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If the integral I is invariant with respect to a Gρ, then ρ linearly independent combinations of the Lagrange expressions become divergences – and from this, conversely, invariance of I with respect to a Gρ will follow. The theorem holds good even in the limiting case of infinitely many parameters.
Mathematics is a sensory perception organ for those who learn how to use it.
—S. James Gates
The town. The shrubs are white, the trunks a dull copper. She walks alone like a ghost and the wind goes through the grass and trees and whispers and is gone. She is alone and the weeds push up between the stones and the houses are bare and the street empty. The sky is pale grey. She pulls the barrow and sheets of tarp and the metal spokes taken from fences and broken concrete. The device hangs around her neck and now clicks and then again. Pip. If the clicks come quick she will turn and try again. Try the long way round.
No matter. There is time and her alone to fill it. The static click stutters and the barrow squeaks and her heel catches and they are loud in that quiet place. She passes the bus stop and now the shop with the green shutters and the fence and then the school. The lawnmower and the swings. The windows and the film of dirt outside and inside classrooms and desks and dust and fallen plaster. A panel blown free from the roof lying on the pavement ahead. Broken. She tries and the metal folds and breaks along the seam. The flakes like small blades.
Rusted, eaten. She drops it in the overgrowth and crosses over. To the square and the paths around the beds and benches and the statue of the man at the centre. His jacket is open and his shirt buttoned up and he wears a flat cap and his features are strong. Proud and stern in bronze. The eyes stare and they are without pupils and dead and track her as she carries her load. Ahead the buildings are tall along the main road through the town. Looming. She walks further and comes under their shadow and is cold. Wrapping her coat about her.
Carrying on. The cars are empty and they line the street. On either side and abandoned by the lights that do not signal that two months back she stripped for wires. The glass lying in red and yellow shards on the ground below by the leaves and the blocked storm drain. Colours muted by the muck. She stoops and hand on back retrieves the thin cable from the newspaper bundles long rotten and gone. She places it between the plastic sheets and smooths them flat again. Carefully. Moving slowly, though there are yet hours left in the day. Her alone to fill them.
To abide. Picking between the hollow ruins for each needful thing. Some to be found and some patched and hobbled together and most lost to the town slipping forward in time. Buildings leached by air and rain and sun, beams sagging in the houses and windows falling loose from the blocks. Rubble growing at the outskirts. There are places she is not safe to go, where the counter clicks and where the buildings fall. Where there are only the tall thin bones of the naked trees. Copper. The trunks a dull copper and the shrubs around them white. To the south of town.
She walks. Making her way north and along the main avenue. The tall buildings end up ahead and beyond it is not bright, it is brighter and the air is warmer. The sky is pale grey and sunless but it is warmer out from under the buildings along the broad avenue. The barrow squeaks behind her. The device hangs round her neck and counts clicks and they come slow and she does not hurry. The road rises and the hill ahead as the buildings fall away. Dry. The bushes and the grass and the earth in the fields. The whole place is unslaked.
Late again. But the rains will come, and will fall heavy. She knows when the rain comes and when the sun and when the first snow, when the long cold breaks. It is written into the place which is her home and as though it were a part of her body. It is growing within her. The place written into her and she alone in it and pulling the barrow along the wide road. It is steep and the going is hard but she is strong. Older. Much older than she was, not as strong and still pulling. Turning the barrow to push.
Climbing higher. There is a tractor and mixer at the crest. She sets the barrow by the wheel and rests and looks back and sees the pylons and the telephone wires. The railway timber and the tracks warping and the tanker in the grass and the dark earth and fading signs. The dogs have moved on. Their cries replaced by the wind between the trees and the buildings, against the balconies and bare concrete. The town desolate and her now leaving it behind like a ghost. Alone. Carrying plastic sheets and cables and metal spokes in the barrow. Leaning against the tractor wheel.
Looking out. The rubber firm like a stone against her back. She walks round and lifts the bars and sets off again down the wide road pulling the load behind her. The road goes on a half mile and she makes her way and comes to a path and takes it. The path leading down again. Then the hedgerow tall on the left and the steep bank that carries on up to the right. In the basin there is the lake and the waters are still. Silence. Here the wind passes high above and the water is still. The house by the shore.
Her home. On the far side by the reeds and sheeting. Reeds that trap the mist and metal sheets lying ribbed on the shore like waves in the still water breaking. There is a caravan by the house by the shore and the trailer loaded up with two large black drums. There the old diesel generator. She follows the narrow line she has worn from the path through the grass to the porch outside. By the decking she rests the barrow and kicks her shoes clean. Weary. She retrieves the cable and the metal and leaves the tarp. The door closes behind her.
