will say that there are 2 TDCs on 4
stroke engines. While the piston does come to top twice per combustion
cycle, the TDC referred to in all 4 stroke overhaul manuals, whether auto or
lawnmower, is when piston is at the top after compression stroke. Turn the
engine in the normal direction of rotation, watch the rocker arm action to
determine when the Intake Valve, the bottom valve, has just closed, use a screw driver in spark
plug hole to feel when piston is coming up and then at top. (A better idea
is a long pencil with an eraser, put the eraser end in spark plug hole) Per B&S
instructions, continue turning engine until the piston has receded ¼" down
past TDC. To do this, put a mark on screw driver (or pencil) ¼" above highest point
reached to tell when piston has receded ¼". You will need a torques head
screw driver bit for the locking screw on tappets. I would check to see that the Rocker Arm Studs are still tight in the head, they sometimes work loose.

The proper setting is Intake Valve - .003 - .005", Exhaust Valve - .005 -
.007". Use a flat feeler gauge between the Rocker Arm face and the Valve
Stem. These settings are critical as the Intake setting controls the operation of the Compression Release. I usually have to make 2 - 3 attempts before I am satisfied with the setting. When you think you have the proper clearance, take a hold of the Rocker Arm and move it by hand, side to side, etc, as much as you can and recheck the clearance
setting. The nut must be held while Locking Screw is torqued to 60 inch
lbs. I recheck the clearance setting again here. I would re-torque the head bolts while you are there.

I recommend removing and reversing the Intake Valve Push Rod when adjusting
the valves. If you remove the Aluminum Intake Valve Push Rod, you will see a
worn area down a little ways from the Rocker Arm end. This is from riding on
a fiber support rub block and the aluminum push rod will eventually bend or
break. Both ends are the same and reversing will prolong the life of the
push rod.

Note: IF you remove the push rod or any other rocker arm components, there
is a cap that sits on top of each valve stem, take care that you do not
dislodge and lose the cap. It is between this cap and the rocker arm that
you actually take your clearance reading but you will not notice there is a
cap normally. (Apparently, from the Overhaul Manual, ALL engines do not have
this cap)

Replace valve cover with new gasket, torque valve cover to 60 inch lbs.,
replace spark plug.