Dark inside. She reaches the square table and lights the lamp. The light is warm and the place is small and there is an armchair by the stove in the corner. There is dense cardboard stapled to the back of the door where the thin glass sits broken in the wood. Fresh paint on the stairs. There are books and DIY manuals and beside the armchair a chessboard fixed to a stand on wheels. On the board sit screws and bolts in an old tobacco tin. Marbles. Tools and wires and engine parts stowed in green munition boxes. A deck of worn cards.
The kitchen. Jars and tins and plastic pots on the shelves. The sink and vinyl clean and the freezer hooked up to a car battery and the narrow fridge turned off. Check red and white and fading on the wall between the draining board and cupboards, blue flowers on the curtain. The cracks around the window. She searches through the green boxes and finds the tools she needs and takes them and the spokes. Lodging the back door open with a wide log from the pile. Measuring. The cracks growing past the lines marked out in dark pencil. The spokes and the frame.
Hand drill. Through the wall within the cracks and then around. Holding the spokes flat and straight and the brackets and driving screws into the plugs in the wood and brickwork. She fetches the tarp and pins it between a panel and the wall above and angles the last spokes downward. An awning against the rain. To keep the pane dry, if it should come before she has found sealant, or a new window. The lake is still, and on the far side stands a heron. Delicate. Flicking at the dead water, long beak like a mottled blade. Spread of wings and rising.
Impossibly light. The frail wings beating wide and thin and hushed. She watches as the frame of the bird climbs into the sky until it is a line and then gone. There are wolves in the forest but there have been no animals come by the house for a long time. Not since the dogs left. No rabbits breeding in the earth nor fish in the water, no birds nor deer nor wild pigs. Her alone in the house by the lake just north of town. Quiet. Where the wind carries over above and the water is still. The door closes behind her.
The sink. Washing her hands, tap fed by the drums outside. Drying with the cloth she crosses to the stove where she opens the grate and empties the tray of ash. She replaces it and builds a nest of leaves and sticks from the basket which she lights and then blows. The flames lick into life. She waits and watches them grow and feeds them two good thick logs and closes the grate again. The kettle on the stove and the teapot from the drying rack. Sun. Leaves from the foil packet with the sun printed on it. Her usual mug in purple.
Chickpeas soaking. In the pantry where there are spices and garlic. She drains and mashes the tough round pellets in the bowl with her hands and then she dices the clove. Smearing it with the flat of the knife and dicing it twice again so that it becomes an uneven paste. Seasoning with salt and paprika. Mixing in the pan with hot water from the kettle and finding room for both on the stove. Clearing the table as the smell fills the room and is good. Rich. The growing warmth of the stove and the dying light outside.
The stillness of the lake.
Night falls. The kettle slow to boil with the heat split. She pours the water over the leaves loose in the pot which she swirls and sets to rest and brew. After she has eaten she sets the pan and bowl and spoon in the sink and pours herself a cup. She sits in the armchair. In the corner sipping the tea black with her feet on the stool out in front of her. She finishes and places the mug on a manual on the chessboard. Clear. The liquor from the half-empty bottle between the green munition boxes. She measures out two fingers.
She reads. Books from the school library and heavy instruction manuals. How to fix the carburettor in the diesel generator and the parts she will need to source from the cars. An abridged romance and poems, short stories by great writers she had read when she was young and since forgotten. The clock sounds from upstairs. She pours herself another finger and settles back in the chair and reaches and wheels over the board. Scribbling lists on the pad from the pages open on her lap. Bolts. Drawing sketches of gaskets and rings, a note on cleaning fluid. Hose and float and needle.
Dark outside. The waters still in the darkness and no wind. Nothing at all to disturb her, in that lonely place in that small house by the shore by the lake. No cattle grazing nor passing car, only her in the house by the shore by the lake north of town. Her alone by the waters. Reading in the night and by day planting and seeding, sourcing each needful thing and making her repairs. Fixing things and bolting them back to keep them from slipping forward. Decaying. Like the buildings in the town and further to the south. Toward where things fell apart.
The moon. It is drowning in the stillness of the lake. Drowning in the stillness of the waters that are a poison, the lake that in time will kill her too. That is killing her, with the air and the earth and the seeds of the small tomatoes, the misshapen potatoes. The wood from the trees. Metal spokes and plastic sheeting and cables from the rotten newspapers, each needful thing salvaged from the town. Washers and wires and bolts and screws and glass and tinned food. Fatal. Through the days that linger and pass, the days to come. As time slips slowly forward.
Eating her. Disease growing root and branch like a tree inside. Leaching her like the air and rain and sun in the houses and the tall residential blocks in the town. The place that is her home and that is written into her and which is a part of her body. The place growing within her. There are others who will go under drips and sharp light in hospitals a long way from here. Lying in soft bedding and fading out in the blooming of euphoria. Chemicals. Dulling first to the body and then to the world around. Far away from this place.
Two decades. Twenty years ago, then two and her alone since. In the house by the lake after two years, the first ever away and then back and never gone again. Still in the town after twenty, the plant to the south where the buildings fall and the rubble is growing. Where the counter clicks, stutters. Where the trees are naked and the shrubs white and there is rust and the dark dry ground. Where there are the miles of road and then the barbed fence. Meteor. The huge dome like a white comet embedded in the earth. A strange and perfect visitor.
The plant. Under the dome encased in concrete buried in sand. Where fire poured out of a fissure like liquid from a mouth, spitting salt to be carried on the wind. Matter unstructured and undone, the walls molten and swollen like great infected limbs and steaming, bubbling deep into the ground. Smoke and steam billowing, rising. Fire and salt carried on the wind into the burning air and the dark earth and the water. The building blown apart and scattered and melted and smouldering all around. Scorched. The earth scorched and the air too and the water tainted. The water that is poison.
The plume. In the sky for days afterward, a rippling column. In the aftermath rising under a tower of light, like the site were communing with some deep and off-world event. A beam electric and ephemeral, powering into the vastness of the starless sky, breaking out beyond and into the vacuum. The ash like black snow. Carried on the burning wind as salt and fire into the earth and over the waters, the lake. The ash that fell as a blanket in the days, weeks afterwards. Dense. Heavy on the ground, feathered on the face of the buildings. The ash falling thick, black.
An end. Death of a place, and in time its people. So that only ghosts come now, she and time and her alone to fill it as the town slips forward. To abide, picking through the hollows as the buildings fall and weeds bind the cars and park benches and streetlamps. The bones of the trees. The stillness of the water that is poison and in the shallow basin where the wind passes over. Her walking like a ghost, pulling the barrow behind like a plough. Trawling. For metal and tarpaulin, spare parts and tough glass and scrap. Abiding by each needful thing.
She wakes. It is still dark out, the clock sounds upstairs. The empty glass on the floor, the list and the manual open in her lap and the fire burning low. She closes the pages and replaces the bottle quarter-full and picks up the tumbler and places it in the sink. Wet cloth, cleaning the table. Washing up, the cold water raw against her hands and the knuckle swollen sore under her flecked skin. She leaves the list on the table under the old tobacco tin. Tomorrow. All the time she needs to find the tools and parts. She will fix the generator.
The stairs. The fresh paint and in it the candle wavering. Wood of the bannister strong under her hand and weight as she climbs the steps and comes to the landing. The bathroom with the pale green tub facing the window and looking out and toward the ridge and the treeline. Her bedroom and narrow bed. The gas lamp on the side table and metal trunk with her clothes and shoes under the frame. Wallpaper old and yellowing here and there, under a sheen of varnish. Preserved. The paper fixed in time by a veneer of thick glue. Held back from slipping further.
She undresses. Pulling into her long thin shirt and dressing gown. Her body slight and older than it was and still strong and lithe but not as strong as it was. The place written into her body and growing inside, her root and branch like the tall bones of the trees. Mouthless, rust eating through metal. Dull copper and the shrubs stricken white between the buildings that are falling toward the south of town. She climbs into bed and wraps the woollen blanket tight about her. Cold. But growing warm and watching the candlelight play on the ceiling. Shadows dancing on the ceiling.
The window. Clothesline running from sill to the post half buried. In the yard out front the metal sheets lying where the still waters meet the shore and the dark earth. Ribbed waves at the shore of the lake that is still, in it the moon floating perfect as a coin. A circle in perfect white. The reeds sharp and dark with the mist that wallows low and beneath the wind that passes over. The wind that whispers in the town and passes over the basin. Quiet. The place is silent, her in the house by the shore. The house fed by drums.
Darkness inside. She snuffs the candle, rolls over and is still. Her alone in the house, the basin where there is no wind and there are no rabbits in the earth. Where there are wolves in the forest to the west, no dogs in the town nor cattle nor passing car. Near where things fell apart. Where fire poured from the fissure, spitting salt and undoing the all-around and falling thick as black snow. Where buildings fall and the trees are bare and the shrubs white. Alone. Where the ash lay on the water like algae in bloom. The lake in bitter bloom.